Comprehensive Inorganic Chemistry III. Volume 3: Theory and Bonding of Inorganic Nonmolecular Systems [3 ed.] 9780128231449

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COMPREHENSIVE INORGANIC CHEMISTRY III

COMPREHENSIVE INORGANIC CHEMISTRY III EDITORS IN CHIEF

Jan Reedijk Leiden Institute of Chemistry, Leiden University, Leiden, the Netherlands

Kenneth R. Poeppelmeier Department of Chemistry, Northwestern University, Evanston, IL, United States

VOLUME 3

Theory and Bonding of Inorganic Nonmolecular Systems VOLUME EDITOR

Daniel C. Fredrickson Department of Chemistry, University of Wisconsin-Madison, Madison, WI, United States

Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge MA 02139, United States Copyright Ó 2023 Elsevier Ltd. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers may always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-823144-9

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CONTENTS OF VOLUME 3 Editor Biographies

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Volume Editors

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Contributors to Volume 3

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Preface 3.01

xvii Introduction: Theory and bonding of inorganic non-molecular systems Daniel C Fredrickson

1

SECTION 1: MODELS FOR EXTENDED INORGANIC STRUCTURES 3.02

Electronic structure of oxide and halide perovskites Robert F Berger

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3.03

Bonding in boron rich borides Pattath D Pancharatna, Sohail H Dar, and Musiri M Balakrishnarajan

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3.04

The Zintl-Klemm concept and its broader extensions Ángel Vegas and Álvaro Lobato

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3.05

An introduction to the theory of inorganic solid surfaces Émilie Gaudry

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3.06

Bond activation and formation on inorganic surfaces Yuta Tsuji

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SECTION 2: TOOLS FOR ELECTRONIC STRUCTURE ANALYSIS 3.07

Chemical bonding with plane waves Ryky Nelson, Christina Ertural, Peter C Müller, and Richard Dronskowski

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3.08

Chemical bonding analyses using wannier functions Koichi Kitahara

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3.09

Chemical bonding analysis in position space Frank R Wagner and Yuri Grin

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3.10

The structures of inorganic crystals: A rational explanation from the chemical pressure approach and the anions in metallic matrices model J Manuel Recio, Álvaro Lobato, Hussien H Osman, Miguel Ángel Salvadó, and Ángel Vegas

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SECTION 3: PREDICTIVE EXPLORATION OF NEW STRUCTURES 3.11

Energy landscapes in inorganic chemistry J Christian Schön

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3.12

First principles crystal structure prediction Lewis J Conway, Chris J Pickard, and Andreas Hermann

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3.13

Crystal chemistry at high pressure Katerina P Hilleke and Eva Zurek

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SECTION 4: PROPERTIES AND PHENOMENA 3.14

In silico modeling of inorganic thermoelectric materials José J Plata, Pinku Nath, Javier Fdez Sanz, and Antonio Marquez

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3.15

Correlated electronic states in quasicrystals Nayuta Takemori and Shiro Sakai

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3.16

Chemical bonding principles in magnetic topological quantum materials Madalynn Marshall and Weiwei Xie

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EDITOR BIOGRAPHIES Editors in Chief Jan Reedijk Jan Reedijk (1943) studied chemistry at Leiden University where he completed his Ph.D. (1968). After a few years in a junior lecturer position at Leiden University, he accepted a readership at Delft University of Technology in 1972. In 1979 he accepted a call for Professor of Chemistry at Leiden University. After 30 years of service, he retired from teaching in 2009 and remained as an emeritus research professor at Leiden University. In Leiden he has acted as Chair of the Department of Chemistry, and in 1993 he became the Founding Director of the Leiden Institute of Chemistry. His major research activities have been in Coordination Chemistry and Bioinorganic Chemistry, focusing on biomimetic catalysis, molecular materials, and medicinal inorganic chemistry. Jan Reedijk was elected member of the Royal Netherlands Academy of Sciences in 1996 and he was knighted by the Queen of the Netherlands to the order of the Dutch Lion (2008). He is also lifetime member of the Finnish Academy of Sciences and Letters and of Academia Europaea. He has held visiting professorships in Cambridge (UK), Strasbourg (France), Münster (Germany), Riyadh (Saudi Arabia), Louvain-la-Neuve (Belgium), Dunedin (New Zealand), and Torun (Poland). In 1990 he served as President of the Royal Netherlands Chemical Society. He has acted as the Executive Secretary of the International Conferences of Coordination Chemistry (1988–2012) and served IUPAC in the Division of Inorganic Chemistry, first as a member and later as (vice/past) president between 2005 and 2018. After his university retirement he remained active as research consultant and in IUPAC activities, as well as in several editorial jobs. For Elsevier, he acted as Editor-in-Chief of the Reference Collection in Chemistry (2013–2019), and together with Kenneth R. Poeppelmeier for Comprehensive Inorganic Chemistry II (2008–2013) and Comprehensive Inorganic Chemistry III (2019-present). From 2018 to 2020, he co-chaired the worldwide celebrations of the International Year of the Periodic Table 2019. Jan Reedijk has published over 1200 papers (1965–2022; cited over 58000 times; h ¼ 96). He has supervised 90 Ph.D. students, over 100 postdocs, and over 250 MSc research students. Kenneth R. Poeppelmeier Kenneth R. Poeppelmeier (1949) completed his undergraduate studies in chemistry at the University of Missouri (1971) and then volunteered as an instructor at Samoa CollegedUnited States Peace Corps in Western Samoa (1971–1974). He completed his Ph.D. (1978) in Inorganic Chemistry with John Corbett at Iowa State University (1978). He joined the catalysis research group headed by John Longo at Exxon Research and Engineering Company, Corporate Research–Science Laboratories (1978–1984), where he collaborated with the reforming science group and Exxon Chemicals to develop the first zeolite-based light naphtha reforming catalyst. In 1984 he joined the Chemistry Department at Northwestern University and the recently formed Center for Catalysis and Surface Science (CCSS). He is the Charles E. and Emma H. Morrison Professor of Chemistry at Northwestern University and a NAISE Fellow joint with Northwestern University and the Chemical Sciences and Engineering Division, Argonne National Laboratory. Leadership positions held include Director, CCSS, Northwestern University; Associate Division Director for Science, Chemical Sciences and Engineering Division, Argonne National Laboratory; President of the Chicago Area Catalysis Club; Associate Director, NSF Science and Technology Center for Superconductivity; and Chairman of the ACS Solid State Subdivision of the Division of Inorganic Chemistry. His major research activities have been in Solid State and Inorganic Materials Chemistry focusing on heterogeneous catalysis, solid state chemistry, and materials chemistry. His awards include National Science Council of Taiwan Lecturer (1991); Dow Professor of Chemistry (1992–1994); AAAS Fellow, the American Association for the Advancement of Science (1993); JSPS Fellow, Japan Society for the Promotion of Science (1997); Natural Science Foundation of China Lecturer (1999); National Science Foundation Creativity Extension Award (2000

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and 2022); Institut Universitaire de France Professor (2003); Chemistry Week in China Lecturer (2004); Lecturer in Solid State Chemistry, China (2005); Visitantes Distinguidos, Universid Complutenses Madrid (2008); Visiting Professor, Chinese Academy of Sciences (2011); 20 years of Service and Dedication Award to Inorganic Chemistry (2013); Elected foreign member of Spanish National Academy: Real Academia de Ciencia, Exactas, Fisicas y Naturales (2017); Elected Honorary Member of the Royal Society of Chemistry of Spain (RSEQ) (2018); and the TianShan Award, Xinjiang Uygur Autonomous Region of China (2021). He has organized and was Chairman of the Chicago Great Lakes Regional ACS Symposium on Synthesis and Processing of Advanced Solid State Materials (1987), the New Orleans National ACS Symposium on Solid State Chemistry of Heterogeneous Oxide Catalysis, Including New Microporous Solids (1987), the Gordon Conference on Solid State Chemistry (1994) and the First European Gordon Conference on Solid State Chemistry (1995), the Spring Materials Research Society Symposium on Environmental Chemistry (1995), the Advisory Committee of Intense Pulsed Neutron Source (IPNS) Program (1996–1998), the Spring Materials Research Society Symposium on Perovskite Materials (2003), the 4th International Conference on Inorganic Materials, University of Antwerp (2004), and the Philadelphia National ACS Symposium on Homogeneous and Heterogeneous Oxidation Catalysis (2004). He has served on numerous Editorial Boards, including Chemistry of Materials, Journal of Alloys and Compounds, Solid State Sciences, Solid State Chemistry, and Science China Materials, and has been a co-Editor for Structure and Bonding, an Associate Editor for Inorganic Chemistry, and co-Editor-in-Chief with Jan Reedijk for Comprehensive Inorganic Chemistry II (published 2013) and Comprehensive Inorganic Chemistry III (to be published in 2023). In addition, he has served on various Scientific Advisory Boards including for the World Premier International Research Center Initiative and Institute for Integrated Cell-Material Sciences Kyoto University, the European Center SOPRANO on Functional Electronic Metal Oxides, the Kyoto University Mixed-Anion Project, and the Dresden Max Planck Institute for Chemistry and Physics. Kenneth Poeppelmeier has published over 500 papers (1971–2022) and cited over 28000 times (h-index ¼ 84). He has supervised over 200 undergraduates, Ph.D. students, postdocs, and visiting scholars in research.

VOLUME EDITORS Risto S. Laitinen Risto S. Laitinen is Professor Emeritus of Chemistry at the University of Oulu, Finland. He received the M.Sc and Ph.D. degrees from Helsinki University of Technology (currently Aalto University). His research interests are directed to synthetic, structural, and computational chemistry of chalcogen compounds focusing on selenium and tellurium. He has published 250 peer-reviewed articles and 15 book chapters and has edited 2 books: Selenium and Tellurium Reagents: In Chemistry and Materials Science with Raija Oilunkaniemi (Walther de Gruyter, 2019) and Selenium and Tellurium Chemistry: From Small Molecules to Biomolecules and Materials with Derek Woollins (Springer, 2011). He has also written 30 professional and popular articles in chemistry. He is the Secretary of the Division of Chemical Nomenclature and Structure Representation, International Union of Pure and Applied Chemistry, for the term 2016–2023. He served as Chair of the Board of Union of Finnish University Professors in 2007–2010. In 2017, Finnish Cultural Foundation (North Ostrobothnia regional fund) gave him an award for excellence in his activities for science and music. He has been a member of Finnish Academy of Science and Letters since 2003.

Vincent L. Pecoraro Professor Vincent L. Pecoraro is a major contributor in the fields of inorganic, bioinorganic, and supramolecular chemistries. He has risen to the upper echelons of these disciplines with over 300 publications (an h-index of 92), 4 book editorships, and 5 patents. He has served the community in many ways including as an Associate Editor of Inorganic Chemistry for 20 years and now is President of the Society of Biological Inorganic Chemistry. Internationally, he has received a Le Studium Professorship, Blaise Pascal International Chair for Research, the Alexander von Humboldt Stiftung, and an Honorary PhD from Aix-Maseille University. His many US distinctions include the 2016 ACS Award for Distinguished Service in the Advancement of Inorganic Chemistry, the 2021 ACS/SCF FrancoAmerican Lectureship Prize, and being elected a Fellow of the ACS and AAAS. He also recently cofounded a Biomedical Imaging company, VIEWaves. In 2022, he was ranked as one of the world’s top 1000 most influential chemists.

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Zijian Guo Professor Zijian Guo received his Ph.D. from the University of Padova and worked as a postdoctoral research fellow at Birkbeck College, the University of London. He also worked as a research associate at the University of Edinburgh. His research focuses on the chemical biology of metals and metallodrugs and has authored or co-authored more than 400 peer-reviewed articles on basic and applied research topics. He was awarded the First Prize in Natural Sciences from Ministry of Education of China in 2015, the Luigi Sacconi Medal from the Italian Chemical Society in 2016, and the Outstanding Achievement Award from the Society of the Asian Biological Inorganic Chemistry in 2020. He founded Chemistry and Biomedicine Innovation Center (ChemBIC) in Nanjing University in 2019, and is serving as the Director of ChemBIC since then. He was elected to the Fellow of the Chinese Academy of Sciences in 2017. He served as Associated Editor of Coord. Chem. Rev and an editorial board member of several other journals.

Daniel C. Fredrickson Daniel C. Fredrickson is a Professor in the Department of Chemistry at the University of WisconsinMadison. He completed his BS in Biochemistry at the University of Washington in 2000, where he gained his first experiences with research and crystals in the laboratory of Prof. Bart Kahr. At Cornell University, he then pursued theoretical investigations of bonding in intermetallic compounds, the vast family of solid state compounds based on metallic elements, under the mentorship of Profs. Stephen Lee and Roald Hoffmann, earning his Ph.D. in 2005. Interested in the experimental and crystallographic aspects of complex intermetallics, he then carried out postdoctoral research from 2005 to 2008 with Prof. Sven Lidin at Stockholm University. Since starting at UW-Madison in 2009, his research group has created theory-driven approaches to the synthesis and discovery of new intermetallic phases and understanding the origins of their structural features. Some of his key research contributions are the development of the DFT-Chemical Pressure Method, the discovery of isolobal bonds for the generalization of the 18 electron rule to intermetallic phases, models for the emergence of incommensurate modulations in these compounds, and various design strategies for guiding complexity in solid state structures.

Patrick M. Woodward Professor Patrick M. Woodward received BS degrees in Chemistry and General Engineering from Idaho State University in 1991, an MS in Materials Science, and a Ph.D. in Chemistry from Oregon State University (1996) under the supervision of Art Sleight. After a postdoctoral appointment in the Physics Department at Brookhaven National Laboratory (1996–1998), he joined the faculty at Ohio State University in 1998, where he holds the rank of Professor in the Department of Chemistry and Biochemistry. He is a Fellow of the American Chemical Society (2020) and a recipient of an Alfred P. Sloan Fellowship (2004) and an NSF Career Award (2001). He has co-authored two textbooks: Solid State Materials Chemistry and the popular general chemistry textbook, Chemistry: The Central Science. His research interests revolve around the discovery of new materials and understanding links between the composition, structure, and properties of extended inorganic and hybrid solids.

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P. Shiv Halasyamani Professor P. Shiv Halasyamani earned his BS in Chemistry from the University of Chicago (1992) and his Ph.D. in Chemistry under the supervision of Prof. Kenneth R. Poeppelmeier at Northwestern University (1996). He was a Postdoctoral Fellow and Junior Research Fellow at Christ Church College, Oxford University, from 1997 to 1999. He began his independent academic career in the Department of Chemistry at the University of Houston in 1999 and has been a Full Professor since 2010. He was elected as a Fellow of the American Association for the Advancement of Science (AAAS) in 2019 and is currently an Associate Editor of the ACS journals Inorganic Chemistry and ACS Organic & Inorganic Au. His research interests involve the design, synthesis, crystal growth, and characterization of new functional inorganic materials.

Ram Seshadri Ram Seshadri received his Ph.D. in Solid State Chemistry from the Indian Institute of Science (IISc), Bangalore, working under the guidance of Professor C. N. R. Rao FRS. After some years as a Postdoctoral Fellow in Europe, he returned to IISc as an Assistant Professor in 1999. He moved to the Materials Department (College of Engineering) at UC Santa Barbara in 2002. He was recently promoted to the rank of Distinguished Professor in the Materials Department and the Department of Chemistry and Biochemistry in 2020. He is also the Fred and Linda R. Wudl Professor of Materials Science and Director of the Materials Research Laboratory: A National Science Foundation Materials Research Science and Engineering Center (NSF-MRSEC). His work broadly addresses the topic of structure–composition– property relations in crystalline inorganic and hybrid materials, with a focus on magnetic materials and materials for energy conversion and storage. He is Fellow of the Royal Society of Chemistry, the American Physical Society, and the American Association for the Advancement of Science. He serves as Associate Editor of the journals, Annual Reviews of Materials Research and Chemistry of Materials.

Serena Cussen Serena Cussen née Corr studied chemistry at Trinity College Dublin, completing her doctoral work under Yurii Gun’ko. She then joined the University of California, Santa Barbara, working with Ram Seshadri as a postdoctoral researcher. She joined the University of Kent as a lecturer in 2009. She moved to the University of Glasgow in 2013 and was made Professor in 2018. She moved to the University of Sheffield as a Chair in Functional Materials and Professor in Chemical and Biological Engineering in 2018, where she now serves as Department Head. She works on next-generation battery materials and advanced characterization techniques for the structure and properties of nanomaterials. Serena is the recipient of several awards including the Journal of Materials Chemistry Lectureship of the Royal Society of Chemistry. She previously served as Associate Editor of Royal Society of Chemistry journal Nanoscale and currently serves as Associate Editor for the Institute of Physics journal Progress in Energy.

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Rutger A. van Santen Rutger A. van Santen received his Ph.D. in 1971 in Theoretical Chemistry from the University of Leiden, The Netherlands. In the period 1972–1988, he became involved with catalysis research when employed by Shell Research in Amsterdam and Shell Development Company in Houston. In 1988, he became Full Professor of Catalysis at the Technical University Eindhoven. From 2010 till now he is Emeritus Professor and Honorary Institute Professor at Technical University Eindhoven. He is a member of Royal Dutch Academy of Sciences and Arts and Foreign Associate of the United States National Academy of Engineering (NAE). He has received several prestigious awards: the 1981 golden medal of the Royal Dutch Chemical Society; in 1992, the F.G. Chiappetta award of the North American Catalysis Society; in 1997, the Spinoza Award from the Dutch Foundation for Pure and Applied Research; and in 2001, the Alwin Mittasch Medal Dechema, Germany, among others. His main research interests are computational heterogeneous catalysis and complex chemical systems theory. He has published over 700 papers, 16 books, and 22 patents.

Emiel J. M. Hensen Emiel J. M. Hensen received his Ph.D. in Catalysis in 2000 from Eindhoven University of Technology, The Netherlands. Between 2000 and 2008, he worked at the University of Amsterdam, Shell Research in Amsterdam, and Eindhoven University of Technology on several topics in the field of heterogeneous catalysis. Since July 2009, he is Full Professor of Inorganic Materials and Catalysis at TU/e. He was a visiting professor at the Katholieke Universiteit Leuven (Belgium, 2001–2016) and at Hokkaido University (Japan, 2016). He is principal investigator and management team member of the gravitation program Multiscale Catalytic Energy Conversion, elected member of the Advanced Research Center Chemical Building Blocks Consortium, and chairman of the Netherlands Institute for Catalysis Research (NIOK). Hensen was Head of the Department of Chemical Engineering and Chemistry of Eindhoven University of Technology from 2016 to 2020. Hensen received Veni, Vidi, Vici, and Casmir grant awards from the Netherlands Organisation for Scientific Research. His main interests are in mechanism of heterogeneous catalysis combining experimental and computation studies. He has published over 600 papers, 20 book chapters, and 7 patents.

Artem M. Abakumov Artem M. Abakumov graduated from the Department of Chemistry at Moscow State University in 1993, received his Ph.D. in Chemistry from the same University in 1997, and then continued working as a Researcher and Vice-Chair of Inorganic Chemistry Department. He spent about 3 years as a postdoctoral fellow and invited professor in the Electron Microscopy for Materials Research (EMAT) laboratory at the University of Antwerp and joined EMAT as a research leader in 2008. Since 2015 he holds a Full Professor position at Skolkovo Institute of Science and Technology (Skoltech) in Moscow, leading Skoltech Center for Energy Science and Technology as a Director. His research interests span over a wide range of subjects, from inorganic chemistry, solid state chemistry, and crystallography to battery materials and transmission electron microscopy. He has extended experience in characterization of metal-ion battery electrodes and electrocatalysts with advanced TEM techniques that has led to a better understanding of charge–discharge mechanisms, redox reactions, and associated structural transformations in various classes of cathode materials on different spatial scales. He has published over 350 papers, 5 book chapters, and 12 patents.

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Keith J. Stevenson Keith J. Stevenson received his Ph.D. in 1997 from the University of Utah under the supervision of Prof. Henry White. Subsequently, he held a postdoctoral appointment at Northwestern University (1997– 2000) and a tenured faculty appointment (2000–2015) at the University of Texas at Austin. At present, he is leading the development of a new graduate level university (Skolkovo Institute for Science and Technology) in Moscow, Russia, where he is Provost and the former Director of the Center for Energy Science and Technology (CEST). To date he has published over 325 peer-reviewed publications, 14 patents, and 6 book chapters in this field. He is a recipient of Society of Electroanalytical Chemistry Charles N. Reilley Award (2021).

Evgeny V. Antipov Evgeny V. Antipov graduated from the Department of Chemistry at Moscow State University in 1981, received his Ph.D. in Chemistry in 1986, DSc degree in Chemistry in 1998, and then continued working at the same University as a Researcher, Head of the Laboratory of Inorganic Crystal Chemistry, Professor, Head of Laboratory of fundamental research on aluminum production, and Head of the Department of Electrochemistry. Since 2018 he also holds a professor position at Skolkovo Institute of Science and Technology (Skoltech) in Moscow. Currently his research interests are mainly focused on inorganic materials for application in batteries and fuel cells. He has published more than 400 scientific articles and 14 patents.

Vivian W.W. Yam Professor Vivian W.W. Yam is the Chair Professor of Chemistry and Philip Wong Wilson Wong Professor in Chemistry and Energy at The University of Hong Kong. She received both her B.Sc (Hons) and Ph.D. from The University of Hong Kong. She was elected to Member of Chinese Academy of Sciences, International Member (Foreign Associate) of US National Academy of Sciences, Foreign Member of Academia Europaea, Fellow of TWAS, and Founding Member of Hong Kong Academy of Sciences. She was Laureate of 2011 L’Oréal-UNESCO For Women in Science Award. Her research interests include inorganic and organometallic chemistry, supramolecular chemistry, photophysics and photochemistry, and metal-based molecular functional materials for sensing, organic optoelectronics, and energy research. Also see: https://chemistry.hku.hk/wwyam.

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David L. Bryce David L. Bryce (B.Sc (Hons), 1998, Queen’s University; Ph.D., 2002, Dalhousie University; postdoctoral fellow, 2003–04, NIDDK/NIH) is Full Professor and University Research Chair in Nuclear Magnetic Resonance at the University of Ottawa in Canada. He is the past Chair of the Department of Chemistry and Biomolecular Sciences, a Fellow of the Royal Society of Chemistry, and an elected Fellow of the Royal Society of Canada. His research interests include solid-state NMR of low-frequency quadrupolar nuclei, NMR studies of materials, NMR crystallography, halogen bonding, mechanochemistry, and quantum chemical interpretation of NMR interaction tensors. He is the author of approximately 200 scientific publications and co-author of 1 book. He is the Editor-in-Chief of Solid State Nuclear Magnetic Resonance and Section Editor (Magnetic Resonance and Molecular Spectroscopy) for the Canadian Journal of Chemistry. He has served as the Chair of Canada’s National Ultrahigh-Field NMR Facility for Solids and is a past co-chair of the International Society for Magnetic Resonance conference and of the Rocky Mountain Conference on Magnetic Resonance Solid-State NMR Symposium. His work has been recognized with the Canadian Society for Chemistry Keith Laidler Award and with the Gerhard Herzberg Award of the Canadian Association of Analytical Sciences and Spectroscopy.

Paul R. Raithby Paul R. Raithby obtained his B.Sc (1973) and Ph.D. (1976) from Queen Mary College, University of London, working for his Ph.D. in structural inorganic chemistry. He moved to the University of Cambridge in 1976, initially as a postdoctoral researcher and then as a faculty member. In 2000, he moved to the University of Bath to take up the Chair of Inorganic Chemistry when he remains to the present day, having been awarded an Emeritus Professorship in 2022. His research interests have spanned the chemistry of transition metal cluster compounds, platinum acetylide complexes and oligomers, and lanthanide complexes, and he uses laboratory and synchrotron-based X-ray crystallographic techniques to determine the structures of the complexes and to study their dynamics using time-resolved photocrystallographic methods.

Angus P. Wilkinson

Angus P. Wilkinson completed his bachelors (1988) and doctoral (1992) degrees in chemistry at Oxford University in the United Kingdom. He spent a postdoctoral period in the Materials Research Laboratory, University of California, Santa Barbara, prior to joining the faculty at the Georgia Institute of Technology as an assistant professor in 1993. He is currently a Professor in both the Schools of Chemistry and Biochemistry, and Materials Science and Engineering, at the Georgia Institute of Technology. His research in the general area of inorganic materials chemistry makes use of synchrotron X-ray and neutron scattering to better understand materials synthesis and properties.

CONTRIBUTORS TO VOLUME 3 Musiri M Balakrishnarajan Chemical Information Science Lab, Pondicherry University, Puducherry, India

Katerina P Hilleke Department of Chemistry, State University of New York at Buffalo, Buffalo, NY, United States

Robert F Berger Department of Chemistry, Western Washington University, Bellingham, WA, United States

Koichi Kitahara Department of Advanced Materials Science, The University of Tokyo, Kashiwa, Chiba, Japan

Lewis J Conway Department of Materials Science & Metallurgy, University of Cambridge, Cambridge, United Kingdom

Álvaro Lobato MALTA-Consolider Team and Department of Physical and Analytical Chemistry, University of Oviedo, Oviedo, Spain

Sohail H Dar Chemical Information Science Lab, Pondicherry University, Puducherry, India Richard Dronskowski Institute of Inorganic Chemistry, RWTH Aachen University, Aachen, Germany Christina Ertural Institute of Inorganic Chemistry, RWTH Aachen University, Aachen, Germany Javier Fdez Sanz Departamento de Química Física, Facultad de Química, Universidad de Sevilla, Sevilla, Spain Daniel C Fredrickson Department of Chemistry, University of WisconsindMadison, Madison, WI, United States Émilie Gaudry University of Lorraine, CNRS, Institut Jean Lamour, Nancy, France Yuri Grin Max-Planck-Institut für Chemische Physik fester Stoffe, Dresden, Germany Andreas Hermann School of Physics and Astronomy, The University of Edinburgh, Edinburgh, United Kingdom

Antonio Marquez Departamento de Química Física, Facultad de Química, Universidad de Sevilla, Sevilla, Spain Madalynn Marshall Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, NJ, United States Peter C Müller Institute of Inorganic Chemistry, RWTH Aachen University, Aachen, Germany Pinku Nath BCAM - Basque Center for Applied Mathematics, Bilbao, Spain; and Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Sapporo, Japan Ryky Nelson Institute of Inorganic Chemistry, RWTH Aachen University, Aachen, Germany Hussien H Osman Department of Chemistry, Helwan University, Cairo, Egypt Pattath D Pancharatna Chemical Information Science Lab, Pondicherry University, Puducherry, India

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Chris J Pickard Department of Materials Science & Metallurgy, University of Cambridge, Cambridge, United Kingdom; and Advanced Institute for Materials Research, Tohoku University, Sendai, Japan

Nayuta Takemori Research Institute for Interdisciplinary Science, Okayama University, Okayama, Japan; and Center for Emergent Matter Science, RIKEN, Wako, Saitama, Japan

José J Plata Departamento de Química Física, Facultad de Química, Universidad de Sevilla, Sevilla, Spain

Yuta Tsuji Faculty of Engineering Sciences, Kyushu University, Kasuga, Fukuoka, Japan

J Manuel Recio MALTA-Consolider Team and Department of Physical and Analytical Chemistry, University of Oviedo, Oviedo, Spain

Ángel Vegas Hospital del Rey, University of Burgos, Burgos, Spain

Shiro Sakai Center for Emergent Matter Science, RIKEN, Wako, Saitama, Japan Miguel Ángel Salvadó MALTA-Consolider Team and Department of Physical and Analytical Chemistry, University of Oviedo, Oviedo, Spain J Christian Schön Max Planck Institute for Solid State Research, Stuttgart, Germany

Frank R Wagner Max-Planck-Institut für Chemische Physik fester Stoffe, Dresden, Germany Weiwei Xie Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, NJ, United States Eva Zurek Department of Chemistry, State University of New York at Buffalo, Buffalo, NY, United States

PREFACE Comprehensive Inorganic Chemistry III is a new multi-reference work covering the broad area of Inorganic Chemistry. The work is available both in print and in electronic format. The 10 Volumes review significant advances and examines topics of relevance to today’s inorganic chemists with a focus on topics and results after 2012. The work is focusing on new developments, including interdisciplinary and high-impact areas. Comprehensive Inorganic Chemistry III, specifically focuses on main group chemistry, biological inorganic chemistry, solid state and materials chemistry, catalysis and new developments in electrochemistry and photochemistry, as well as on NMR methods and diffractions methods to study inorganic compounds. The work continues our 2013 work Comprehensive Inorganic Chemistry II, but at the same time adds new volumes on emerging research areas and techniques used to study inorganic compounds. The new work is also highly complementary to other recent Elsevier works in Coordination Chemistry and Organometallic Chemistry thereby forming a trio of works covering the whole of modern inorganic chemistry, most recently COMC-4 and CCC-3. The rapid pace of developments in recent years in all areas of chemistry, particularly inorganic chemistry, has again created many challenges to provide a contemporary up-to-date series. As is typically the challenge for Multireference Works (MRWs), the chapters are designed to provide a valuable long-standing scientific resource for both advanced students new to an area as well as researchers who need further background or answers to a particular problem on the elements, their compounds, or applications. Chapters are written by teams of leading experts, under the guidance of the Volume Editors and the Editors-inChief. The articles are written at a level that allows undergraduate students to understand the material, while providing active researchers with a ready reference resource for information in the field. The chapters are not intended to provide basic data on the elements, which are available from many sources including the original CIC-I, over 50-years-old by now, but instead concentrate on applications of the elements and their compounds and on high-level techniques to study inorganic compounds. Vol. 1: Synthesis, Structure, and Bonding in Inorganic Molecular Systems; Risto S. Laitinen In this Volume the editor presents an historic overview of Inorganic Chemistry starting with the birth of inorganic chemistry after Berzelius, and a focus on the 20th century including an overview of “inorganic” Nobel Prizes and major discoveries, like inert gas compounds. The most important trends in the field are discussed in an historic context. The bulk of the Volume consists of 3 parts, i.e., (1) Structure, bonding, and reactivity in inorganic molecular systems; (2) Intermolecular interactions, and (3) Inorganic Chains, rings, and cages. The volume contains 23 chapters. Part 1 contains chapters dealing with compounds in which the heavy p-block atom acts as a central atom. Some chapters deal with the rich synthetic and structural chemistry of noble gas compounds, low-coordinate p-block elements, biradicals, iron-only hydrogenase mimics, and macrocyclic selenoethers. Finally, the chemistry and application of weakly coordinating anions, the synthesis, structures, and reactivity of carbenes containing non-innocent ligands, frustrated Lewis pairs in metal-free catalysis are discussed. Part 2 discusses secondary bonding interactions that play an important role in the properties of bulk materials. It includes a chapter on the general theoretical considerations of secondary bonding interactions, including halogen and chalcogen bonding. This section is concluded by the update of the host-guest chemistry of the molecules of p-block elements and by a comprehensive review of closed-shell metallophilic interactions. The third part of the Volume is dedicated to chain, ring and cage (or cluster) compounds in molecular inorganic chemistry. Separate

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chapters describe the recent chemistry of boron clusters, as well as the chain, ring, and cage compounds of Group13 and 15, and 16 elements. Also, aromatic compounds bearing heavy Group 14 atoms, polyhalogenide anions and Zintl-clusters are presented. Vol. 2: Bioinorganic Chemistry and Homogeneous Biomimetic Inorganic Catalysis; Vincent L. Pecoraro and Zijian Guo In this Volume, the editors have brought together 26 chapters providing a broad coverage of many of the important areas involving metal compounds in biology and medicine. Readers interested in fundamental biochemistry that is assisted by metal ion catalysis, or in uncovering the latest developments in diagnostics or therapeutics using metal-based probes or agents, will find high-level contributions from top scientists. In the first part of the Volume topics dealing with metals interacting with proteins and nucleic acids are presented (e.g., siderophores, metallophores, homeostasis, biomineralization, metal-DNA and metal-RNA interactions, but also with zinc and cobalt enzymes). Topics dealing with iron-sulfur clusters and heme-containing proteins, enzymes dealing with dinitrogen fixation, dihydrogen and dioxygen production by photosynthesis will also be discussed, including bioinspired model systems. In the second part of the Volume the focus is on applications of inorganic chemistry in the field of medicine: e.g., clinical diagnosis, curing diseases and drug targeting. Platinum, gold and other metal compounds and their mechanism of action will be discussed in several chapters. Supramolecular coordination compounds, metal organic frameworks and targeted modifications of higher molecular weight will also be shown to be important for current and future therapy and diagnosis. Vol. 3: Theory and Bonding of Inorganic Non-molecular Systems; Daniel C. Fredrickson This volume consists of 15 chapters that build on symmetry-based expressions for the wavefunctions of extended structures toward models for bonding in solid state materials and their surfaces, algorithms for the prediction of crystal structures, tools for the analysis of bonding, and theories for the unique properties and phenomena that arise in these systems. The volume is divided into four parts along these lines, based on major themes in each of the chapters. These are: Part 1: Models for extended inorganic structures, Part 2: Tools for electronic structure analysis, Part 3: Predictive exploration of new structures, and Part 4: Properties and phenomena. Vol. 4: Solid State Inorganic Chemistry; P. Shiv Halasyamani and Patrick M. Woodward In a broad sense the field of inorganic chemistry can be broken down into substances that are based on molecules and those that are based on extended arrays linked by metallic, covalent, polar covalent, or ionic bonds (i.e., extended solids). The field of solid-state inorganic chemistry is largely concerned with elements and compounds that fall into the latter group. This volume contains nineteen chapters covering a wide variety of solid-state inorganic materials. These chapters largely focus on materials with properties that underpin modern technology. Smart phones, solid state lighting, batteries, computers, and many other devices that we take for granted would not be possible without these materials. Improvements in the performance of these and many other technologies are closely tied to the discovery of new materials or advances in our ability to synthesize high quality samples. The organization of most chapters is purposefully designed to emphasize how the exceptional physical properties of modern materials arise from the interplay of composition, structure, and bonding. Not surprisingly this volume has considerable overlap with both Volume 3 (Theory and Bonding of Inorganic NonMolecular Systems) and Volume 5 (Inorganic Materials Chemistry). We anticipate that readers who are interested in this volume will find much of interest in those volumes and vice versa Vol. 5: Inorganic Materials Chemistry; Ram Seshadri and Serena Cussen This volume has adopted the broad title of Inorganic Materials Chemistry, but as readers would note, the title could readily befit articles in other volumes as well. In order to distinguish contributions in this volume from

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those in other volumes, the editors have chosen to use as the organizing principle, the role of synthesis in developing materials, reflected by several of the contributions carrying the terms “synthesis” or “preparation” in the title. It should also be noted that the subset of inorganic materials that are the focus of this volume are what are generally referred to as functional materials, i.e., materials that carry out a function usually through the way they respond to an external stimulus such as light, or thermal gradients, or a magnetic field.

Vol. 6: Heterogeneous Inorganic Catalysis; Rutger A. van Santen and Emiel J. M. Hensen This Volume starts with an introductory chapter providing an excellent discussion of single sites in metal catalysis. This chapter is followed by 18 chapters covering a large part of the field. These chapters have been written with a focus on the synthesis and characterization of catalytic complexity and its relationship with the molecular chemistry of the catalytic reaction. In the 1950s with the growth of molecular inorganic chemistry, coordination chemistry and organometallic chemistry started to influence the development of heterogeneous catalysis. A host of new reactions and processes originate from that time. In this Volume chapters on major topics, like promoted Fischer-Tropsch catalysts, structure sensitivity of well-defined alloy surfaces in the context of oxidation catalysis and electrocatalytic reactions, illustrate the broadness of the field. Molecular heterogeneous catalysts rapidly grew after high-surface synthetic of zeolites were introduced; so, synthesis, structure and nanopore chemistry in zeolites is presented in a number of chapters. Also, topics like nanocluster activation of zeolites and supported zeolites are discussed. Mechanistically important chapters deal with imaging of single atom catalysts. An important development is the use of reducible supports, such as CeO2 or Fe2O3 where the interaction between the metal and support is playing a crucial role.

Vol. 7: Inorganic Electrochemistry; Keith J. Stevenson, Evgeny V. Antipov and Artem M. Abakumov This volume bridges several fields across chemistry, physics and material science. Perhaps this topic is best associated with the book “Inorganic Electrochemistry: Theory, Practice and Applications” by Piero Zanello that was intended to introduce inorganic chemists to electrochemical methods for study of primarily molecular systems, including metallocenes, organometallic and coordination complexes, metal complexes of redox active ligands, metal-carbonyl clusters, and proteins. The emphasis in this Volume of CIC III is on the impact of inorganic chemistry on the field of material science, which has opened the gateway for inorganic chemists to use more applied methods to the broad areas of electrochemical energy storage and conversion, electrocatalysis, electroanalysis, and electrosynthesis. In recognition of this decisive impact, the Nobel Prize in Chemistry of 2019 was awarded to John B. Goodenough, M. Stanley Whittingham, and Akira Yoshino for the development of the lithium-ion battery.

Vol. 8: Inorganic Photochemistry; Vivian W. W. Yam In this Volume the editor has compiled 19 chapters discussing recent developments in a variety of developments in the field. The introductory chapter overviews the several topics, including photoactivation and imaging reagents. The first chapters include a discussion of using luminescent coordination and organometallic compounds for organic light-emitting diodes (OLEDs) and applications to highlight the importance of developing future highly efficient luminescent transition metal compounds. The use of metal compounds in photo-induced bond activation and catalysis is highlighted by non-sacrificial photocatalysis and redox photocatalysis, which is another fundamental area of immense research interest and development. This work facilitates applications like biological probes, drug delivery and imaging reagents. Photochemical CO2 reduction and water oxidation catalysis has been addressed in several chapters. Use of such inorganic compounds in solar fuels and photocatalysis remains crucial for a sustainable environment. Finally, the photophysics and photochemistry of lanthanoid compounds is discussed, with their potential use of doped lanthanoids in luminescence imaging reagents.

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Vol. 9: NMR of Inorganic Nuclei; David L. Bryce Nuclear magnetic resonance (NMR) spectroscopy has long been established as one of the most important analytical tools at the disposal of the experimental chemist. The isotope-specific nature of the technique can provide unparalleled insights into local structure and dynamics. As seen in the various contributions to this Volume, applications of NMR spectroscopy to inorganic systems span the gas phase, liquid phase, and solid state. The nature of the systems discussed covers a very wide range, including glasses, single-molecule magnets, energy storage materials, bioinorganic systems, nanoparticles, catalysts, and more. The focus is largely on isotopes other than 1H and 13C, although there are clearly many applications of NMR of these nuclides to the study of inorganic compounds and materials. The value of solid-state NMR in studying the large percentage of nuclides which are quadrupolar (spin I > ½) is apparent in the various contributions. This is perhaps to be expected given that rapid quadrupolar relaxation can often obfuscate the observation of these resonances in solution. Vol. 10: X-ray, Neutron and Electron Scattering Methods in Inorganic Chemistry; Angus P. Wilkinson and Paul R. Raithby In this Volume the editors start with an introduction on the recent history and improvements of the instrumentation, source technology and user accessibility of synchrotron and neutron facilities worldwide, and they explain how these techniques work. The modern facilities now allow inorganic chemists to carry out a wide variety of complex experiments, almost on a day-to-day basis, that were not possible in the recent past. Past editions of Comprehensive Inorganic Chemistry have included many examples of successful synchrotron or neutron studies, but the increased importance of such experiments to inorganic chemists motivated us to produce a separate volume in CIC III dedicated to the methodology developed and the results obtained. The introduction chapter is followed by 15 chapters describing the developments in the field. Several chapters are presented covering recent examples of state-of-the-art experiments and refer to some of the pioneering work leading to the current state of the science in this exciting area. The editors have recognized the importance of complementary techniques by including chapters on electron crystallography and synchrotron radiation sources. Chapters are present on applications of the techniques in e.g., spin-crossover materials and catalytic materials, and in the use of time-resolved studies on molecular materials. A chapter on the worldwide frequently used structure visualization of crystal structures, using PLATON/PLUTON, is also included. Finally, some more specialized studies, like Panoramic (in beam) studies of materials synthesis and high-pressure synthesis are present. Direct observation of transient species and chemical reactions in a pore observed by synchrotron radiation and X-ray transient absorption spectroscopies in the study of excited state structures, and ab initio structure solution using synchrotron powder diffraction, as well as local structure determination using total scattering data, are impossible and unthinkable without these modern diffraction techniques. Jan Reedijk, Leiden, The Netherlands Kenneth R. Poeppelmeier, Illinois, United States March 2023

3.01

Introduction: Theory and bonding of inorganic non-molecular systems

Daniel C. Fredrickson, Department of Chemistry, University of WisconsindMadison, Madison, WI, United States © 2023 Elsevier Ltd. All rights reserved.

References

3

Abstract The term “non-molecular inorganic systems” encompasses a remarkable fraction of the compounds and materials that can be created from the elements of the periodic table. It includes all of the minerals on display at natural history and geological museums, the metals and alloys that accompanied the rise of civilization to the modern day, zeolites and heterogeneous catalysts that underlie much of industrial chemistry, and the wide range of other solid state compounds and intermetallic phases that have been created synthetically. In addition, surfaces, inorganic polymers, and 2D materials would also fit into this category. A comprehensive coverage of the electronic structure and bonding of all of these materials would seem to be a daunting task, especially as in many cases these compounds represent an open frontier for the development of new chemical concepts. Instead, this volume is designed to explore the ways in which models, theoretical analysis, and computational algorithms can be used to elucidate and predict the structures and properties of inorganic extended structures, and do so in terms that invite the participation of inorganic chemists in the field.

The term “non-molecular inorganic systems” encompasses a remarkable fraction of the compounds and materials that can be created from the elements of the periodic table. It includes all of the minerals on display at natural history and geological museums, the metals and alloys that accompanied the rise of civilization to the modern day, zeolites and heterogeneous catalysts that underlie much of industrial chemistry, and the wide range of other solid state compounds and intermetallic phases that have been created synthetically. In addition, surfaces, inorganic polymers, and 2D materials would also fit into this category. A comprehensive coverage of the electronic structure and bonding of all of these materials would seem to be a daunting task, especially as in many cases these compounds represent an open frontier for the development of new chemical concepts. Instead, this volume is designed to explore the ways in which models, theoretical analysis, and computational algorithms can be used to elucidate and predict the structures and properties of inorganic extended structures, and do so in terms that invite the participation of inorganic chemists in the field. The electronic structures of periodic systems are, in fact, uniquely accessible to those trained in inorganic chemistry, with its emphasis on symmetry and crystallography. Band theory represents an extension of the group theoretical analysis we are familiar with for molecular complexes to include translations as symmetry operations. Whereas in the molecular case, a point group will have a limited number of irreducible representations, the essentially infinite number of lattice translations that can be carried out on a periodic structure leads to a continuum of irreducible representations that are best mapped out in reciprocal space, the same domain in which the diffraction patterns of crystals are analyzed, with a vector usually called k. This connection can be seen in the usual generation of symmetry adapted linear combinations (SALCs) with the projector operator1: 1 X  ^ fSALC;k ¼ pffiffiffiffi c ^ Rfstart (1) N R^ k;R where the SALC transforming as an irreducible representation k is obtained by applying all of the symmetry operations of a point group to a starting basis orbital, fstart, multiplying the results by the complex conjugates for the characters of the operations, c , k;b R and collecting the transformed orbitals into a single function. Analogous SALCS can be made for translationally-equivalent atomic orbitals in a crystal: fSALC;k;j ðrÞ ¼

XXXeik$ðna aþnb bþnc cÞ pffiffiffiffi fj ðr  na a  nb b  nc cÞ N na nb nc

(2)

where e ik $ (naa þ nbb D ncc) is the character for a translation of the lattice by naa þ nbb þ ncc for the reciprocal space point k, and j goes over the atomic orbitals within the unit cell in a crystal containing N unit cells (with N ➔ N). Functions with different k values transform as different irreducible representations of the translational symmetry, and thus do not interact with each other. The full the 1-electron wavefunctions for a 3D crystal then have the form ) ( X XXXeik$ðna aþnb bþnc cÞ pffiffiffiffi jk;n ðrÞ ¼ ck;n;j (3) fj ðr  na a  nb b  nc cÞ N na nb nc j

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1

2

Introduction: Theory and bonding of inorganic non-molecular systems

which are simply linear combinations of SALCs with the same k value. This volume consists of 15 chapters that build on this symmetry-based expression for the wavefunctions of extended structures toward models for bonding in solid state materials and their surfaces, algorithms for the prediction of crystal structures, tools for the analysis of bonding, and theories for the unique properties and phenomena that arise in these systems. The volume can be divided into four parts along these lines, based on major themes in each of the chapters: Section 1: Models for extended inorganic structures Chapter Chapter Chapter Chapter Chapter

3.02: 3.03: 3.04: 3.05: 3.06:

Electronic structure of oxide and halide perovskites Bonding in boron-rich borides The Zintl-Klemm concept and its broader extensions An introduction to the theory of inorganic solid surfaces Bond activation and formation on inorganic surfaces

Section 2: Tools for electronic structure analysis Chapter 3.07: Chemical bonding with plane waves Chapter 3.08: Chemical bonding analyses using wannier functions Chapter 3.09: Chemical bonding analysis in position space Chapter 3.10: The structures of inorganic crystals: A rational explanation from the chemical pressure approach and the anions in metallic matrices model Section 3: Predictive exploration of new structures Chapter 3.11: Energy landscapes in inorganic chemistry Chapter 3.12: First principles crystal structure prediction Chapter 3.13: Crystal chemistry at high pressure Section 4: Properties and phenomena Chapter 3.14: In silico modeling of inorganic thermoelectric materials Chapter 3.15: Correlated electronic states in quasicrystals Chapter 3.16: Chemical bonding principles in magnetic topological quantum materials Section 1 begins with an overview of the band structures of perovskites (see Chapter 3.02), one of the most intensively studied families of oxides. The simple assignment of oxidation states from a classical inorganic electron counting gives way to a much richer story, especially as the chapter explores how the common distortion modes away from the cubic symmetry of the parent geometry influence the electronic picture. The coverage of selected families of inorganic materials continues with borides (see Chapter 3.03). Here, complex structures based on the overlapping boron cages are elegantly rationalized with an extension of the Wades-Mingos rules2 for borohydrides. The power of such molecular-style electron-counting schemes if further developed in Chapter 3.04, as the Zintl-Klemm concept3 is first described in its traditional application to intermetallic phases and then shown to form a framework to account for the structure of countless inorganic structures via the Anions in Metallic Matrices model.4 In Chapters 3.05 and 3.06, the discussion moves beyond bulk solid state materials to the bonding at their surfaces, the locations that are of the most interest for chemical reactivity and catalysis. This begins with a broad and thorough overview of the theory of inorganic surfaces (see Chapter 3.05), which flows naturally into a practical account of how bond activation on surfaces can be studied computationally (see Chapter 3.06). The theme of models for understanding structural chemistry continues in the later chapters beyond Section 1. For example, an extension of the 18 electron rule to intermetallics will emerge in the demonstration of Wannier analysis in Chapter 3.08, while the Anions in Metallic Matrices model is elaborated with chemical pressure analysis in Chapter 3.10. Similarly, principles of high pressure chemistry form a core component of Chapter 3.13. In Section 2, we turn to the theoretical tools that have been developed to analyze bonding in extended structures, and thus support the creation of new models. Chapter 3.07 introduces the Local-Orbital Basis Suite Towards Electronic-Structure Reconstruction (LOBSTER),5 which allows the results of planewave DFT calculations to be analyzed in terms of atomic orbitals and their interactions. Here, classic tools popularized by and extensively applied to solid state structures with extended Hückel calculations6dprojected density of states distributions, crystal orbital overlap populations, crystal orbital Hamilton populations and Mulliken charges are made available for state-of-the-art DFT electronic structures. The capabilities of these methods to elucidate an extensive range of chemical problems are also reviewed. In Chapter 3.08, the Wannier analysis7 is introduced, an approach that takes the idea of localized MOs to an extreme level: a full band structure can be transformed into localized functions in individual unit cells. In this way, surprisingly simple bonding schemes can be derived for some intermetallic phases with coordination numbers that far beyond those commonly encountered in molecules. Next, Chapters 3.09 and 3.10 cover approaches to understanding structure and bonding not with the wavefunctions themselves, but with functions that integrate the information contained in them into useful forms. Chapter 3.09 illustrates how this can be done elegantly in the context of the Quantum Theory of Atoms in Molecules8 by investigating the topography of the electronic density and measures of electron localization9,10 in parallel with each other. In Chapter 3.10, the chemical pressure method11,12 is

Introduction: Theory and bonding of inorganic non-molecular systems

3

introduced, in which maps of the local pressures within structures are constructed from DFT calculations, revealing forces underlying chemical bonding. With the chemical pressure approach, the anions in metallic lattices model is vividly affirmed as regions of negative pressure in metallic crystal structures anticipate the anion positions in inorganic salts. Intriguing hints at correspondences between physical and chemical pressure also arise here. Section 3 turns to the historically vexing challenge of crystal structure prediction13 and the breathtaking progress that has been made in this area in recent years. The nature and magnitude of the problem is expounded in Chapter 3.11, with a comprehensive overview of energy landscapes in inorganic systems and approaches to investigating them. Then, in Chapter 3.12, the ingenious methods for structure prediction available to the community are covered, such as evolutionary algorithms. This overview then culminates in Chapter 3.13, which demonstrates how the use of these tools can guide the exploration of high-pressure materials. Altogether, these three chapters offer a comprehensive resource to researchers interested in exploring potential crystal structures computationally. This volume concludes with selected topics on how theory and electronic structure calculations can be applied to understanding and predicting the physical properties of solid state compounds. Thermoelectric behavior (see Chapter 3.14), correlated electronic states (e.g. superconductivity) in quasicrystals (see Chapter 3.15), and topological phenomena (see Chapter 3.16) are chosen as they highlight different issues in modeling materials properties. To predict the thermoelectric efficiency of a compound, one must integrate calculations of the electrical conductivity for different carrier types and thermal conductivity. As such, it represents the net effect of several more basic properties that are closely tied to the features of a crystal structure, as well as its electronic and phonon band structures. Meanwhile, superconductivity in quasicrystals offers the chance to understand this fascinating quantum phenomenon outside of the usual framework of the reciprocal lattice of a periodic structure. Finally, in topological materials, one sees how the extensions of group theory to band structures sets the stage for new quantum states, the discovery of which can be led by chemical principles. Throughout these chapters, common themes are the open questions and challenges in the theory and computational modeling of inorganic non-molecular systems, with their remarkable diversity. From bonding models for diverse compounds and their surfaces, to tools for bonding analysis and structure prediction, and finally to the anticipation and calculation of physical properties, this volume represents an invitation to join in the pursuit of the vast opportunities offered by this field.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, 1990. Burdett, J. K. Molecular Shapes: Theoretical Models of Inorganic Stereochemistry, Wiley: New York, 1980. Kauzlarich, S. M., Ed.; Chemistry, Structure, and Bonding of Zintl Phases and Ions, VCH: New York, 1996. Vegas, Á.; Santamaria-Pérez, D.; Marqués, M.; Flórez, M.; García Baonza, V.; Recio, J. M. Anions in Metallic Matrices Model: Application to the Aluminium Crystal Chemistry. Acta Crystallogr. B 2006, 62, 220–227. Maintz, S.; Deringer, V. L.; Tchougréeff, A. L.; Dronskowski, R. LOBSTER: A Tool to Extract Chemical Bonding from Plane-Wave Based DFT. J. Comput. Chem. 2016, 37, 1030–1035. Hoffmann, R. Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures, VCH Publishers: New York, NY, 1988. Marzari, N.; Mostofi, A. A.; Yates, J. R.; Souza, I.; Vanderbilt, D. Maximally Localized Wannier Functions: Theory and Applications. Rev. Mod. Phys. 2012, 84, 1419–1475. Bader, R. F. W. Atoms in Molecules: A Quantum Theory, Oxford University Press: Oxford, England, 1990. Savin, A.; Nesper, R.; Wengert, S.; Fässler, T. F. ELF: The Electron Localization Function. Angew. Chem. Int. Ed. 1997, 36, 1808–1832. Kohout, M.; Pernal, K.; Wagner, F. R.; Grin, Y. Electron Localizability Indicator for Correlated Wavefunctions. I Parallel-spin Pairs. Theor. Chem. Acc. 2004, 112, 453–459. Osman, H. H.; Salvadó, M. A.; Pertierra, P.; Engelkemier, J.; Fredrickson, D. C.; Recio, J. M. Chemical Pressure Maps of Molecules and Materials: Merging the Visual and Physical in Bonding Analysis. J. Chem. Theory Comput. 2018, 14, 104–114. Lu, E.; Van Buskirk, J. S.; Cheng, J.; Fredrickson, D. C. Tutorial on Chemical Pressure Analysis: How Atomic Packing Drives Laves/Zintl Intergrowth in K3Au5Tl. Crystals 2021, 11, 906. Gavezzotti, A. Are Crystal Structures Predictable? Acc. Chem. Res. 1994, 27, 309–314.

3.02

Electronic structure of oxide and halide perovskites

Robert F. Berger, Department of Chemistry, Western Washington University, Bellingham, WA, United States © 2023 Elsevier Ltd. All rights reserved.

3.02.1 3.02.2 3.02.2.1 3.02.2.2 3.02.3 3.02.4 3.02.4.1 3.02.4.2 3.02.5 3.02.5.1 3.02.5.2 3.02.5.3 3.02.6 3.02.6.1 3.02.6.2 3.02.6.3 3.02.7 3.02.7.1 3.02.7.2 3.02.8 Acknowledgment References

Introduction Perovskite structure and composition Cubic perovskite structure and variety in elemental composition Common perovskite distortions Computational methods for perovskite electronic structure Conceptualizing cubic perovskite band structures SrTiO3, a d0 oxide perovskite CsPbI3, a halide perovskite photovoltaic Effect of composition on band structure Effect of the X-site anion Effect of the B-site cation Effect of the A-site cation Effects of distortions on band structure Ferroelectric distortion Octahedral rotation Unifying messages Other strategies for tuning the band structure High pressure and epitaxial strain Superstructures with reduced dimensionality Concluding remarks

4 5 5 6 8 9 9 12 13 14 14 15 15 16 17 17 18 19 19 20 21 21

Abstract Compounds crystallizing in the ABX3 perovskite structure are studied for a remarkable variety of technologies. Particularly for applications such as photovoltaics and photocatalysis, it is crucial to understand the key features of perovskite electronic structure and how they can be tuned by modifying the composition and crystal structure. This chapter begins with an overview of the compositional and structural diversity of perovskites. Then, density functional theory-based computational methods that have been used to study perovskite compounds are described. Next, the electronic band structures of an undistorted oxide (SrTiO3) and halide (CsPbI3) perovskite are explained in detail, merging the viewpoints of crystal wavefunctions as both linear combinations of atomic orbitals and perturbed plane waves. Finally, routes toward the tunability of perovskite electronic structure and properties are reviewed for various modifications: changes in elemental composition, various modes of geometric distortion, the application of high pressure or strain, and the formation of superstructures with reduced dimensionality. While the concepts and discussion herein are relevant to all perovskite compounds, the examples described in this chapter are mainly d0 oxide perovskite photocatalysts and halide perovskite photovoltaics.

3.02.1

Introduction

Perovskite compounds exhibit remarkable and ever-increasing variety in both their chemical composition and technological applications. There are active fields of research in designing and optimizing perovskite-related compounds for applications including ferroelectricity,1–3 piezoelectricity,4–6 high-temperature superconductivity,7–9 and photocatalysis.10–14 Perhaps most notably in the past decade, lead-halide perovskites have rapidly been developed as light-absorbing materials in photovoltaic technologies with efficiencies comparable to silicon solar cells.15–19 While each of these applications is distinct, each is governed by the complex relationships among perovskite composition, atomic structure, and electronic structure. The overarching goal of this chapter is to organize and rationalize these relationships, developing a cohesive intuitive picture of the electronic band structure of perovskites and how it might be modified. The discovery and structural exploration of perovskite compounds, which is covered thoroughly in a historical review by Bhalla et al.,2 goes back long before interest in or ability to compute their electronic properties. “Perovskite” originally referred to the naturally-occurring mineral CaTiO3, discovered in 1839. The term now classifies a large number of solids with the same 1:1:3

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Electronic structure of oxide and halide perovskites

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stoichiometry and similar crystal structures, described in Section 2. Much of the pioneering work to expand and describe this class of compounds was done in the 1920s by Victor Goldschmidt. In 1926, Goldschmidt published a simple framework based on the packing of hard spheres to rationalize the structural stability of perovskites.20 The framework explains not only why certain combinations of elements form perovskite compounds, but also the ways in which these structures tend to distort. Implicit in Goldschmidt’s work is perhaps the most important contributor to the incredible versatility of perovskites in a diverse range of applications: that small changes in composition and conditions can access degrees of subtle geometric tunability. Put another way, perovskites have shallow potential energy landscapes, and are always on the brink of distortion. This is why, for example, BaTiO3 is a ferroelectric material with a remarkably high dielectric constant, used in capacitors.1 It is also why PbZrxTi1–xO3 is a piezoelectric material, distorting under mechanical strain.4 In perovskite photovoltaics (e.g., CH3NH3PbI3) and photocatalysts (e.g., SrTiO3) as well, changes in composition can tune structural parameters, allowing for the optimization of band gaps, band-edge energies, and solar energy conversion efficiency. Over time, synthetic chemists and materials scientists have expanded the toolset with which they can tune the composition and crystal structure of perovskites. Common approaches include doping and elemental substitution, engineering various types and concentrations of defects,21–24 applying pressure25–27 or anisotropic strain,28–35 and designing layered superstructures and nanostructures.36–40 All told, the compositional and structural parameter space of perovskites is virtually infinite, presenting both an infinite opportunity and an infinite challenge. The opportunity is that the vast array of perovskite-related materials that have been explored to date are surely just a fraction of those that are possible. Discovery of new compounds and tuning of existing ones will continue. The challenge is the exploration of this vast phase space runs the risk of being inefficient without a deep understanding of how perovskite composition and atomic structure govern the properties of interest. Computation can be a valuable tool in predicting the properties and potential utility of perovskite materials. In our view, whether computational predictions are based on density functional theory (DFT) calculations or machine learning approaches, they become even more powerful when rooted in intuitive patterns. We begin this chapter with an overview of the compositional and structural diversity of perovskites, which may be review for readers already familiar with this class of materials. Then, we discuss the DFT-based computational methods that have typically been used to calculate perovskite compounds and their electronic properties. Next, we take a detailed conceptual tour of the electronic band structures of an oxide (SrTiO3) and a halide (CsPbI3) perovskite, bringing together the chemical (i.e., linear combinations of atomic orbitals) and physical (i.e., perturbed plane waves) viewpoints. Finally, informed by past experimental and computational work, we describe the electronic tunability brought about by various modifications to perovskite compounds: changes in elemental composition, various modes of geometric distortion, the application of high pressure or strain, and the formation of superstructures with reduced dimensionality. This chapter is written with two audiences in mind. For a student entering the field with a background in undergraduate chemistry and physics, it is intended to provide an introduction to the relationships among the atomic and electronic structure of perovskite compounds. For an expert in the field, it is intended to provide an organizational framework for viewing perovskite structure/ property relationships in new ways. We hope all readers come away with a clearer understanding of the electronic structure of these remarkable materials.

3.02.2

Perovskite structure and composition

3.02.2.1

Cubic perovskite structure and variety in elemental composition

 is shown in Fig. 1A and B. Perovskite compounds have the general The prototypical cubic perovskite structure (space group Pm3m) stoichiometry ABX3, in which an A-site cation lies at unit cell vertices, a B-site cation lies at the body center, and X-site anions lie at the face centers. While Fig. 1A shows all ions explicitly, Fig. 1B is a common representation in which a B-site cation and its surrounding octahedron of X-site nearest neighbors are shown as a solid unit. Perovskites are often discussed as two main classes of compounds: those with oxide anions (O2
) at the X site, and those with halide anions (F
, Cl
, Br
, I
). In both classes, there is significant variety in the elemental composition of the A and B sites. To balance the charge of three oxide anions per formula unit, an oxide perovskite typically has the form AþB5þ(X2
)3, A2þB4þ(X2
)3, or A3þB3þ(X2
)3. Fig. 1C shows the range of elements occupying the A and B sites of known oxide perovskites, based on the list of 254 experimentally observed oxide perovskites gathered in the work of Balachandran et al.41 The A site may be occupied by an alkali or alkaline earth element, a lanthanide element, or another monovalent, divalent, or trivalent cation. The B site is often occupied by a transition metal, but may be a main-group, lanthanide, or actinide element. Halide perovskites usually have the form AþB2þ(X
)3. Their variety of A- and B-site cations is shown in Fig. 1D, based on the list of 76 inorganic halide perovskites known to be thermodynamically stable at room temperature and pressure gathered in the work of Travis et al.42 In inorganic halide perovskites, the A site is occupied by an alkali metal or other monovalent cation. Even beyond these, it is important to note that many of the most widely studied halide perovskites (e.g., light-absorbing lead-halides for photovoltaic applications) are hybrid organic-inorganic compounds, which have polyatomic cations such as methylammonium (CH3NH3þ) or formamidinium (CH(NH2)2þ) at the A site. The B site in halide perovskites is occupied by an alkaline earth, transition metal, or other divalent cation.

6

Electronic structure of oxide and halide perovskites

Fig. 1 The undistorted cubic perovskite (ABX3) unit cell, shown (A) in ball-and-stick form and (B) with a B–X octahedron. A wide variety of elements are found in bulk inorganic ABX3 (C) oxide and (D) halide perovskites. Panel C is based on the list of 254 experimentally observed oxide perovskites considered in the work of Balachandran et al.41 Panel D is based on the list of 76 experimentally observed (at room temperature and pressure) inorganic halide perovskites considered in the work of Travis et al.42

The compositional variety shown in Fig. 1C and D is truly only the tip of the iceberg, as it represents only thermodynamically stable bulk inorganic ABX3 perovskites. In addition, perovskite composition may be tuned by doping small concentrations of other elements at each site, as in oxynitrides of the form ABO3–xNx.43–45 Superstructures exist with ordered arrangements of multiple elements at a given site, such as double perovskites with the formula A2B0 BX6.14,46–51,52 Polyatomic organic cations (rather than monatomic inorganic ones) may occupy the A site. Metastable perovskite compounds may be synthesized even if they are not thermodynamically stable at room temperature. While some examples of these modifications will be discussed, our goal in this chapter is not to enumerate the massive compositional variety of perovskite compounds, but rather to identify and rationalize the key features of their electronic structure. We therefore choose to focus much of our analysis on two specific groups of compounds: d0 oxide perovskites (e.g., SrTiO3) and group IV halide perovskites (e.g., CsPbI3). These groups are both of interest in solar energy conversion (the oxides for photocatalysis, the halides for photovoltaics), and they both have closed-shell electron configurations and band gaps that make their electronic structures simpler to conceptualize. Importantly, the key points of our discussion of the electronic structure of these specific groups of perovskites can be extended to other oxide and halide perovskites with different electron counts.

3.02.2.2

Common perovskite distortions

At this point, we must address the important fact that most perovskites are not cubic at room temperature. While a few compounds (e.g., SrTiO3, KTaO3, and CH3NH3PbBr3) are cubic at room temperature and others become cubic when their temperatures are

Electronic structure of oxide and halide perovskites

7

elevated by a few hundred degrees, structural distortions are prevalent in most perovskites and play a key role in tuning their electronic structure. Perovskite distortions tend to fall into two main classes, illustrated in Fig. 2. In one class (Fig. 2A), the cation and anion sublattices translate relative to each other. Such distortions may result in a compound having a dipole moment, as in ferroelectric distortions in BaTiO3 and CsGeI3. Alternatively, the movements of individual ions may cancel, as in the nonpolar antiferroelectric distortion in NaNbO3. These relative translations of ions may occur along various axes within a perovskite. For example, translations in BaTiO3 are oriented along a cubic unit cell axis (001), while those in CsGeI3 are oriented along the body diagonal (111). In the other class of perovskite distortions (Fig. 2B), B–X octahedra exhibit patterns of collective rotation, as in CaTiO3 and CsPbI3. These octahedral rotations can be oriented around various perovskite axes, and can differ in whether consecutive layers rotate in the same or opposite sense. Modes of octahedral rotation are often described using Glazer notation,53,54 which indicates whether consecutive layers along a particular axis rotate in the same sense (þ), opposite sense (
), or not at all (0). For example, the rotation pattern in both CaTiO3 and CsPbI3 is notated as aþ b
b
. Two rotation patterns that are easier to visualize, a0a0cþ (observed in high-temperature CsSnI3) and a0a0c
(observed in low-temperature SrTiO3), are illustrated in Fig. 2B. Often, a given compound has a series of multiple phase transitions from lower symmetry to higher symmetry as its temperature rises. Though perovskite compounds generally possess shallow, complex structural energy landscapes, their distortions are largely driven by simple considerations of ionic size. For almost a century, the Goldschmidt tolerance factor20 and modern updates to it55–57 have been used as approximate predictors of whether and how a given combination of elements is likely to distort from the cubic perovskite structure. The classic definition of the tolerance factor t is given in Eq. (1): rA þ rX t ¼ pffiffiffi 2 ðrB þ rX Þ

(1)

where the r’s are the ionic radii of the A-, B-, and X-site elements. If the tolerance factor is close to 1, then a set of elements is essentially able to pack together like marbles in the perovskite structure, adopting stable bond distances without distortion. If, however, the tolerance factor deviates from 1, then energetic stability may be gained through distortion. If t > 1 (as in BaTiO3), then a compound has relatively small B-site cations, and is stabilized when those cations move toward one or more of their X-site neighbors. If t < 1 (as in CsPbI3), then a compound has relatively small A-site cations, and is stabilized when the B–X octahedra collectively rotate to fill that empty space. If a combination of elements has a tolerance factor much larger or much smaller than 1, it will likely not form a perovskite. For a variety of oxide and halide perovskites, Table 1 shows how tolerance factors relate to the types of distortions in the roomtemperature crystal structure. Tolerance factors are computed based on Shannon crystal radii,58 where available. Both oxide and halide compounds exhibit both ferroelectric distortions and octahedral rotations. Given their simplicity and reliance on hardsphere-based arguments, tolerance factors do a reasonable job rationalizing the room-temperature distortions of this set of compounds. For the most part, larger tolerance factors lead to ferroelectric distortion and smaller tolerance factors lead to octahedral rotation.

Fig. 2 (A) Ferroelectric distortions in BaTiO3 (left) and CsGeI3 (right), with ionic movements along the 001 and 111 axes, respectively. (B) Octahedral rotations corresponding to a0a0cþ (left) and a0a0c
(right) in Glazer notation, observed in high-temperature CsSnI3 and low-temperature SrTiO3, respectively. Amplitudes of the distortions are exaggerated for clarity.

8

Electronic structure of oxide and halide perovskites Table 1

Tolerance factors and room-temperature distortions and space groups of a variety of oxide and halide perovskites.

Compound Tolerance factor Room-temperature distortions CaTiO3 SrTiO3 BaTiO3 CaZrO3 SrZrO3 BaZrO3 CaHfO3 SrHfO3 BaHfO3 NaNbO3 KNbO3 NaTaO3 KTaO3 CsGeCl3 CsGeBr3 CsGeI3 CsPbCl3 CsPbBr3 CsPbI3

0.97 1.00 1.06 0.91 0.95 1.00 0.92 0.95 1.01 0.97 1.05 0.97 1.05 1.03 1.01 0.98 0.87 0.86 0.85

Rotation (aþb
b
) None Ferroelectric (001) Rotation (aþb
b
) Rotation (aþb
b
) None Rotation (aþb
b
) Rotation (aþb
b
) None Rotation (a
a
a
) and antiferroelectric Ferroelectric (110) Rotation (aþb
b
) None Ferroelectric (111) Ferroelectric (111) Ferroelectric (111) Rotation (aþb
b
) Rotation (aþb
b
) Rotation (aþb
b
)

Space group Pnma59 Pm 3m 60 P4mm61 Pnma59 Pnma62 Pm 3m 63 Pnma64 Pnma64 Pm 3m 64 Pbcm65 Amm266 Pnma67 Pm 3m 68 R3c69 R3c69 R3c69 Pnma70 Pnma71 Pnma72

Tolerance factors are computed based on Shannon crystal radii. Shannon, R. Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Cryst. A 1976, 32, 751–767, doi:10.1107/S0567739476001551.

3.02.3

Computational methods for perovskite electronic structure

The numerous computational studies of the atomic and electronic structure of oxide and halide perovskites have primarily used DFT-based methods.73,74 Within DFT, however, there is considerable variety in the physics included in these calculations, the computational expense incurred, and the quantitative agreement of computed results with experiments. In general, the appropriate choice of computational methodology depends on chemical composition, property of interest, and relative need for quantitative accuracy. At the bottom of the so-called “Jacob’s ladder” of DFT methods75 lie the simplest functionals: the local density approximation (LDA) and generalized approximations (GGAs), which are desirable for their low computational expense. Although LDA tends to underestimate unit cell size relative to experiment and GGAs tend to overestimate it, these methods are often sufficient to capture the shallow and complex potential energy landscapes of perovskites with respect to structural distortion. For oxide perovskites, LDA76,77 and GGAs78,79 have often been used for structural energy comparisons, though stepping up the ladder to hybrid functionals (which incorporate a fraction of exact exchange) has been shown to more precisely capture experimental geometric parameters.80 For halide perovskites, GGAs have been most prevalent in past structural energy comparisons, with some debate as to the importance of explicit van der Waals corrections in compounds with organic A-site cations.81 While van der Waals corrections counteract the tendency of GGAs to overestimate unit cell size,82–84 they generally do not affect the relative energies of different phases.85,86 Furthermore, because these methods reliably capture crystal orbital symmetry and character, they typically provide reliable qualitative trends in band energies and band gaps. That is, LDA and GGAs are sufficient to predict whether perovskite band energies and band gaps increase or decrease with changes in elemental composition, structural distortion, strain, etc.77,79 However, it is well documented for many classes of compounds that these simplest DFT functionals tend to severely underestimate band gaps relative to experiment. For this reason, if one hopes to compute quantitatively accurate band gaps, it is necessary to use a method that incorporates more physics, likely at greater computational expense. For oxide perovskites, there have been numerous demonstrations of hybrid functionals,80,87 DFT þ U calculations,88,89 and many-body GW corrections90–94 bringing band energies and band gaps into closer agreement with experiment. To illustrate the effect of hybrid functionals on perovskite oxide band gap calculations, Table 2 shows PBE (a GGA), HSE06 (a hybrid functional), and experimental optical band gaps of three representative d0 oxide perovskites that are cubic at room temperature. As is typical, PBE severely underestimates experimental band gaps by > 1 eV, but more accurately identifies the differences in gaps among the compounds. At greater computational expense, the HSE06 hybrid functional brings the band gaps of all three compounds into reasonable agreement (within 0.2 eV) with experiment. The situation is somewhat more complicated for halide perovskite photovoltaics. Simple GGAs such as the PBE functional match the experimental band gaps of lead-halide perovskites such as CsPbI3 surprisingly well. Importantly, this does not illustrate the

Electronic structure of oxide and halide perovskites Table 2

9

Comparison of computed and experimental band gaps of cubic d0 oxide perovskites SrTiO3, BaZrO3, and KTaO3.

Method

SrTiO3

BaZrO3

KTaO3

PBE HSE06 Experiment

1.80 eV 3.26 eV 3.2 eV95

3.12 eV 4.61 eV 4.8 eV96

2.11 eV 3.43 eV 3.6 eV97

accuracy of the method, but rather a cancelation of significant errors. As shown in Table 3, the hybrid PBE0 functional widens the computed band gaps of cubic iodide perovskites relative to PBE. In past computational halide perovskite literature, these gap widening effects have been observed for both hybrid functionals and GW calculations.98–103 However, when computing halide perovskites, there is an additional methodological concern: the inclusion of spin-orbit coupling. The results in Table 3 show that spin-orbit coupling reduces the computed band gaps of lead-halide perovskites by approximately 1 eV, and those of tinand germanium-halide perovskites by significantly less. This too is consistent with past literature.104–106 Taken together, these two errors – the tendency to GGA to underestimate band gaps and the neglect of spin-orbit coupling – approximately cancel for lead-halide perovskites, but not all halide perovskites. Therefore, both types of corrections are needed to bring electronic structure calculations of this class of compounds into close, consistent agreement with experiment.98,99,101,103 Where our own calculations are presented in this chapter, the VASP package107–110 is employed with PAW potentials.111 The PBE functional112 is used to compute electronic band structures. For more quantitatively accurate band gaps, d0 oxide gaps are computed with the hybrid HSE06 functional,113 while halide gaps are computed with the hybrid PBE0 functional114 and spin-orbit coupling.115 All images of crystal structures in this chapter are made using VESTA.116

3.02.4

Conceptualizing cubic perovskite band structures

When presented with an electronic band structure, chemists and physicists often approach it from different viewpoints and with different language. These differences are obstacles that must be overcome for effective communication between the inorganic chemistry and condensed matter physics communities. Furthermore, the ability to think about band structures from these different viewpoints helps one find underlying order and patterns in the electronic structure. An inorganic chemist may prefer to view electronic states in terms of linear combinations of atomic orbitals, and feel more at home with discrete molecules than periodic solids. A condensed matter physicist may prefer to view electronic states in terms of nearly-free electrons, perturbed by the periodic array of nuclei in a crystal.117 In this section, we present the near-gap band structures of prototypical cubic perovskite semiconductors through the lenses of both linear combinations of atomic orbitals (the “chemist’s view”) and perturbed plane waves (the “physicist’s view”). Depending on your background, some parts of these descriptions may be quite familiar, and others more foreign. We encourage you to try to recognize the value and beauty of both viewpoints, which are consistent with and complementary to each other. By doing so, it is possible to develop a deeper qualitative understanding of perovskite band structures and why they look the way they look.

3.02.4.1

SrTiO3, a d0 oxide perovskite

 perovskite at room temperature, from the perspective of linear combiWe begin by looking at SrTiO3, a prototypical cubic ðPm3mÞ nations of atomic orbitals. Experimentally, SrTiO3 has an indirect optical band gap of 3.2 eV,95 and has shown promise as a light absorber in photocatalytic applications despite having a gap too large for its electrons to be excited by most of the solar spectrum.118,119 This compound can be viewed as largely ionic, with balanced ionic charges of Sr2þ, Ti4þ, and 3  O2
(each of which has a noble gas configuration). The highest-energy filled crystal orbitals in SrTiO3 are comprised of the oxide anion 2p orbitals. Table 3

Band gaps of cubic iodide perovskites computed with a generalized gradient approximation (PBE) and a hybrid functional (PBE0), without and with spin-orbit coupling (SOC). Computed band gap

Method

CsGeI3

CsSnI3

CsPbI3

PBE PBE þ SOC PBE0 PBE0 þ SOC

0.64 eV 0.35 eV 1.56 eV 1.47 eV

0.46 eV 0.06 eV 1.34 eV 1.23 eV

1.48 eV 0.29 eV 2.57 eV 1.54 eV

10

Electronic structure of oxide and halide perovskites

Because there are nine of these orbitals in a cubic unit cell (three on each of the three oxide anions), there is a block of nine valence bands just below the band gap (Fig. 3). Because Ti4þ is formally d0, the lowest-energy unfilled crystal orbitals in SrTiO3 are comprised of the titanium 3d orbitals. While there are five 3d orbitals on the titanium cation, they are split in energy when surrounded by an octahedron of six oxide anions. As in an octahedral transition metal complex, the three of those d orbitals whose lobes point between the surrounding anions (the t2g orbitals) are lower in energy than the two whose lobes point at the surrounding anions (the eg orbitals). In the band structure of SrTiO3, this means there is a block of three conduction bands, the t2g bands, just above the band gap (Fig. 3). Because the strontium cations have filled orbitals well below the band gap and unfilled orbitals well above the band gap, they do not contribute significantly to the near-gap band structure. Though Fig. 3 begins to find order in the band structure of cubic SrTiO3 in terms of atomic orbitals, the qualitative shapes of the bands are at first glance more mysterious. In order to rationalize those, we must bring in the concepts of reciprocal space, k-points, and free electrons. There exist various pedagogical references that present these concepts, which we briefly describe here, to a chemistry audience.120,121     Every crystal orbital j! ! r can be represented by a Bloch function (Equation 2) in which some function u ! r with periodk ! icity matching the unit cell is multiplied by a phase factor that includes a vector k .       !! r eiðk1 xþk2 yþk3 zÞ (2) r ¼u ! r ei k $ r ¼ u ! j! ! k

! The coordinates of k specify how the sign of a wavefunction changes from one unit cell to the next. While wavefunctions in a crystal are in general represented by complex numbers, they take on real values at high-symmetry k-points. For a cubic crystal with unit cell of edge length a (e.g., the cubic perovskite structure), these high-symmetry k-points occur where the coordinates of ! ! k are each 0 or p/a. If a coordinate of k is 0, a crystal orbital has the same sign from one unit cell to the next along a given ! axis. If a coordinate of k is p/a, the positive and negative signs within a crystal orbital alternate from one unit cell to the next along a given axis. In electronic band structures, each high-symmetry k-point is assigned a Latin or Greek letter. In a crystal structure with a primitive cubic Bravais lattice: R ¼ (p/a, p/a, p/a), G ¼ (0,0,0), X ¼ (p/a, 0, 0), and M ¼ (p/a, p/a, 0). From this information, some key features of the SrTiO3 band structure can be deduced. For example, we expect the highestenergy filled orbitals in SrTiO3 to be the most anti-bonding combinations of oxide 2p orbitals, and the lowest-energy unfilled orbitals to be titanium t2g orbitals that are forbidden by symmetry from forming antibonding interactions with oxide 2p orbitals. These expectations are consistent with the computed band-edge electron densities shown on the left side of Fig. 3. The valence band maximum (VBM) lies at k-point R, while the conduction band minimum (CBM) lies at k-point G. Furthermore, both the valence band maximum and conduction band minimum are threefold-degenerate, as crystal orbitals residing in the planes perpendicular to the 100, 010, and 001 axes are all symmetry-equivalent in a cubic structure. Similar considerations of symmetry and atomic orbital mixing can reveal features of the band shapes. However, a coherent overall picture of the near-gap band structure emerges more clearly when viewing it in terms of nearly-free electrons. In the free-electron approximation, wave-functions are assumed to exist in a region of constant electrostatic potential (rather than an array of localized nuclei). The three-dimensional time-independent Schrödinger equation with constant electrostatic potential (defined as zero) is given in Eq. (3):       Z2 v2 v2 v2 (3) r þ 2þ 2 j !
r ¼ Ej ! 2 2m vx vy vz

Fig. 3 Near-gap electronic band structure of cubic SrTiO3, computed using DFT-PBE. Nine valence bands (consisting primarily of oxide 2p states) and three conduction bands (consisting primarily of titanium 3d states of t2g symmetry) are shown. The valence band edge is defined as zero energy. Charge densities corresponding to one of each of the threefold-degenerate band edges are illustrated on the left (VBM ¼ valence band maximum, CBM ¼ conduction band minimum).

Electronic structure of oxide and halide perovskites

11

For an electron in a free-space cubic unit cell of edge length a, the wavefunctions that satisfy this equation have the form of plane ! waves with wavefunctions and energies described by Eq. (4) and (5) (with no boundary conditions constraining wavevector k , and where V is a macroscopic volume inside which wavefunctions are periodic):   1 !! 1 j! ! (4) r ¼ pffiffiffiffiei k , r ¼ pffiffiffiffieiðk1 xþk2 yþk3 zÞ k V V E! ¼ k

  Z2 !2 k 2m 

(5)

! ! To facilitate the visualization of plane waves in this section, degenerate wavefunctions corresponding to k and
k will be combined into real, sinusoidal wavefunctions. We have adopted the common but potentially confusing usage of the symbol k to refer to two related but distinct things: the reciprocal-space vector representing the phase of any crystal orbital from one unit cell to the next, and the wavevector of a free-electron wavefunction. In order to begin to connect the atomic orbital view of SrTiO3 with the free-electron view, consider the illustrations of one of the titanium t2g orbitals at k-point G (i.e., the conduction band minimum) in Fig. 4. Images show both the DFT-PBE-computed electron density of each band-edge orbital (left) and a more schematic view (right). On the right side of the figure, a linear combination of titanium 3d orbitals is shown in the foreground, and a plane wave in the background. These two representations have matching signs and nodal structure. Thus, what appears from one viewpoint to be a linear combination of atomic orbitals can alternatively be seen as a plane wave, perturbed by the presence of nuclei in the crystal. In the case of the conduction band minimum of SrTiO3, ! the orbital lying in the xy-plane corresponds to wavevector k ¼ ð2p=a; 2p=a; 0Þ. The degenerate orbitals in the xz- and yz- planes ! have wavevectors k ¼ ð2p=a; 0; 2p=aÞ and (0, 2p/a, 2p/a). In Fig. 5, the computed DFT-PBE (Fig. 5A) and free-electron (Fig. 5B) band structures of the three t2g conduction bands are shown side-by-side, with pictures of the wavefunctions at high-symmetry k-points. The similarity between the two is clear, both in the appearance of the wavefunctions and in the shapes of the bands. For example, in both panels, threefold degeneracies are seen at k-points G and R, with the conduction band minimum lying at G and the highest-energy states at R. Depending on one’s viewpoint, the rising and falling of band energies can be attributed either to bonding and antibonding interactions among atomic orbitals or to the changing wavelength of plane waves. These viewpoints are connected in the schematic orbital pictures in Fig. 5B, which show linear combinations of atomic orbitals against the backdrop of free-electron plane waves. Both views are valid, and both are helpful in conceptualizing the key features of the conduction band structure of oxide perovskites. Naturally, the bands in a band structure that most closely resemble free-electron behavior are those for which the orbitals are most delocalized and wavelike. The three conduction bands in Fig. 5 fit this description because, while titanium 3d orbitals are relatively diffuse, t2g are forbidden by symmetry from forming localized s bonds and antibonds with the surrounding oxide anions. The same cannot be said of the nine highest-energy valence bands in SrTiO3. While orbitals within those bands often have the appearance of waves, their energies are highly influenced by the presence of localized s bonding and antibonding interactions between neighboring oxide and titanium ions. For example, Fig. 6A illustrates one of the valence band maximum orbitals at k-point R, while Fig. 6B shows another orbital at k-point R more than 4 eV lower in energy. While both resemble free-electron wavefunctions of the same periodicity, the energy of the latter is lowered by s bonds between oxide 2p orbitals and titanium 3d orbitals. So, while some features of the valence bands can indeed be rationalized based on free-electron waves, they are more influenced by localized covalent bonds than are the t2g conduction bands.

Fig. 4 Computed and schematic illustrations of one of the threefold-degenerate conduction band minimum orbitals in cubic SrTiO3. The DFT-PBEcomputed charge density corresponding to the orbital is shown on the left, while a schematic picture connecting a linear combination of atomic orbitals (LCAOs) to free-electron plane waves is shown on the right.

12

Electronic structure of oxide and halide perovskites

Fig. 5 Electronic band diagrams showing the titanium t2g bands (i.e., the three lowest-energy conduction bands) of cubic SrTiO3, computed using (A) DFT-PBE and (B) the assumption of free electrons. The DFT-PBE valence band edge is defined as zero energy. In the free-electron band diagram, schematic orbital pictures are shown at high-symmetry k-points, connecting linear combinations of atomic orbitals (LCAOs) to free-electron plane waves. Under each orbital picture, the corresponding free-electron wavevectors are given.

3.02.4.2

CsPbI3, a halide perovskite photovoltaic

The next compound we will examine is CsPbI3, a halide perovskite with a direct room-temperature band gap of 1.7 eV122,123 that has been explored for photovoltaic applications. While the PbeI octahedra within this compound exhibit an aþb
b
rotation pattern (Pnma) at room temperature, examination of the undistorted cubic structure serves as a useful starting point for our understanding. CsPbI3 can be viewed as largely ionic, with balanced ionic charges of Csþ, Pb2þ, and 3  I
. The highest-energy filled crystal orbitals are comprised of the highest-energy filled atomic orbitals in these ions: primarily iodide 5p orbitals, with some mixing of lead 6 s orbitals. Similar to the case of SrTiO3, because there are nine iodide 5p orbitals in a cubic unit cell (three on each of the three iodide anions), there is a block of nine valence bands just below the band gap (Fig. 7). The lowest-energy unfilled crystal orbitals in CsPbI3 are comprised of lead 6p orbitals. These three bands (representing of the set of three 6p orbitals on the lead cation) are not completely separated in energy from the higher-energy conduction bands. The A-site cesium cations do not make significant electronic contributions to the near-gap band structure. As in the titanium t2g bands of SrTiO3, the three lead 6p bands of CsPbI3 bear some resemblance to free electrons in both their energies and the appearance of their wavefunctions (Fig. 8). The threefold-degenerate conduction band minimum at k-point R consists mostly of nonbonding lead 6p states, with minor antibonding interactions with neighboring iodide 5s states. These wave! functions resemble free-electron waves with wavevectors k ¼ ð3p=a; p=a; p=aÞ, (p/a, 3p/a, p/a), and (p/a, p/a, 3p/a). With changing k-point, the energies of these bands rise and fall due to, depending on one’s viewpoint, varying degrees of antibonding interactions with iodide 5p states or changes in the periodicities of the oscillating wavefunctions. The end result is that the lowest-energy conduction bands as computed by DFT-PBE (Fig. 8A) resemble the shapes of the corresponding free-electron bands (Fig. 8B), though the tops of the DFT bands are somewhat obscured by the bands above them. It is interesting to note that the shapes of the three lowest-energy conduction bands of CsPbI3 (Fig. 8B, with their lowest energy at k-point R and their highest energy at k-point G) are a sort of inversion of the three lowest-energy conduction bands of SrTiO3 (Fig. 5B, with lowest energy at G and highest energy at R).

Fig. 6 Schematic illustrations of (A) one of the threefold-degenerate valence band maximum (VBM) orbitals in cubic SrTiO3, and (B) another orbital at the same k-point that is more than 4 eV lower in energy. The orbitals resemble free-electron plane waves of the same periodicity, but their energies are strongly affected by their suitability for bonding with titanium 3d orbitals.

Electronic structure of oxide and halide perovskites

13

Fig. 7 Near-gap electronic band structure of cubic CsPbI3, computed using DFT-PBE. Nine valence bands (consisting primarily of iodide 5p states) and several conduction bands (the lowest-energy of which consist primarily of lead 6p states) are shown. The valence band edge is defined as zero energy. Charge densities corresponding to the band edges are illustrated on the left (VBM ¼ valence band maximum, CBM ¼ conduction band minimum).

While the nine highest-energy valence bands of CsPbI3 are comprised primarily of iodide 5p states, lead 6s plays a major role as well, particularly in the valence band maximum itself. The valence band maximum (Fig. 7) is the most antibonding combination of iodide 5p and lead 6s states, in which the atomic orbitals on each pair of neighboring atoms form a s antibonding interaction. As in the case of SrTiO3, a free-electron treatment of the CsPbI3 valence bands is not particularly illuminating, as their shapes and energies are heavily influenced by localized bonding and antibonding interactions. Nonetheless, a clear connection between the shapes of the valence and conduction bands – beautifully described as a “mirror of bonding” in previous work124 – is evident.

3.02.5

Effect of composition on band structure

Among both oxide and halide perovskites, a variety of elements can be substituted at all sites. These substitutions can take the form of doping at low concentration, partial substitution in either ordered or disordered arrangements, or full substitution. For simplicity, this chapter focuses mainly on bulk ABX3 perovskites with various elements at the A, B, and X sites, as times commenting on the effects of doping and mixed occupation. To a large degree, the qualitative features of the band structures discussed in Section 4 (e.g., the character of band-edge orbitals, the k-points at which the band edges lie, and the shapes of the valence and conduction bands) remain the same when elements are substituted within a column of the periodic table. Still, changes in elemental identity can tune the electronic band structure of a perovskite in two main ways. First, because different elements within a column of the periodic table have different electronegativities and therefore different atomic orbital energies, valence and conduction bands tend to shift in energy when substitutions are made. Second, because of differences in ionic radius, substitution may change the size of the unit cell and/or induce structural distortions, both of which have an impact on crystal orbital energy and character. The

Fig. 8 Electronic band diagrams showing the conduction bands of cubic CsPbI3, computed using (A) DFT-PBE and (B) the assumption of free electrons. The DFT-PBE valence band edge is defined as zero energy. In the free-electron diagram, schematic orbital pictures are shown at highsymmetry k-points, connecting linear combinations of atomic orbitals (LCAOs) to free-electron plane waves and their corresponding wavevectors.

14

Electronic structure of oxide and halide perovskites

remainder of this section focuses on the effects of atomic orbital energy and unit cell size, while the effects of structural distortion are the subject of Section 6.

3.02.5.1

Effect of the X-site anion

As discussed in Section 4, in both d0 transition metal oxides and halide photovoltaics, the valence bands are comprised primarily of X-site p orbitals. Therefore, the identity of the X-site element plays a key role in the determining the valence band energies and band gaps. In general, the more electronegative the X-site element and the lower in energy its atomic orbitals, the lower in energy the valence bands and the larger the band gap. In oxide perovskites, the X site is occupied by oxide anions that can often be doped or partially substituted. From the standpoint of photocatalysis, oxide 2p orbitals are lower in energy than necessary to facilitate the oxidation half-reaction in processes such as water splitting. This means that band gaps of d0 oxide perovskites are larger than necessary, and their ability to absorb photons in the solar spectrum could potentially be improved by raising the energy of the valence band edge. Attempts to improve solar absorption have often involved partial substitution of oxide anions with a less electronegative element such as nitrogen or sulfur. Indeed, there have been numerous examples of oxynitride (i.e., nitrogen-doped or codoped) perovskites with promising light-absorption and photocatalytic behavior.43-45,125 Sulfur doping126,127 has been less common because sulfide anions are much larger than oxide, but there do exist several distorted sulfide perovskites.128–130 In halide perovskites, the X site is commonly occupied by chloride, bromide, or iodide, or a combination thereof. Throughout the computational literature, the expected trend is observed,81,85,131–134 with higher X-site electronegativity leading to larger band gaps: X ¼ I
< Br
< Cl–. Our band gap calculations of cubic CsBX3 halide compounds in Table 4 confirm that, for a given B-site cation, this trend indeed holds. It is also possible to tune the band gap (and photovoltaic efficiency) more finely by mixing multiple halide anions at the X site, which has been demonstrated experimentally for mixtures of iodide and bromide,135–138 and computed for all halides.85,131

3.02.5.2

Effect of the B-site cation

In both oxide and halide perovskites, the lowest-energy conduction bands have primarily B-site character. In halides, the valence band edge has significant B-site character as well. The identity of the B-site element therefore has a significant impact on the near-gap electronic structure of both classes of compounds. In d0 oxide perovskites, the conduction band edge energy (and therefore the band gap) varies with B-site electronegativity. As the B-site element is more electronegative and its atomic orbitals lower in energy, the conduction bands are lower in energy and the band gap smaller. This is demonstrated for the computed band gaps of several compounds in Table 5. Consistent with past experimental and computational literature, greater B-site electronegativity leads to smaller band gaps,139,140 over a large range of 2–3 eV. There have also been numerous examples in which small concentrations of various B-site metal cations are doped at the B site,141–144 as well as classes of superstructures such as double perovskites (A2B0 BO6) that are ordered (in either a rock salt or layered arrangement52) at the B site,14,46,50,51 shown to tune the light absorption behavior of oxide perovskite photocatalysts. The B-site dependence of band gaps is somewhat more complex in halide perovskite photovoltaics than in d0 oxides. Because the valence band edge in halide perovskites has B-site s character and the conduction band edge has B-site p character, the energies of both – and specifically the difference between the B-site s- and p-orbital energies – matter. If the B-site element is more electronegative, both band edges are lower in energy. For a given X-site anion, the observed band gap trend in Table 4 (B ¼ Sn2þ < Ge2þ < Pb2þ) runs opposite what was seen for oxides, with more electronegative B-site elements leading to larger band gaps in halides. This suggests that the valence band edge in halide compounds varies more with B-site electronegativity, which has the dominant effect on band gap. Again, combinations of elements have been demonstrated to tune halide perovskite band gaps and band edges, both in disordered mixtures of lead and tin145,146 and in ordered lead-free A2B0 BX6 double perovskites.47–49 Table 4

DFT-PBE0 þ SOC-computed band gaps of cubic halide perovskites, compared to the B-site and X-site Pauling electronegativities (EN).

Cubic compound

B-site EN

X-site EN

DFT-PBE0 + SOC band gap

CsGeCl3 CsGeBr3 CsGeI3 CsSnCl3 CsSnBr3 CsSnI3 CsPbCl3 CsPbBr3 CsPbI3

2.01 2.01 2.01 1.96 1.96 1.96 2.33 2.33 2.33

3.16 2.96 2.66 3.16 2.96 2.66 3.16 2.96 2.66

2.19 eV 1.78 eV 1.47 eV 1.92 eV 1.50 eV 1.23 eV 2.53 eV 1.99 eV 1.54 eV

Electronic structure of oxide and halide perovskites Table 5

3.02.5.3

15

DFT-HSE06-computed band gaps of cubic d0 oxide perovskites, compared to the B-site Pauling electronegativity (EN).

Cubic compound

B-site EN

DFT-HSE06 band gap

SrTiO3 SrZrO3 SrHfO3 KNbO3 KTaO3

1.54 1.33 1.3 1.6 1.5

3.26 eV 4.82 eV 5.25 eV 2.76 eV 3.43 eV

Effect of the A-site cation

In oxide and halide perovskites, neither the highest-energy valence bands nor the lowest-energy conduction bands have significant A-site character. Instead, the identity of the A-site element (or polyatomic cation) has only a secondary effect on the band-edge energies by influencing the size of the unit cell and the amplitudes of structural distortions. While structural distortions will be discussed in detail in Section 6, the effect of unit cell size is described here. In cubic d0 oxide perovskites, larger A-site cations and the accompanying larger unit cells lead to band gaps that are smaller by as much as a few tenths of an eV. This trend is illustrated in Table 6 for DFT calculations of the series of cubic titanates and tantalates. To demonstrate that the differences among the computed band gaps with varying A-site cation are primarily an effect of unit cell size and not electronic involvement of the A site in band-edge orbitals, we have also computed the band gaps of cubic SrTiO3 and KTaO3 constrained to the unit cell sizes of compounds with different Asite cations. These constrained SrTiO3 and KTaO3 calculations have band gaps within a few hundredths of an eV of those with the comparable A-site cations, confirming that the importance of the A-site element is indeed its effect on geometry, rather than any involvement in near-gap orbital character. For halide perovskites, the opposite trend is observed, with larger unit cells leading to larger band gaps. Past calculations have shown that, for the three most commonly used A-site cations, the trends in both ionic size and band gap are A ¼ Csþ < CH3NH3þ < CH(NH2)2þ.131,132,147 Our calculations in Table 7 further illustrate this by showing that the computed band gaps of cubic CsGeI3, CsSnI3, and CsPbI3 increase when the compounds are given larger cubic unit cells. Overall, the effect of A-site identity on the electronic structure of oxide and halide perovskites is less significant than the effects of the B and X sites. Still, for both classes of compounds, the ability to fine-tune electronic structure by mixing different A-site cations has proven useful in optimizing solar energy conversion.143,148,149

3.02.6

Effects of distortions on band structure

 crystal structure, the movements of the ions are fairly small. Consequently, the elecWhen perovskites deviate from the cubic Pm3m tronic band structure of a distorted perovskite still resembles that of a cubic perovskite, but with subtle shifts in electronic energies and breaking of degeneracies. Nonetheless, if the minor tunability that results from a distortion (e.g., a 0.2 eV change in the band gap of a light-absorbing material) can be harnessed, it can help to optimize the functionality of perovskite materials. In this section, we explore the effects of structural distortions on the electronic band structures and band gaps of oxide and halide perovskites. Starting from the intuition already developed for cubic perovskites in Section 4, we examine how small movements of the atoms break Table 6

DFT-HSE06-computed band gaps of cubic d0 oxide perovskites, compared to the A-site Shannon crystal radius.58

Cubic compound

A-site Shannon crystal radius

1.48 Å CaTiO3 SrTiO3 1.58 Å BaTiO3 1.75 Å SrTiO3 in CaTiO3 unit cell SrTiO3 in BaTiO3 unit cell NaTaO3 1.53 Å 1.78 Å KTaO3 RbTaO3 1.86 Å KTaO3 in NaTaO3 unit cell KTaO3 in RbTaO3 unit cell

DFT-HSE06 band gap 3.34 eV 3.26 eV 3.10 eV 3.36 eV 3.08 eV 3.68 eV 3.43 eV 3.32 eV 3.60 eV 3.34 eV

16

Electronic structure of oxide and halide perovskites Table 7

DFT-PBE0 þ SOC-computed band gaps of cubic iodide perovskites, with unit cell parameters shortened or lengthened by 1% of the respective relaxed structures. DFT-PBE0 þ SOC band gap

Cubic compound


1% size

Relaxed

þ1% size

CsGeI3 CsSnI3 CsPbI3

1.39 eV 1.15 eV 1.46 eV

1.47 eV 1.23 eV 1.54 eV

1.56 eV 1.31 eV 1.61 eV

symmetries, strengthen or weaken bonding and antibonding interactions within crystal orbitals, and consequently alter electronic properties. We attempt to paint a coherent picture of these relationships among atomic and electronic structure, referring to the body of past work in this area.

3.02.6.1

Ferroelectric distortion

As discussed in Section 2, perovskites with relatively large A-site cations and/or small B-site cations often undergo ferroelectric distortion, in which the cation and anion sublattices translate relative to each other. For example, at room temperature, BaTiO3 distorts along the 001 axis and CsGeI3 distorts along the 111 axis. Naturally, such distortive modes affect band gaps and bandedge orbital symmetry and character, as illustrated in Fig. 9. The top panels of Fig. 9 show the near-gap electronic band structure of BaTiO3 without (Fig. 9A) and with (Fig. 9B) ferroelectric distortion along the 001 axis. The band structure of the undistorted cubic phase is shown from k-point R ¼ (p/a, p/a, p/a) to G ¼ (0, 0, 0). Because ferroelectric BaTiO3 has tetragonal P4mm symmetry, its analogous k-points are named differently (A and G), but represent the same periodicity of the wavefunctions. As seen earlier for cubic SrTiO3 (Fig. 3), the band gap of cubic BaTiO3 (Fig. 9A) is bracketed by a threefold-degenerate valence band maximum at k-point R consisting of antibonding combinations of oxide 2p states, and a threefold-degenerate conduction band minimum at k-point G consisting of nonbonding titanium t2g states. These degeneracies arise from the fact that the three Cartesian axes are equivalent in a structure with cubic symmetry. When

Fig. 9 Comparison of the near-gap DFT-PBE band structures of BaTiO3 and CsGeI3 without and with ferroelectric distortion. When (A) cubic BaTiO3 distorts along the 001 axis to the (B) tetragonal phase, degeneracies at both band edges break. When (C) cubic CsGeI3 distorts along the 111 axis to the (D) tetragonal phase, the degeneracy at the conduction band edge breaks. The valence band edge of each phase is defined as zero energy.

Electronic structure of oxide and halide perovskites

17

distortion lowers the symmetry to tetragonal, one of those axes becomes distinct from the other two, breaking the band-edge degeneracies as shown in Fig. 9B. Defining the axis of distortion as the z-axis, the band-edge orbitals lying the xz- and yz-planes are pushed apart in energy by several tenths of an eV as oxide 2p states and titanium 3d states mix. However, the band-edge orbitals lying the xyplane remain nearly unchanged because distortion along the z-axis has little effect on bond lengths in the xy-plane. As a result, the computed band gap of ferroelectric BaTiO3 is only 0.07 eV larger than that of cubic BaTiO3, and remains indirect at the same kpoints. The bottom panels of Fig. 9 show the near-gap electronic band structure of CsGeI3 without (Fig. 9C) and with (Fig. 9D) ferroelectric distortion along the 111 axis. Because ferroelectric CsGeI3 has rhombohedral R3c symmetry, its k-points in the band structure (T and G) are named differently from those in the cubic band structure. As seen earlier for cubic CsPbI3 (Fig. 7), cubic CsGeI3 (Fig. 9A) has a direct band gap at k-point R bracketed by a valence band maximum consisting of an antibonding combination of iodide 5p and germanium 4s states, and a threefold-degenerate conduction band minimum consisting mainly of nonbonding germanium 4p states. When distortion lowers the symmetry to rhombohedral, all three conduction band edge states rise in energy as the germanium 4p states develop antibonding interactions with nearby iodide orbitals. Furthermore, the degeneracy of that band edge splits into one orbital oriented along the axis of distortion, and two oriented perpendicular to that axis. The computed band gap of ferroelectric CsGeI3 is 0.45 eV larger than that of cubic CsGeI3, and remains direct at the same k-point.

3.02.6.2

Octahedral rotation

The other common class of perovskite distortion, B–X octahedral rotation, typically occurs in compounds with relatively small Asite cations and/or large B-site cations (i.e., tolerance factors less than 1). For example, SrTiO3 adopts an a0a0c
pattern of rotation in its low-temperature phase (tetragonal I4/mcm), while CsSnI3 adopts an a0a0cþ pattern in one of its high-temperature phases (tetragonal P4/mbm). As seen in the previous subsection for ferroelectric distortion, octahedral rotation lowers the symmetry of a compound, thereby tuning band gaps and band edges by allowing new orbital mixings (Fig. 10). Fig. 10A illustrates the evolution of the near-gap band structure of SrTiO3 from the room-temperature cubic phase to the tetragonal phase, which exists below 103 K.60 On the left, we again see the familiar cubic band structure, with an indirect R / G gap and threefold degeneracy at both band edges. The center panel of Fig. 10A also shows cubic SrTiO3, but with its unit cell size doubled. This change in unit cell causes the bands to fold back on themselves in k-space, doubling the number of bands at each k-point. While enlarging the unit cell may seem arbitrary, it allows for direct comparison to the distorted phase, which requires a 10-atom unit cell. When octahedral rotations are introduced (Fig. 10A, right), the degeneracies at both band edges break as the xy-plane becomes distinct from the xz- and yz-planes. The main effect is that, as pairs of oxide anions in the xy-plane move closer together, their 2p orbitals develop bonding interactions that lower the energy of the valence band maximum state lying in the xy-plane. Because the effects on other band-edge orbitals are minor, these octahedral rotations widen the computed band gap of SrTiO3 by only 0.08 eV. It is interesting to note that, due to the doubling of the unit cell and subsequent folding of k-space, low-temperature SrTiO3 has a direct band gap at k-point G, in contrast to the indirect band gap of room-temperature SrTiO3. Fig. 10B traces the band structure of CsSnI3 in a similar way, from the cubic phase that exists above 425 K to the tetragonal phase that exists between 425 K and 351 K.150 The familiar band structure of the cubic phase (Fig. 10B, left) takes on twice the number of bands when its unit cell is enlarged to accommodate double the number of atoms (Fig. 10B, center). When SneI octahedra rotate in an a0a0cþ pattern, the threefold degeneracy of the conduction band minimum breaks as the tin 5pz orbital becomes distinct from 5px and 5py. While 5px and 5py are raised in energy by new antibonding interactions that accompany lower symmetry, 5pz remains mostly unchanged, and the computed band gap widens by only 0.13 eV. The band gap of the tetragonal phase remains direct, though the name of the k-point at which the gap resides (Z) differs from the cubic phase (R) due to their different space group symmetries.

3.02.6.3

Unifying messages

Throughout this section, we have seen common themes in the ways that distortions of the atomic structure tune the electronic band structure of perovskites. All distortions, whether ferroelectric distortions or octahedral rotations, lower the structural symmetry in ways that break degeneracies and allow new orbital mixings, which generally widen band gaps by pushing the band-edge orbitals apart in energy. Past literature consistently supports this idea that structural distortions widen perovskite band gaps. Given the current technological relevance of perovskite photovoltaics, this phenomenon has been reported most often (computationally100,131,147,151,152 and experimentally153) for octahedral rotations of lead- and tin-halide perovskites. Evidence of and rationale for distortions widening band gaps has also been observed in computational work on oxide perovskites.77,79,154 Beyond the general observation that distortions tend to widen band gaps, we can draw more detailed conclusions about the extent to which the band gaps are widened. Several of the examples in this section – ferroelectric BaTiO3 (Fig. 9B), rotated SrTiO3 (Fig. 10A), and rotated CsSnI3 (Fig. 10B) – feature phases in which distortions lie along or around the 001 axis (i.e., a B–X perovskite bond axis). In each of these cases, some band-edge orbitals mix strongly and are pushed apart in energy, but others are largely unaffected by the distortions. Thus, distortions along or around the 001 axis lead to relatively small changes in band gap, typically on the order of 0.1 eV. In contrast, the remaining example in this section – ferroelectric CsGeI3 (Fig. 9D) – features a polar distortion along the 111 axis, the body diagonal of a primitive perovskite unit cell. In other words, the distortive mode in CsGeI3 is a sum of

18

Electronic structure of oxide and halide perovskites

Fig. 10 Comparison of the near-gap DFT-PBE band structures of SrTiO3 and CsSnI3 without and with octahedral rotations. (A) When cubic SrTiO3 adopts an a0a0c
pattern of octahedral rotation, degeneracies at both band edges break and the band gap becomes direct. (B) When cubic CsSnI3 adopts an a0a0cþ pattern of octahedral rotation, the degeneracy at the conduction band edge breaks. The valence band edge of each phase is defined as zero energy.

distortions along each of the three B–X perovskite bond axes. As a result, all band-edge orbitals are pushed apart in energy through mixing when the structure distorts, and the band gap widens several times more than those of the other examples. This conclusion can be generalized. When a perovskite compound undergoes a distortion that is a combination of individual distortive modes – either a combination of ferroelectric distortion and octahedral rotation, or a distortion along or around an axis other than 001 – its band gap widens significantly more. This idea has been observed in the computational band gap engineering of ferroelectric oxide perovskites distorted along different axes,79 d0 oxide perovskites with combinations of distortive modes,154 and halide perovskites with various temperature-controlled octahedral tilt patterns.147,152

3.02.7

Other strategies for tuning the band structure

Sections 4–6 of this chapter have described the band structures of cubic oxide and halide perovskites, and the ways in which they are altered by changing the identities of the elements and positions of the atoms. Having laid this groundwork, we now turn to some practical routes by which synthetic scientists can further tune the electronic structure of perovskites: high pressure, epitaxial strain, and reduction in dimensionality. Because these modes of structural modification are all intertwined with the issues of composition and distortion already covered, their effects on electronic structure vary from compounds to compound. Rather than attempting to enumerate what is known about the tunability of every perovskite compound, we instead organize our discussion around general guiding principles for band structure engineering.

Electronic structure of oxide and halide perovskites

19

It is worth noting that in addition to the approaches covered in this section, perovskite electronic structure may also be tuned via the engineering of defects, which has a complex literature unto itself.155–157 In both oxide and halide perovskites, various types of defects can introduce in-gap or near-gap donor or acceptor electronic states that affect the compounds’ light absorption and charge carrier separation behavior.21–24 As our focus in this chapter remains on the band structures of bulk perovskite compounds, we will not focus on defects in detail. However, defects are consequential in these classes of compounds, and represent yet another adjustment knob with which the electronic structure of perovskites may be tuned.

3.02.7.1

High pressure and epitaxial strain

Oxide and halide perovskite unit cells can be resized and reshaped by applying high pressure or epitaxial strain. High pressure, typically applied using a diamond anvil cell, reduces the size of the unit cell and may induce or tune the amplitudes of the types of structural distortions described earlier. Epitaxial strain, achievable using various layer-by-layer thin film growth techniques, compresses or stretches a structure in two dimensions while allowing the third axis to relax, which again may affect the presence or amplitude of structural distortions. As we have seen throughout this chapter, such changes in unit cell size, shape, and symmetry often lead to changes in perovskite band gaps and band-edge orbital character. High pressure and epitaxial strain have therefore been used to engineer the electronic band structure of oxide and halide perovskites, with effects that strongly depend on the compounds’ tolerance factors and tendency to distort. Several notable examples are described in this sub-section, in which band gaps and band edges may be tuned by up to several tenths of an eV. A variety of halide perovskites have been studied experimentally under high pressure,27,158–162 with the effects on their band gaps compared in recent reviews.163,164 While the specific behavior is different for each compound due to differences in tolerance factor and distortive modes, band gap often decreases as unit cell size is compressed until the onset of a structural distortion causes the band gap to increase. This is consistent with the halide perovskite band gap trends described earlier in this chapter with respect to unit cell size (Table 7) and distortion (Figs. 9D and 10B). In the area of epitaxial strain, oxide perovskites have a longer history than halides. Many oxide perovskites have been engineered by the growth of epitaxial thin films, strained on the order of 2–3% of their in-plane unit cell axis lengths.28,29,33 While epitaxial strain of halide perovskites has proven more challenging, there has been recent progress in that area.30-32,34,35 For both oxides and halides, consistent with trends described earlier in this chapter, DFT predictions in the literature show a complex landscape of small shifts in band gap with changing unit cell size and shape, and larger shifts in band gap with the onset of structural distortions.77,152,165 To date, epitaxial strain has primarily been applied perpendicular to the 001 perovskite axis (i.e., the B–X bond axis). Typically, when strain is applied, a crystal structure responds in such a way that physically reasonable bond distances are preserved as much as possible. Therefore, when compressive strain is applied perpendicular to 001, which would shorten B–X bond distances in the absence of further distortion, structures may be driven to adopt ferroelectric distortion along 001 or octahedral rotation around 001, in order to provide more space for B–X bonds. In contrast, tensile strain perpendicular to 001 often enhances distortions along and around in-plane axes (e.g., 100 or 110). Strain perpendicular to axes other than 001, via the growth of perovskite thin films on different substrate surfaces, may provide opportunities to access a greater variety of distortive modes with different effects on the electronic structure.

3.02.7.2

Superstructures with reduced dimensionality

So far, all compounds in the chapter, whether cubic, distorted, or strained, have maintained a three-dimensionally connected perovskite network. Though alterations of the unit cell size and shape and atomic positions have tuned interactions within crystal orbitals, there have been no fundamental changes in which atoms are connected. We now shift our discussion to perovskite-related superstructures of reduced dimensionality, in which connections among the atoms do change, leading to quantum confinement effects and consequent opportunities to tune the electronic structure.166,167 There are many examples in which regions of both oxide and halide perovskite compounds have been confined to lower dimensions. In addition to nanostructures in two, one, or zero dimensions (e.g., nanosheets, nanorods, and quantum dots),168–171 a variety of bulk phases have been synthesized in which perovskite regions are effectively separated from one another. In oxide compounds, two-dimensional perovskite regions of tunable thickness are seen in Ruddlesden-Popper,172,173 Aurivillius,174,175 and Dion-Jacobson176,177 phases. In halides, layers of organic cations can separate two-dimensional perovskite regions in these same series39,40,178,179 and in other phases.36 While two-dimensional perovskite regions are usually confined in the 001 direction (i.e., along a B–X bond axis), halide superstructures have been synthesized with layers confined in the 110 direction as well.37 Ruddlesden-Popper (RP) compounds provide a clear illustration of how reduced dimensionality impacts the electronic structure of perovskites. The RP phases (Fig. 11A) have a general formula Anþ1BnX3nþ1, in which n is a positive integer. These compounds essentially consist of blocks of n ABX3 perovskite layers spaced by an extra AX layer. The thickness of the perovskite region is tunable from a single perovskite layer (n ¼ 1) to the bulk perovskite structure itself (n ¼ N). RP halides have recently shown promise in photovoltaic applications,39,40 and both oxides and halides have band gaps that widen as perovskite regions are narrowed to a single layer, as a result of quantum confinement of the electronic states.38,180,181 Interestingly, RP sulfides sometimes show the opposite trend, as the high degree of distortion in ABS3 perovskites (and consequent widening of the band gap) may outweigh the effect of quantum confinement.181

20

Electronic structure of oxide and halide perovskites

Fig. 11 (A) Unit cells of the Ruddlesden-Popper phases (Anþ1BnX3nþ1) corresponding to n ¼ 1 and 2 and the perovskite structure (n ¼ N). The band-edge charge densities of two n ¼ 1 phases, (B) Sr2TiO4 and (C) Cs2PbI4, resemble those of the corresponding perovskites, except that they are confined to the two-dimensional perovskite-like regions of the structures.

Table 8

DFT-PBE-computed band gaps of undistorted Ruddlesden-Popper phases of Srnþ1TinO3nþ1 and Csnþ1PbnI3nþ1.

Phase

Srnþ1TinO3nþ1

Csnþ1PbnI3nþ1

n¼1 n¼2 n¼N

1.94 eV 1.88 eV 1.80 eV

1.90 eV 1.78 eV 1.48 eV

The quantum confinement effects on the band edges of an oxide (Sr2TiO4) and a halide (Cs2PbI4) n ¼ 1 RP phase are shown in Fig. 11B and C, respectively. The charge densities of the band-edge orbitals resemble those of the corresponding perovskite compounds (Figs. 3 and 7), except that the RP band edges are confined to the two-dimensional perovskite regions. Because the perovskite networks in RP phases are not fully delocalized in three dimensions, atoms within the perovskite regions of RP phases have fewer neighbors with which to share orbital overlap. As a result, the valence and conduction bands in RP phases are not as wide as those in perovskites, causing RP phases to have larger band gaps. The computed band gaps of the n ¼ 1, 2, and N phases of Srnþ1TinO3nþ1 and Csnþ1PbnI3nþ1 are given in Table 8, confirming the trend seen in the literature38,180,181 of larger band gaps with small n.

3.02.8

Concluding remarks

Rather than attempting to catalog the electronic properties of many perovskite compounds, this chapter has described a conceptual framework for thinking about perovskite electronic structure and how it can be tuned. With a focus on band gaps and the character of band-edge orbitals, the electronic band structures of perovskites have been presented in a manner that we hope is both accessible to chemists and generalizable to classes of perovskites apart from those covered in this chapter. The two classes of compounds discussed, d0 oxide perovskite photocatalysts and halide perovskite photovoltaics, are not only relevant toward technological applications that depend on features of their electronic band structure, but also exhibit structural distortions and orbital interactions that are

Electronic structure of oxide and halide perovskites

21

representative of perovskite compounds as a whole. While this chapter has high-lighted a number of differences between oxide and halide perovskites, it has also showed many similarities and has demonstrated that their features can be understood based on the same considerations of structural symmetry and orbital character. Finally, by discussing and rationalizing the ways in which perovskite electronic structure can be tuned via changes in composition, temperature, pressure, strain, and reduced dimensionality, we hope the insights in this chapter can aid synthetic scientists in using their extensive toolbox to optimize perovskite properties of interest.

Acknowledgment We gratefully acknowledge the Research Corporation for Science Advancement for financial support through a Cottrell Scholar Award.

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3.03

Bonding in boron rich borides

Pattath D. Pancharatna, Sohail H. Dar, and Musiri M. Balakrishnarajan, Chemical Information Science Lab, Pondicherry University, Puducherry, India © 2023 Elsevier Ltd. All rights reserved.

3.03.1 3.03.2 3.03.2.1 3.03.2.2 3.03.2.3 3.03.2.4 3.03.3 3.03.4 3.03.4.1 3.03.4.2 3.03.4.3 3.03.4.4 3.03.5 3.03.6 3.03.7 3.03.7.1 3.03.7.2 3.03.8 3.03.9 3.03.10 References

Introduction Classical bonding Borides with sp hybridized boron Borides with sp2 hybridized boron Borides with sp3 hybridized boron Miscellaneous systems Localized 3ce2e bonding Delocalized monopolyhedral bonding Borides with B6 units Borides with B12 units Hybrid networks with B12 and B6 units Borides with other Bn deltahedra Bonding in condensed polyhedra Bonding in single-vertex sharing macropolyhedra Perturbations in polyhedral bonding Borides with missing vertices Borides with capping vertices Bonding in non-deltahedral borides Bonding in elemental boron polymorphs Concluding remarks

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Abstract The crystal structures of boron-rich borides are often large and complex and show significant variations in compositions with abundant partial and mixed occupancies. From a chemical viewpoint, they form periodic covalent boron networks with some occasional hetero atoms while hosting the metal ions in the voids of the network. Understanding the chemical bonding in the covalent network requires a hybrid blend of both VB and MO theories, using hybridization based on local symmetry, to distribute the available electrons into discreet localized and delocalized bonding regions. In addition to the classical bonds i.e., single, double, triple, and their delocalized variants, there are multicentered bonding units required. These are localized 3c-2e bond and delocalized n-centered bond with specific electronic requirements depending on the overall polyhedra, which can also be generalized eventually. For a n-vertex polyhedra involving n tangential and one radial bonding molecular orbitals (MOs), nþ1 electron-pairs are needed. The n-vertex macropolyhedra with m discreet polyhedra sharing one or more edges or a rhombic linkage require “nþm electron pairs"; as it encompasses m distinct radial bonding MOs, but has n tangential MOs shared across the entire condensed polyhedra. Single-vertex sharing requires additional electron-pairs (mþnþo, where o is the number of single-vertex sharing atoms), as each polyhedral subunit has their own distinct tangential MOs too. The exo-polyhedral bonds are treated as two-electron bonds involving two or three centers. Perturbations caused by missing and capping vertices inside/outside the surface of the polyhedra keep the ideal electronic requirement for optimal bonding intact. The principle of ring-cap matching based on overlap control explains the relative stabilities of the deltahedral boron clusters. Large polyhedra with multiple vertices capping a hexagonal ring prefer being adjacent and cleave the connecting bonds resulting in non-deltahedral clusters observed exclusively in solids.

3.03.1

Introduction

Boron forms a wide variety of boron rich compounds with both metals and non-metals as expected from its metalloid character. Intriguingly, most of them can be formally referred to as boride since their chemical description of bonding involves anionic boron atoms. This tendency of boron to steal electrons even from elements of higher electronegativity arises from its large ionization energies, higher than any other element with less than half-filled valence shell. Much of the conceptual development of bonding in boron rich borides were developed using frontier orbital interaction theory.1 This is because boron frequently defies the central guiding principle of classical chemical structure theory based on Lewis’ electron-pair bonding2 and octet rule.3 The clarity in the concept of multi-centered bonding, the governing principles behind the different structural motifs and their electronic

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Comprehensive Inorganic Chemistry III, Volume 3

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27

requirements; evolved albeit slowly and is still evolving. Though technological advances and the development of efficient algorithms allow the quantum mechanical probing of their electronic structure with the desired accuracy, chemical bonding in many of these boron rich borides remain obscure. Since boron prefers to bond in a covalent fashion, bonding in boron rich borides is understood by identifying the molecular motifs. Before attempting the analysis of chemical bonding in boron rich systems, it is imperative to understand what typically constitutes ‘chemical bonding’ in sp elements. From the perspective of quantum mechanics that defines molecules with the wavefunction, which is a function of nuclear/electron coordinates and spin, a common chemist’s visualization of chemical bonding as a “collection of atoms bound together by different kinds of bonds” is an atrocious approximation. Atoms and bonds have no generally agreed universal definition and remain abstract concepts despite several attempts to define them. Most electronic structure calculations on boron rich borides are currently performed with plane-wave pseudo potential methods using density functional theory. However, quantum mechanical methods employing variational trial wave functions constructed from approximate atomic wavefunctions of the constituent atoms such as Valence bond (VB) and Molecular Orbital (MO) theories are hugely popular among chemists as they allow cognition of atoms in molecule. MO theory that uses the linear combinations of atomic orbitals posited on individual atoms (LCAO) as trial wave functions delocalizes electrons typically over the entire molecule and does not lend itself to an easy recognition of chemical bonds beyond homonuclear diatomics. Identifying chemical bonds around atoms in the classical systems typically utilizes the concept of hybridization, originally an off-shoot of valence bond (VB) theory. Initially introduced to explain Lewis electron-pair bonding in tetrahedral carbon,4 it was eventually adapted and nurtured even by popular MO theoreticians5 to escape the gruelling delocalized nature of canonical molecular orbitals. Though an artefact that ignores the fine differences in energy/diffuseness between the valence s and p orbitals, it forms the indispensable classical backbone behind the modern chemical structure theory of sp elements. Hybridization is an effective heuristic for identification of regions of space around each atom where its valence electrons are assumed to be paired and localized. These may be either involved with other atoms forming 2ce2e (two centered two electron) bonds or stand in isolation generating lone pairs. Depending on the type of hybridization, the electrons can be in the s or p framework; the p framework is usually formed from the unhybridized p orbitals. The concept of p-delocalization over a specific region of space is invoked to explain the electronic effects such as conjugation, aromaticity etc., using resonance or frontier orbital interactions theory. Huckel’s 4n þ 2 p-electron rule, theory of aromatic sextets etc., are introduced where the 2ce2e p-localization is partially replaced by delocalization involving multiple centers; a partially delocalized MO description is clearly favored. The classical octet rule that predicts higher stability for p block elements having eight electrons in their valence shell is the first electron counting rule ever proposed that was effectively justified using hybridization concepts. This is termed classical as it is originally proposed in the era of old quantum theory, much before the advent of Schrodinger’s atom, the discovery of electron spin and its dyadic nature. Isovalent hybridisation,6 VSEPR theory7 etc. were further improvement over simple hybridization concept that were employed for explaining or arriving at the observed geometry around each atom. Modern computational quantum chemistry led to the development of electron/energy partitioning schemes and other tools, like ELF, AIM etc.,8,9 to resolve possible ambiguities and quantify the strengths of individual bonds. Usually, chemical bonding in sp elements other than carbon is understood by comparing them with the appropriate isoelectronic organic analogue and recognizing their similarities and differences. In boron chemistry, octet obedient classical bonding is less ubiquitous and much of the difficulty in understanding their structure and bonding arise from its inherent tendency to have more than four neighbors. This leads to more bonding interactions than that is expected from the available valence orbitals and electrons, leading to the non-classical regime. Generally referred to as ‘electron-deficient bonding’, they fail to fit into the classical framework of the hybridization and 2ce2e bonding; diborane and other polyhedral clusters forming early examples. The introduction of localized three-centered two-electron (3ce2e) bond,10 that are usually drawn with more than one line, marked the birth of multi-centered, ‘electron-deficient’ bond. This helped resolve bulk of the mystery surrounding bonding in diborane and related compounds. The development of polyhedral bonding and molecular electron counting rules11,12 that allocates several electron pairs collectively to the entire bonding of the polyhedral cluster as a singular unit helps understand the structural motifs of the borane polyhedra. Their relative stabilities are typically explained from the principle of maximal overlap between fragment orbitals of individual vertices and rest of the unit with the open face. Note that the usage of the term ‘electron deficient bonding’ is to emphasize that the multiple lines drawn around boron do not represent an electron pair bond or valence; it does not necessarily imply electrondeficient reactivity as Lewis acids. Since boron rich borides typically involve infinite covalent network of boron and occasional non-metals with directional bonding, bulk of the bonding motifs established in its molecular world can be seamlessly exported to boron rich borides. Unlike metalloboranes, metal atoms are seldom found to be an integral part of polyhedral unit in boron rich borides and thus can be treated separately as innocent bystanders in a Zintl way, just providing the electrons required by the covalent boride lattice. Still there are several network motifs found only in boron rich borides that are not observed in the molecular world, making the interpretation of their chemical bonding challenging. In addition to the complexities arising from the non-classical structures favored by boron, boron rich borides also pose additional difficulties by their penchant for disorder. The prevalent disorder arises from the partial occupancy of the non-boron sites, predominantly involving metals and occasionally even boron sites. Disorder is also induced by mixed occupancy of boron sites with other elements like carbon, beryllium and silicon, posing special problems for unambiguous structural characterization. Often, the structure prescribed are idealized models for the experimental non-stoichiometric compound that lacks strict translational symmetry. Further, disorder in the site occupancy due to the availability of multiple ‘symmetrically equivalent’ sites occur in large

28

Bonding in boron rich borides

unit cells, compromising their structural uniqueness. Modern refinements of the structure using state-of-art XRD and neutron diffraction methods has partially addressed these occupancy problems, though there are several boron rich borides where such a clarity is still lagging. Multitude of these complications coupled with the difficulties in experimental isolation makes boron rich borides the most formidable structures. A comprehensive understanding of the electronic structure and chemical bonding in these systems has always been a challenge. Due to the light weight of boron and strong directional bonding underlying its covalent lattice, boron rich borides have found variety of applications that are established and envisaged. Their ability to withstand extreme temperatures, enormous pressures, high hardness comparable to that of diamond, light weight, chemical inertness and semiconducting nature makes them ideal for abrasives, body armor,13 hot cathodes, turbine blade, rocket nozzles,14etc. There is continuous research in enhancing their performance in all these applications. Metal borides of s-block have established and promising applications as superconductors,15 p-block borides serves as inexpensive light/hard materials16,17 while d/f block metal borides are useful as monochromators for soft X-ray synchrotron radiation,18 their widely varying thermal conductivity and dilute magnetic semiconducting19 nature offer additional promises. Quantum mechanical calculations and simulations, particularly from state of the art density functionals, provides reasonably accurate information about the electronic structure and other physical observables. However, rational interpretation of the computed numbers and making comparisons across related systems to explain the observed trends in the observables necessitates bonding insights from a chemical perspective. In the next section, we consolidate some of the established bonding paradigms that are unveiled so far that aids in rationalizing chemical bonding in these extended boride networks. Though boron rich borides are typically referred as having 80% or more of boron (ABn, n  4), we keep chemical bonding as the primary motive and utilize structurally well-characterized borides having infinite covalent boride network in the discussion. Boron exhibits both classical and non-classical electron-deficient bonding in borides based on the availability of electrons. In localized bonding, the classical 2ce2e single bonds and non-classical 3ce2e bonds are common while double and triple bonds are rare. In delocalized bonding, it exhibits two-dimensional electronic delocalization involving p-electrons of the boron rings as well as non-classical three-dimensional electronic delocalization in deltahedral clusters of varying sizes and shapes. In addition, it also extends delocalization across rings and clusters in polycyclic and macropolyhedral systems; the latter involves sharing of up to four atoms between polyhedra, analogous to polycyclic aromatic hydrocarbons. In short, boron exhibits a rich and diverse variety of bonding environments in molecules. Identifying these molecular motifs in boron rich borides facilitates the analysis of chemical bonding in these systems effortlessly. In the next section, a brief overview of bonding motifs in boron rich molecular systems is presented. Starting from electron precise classical bonding, it explores the various strategies envisaged by boron to address the increase in electron deficiency along with its structural preferences.

3.03.2

Classical bonding

Ideally, boron needs one more electron to become formally B that can exhibit classical bonding forming four 2ce2e bonds to satisfy the octet rule. In this electron precise situation, boron is more like carbon with four or fewer neighboring atoms that can typically be another boron or some non-metal lying within its covalent radius. In these instances, spn (n ¼ 1–3) hybridization in principle can be used directly as a guiding principle. Once hybridization of the atom is unambiguously identified based on its local geometry, nature of its bonds as either single, double, or triple, and its delocalized variants can be construed based on the bond length variations. Though boron is found to exhibit all the three forms of hybridization among molecules, in solids the classical octet obedient bonding between boron atoms (MBn, with n  4) are relatively rare due to the building up of negative charge. Among the experimentally reported classical borides, delocalized p-bonding with sp2 hybridized boron are ubiquitous with MB2 structure type. Localized p bonding between boron atoms are relatively rare as in molecules. The next most abundant classical structures are with sp3 hybridized boron as they form 3-D networks, 2D- layered structures and also 1D- zig-zag chains with transition metals. Though hybridization helps understand the bonding in the boron sublattice, their perceived electronic requirements are often dictated by MeB and MeM interactions that dominate the frontier, often leading to metallicity. The active involvement of transition metals in stabilizing the boron network makes their bonding direly complicated. Despite its rarity even in molecules, boron does have one exceptional sp hybridized network available in LiB. In these classical covalent networks, partial occupancies are relatively rare, though doping allows the electron count variations.

3.03.2.1

Borides with sp hybridized boron

Extended network with sp hybridization is ideally expected to form a 1-D chain, either with cumulenic ]B] units involving uniform bond lengths or with ^Be units that exhibit bond alternation. In one dimensional carbon chains, the alternating single and triple bonds are thermodynamically more stable as it opens up the band gap due to the Peierls distortion of the cumulenic double bonds.20 For boron, due to the weakened p-overlap between the unhybridized 2p orbitals, it is not straightforward to assess whether boron chains will prefer bond alternation like carbon as the molecular systems are not known experimentally. Though sp hybridized chains are experimentally characterized in one of its binary phases with lithium metal (Fig. 1A), excessive disorder in the boron atom positions makes it impossible to assess the nature of BeB chains. The origin of the disorder can be variously attributed

Bonding in boron rich borides

29

Fig. 1 LiB structures: (A) Experimentally characterized (P63/mmc) and (B) DFT optimized structure (Pnma). The bond angles around Li and the LieB close contacts are given.

to (a) bond alternation (b) rigid lithium network arising from finite covalent bonding between lithium atoms that is incommensurate with the boron sublattice (c) lack of translational periodicity across different boron chains. Assuming stoichiometric composition with uniform BeB distances leads to the BeB bond length of 1.39 Å, too short even for a B^B bond.21 Alternatively, the BeB distance of 1.58 Å is reported if one assumes non-stoichiometry phase with the composition LiBx (x ¼ 0.82–1.0).22,23 Both Li5B4 and Li7B6 reported earlier has powder pattern matching with the present P63/mmc. However, all these crystal structure reports tend to support uniform BeB distances with cumulenic double bonds and the experimentally observed metallic conductivity supports this conclusion. Multiple electronic structure calculations fail to conclude the nature of the BeB chains due to the difficulties in modelling the disorder.24 The computed band structure shows multiple bands crossing the Fermi level for the ideal LiB composition adding to the mystery.25 Though there is a general agreement that the bonding is cumulenic, the diffused scattering that happens perpendicular to the direction of the boron chain also points to the possibility of non-linearity and bending.23 Indeed, such a bent structure (Fig. 1B) is computed to be thermodynamically stable by one of the reports based on supercell DFT calculations.26 Lack of molecular analogs forbid making definite conclusions and currently, the nature of bonding between boron chains remains largely inconclusive.

3.03.2.2

Borides with sp2 hybridized boron

Due to the poor p-overlap between sp2 hybridized boron atoms, B]B bonds are quite scarce in molecular systems.27 Extended conjugation and cyclic conjugation between B]B bonds are even more scarce. However, in the condensed phase, boron readily forms planar cyclic networks. Such networks give rise to a layered structure with varying ring sizes and metals to compensate for the charge requirement (Fig. 2). Metals are usually seen as capping the boron rings and lying sandwiched between the boron layers.28 Simplest among this kind of network is the MB2 type lattices, where a 2-D graphene layer of boron atoms alternates with the metal layer (P6/mmm).29 The common examples include AlB2, the superconducting MgB2, most of the first row and early transition metals. Applying Zintl reasoning, MgB2 is expected to be a zero-bandgap semiconductor as its boride lattice is isoelectronic to graphene. However, all these layered borides including MgB2, are invariably metallic. The frontier bands in MgB2 are very different

Fig. 2

Crystal structures of layered borides: (A) AlB2 (P6/mmm) (B) Y2ReB6 (Pbam) and (C) YReB4 (Pbam).

30

Bonding in boron rich borides

from graphene, the doubly degenerate s-band originating from B(px,py) orbitals lie above the occupied valence p-band at Gamma and crosses the Fermi in several points.30 The metallic nature of the hexagonal boron lattice allows enormous flexibility in accommodating a wide range of electron counts that also encourages partial occupancies for the metal sites.31,32 Comparing MgB2 with AlB2, the increased electron count weakens the BeB bonding, but the metal-boron interactions become increasingly covalent.33,34 In the case of transition metal borides, charge transfer from metal to boride lattice is not complete since metal d orbitals energies are lying in the boron 2se2p gap. Metal-boron interactions play a dominant role in determining their stability.35 The BeB bonding increases up to d5 count beyond which it is weakened due to the filling of BeB antibonding orbitals. The ‘through space interactions’ between the transition metals across the ring is reported to play a large role in determining the stability of these borides.36 In addition to the ubiquitous hexagonal lattice, sp2 boron atoms also exhibit layered structures comprising of five and seven membered rings with translational periodicity while hosting rare earth elements as illustrated in the ternary Y2ReB6 and YReB4 borides.37,38 This is mostly driven by the size of metals, as larger heptagonal ring are better hosts for the rare earths. Their presence forces the lattice to have five membered rings to attain translational periodicity. The structural resemblance between the classically bonded graphene and hexagonal layered boron is merely an illusion and does not reflect in its electronic structure or bonding in any of these metal borides.39

3.03.2.3

Borides with sp3 hybridized boron

Single bonds between sp3 hybridized boron atoms are thermodynamically more stable than multiple bonds as the p-overlaps are weak. Hence, they are relatively more abundant in molecular systems, but rare in boron rich borides. Borides having weakly coordinating metal cations like heavy s block elements and other counter cations are conspicuously absent, indicating their relative instability. Borides hosting exclusively sp3 hybridized 3D-network is characterized experimentally in CrB4 and MnB4, indicative of the active role of metal-boron interactions in stabilizing the network. The structure of boron sublattice (Fig. 3) resembles the theoretically predicted tetragonal carbon polymorph.40 As in the case of layered MB2 type transition metal borides, the charge transfer is not complete. Even though the boron sublattice is strained hosting four membered rings and eclipsed orientation around some BeB bonds, a detailed bonding analysis indicates that this framework is ideal for complexation with transition metals.41 Chromium is found to have the ideal electron count in maximizing the bonding across BeB, MeB as well as MeM bonds. CrB4, though reported earlier in the Immm space group which involves interconnected B4 rectangles, was later refined to exist in Pnnm symmetry with B4 parallelograms (opposite sides equal).42,43 MnB4, on the other hand, crystallizes in a lower symmetry of P21/c; the further lowering in symmetry is presumed to be arising from Peierl’s distortion of the otherwise equivalent MeM bonds in the metal chain as this opens up a pseudo-gap for MnB4.44 The distortion also has an impact on the B network with the B4 ring appearing as a rhomboid ring with all different BeB bond lengths. The corrugated phases of the MB2 type with the layered B network adopting chair and boat conformations have bonding features of a sp3 hybridized boron with BeB bond lengths around 1.8 Å45 indicating that the boride lattice has more than one electron at its disposal leading to boron lone-pairs. The chair form crystallizes in P63/mmc while the boat structure has Pmmn symmetry and are known with late transition metals, due to the increased availability of electrons. Metal-boron interactions play a vital role in their stability as it is also clear from the geometry of the three phases that metal-boron interactions are distinct (Fig. 4A and B). Unlike planar sheets where metal caps a hexagonal ring, it has a h2 type interaction with the BeB bond in the boat conformer while in the chair conformer there are two short metal-boron interactions that are trans to each other and with additional close contacts results in an overall coordination number of 8. Comparison of bonding across these three types of MB2 phases using transition metals Ti, Re and Os confirms that the chair form with Re involves an optimum bonding across B atoms as well as between Re and B imparting high hardness to the material.46 WB2 (P63/mmc) and MoB2 (R-3 m) are layered structures where the planar and chair-type puckered forms alternate with metals in between them (Fig. 5).47 The metal -boron interactions are same as that observed in the corresponding phases discussed above.

Fig. 3

The crystal structure of (A) CrB4 (Pnnm) (B) MnB4 (P21/c) lattice. The bond lengths within the B4 ring is given.

Bonding in boron rich borides

Fig. 4

31

Two different orientations of the crystal structures of (A) ReB2 and (B) RhB2.

3.03.2.4

Miscellaneous systems

Transition metals also exist in a 1:1 stoichiometric phase with boron (Fig. 6A). But the boron chains here show bending with bond angles 110 degrees indicative of sp3 hybridized BeB single bonds and have identical bond lengths of  1.7 Å which are much larger.48 Metal-metal distances are also suggestive of bonding interactions and forms a cluster with boron embedded inside.49 Most transition metals are known to exhibit this metallic phase though analysis of bonding of their electronic structure show

Fig. 5

Crystal structures of (A) WB2 (B) MoB2.

(A)

(C)

(B)

(E)

(D)

Fig. 6 The crystal structure of metal-rich borides showcasing various bonding patterns of boron with illustrative examples. (A) MB (B) MAB phase (C) Ru11B8 (D) Cr3B4 (E) V5B6.

32

Bonding in boron rich borides

substantial covalent character for BeB as well as MeB bonds.50 MeM interactions dominate the frontier, accounting for their metallicity. These borides are of great interest due to their ultra-compressibility, high hardness, and electrical conductivity. The hardness is believed to be originated from the covalent network of boron as well as metal atoms. Similar motifs are observed in MAB-phases (Fig. 6B) with transition metals and aluminium (M2AlB2; M]Fe, Cr, Mn/MAlB; M] Mo, W)51 and other ternary phases such as Ru9Al3B8.52 Here, the network of transition metals is formed by the condensation of trigonal prismatic structures with boron occupying the center of the prism. Two such layers are connected through aluminium atom which lie in the plane of boron atoms. Branched chain of boron atoms are observed in Ru11B8 with strong interaction between Ru and the B atoms that point outward (2.067 Å).53,54 Boron zig-zag chains are also found alongside a 1-D network of edge shared hexagons of boron atoms; both structures alternating with transition metal layer between them; e.g., V5B6, Ta5B6.55 In Cr3B4 type structure B atoms form a 1-D chain of edge shared hexagons with the outer B atoms having direct interaction with Cr atoms.56 Similar kind of boron atoms is also present in a 3-D network where boron forms five-membered rings that are interconnected within a layer, except for one which has direct overlap with the metal (e.g., Y3ReB7; Cmcm).37,57 These layers are connected through a tetrahedral boron atom making the whole network three dimensional (Fig. 7). No chemical bonding studies has been attempted so far in any of these metal-rich borides. Metals are found to coordinate to divalent boron atoms and its impact on the covalent bonding in the boron sub-lattice need to be explored.

3.03.3

Localized 3ce2e bonding

If the one-electron requirement of boron atom is not fulfilled as is often the case, boron slips into the non-classical regime involving multicentered bonding in which several boron atoms group themselves into clusters. For a grouping of three boron atoms, the concept of three centered two electron (3ce2e) bond initially introduced for diborane58 comes in handy for electron counting purposes. Unlike diborane and related systems where the line connecting the boron atoms is duteously ignored due to historical reasons,59 most of the boride frameworks hosting 3ce2e bonds are depicted with a line connecting all the three boron atoms involved even if they are more than 2.05 Å. The structure of a-rhombohedral boron and other related borides exhibit 3ce2e bonds as one of the important mechanisms for handling electron deficiency. These 3ce2e bonds typically involve one sp hybrid from each of the three boron atoms, leading to only one bonding molecular orbital that needs to be filled. Grouping of four atoms lead to large variation in their shapes, and the high symmetric tetrahedral arrangement is only observed experimentally for substituents other than hydrogen. The concept of resonance between the localized boron bonds, four 2ce2e bonds or alternatively three 2ce2e bonds and 3ce2e bonds are invoked to explain the bonding in B4 tetrahedra. Though strictly tetrahedral B4 units are not present in boron rich borides, there is evidence for its presence in metal-rich extended systems.60 The most ubiquitous motif of B4 units involves rhomboid geometry that shows wide variations in their electronic requirement, more than any other boron cluster.61 This mode is preferred in interlinking of B10 clusters as known experimentally in B20H182 molecule and present in the Na3B20 boride, but with subtle complications, which will be discussed later.

3.03.4

Delocalized monopolyhedral bonding

Bonding of more than four atoms lead to the ubiquitous deltahedral bonding with spherical (closo) and partially spherical (nido, arachno, etc.) systems. Attempted explanation of resonance between the localized bonds akin to the resonance of 2ce2e p-bonds of conjugated hydrocarbons led to colossal difficulties and erroneous prediction of their electronic requirement. Even the dianionic requirement of these closo borane clusters are hard to interpret from the delocalization of 3ce2e bonds. Three-dimensional delocalization of electrons over the entire cluster is dominantly employed in explaining their polyhedral bonding which requires molecular orbital treatment.

Fig. 7

The crystal structure of Y3ReB7.

Bonding in boron rich borides

33

From simple tight binding calculations on the platonic octahedral B662 and icosahedral B1263 units reported in borides, LonguetHiggins proposed that the exo-polyhedral bonds of these polyhedra can be treated as regular 2ce2e bonds. He used linear combination of valence orbitals and a group theoretical analysis of cluster bonding and found that the exo-bonds are formed by the outward pointing hybrid orbitals on account of its symmetry allowed sp mixing. By doing so, he segregated the two tangential p orbitals and inward pointing hybrid orbitals as responsible for bonding within the polyhedra and predicted the dianionic nature from the high degeneracies of their HOMO. Soon after, the experimental verification of this dianionic requirement came from their borane derivatives64; the 2ce2e nature of exo-polyhedral bonds were confirmed as terminal hydrogens are their de-facto standard for single bonds. The segregation of the exo-polyhedral bonds as 2ce2e bonds and conceptually treating the bonding in whole polyhedra as singular entity marked the birth of polyhedral cluster bonding that paved way for further developments. When bulk of the closo polyhedral boranes BnHn2 (n ¼ 5–12, Fig. 8) and several associated open systems are characterized experimentally, attempts were made to extend the concept of resonance to 3ce2e bonds but was met with limited success.65,66 To alleviate the complexities, Wade put forward a qualitative fragment MO model of bonding to explain the dianionic requirement for all closo-boranes by introducing the concept of ‘sp hybridization’ for boron atoms in polyhedral bonding.67 All boron atoms in the convex polyhedra, irrespective of their symmetry are assumed to exhibit sp hybridization. The usual sp hybridization leads to two sigma bonds from each of these sp hybrids, leaving the two p orbitals for delocalized bonding. Instead, here the inward pointing sp hybrid along with the two unhybridized p orbitals tangential to the sphere are involved in the delocalized polyhedral bonding. Only the outward pointing sp hybrid is assumed to be involved in the formation of localized exopolyhedral 2ce2e bond (Fig. 9). In an n-vertex closo polyhedron, these 2n unhybridized p orbitals lead to n bonding MO and n antibonding MOs, while the inward pointing radial sp hybrids from every boron lead to just one bonding MO. This led to the n þ 1 electron pair rule that fits the experimentally observed dianionic nature as every BH contribute one electron pair to cluster bonding. Though this qualitative explanation involves separation of radial and tangential molecular orbitals, no such clear distinction can be made based on symmetry grounds. An in-depth analysis of the orbital interactions even in selected high-symmetry systems shows the favorable mixing of sp hybrids and tangential unhybridized p orbitals, but keeps the n þ 1 count intact in most clusters.68 The proposition of ‘n tangential bonding MOs and one radial bonding MO requiring n þ 1 electron pair’ is strictly for electron counting purposes. It neither implies localization of electron density nor degeneracy of the MOs and offer little help in understanding the strength of the cluster bonding within or across polyhedra. The parallelism between the n þ 1 rule of polyhedral boranes and Huckel’s 2n þ 1 electron pair rule for CnHn annulenes established later led to the extension of the aromatic delocalization concept to the third dimension.69 This time it also involves the delocalization of one of the sp hybrids that is pointing inward along with the two unhybridized tangential p orbitals at every boron atom. This forms a ‘n centeredd2n þ 2 electron bonding’ that encompasses the entire intra-polyhedral interactions as a singular unit. Since the n þ 1 rule for closo borane is not strictly rooted in group theory it is not always upheld by the existence of higher degeneracies of the HOMO for some of the less symmetric closo boranes. This leads to some exceptions referred variously as ‘hypercloso’ or ‘hypocloso’ systems that require electrons greater or lesser than the optimal electron count suggested by n þ 1 rule.70,71 Treating the interaction between all the vertices in the polyhedral unit as a singular entity conceals the variation in the strengths of individual interactions, especially in the irregular polyhedra. The relative stabilities of these closo borane clusters can be explained using the ‘ring-cap matching principal’ of maximum overlap. When every boron atom is visualized as capping a ring of boron atoms, Jemmis found that the size of the ring hosting the capping boron or other heteroatom is found to determine the overall stability of the interaction.72 This successfully explained the relative stabilities and abundance of different polyhedral clusters and heteroboranes. The frontier MO interactions between the ring and cap typically involve the doubly degenerate perpendicular p orbitals of the ring with the doubly degenerate unhybridized (tangential) p orbitals of the cap. The in-phase combination of one of the composite MO that is filled is given in Fig. 10 for various ring sizes. Here, the boron atoms of the ring fragment are conveniently assumed to be sp2 hybridized, thereby focusing on the MOs arising from the perpendicular p orbitals that lie in the frontier. Though there are finite

Fig. 8

Experimentally known closo borane dianions.

34

Bonding in boron rich borides

Fig. 9

The sp hybridized atomic orbitals that form the basis for Wade’s polyhedral bonding in boranes.

interactions between the ring and cap in the omitted tangential set, it is substantially weak compared to the p-set. Perpendicular porbitals of the ring are more directional towards the cap atom which is also influenced by the movement of hydrogens away from the plane. The ring-cap matching principle is rooted on arguments based on symmetry and overlap control. Hence it also holds good in systems where the rings deviate from planarity. Ring atoms hosting the cap often resort to non-planarity to maximize the overlap and this distortion increases with the size of the ring, as expected. Based on this principle, three membered rings are found to be too small to host a cap as evidenced by the enormous elongation of the BeB bonds in the equilateral three-membered ring in trigonal bipyramidal closo-B5H52 borane. It is seldom found in extended boride network with a singular exception of b-rhombohedral boron structure type, discussed in Section 3.03.9. Ring size four is found to be ideal for effective overlap as found in the ubiquitous octahedral B6 cluster where the high symmetry makes all vertices to cap a similar planar four membered ring. The octahedral symmetry also is favorably extended in all the three dimensions, making it a common building block in many of the popular metal borides. However, a B10 cluster despite having fourmembered rings to cap has reduced ring-cap overlap. This is due to the perturbation on the exo-bonds of the ring atoms changing the inclination of the perpendicular p-orbitals. This reduces its abundance compared to B6 clusters in molecules; in borides only very few examples are known. Pentagonal boron rings are relatively large to interact with the boron cap. This is well illustrated in the compressed BeB bonds of the pentagonal bipyramidal structure of B7 cluster, the smallest polyhedra having pentagonal caps. Besides, the incompatibility of fivefold rotation with translational symmetry makes it impossible for B7 units to exist as discrete units in boron rich borides. However, they are present as a subunit of a macropolyhedral system that will be discussed in Section 3.03.6. The overlap between the ring and cap gets better by the inclination of the tangential p orbitals towards the cap as found in the ubiquitous B12 cluster. This angle of inclination is reflected in the deviation of the exo-polyhedral bonds away from the ring plane. Here, the ideal icosahedral symmetry makes all the vertices equivalent, as in the case of octahedral B6. The exceptional stability of the B12 unit arises from the enhanced ring-cap interaction due to favorable bending of the tangential p orbitals of the perceived ring atoms towards the cap and the large size of the cluster that cushions the dianionic charge. This makes it the most ubiquitous building block for boron rich borides, indeed by symmetry lowering through distortion. Six membered rings are too large to have a capping boron in an ideal environment having equal BeB bonds, the cap cannot exist as it is in the same plane with the ring. This explains the lack of rigidity of the B11H112, the only closo borane hosting hexagonal cap in which the ring BeB bonds shrink considerably. In addition to the inclination of the perpendicular p orbitals of the ring, the ring boron atoms deviate from planarity to assist capping. This C2v symmetric B11 cluster that hosts a hexagonal cap, is not experimentally characterized in boron rich borides. However, heteroboranes having larger atoms like Be, Si and transition metals as hexagonal caps are capable of stabilizing the cluster.73 Mono-polyhedral boranes with more than 12 vertices made exclusively of boron atoms are yet to be characterized experimentally, though predicted by computational studies.74 Referred as supraicosahedral boranes, these polyhedra ideally require hex-caps by geometrical constraints and their number increases with its size. These hex-caps are ideal for the substitution of larger atoms as exemplified in metallaboranes with more than 12 vertices. In borides, 15-vertex monopolyhedral clusters are found with silicon and beryllium as the heteroatom; e.g., Si3B12 in SiB6 and Be3B12 in BeB3 phases (Fig. 11). The combination of n þ 1 rule and ring-cap matching principle allows for a successful interpretation of chemical bonding in mono-polyhedral boranes. This largely aids in understanding the electronic structure and physicochemical properties of boron rich borides, which is illustrated below with representative examples.

Fig. 10

Illustrating ring-cap overlap matching principle for various ring sizes.

Bonding in boron rich borides

Fig. 11

3.03.4.1

35

The computationally characterized structures of suprapolyhedral boranes that are also found in borides.

Borides with B6 units

The electronic requirement of  2 for an octahedral B6 cluster with triply degenerate frontier MOs is adequately compensated by the metal ions present in metal borides. However, hosting the large dianionic charge on a small cluster, works against its stability as it reduces the HOMO-LUMO gap. As a result, these borides exhibit narrow band-gaps for the ideal electron count and is flexible to accommodate variations in the electron count exhibiting metallicity. Octahedral B6 clusters, linked by 2ce2e bonds to form a cubic 3-D network (Fig. 12) is commonly referred as the CaB6 structural prototype (Pm-3m).75 This network shows enormous flexibility to accommodate large variations in the electron count and varying metal sizes. Though the band structure calculations of electron precise CaB6 show a definite gap (< 1 eV) as expected by the electron counting rules, experimentally they are metallic, which is attributed to impurities and/or temperature factors. While the smaller magnesium leads to metastable phases,76 larger alkaline earth metals till barium are known. The expansion of the cavity to accommodate larger metal ions is achieved mostly at the expense of inter-polyhedral B6eB6 bonds that turns them metallic.77 The network also hosts a variety of rare earths and yttrium, that provide more electrons than required resulting in metallicity. But unlike alkaline earth borides, there is little change in the inter-polyhedral bonds; instead, intra-polyhedral bonds of B6 expands and the fermi level is populated with boron p orbitals and metal d/f orbitals.78 KB6 with an electron-deficient network have Fermi level that lies well below the valence band as expected, revealing its metallic character in band structure calculations. With an odd electron unit cell, it shows non-magnetic metallicity in DFT calculations but does not behave like a normal metal as indicated by different experimental measures, including resistivity, EPR and magnetic susceptibility. The various observations point towards spin localization with the formation of bipolarons through local electronphonon interactions. From the bonding perspective, since a pair of linked octahedra have their spins anti-ferromagnetically coupled to form spin-paired dimers, this leads to intra-polyhedral p-bonding. This is also reported to be associated with the partial occupancy of the metal site (K 0.95B6).79 Another alkali metal to form a similar phase with interlinked B6 octahedra is Li, which exist as an electron precise boride with two Li atoms per B6 unit (Li2B6; P4/mbm).80 The presence of two Li atoms reduces the symmetry and has an impact on the boride network as well. The local symmetry of B6 is reduced to D4h resulting in two distinct intrapolyhedral bonds (Fig. 13). The overall boride lattice expands to incorporate two metal atoms that is reflected in the BeB bond lengths; the B6 polyhedra expands while the intra-polyhedral BeB bond lengths remain shorter. This leads to a large band gap (> 2 eV). In the 3-D framework of UB4 type borides (P4/mbm), the B6 octahedra is interspersed by B2 units forming a layered structure and have direct B6eB6 connections across the layers to form the covalent 3D network (Fig. 12). The additional B2 units expand the cavity as the metal channels are made up of heptagonal rings perpendicular to the B6eB2 layer, capable of accommodating larger actinides.

Fig. 12

The structure of CaB6 and Li2B6.

36

Bonding in boron rich borides

Fig. 13

UB4 type lattice in two different perspectives.

The unit cell consists of two B6 units and two sp2 hybridized > B]B< units requiring a total of eight electrons that can be compensated by four divalent ions.81,82 Though CaB4 is computed to have a small but definite, indirect gap by semi-empirical calculations, only carbon-doped CaB4 has been synthesised.83 However, DFT results show the valence bands and conduction bands to traverse the Fermi leading to metallic behavior supporting experiments.84 The network is more ubiquitous with rare earths despite their preference for þ 3 oxidation state. Though band structures have separation of bonding and antibonding bands of the B6 units, there is substantial mixing of metal d bands with B2 units lying at the Fermi level, causing metallicity.85

3.03.4.2

Borides with B12 units

The exceptional stability of the B12 unit arises from the enhanced ring-cap interaction due to favorable bending of the tangential p orbitals of the perceived ring atoms towards the cap, high symmetry and the large size that cushions the dianionic charge. However, to fit into a periodic 3D-lattice, the icosahedral symmetry of the B12 unit must undergo some finite distortion reducing its symmetry. The icosahedral B12 unit has large gap (> 5 eV) between its fourfold degenerate HOMO (gu) and LUMO (gg) exclusively arising from the linear combination of tangential p orbitals.86 Owing to this large gap, the reduction in symmetry forced by crystallographic translation does not adversely affect its optimal electron count. B12 units are frequently exo-bonded to one or more non-polyhedral main group elements, and along with interpolyhedral B12eB12 bonds, form the periodic covalent network. Many of the boron rich borides exhibit only moderate deformation of icosahedral B12 unit. Lattices with various local symmetries of B12 unit such as D3d, C2h, C2v, etc., are found. The associated distortion of the B12 polyhedra hardly affects its dianionic requirement if their exopolyhedral bonds are localized. The most ubiquitous choice of B12 units is the rhombohedral B12Xn with R-3m space group that has B12 in a D3d symmetry.87 It has a layered structure of B12 units where the adjacent layers are connected by B12eB12 2ce2e bonds through their polar B atoms (pink in Fig. 14A). The six equatorial boron atoms (purple in Fig. 14A) of the B12 are linked through non-polyhedral p-block element which are in sp3 hybridized environment. Binding three B12 units together, the fourth valence of the linking atom is often extended to form a linear 2ce2e bond giving rise to a side chain or sometimes carry a lone-pair. The distance between the layers of B12 has to be matched with the size of the linking atoms to determine the nature of interaction across them. Usually, smaller atoms in this non-polyhedral site are involved in compressed three-atom chains or lone-pairs while larger atoms form two atom chains parallel to the C3 axis. The typical linkers are three-atom side chains (Fig. 15A) such as ÑC-B+-C , ÑN+-B-N+ , two-atom side chains (Fig. 16A)

ÑB--S2+ , ÑP+-P+ , ÑAs+-B- , ÑSe2+-B- , etc., and isolated atoms like N or O (Fig. 16B). These side

chains are anticipated to provide two electrons for the B12 unit and is expected to have a semiconducting gap. Large ambiguities exist in characterizing the nature of side chains in many of these phases due to the mismatch in the electronic requirement. This leads to the unavailability of single crystals with strict translational symmetry due to mixed and/or partial occupancies. Primary example is the long-known and well-studied boron carbide, commonly referred to as B4C that exemplifies structural ambiguities with a widely varying carbon concentration. The idealized phase of boron carbide with the required symmetry consists

Fig. 14 The most common local symmetry of the B12 unit (A) D3d, (B)C2h and (C) C2v, with symmetrically distinct atoms indicated with different colors. The exo-bonded atoms are boron (green) and side chain atom (grey; mostly a hetero-atom).

Bonding in boron rich borides

Fig. 15

37

The structure of (A) ideal B13C2 and (B) LiB13C2.

of (B12)2 and

ÑC-B+-CÒ units and should be metallic. However, mixed occupancies are suspected in all the atomic sites,

although it is experimentally proved to be a semiconductor with all its varying carbon concentration. The partial oxidation of the phase through disorder is found to weaken the

ÑC-B+-CÒ

chains while maximizing the

strength of the exopolyhedral BeC bonds from partial p-bonding. This is a violation of the established conviction that exopolyhedral links are always 2ce2e bond, computationally predicted in molecules.88,89 The semiconducting nature is attributed to the localization of disorder. An electron precise phase with a similar network is found in LiB13C2 (Fig. 15B), where Li atoms compensate 86

for the additional charge requirement of B12. Here, B12 units are in a lower C2h symmetry since the

ÑC-B+-CÒ chains are

marginally deviating from linearity. This structure appears to overcome all the partial/mixed occupancies and is a semiconductor with an indirect band gap of 2.3 eV by DFT calculations. The linkers with diatomic chains and isolated atoms are exemplified by the structures of B12P2 and B12O2 where the tetravalent phosphorous and trivalent oxygen donate one electron to satisfy the dianionic requirement of the B12 unit. Though their stoichiometric phases are electron-precise, partial occupancies are reported for the hetero-atom sites for both these borides. Their absence leads to long 3ce2e bonds across the equatorial B atoms that still satisfies the electronic requirement but give rise to localized defects and disorder. When the non-polyhedral linkers are formally neutral, incorporation of metals is necessitated to provide electrons for the cluster

ÑSi-SiÒ chains have tetrahedral Si atoms that are neutral, the two Li atoms provide the electrons for the B12 cluster. Replacement of Si2 units with ÑC-P+Ò units are also known with the associated reducbonding as exemplified in Li2B12Si2.90 Since the

tion in the Li concentration (LiB12CP).91 These networks with B12s in C2h symmetry are also found to be narrow band-gap semiconductors with complete occupancies though the side chains have mixed occupancy in case of

ÑC-P+Ò.

When the side chains are ÑB--B-Ò units, the electronic requirement of the overall lattice increases as such units require 2charge to maintain the octet configuration of the tetrahedral boron atoms. Such a covalent network is found in the borides of the general formula MAlB14 where M is drawn from different blocks of periodic table like Li, Na, Mg, Y, and a variety of other lanthanides in the body centered orthorhombic lattice (Imam); the B12s has a local C2h symmetry (Fig. 17).

Fig. 16

The ideal structure of (A) B13P2 (B) B13O2.

38

Bonding in boron rich borides The four-electron requirement of the network units (B122; ÑB--B-Ò) is perfectly balanced by the stoichiometric Mg2B14.92

However, this network appears to be particularly stabilized by the presence of aluminium closer to the B2 units, presumably due to definite covalent interactions. Since metals in different ternary phases are having variable oxidation states, they tend to be partially occupied in such a way that the overall structure is always slightly less than that is required by the ideal boride network, as in boron carbide. Experimentally, these borides are all semiconductors with definite gaps ( 2 eV).93 The icosahedral B12 is also found to be supported by unsaturated side chains in a reduced C2v environment. In Li2B12C2, the boride network is slightly different as every B12 has only four connections to carbon atoms of the > C]C< chain. The rest of them form inter-polyhedral 2ce2e bonds. Out of these, four of the interpolyhedral bonds connect to the separate polyhedra, whereas the other two pairs are connected to the same B12 unit with adjacent boron atoms forming strained rectangles (Fig. 18). This reduces the local symmetry of B12 to C2v point group. The 2- charge requirement of the cluster is met from two Liþ ions. Despite the strain associated with lower symmetry of this network, Li2B12C2 is a semiconductor with a narrow gap ( 2.5 eV). Isoelectronic phase where two Li atoms in Li2B12C2 is replaced by 1 Mg (MgB12C2) is also known.

3.03.4.3

Hybrid networks with B12 and B6 units

Covalent networks comprising layers of D3d symmetric B12 units and B6 units directly connected along the C3 axis are also known with interlinking non-polyhedral atoms or chains. The simplest of this type is observed in the rhombohedral MgNB9 (R-3m) where layers of B6 and B12 units alternate forming a three-dimensional network.94 The exopolyhedral bonds of the equatorial boron atoms of the B12 unit are connected by a trivalent nitrogen atom (Fig. 19A). The two-electron requirement of the B12 and B6 units are available from the Mg atoms, making this network electron precise. The structure is stoichiometric without any disorder unlike the rhombohedral B12N2 which is riddled with disorder due to electron deficiency. The pyramidalization around nitrogen atoms are reduced, presumably due to the steric problems associated with the bulky B12 substituents. The structure is computed to have a semiconducting gap of 1.76 eV by DFT calculations.95 The interleaving of B6 layers along multiple B12 layers are also reported in the trigonal rare-earth borides.96 A well characterized example is the YB15.5CN (P-3m1) that has fully ordered covalent network of two B12 layers, one B6 layer alternating along the c axis (Fig. 19B). The ÑC-B+-CÒ chains are interlinking within and across B12 layers as in boron carbide while nitrogen atoms interlinking the left out equatorial atoms of the B12 unit. The ÑC-B+-CÒ chains provide only one electron to the lattice, rest of the five electrons needed for polyhedral units (2B122 and 1B6 2) are made available from yttrium. Since only five electrons are required, yttrium sites are forced to be partially occupied and is confirmed by the experimental data. Borides having three and four layers of B12 units intertwined with a monolayer of B6 units are also reported to exist with rare earths (REB28.5C4, REB22C3). However, there are partial occupancies associated with the carbon atoms and their structural characterization is not adequately established. Further experimental and theoretical studies are needed to unambiguously understand their electronic structure and bonding. Resistivity measurements point to semiconducting nature of all these rare-earth borides.

3.03.4.4

Borides with other Bn deltahedra

Boron rich borides are dominated by B12 and B6 units, presumably due to their high symmetry and favorable ring-cap matching. With a single exception of B11H112 that hosts a hexagonal boron cap, other polyhedral framework known in closo borane dianions are also found as building blocks in boron rich borides. Due to their reduced symmetry and the necessity to have localized exopolyhedral two-electron bonds, they often involve complex structures with large unit cells. The structure of Li3B14 (I-42d) is probably

Fig. 17

(A) The structure of MAlB14 and (B) the aluminium coordination environment.

Bonding in boron rich borides

Fig. 18

39

The structure of (A) Li2B12C2 and (B) B12 environment.

the simplest boride reported that has B8 and B10 units occurring together.97 The B10 and B8 exist in the ratio 2:1 in the lattice with every vertex in B8 connected to B10 units. Every B10 unit is bonded to four B8 clusters and the rest to the equivalent B10 polyhedra. Interestingly, its covalent framework is exclusively made up of polyhedral units and does not have any non-polyhedral atoms (Fig. 20A). Lithium atoms are partially occupied due to two equivalent sites though their combined occupancy is close to being stoichiometric. The six electrons requirement of two B10 and one B8 units are available from the six lithium atoms. The computed band gap of  2 eV is consistent with the experimental observations.98 A B12eB9 and B12eB10 interconnected units are observed in a complex quaternary boron rich solid, scandium borocarbosilicide Sc3.67  xB41.4  y  zC0.67 þ zSi0.33  w (x ¼ 0.52, y ¼ 1.42, z ¼ 1.17, w ¼ 0.02). In addition, it has an irregular B16 polyhedron with overall available B atoms amounting to 10.7 only.99 A B8eB12 and B9eB12 connected network is also observed in the ternary scandium borocarbide phase Sc4.5  xB57  y þ zC3.5  z (x ¼ 0.27, y ¼ 1.1, z ¼ 0.2).100 Boride lattices and scandium are particularly involved in several ternary and quaternary phases with more complex networks hosting an array of polyhedral boron clusters. Since growing large and good quality single crystals of these phases are very challenging, no physical properties of these higher borides have been investigated so far.93 The structure of orthorhombic SiB6 (Pnnm) is found to have a 15 vertex polyhedron resembling the structure of D3d symmetric B15H152 with three hex caps (Fig. 20B). It has a large unit cell having approximately 232 boron atoms and 32 silicon atoms. Silicon atoms are found predominantly in the hexagonal caps of the B15 polyhedra giving rise to the cluster composition Si3B12.101 However, these borides host an array of partially occupied interstitial atoms that complicates the evaluation of ideal electron count. Analysis of chemical bonding in these structures requires a comprehensive understanding of the impact of various perturbations caused to the polyhedral cluster and has not yet been done.

Fig. 19

The structure of (A) MgNB9 (B) YB15.5CN.

40

Fig. 20

3.03.5

Bonding in boron rich borides

The structure of (A) Li3B14 and (B) SiB6.

Bonding in condensed polyhedra

Since the electron deficiency per boron atoms reduces with cluster size, it is natural for boron rich borides to choose larger polyhedra with the decrease in electrons from metals or other sources. However, moving beyond B12 proved to be destabilizing due to the necessity for hexagonal caps that flattens the surface due to poor ring-cap matching. Forming macropolyhedral systems that shares several vertices between multiple polyhedral clusters is found to be one of the prominent ways for boron to reduce its electronic requirements. Fortunately, their electronic requirements are deceptively simple as in the case of mono-polyhedral clusters. The associated electron counting rules developed have only few exceptions, that too when transition metals form part of the polyhedral framework. Since transition metals are seldom part of the polyhedra in boron rich borides, these rules help understand the bonding in networks within extremely electron poor environment. Reduction in the availability of additional electrons primarily lead to condensed polyhedral clusters, in which one or more edges are shared between polyhedral units and can be construed as the three-dimensional analogs of polycyclic aromatic hydrocarbons. Analysis of frontier molecular orbitals using extended Huckel calculations show that the n þ 1 electron pair requirement of closo mono-polyhedral clusters must be recast as n þ m electron pairs for condensed polyhedra where m refers to the number of individual polyhedra.102 This leads to the ‘n-centered n þ m electron pair’ bonding with electrons delocalized over the entire condensed polyhedron despite having concave surfaces at the fused site. Shared vertices are readily perceived since they lack exo-polyhedral 2ce2e bonds and helps identify the individual clusters in the condensed polyhedra. It is interesting to note that this formula does not explicitly depend on the total number of shared edges, something that is difficult to decide in the case of long BeB distances. In the case of condensed closo borane clusters (Fig. 21), the n þ m rule implies that edge-sharing retains the dianionic charge irrespective of the number of individual polyhedral units involved. In other words, an edge-shared polymer has zero charge per monomeric unit, while sharing a deltahedral face reduces the charges by one electron per unit and four atom sharing reduces by two. Condensed polyhedra can share only up to a maximum of four atoms to maintain the required concavity of the surface near the condensation. Though condensation serves as an efficient mechanism to reduce the building up of charges, the nonbonding 1,3 interactions across the shared edge(s) destabilize the system. Because of these instabilities, condensed polyhedra are relatively rare in boron rich borides and are observed only in extremely electron poor systems. Though four vertex sharing is unique in the molecule B20H16, they are yet to be characterized as part of the extended boride network. Face sharing is relatively more common in boron rich borides. The hexagonal beryllium boride of composition BeB3 (P6/mmm) is one of the experimentally well characterized covalent network103 that shows face-sharing between two closo-B16 polyhedra though the hexagonal boron sites are partially occupied by larger beryllium atoms (Fig. 22). The covalent network of the unit cell contains three B12 units, two Be3B12 supraicosahedral units and a face sharing Be8B21 condensed polyhedra interconnected to form the extended network. The beryllium atoms in the 15 vertex polyhedra are found to exactly occupy the hex-cap positions without any partial or mixed occupancies due to proper ring-cap matching. The face sharing Be8B21 cluster however show mixed occupancies (50%) in six of the symmetrically equivalent beryllium sites with boron atoms, presumably due to the imposition of high symmetry.

Fig. 21

Some examples of condensed macropolyhedra from sharing of (A) four-atom (B) three-atom and (C) edge.

Bonding in boron rich borides

41

Applying the electron counting rules allows the charges of these individual cluster per unit cell as 3B122, 2(B12Be3)5 and a (Be8B21)9 since eBe is isoelectronic to eB. The overall 25 electrons requirement for this unit cell is exactly met out by 12 symmetrically equivalent Be2þ ions which are not part of the polyhedra, and four other Be2þ ions that are reported to have a partial occupancy of 0.125. Though electron counting rules help in understanding chemical bonding and implies semiconducting nature, it is not verified by experiments or band structure calculations. Similar structures with face sharing 18-vertex closo polyhedra is reported in ScB17C0.25.104

3.03.6

Bonding in single-vertex sharing macropolyhedra

Unlike the condensed polyhedral systems that reduces the electronic requirements, sharing of a single vertex is found to increase the requirement of electrons as the tangential bonding molecular orbitals are not shared between the polyhedra and they behave like discreet units. To address this variation, a generalized mno rule was introduced that governs the macropolyhedral boron clusters in all its avatars.105 Here ‘o’ refers to the number of instances of single vertex sharing in the macropolyhedral cluster. The increased steric repulsion between the atoms proximate to the shared vertex makes it impossible for boron to be in the shared position and is known only with larger atoms in molecules. However, it is present in the covalent network of rhombohedral boron in the form of trigonal pyramidal B5 units sharing its apical boron atom in a complex macropolyhedral environment which will be discussed in the last section. The refined structure of NaB6, with the updated stoichiometry of Na3B20 has pentagonal bipyramids delocalized through the rhombic motif, that shares a single vertex forming linear chains. The orthorhombic Na3B20 (Cmmm) has a layered structure consisting of inter-connected B6 units and B7-based macropolyhedral polymeric chains (Fig. 23). These layers are linked through exopolyhedral BeB bonds through the apical boron atoms of the B7 and B6 units.106–108 Application of mno rule indicates that the stoichiometric structure lacks one electron per unit cell for optimal bonding. Band structure calculations confirm the presence of a semiconducting gap with an additional electron which is otherwise metallic.61 Since partial occupancies and carbon impurities were ruled out, and the structure needs to be metallic with an odd electron unit-cell, which is experimentally confirmed from conductivity measurements and its temperature dependence.109

3.03.7

Perturbations in polyhedral bonding

Though qualitative, the cluster bonding model suggests that the n þ 1 rule of closo polyhedra remain unaltered even if there are minor perturbations on its skeletal framework. Frontier orbital interaction theory provides versatile explanation that the perturbations caused by the removal of one or more vertices (non-adjacent) on its closo surface or adding an additional vertex on its surface from inside (endohedral capping) or outside (exopolyhedral capping) is not expected to drastically change the ideal n þ 1 electron pair requirement of the parent closo borane.73 The usual high stability associated with multicentered cluster bonding leads to large HOMO-LUMO gaps and accounts for this invariance. This implies that the absence of the surface atoms will increase the electron

Fig. 22

The structure of (A) BeB3 (B) the supraicosahedral Be3B12 unit and (C) condensed polyhedral Be8B21 unit.

42

Bonding in boron rich borides

Fig. 23 (A) model of the trigonal bipyramidal sharing a vertex (B) Pentagonal bipyramidal chains with single-vertex sharing (C) the structure of Na3B20.

count while capping and stuffing of additional atom will decrease it. However, these are merely rules, not theorems and exceptions occur if there are increased perturbations happening in proximity. Regardless of these shortcomings, this bonding model is proved to be the most useful paradigm for interpreting bonding in the chemistry of polyhedral boranes. In macropolyhedral boron clusters, missing vertices do occur more frequently and are usually adjacent to the shared boron atoms. The change in the electronic requirement due to these missing vertices and other defects are treated in the similar way as in mono polyhedral clusters and had been corroborated extensively with conjucto boranes. Perturbations arising from endohedral capping are extremely rare110 as the polyhedra need to be big enough to accommodate atoms inside the cluster.111 Exopolyhedral capping is relatively common and they frequently are partially occupied, providing additional electrons to the boride lattices with metal poor compositions.

3.03.7.1

Borides with missing vertices

Monopolyhedral boron clusters with missing vertices are yet to be characterized in boron rich borides as it increases the electronic requirement. However, it is ubiquitous in the condensed polyhedral environment mostly to reduce the non-bonding 1,3 interactions between boron atoms linked to the shared edge.112 The simplest macropolyhedral boron cluster that has one missing vertex is observed in the orthorhombic MgB12 (Pnma) (Fig. 24). It has a large 390 atom unit cell (z ¼ 30) where the covalent network is formed by B12 units and macropolyhedral B20 units and an additional non-polyhedral tetravalent boron atom which initially lead to ambiguities in describing its bonding.113 The B20 units (Fig. 23B) has to be treated as a condensed boron cluster of two face sharing B12 units (B21) with one vertex missing. Since B21 cluster, which is a closo need one electron, B20 with one vertex missing needs three electrons. There are sixteen [B12]2, eight [B20]3 units and eight tetravalent boron atoms. The four bonds of these tetravalent boron atoms connect to two B12 units and two vertices of the B20 unit, which are incident on either side of shared boron atoms. The short BeB distance between these two non-bonded boron atoms (1.97 Å) points to the possibility of 3ce2e bonding, though the other two bonds are in the 2ce2e range. Assuming all the bonds as 2ce2e, the tetrahedral B atoms will add up to a total of 64 electrons per unit cell. The 30 magnesium atoms nearly but not completely compensate this requirement ruling out 3ce2e bonding. No conductivity studies or electronic structure calculations are reported so far for this structure. Since partial oxidation and disorder is ubiquitous in B12 based borides, we anticipate semiconducting nature even though band structure may show a band gap only for the ideal electron count.

Fig. 24

(A) The unit cell of MgB12 (B) the structure of the B20 unit bonding to the tetravalent boron.

Bonding in boron rich borides

43

The covalent network of tetragonal AlB12 in both its a and g-form contain condensed B19 units which has two vertices missing from the B21 unit giving a charge of 5-. The simplest among these borides is the a-AlB12 structure in which B19 polyhedra has a C2 symmetry.114 The covalent network of a-AlB12 (Fig. 25) is made up of eight B12 units, four B19 units and four tetravalent B non-polyhedral atoms in its unit cell, together requiring 40 electrons. There are five aluminium sites, all of them showing partial occupancies (0.75, 0.02, 0.246, 0.48, 0.162) when summed up, giving 13.264 aluminium atoms as there are eight symmetrically equivalent positions in the unit cell. This amounts to 39.792 electrons, less than expected from the electron counting rules only by a fraction of an electron. Existence of a band-gap is expected but not confirmed either by theory or experiments so far. Partial replacement of aluminium with metals like Be, B, etc., are also known with this lattice.115 In the g form of AlB12, there are two different isomers of condensed B19 units present together (Fig. 26). In addition to the C2 symmetric one present in a-AlB12, there is also a Cs symmetric B19 unit where the two missing vertices are from the same side of the face-sharing B12 subunit.114 With all the boron sites fully occupied, the orthorhombic unit cell has sixteen B122 units, eight B195 units, and eight tetravalent  B atoms requiring a total of eighty electrons. Since all the 11 distinct aluminium sites are partially occupied, the ideal electron precise composition will be Al26.66B352. The electronic requirement of the lattice is computed earlier using molecular orbital calculations and assigned B206 and B122 before the advent of n þ m rule. Though the assigned charges tally with electron counting rules, the bonding in B20 unit has to be described as not a single delocalized system but rather as a B195 condensed polyhedra capped by a tetravalent B atom. In the experimental sample, the different aluminium sites together give  82.84 electrons, little more than that is required by this large unit cell having 352 boron atoms. However, the corresponding paper reports the formula unit as Al6.3B88 (Z ¼ 4) that leads to 75.54 electrons.116 The existence of a definite band gap is neither confirmed by band structure calculations nor by experimental data.

3.03.7.2

Borides with capping vertices

In boron rich borides, when the electronic requirement is not effectively compensated by the metal ions, the lattice often resorts to having additional boron atoms bridging the one or more of the inter or intra polyhedral bonds. Since the bonding states of the polyhedra are lying relatively low, these additional boron atoms act as electron sources, just like metals, despite their strong covalent bonding to the polyhedra. Even though they were treated as ionic for electron counting purposes, they have strong directional bonds. These are also the major sources of disorder as their physical presence is not inherently required for the stability of the polyhedral bonding and serve only to address the electron deficiency in the lattice. The controversies associated with the structure of the sodium boride, variously described as NaB15 and Na2B29 serves as a perfect illustration of the complexities surrounding exo-polyhedral capping. The covalent network is like rhombohedral borides, having B12 units interconnected by BeB chains in a sp3 hybridized environment. Both these structural units need two electrons and the sodium atoms provide only half of the requirement. The boride network hosts interstitial B atoms that act as a trivalent cation donating its electrons to the cluster in addition to the Naþ ions (2B122, 2B22, 2Naþ, 2B3þ). These boron atoms are initially reported to be capping the B2 unit in Imma symmetry (Fig. 27A).117 Later, the composition was refined to be Na2B29 with a monoclinic lattice (Cmmm) where alternate interstitial B sites remain unoccupied.118 Recent theoretical investigations point towards the higher stability of Na2B30 over Na2B29 from their formation energies.119 DFT calculations show that bridging the B2 units gives a semimetallic band structure, while an alternate interstitial position where it is found to cap two different B2 units as well as B12 polyhedra (I212121 Fig. 27B) has an indirect band gap of 1.6 eV and thermodynamically more stable. Phonon calculations show both Imma and I212121 structures have dynamic stability for Na2B30 phase. Since all have comparable XRD patterns, it is speculated that these different phases including Na2B29 may coexist or may have occupancy disorders and explains the non-stoichiometry revealed by chemical composition analysis.

Fig. 25 (A) The unit cell of a-AlB12 (B) the structure of the face sharing condensed B19 unit missing two vertices along with the non-polyhedral tetravalent boron.

44

Fig. 26

3.03.8

Bonding in boron rich borides

(A) The unit cell of g-AlB12 (B) B19 unit (Cs) (C) B19 unit (C2); along with the non-polyhedral tetravalent boron.

Bonding in non-deltahedral borides

The preference of deltahedral clusters for boron is well established in the polyhedral borane chemistry though the bond lengths vary from 1.65 Å to 2 Å in different polyhedra. Despite these wide variations, structures other than deltahedra are seldom reported experimentally for closo polyhedra and if they occur, they are usually associated with missing vertices that requires additional electrons. In boron rich borides, there are instances where polyhedral surfaces having square faces are observed with high symmetry. The Oh symmetric B12 units and O symmetric B24 units are the glaring representatives of this phenomenon. Though there are different possible distortions of the B12 icosahedron in various clusters to be compatible with the crystallographic symmetry, the cuboctahedral B12 units cannot be classified as one of them as Oh is not a subgroup of Ih. However, Ih is topologically related to Oh by the transformation of six of the rhomboids on the icosahedral B12 surface to a square face. Despite the severity of this transformation that involves breaking of six of the intra-polyhedral bonds, it is found to be the preferred structural motif for MB12 borides. Transformation of icosahedral B12 to cuboctahedron destroys the fourfold degeneracy of its frontier MOs. The HOMO (gu) of icosahedral B12 transforms into t1u þ a2u, with the t1u enduring extensive destabilization due to the distortion as it was bonding across the broken BeB edges. Similarly, its LUMO gg transforms as t1g þ a2g, t1g getting stabilized as it is antibonding across the broken BeB edges. This leads to the substantial reduction in the HOMO-LUMO gap and subsequent metallicity in the extended lattice even though dianions are still preferred for optimal bonding.65 All the cuboctahedral borides reported so far are metallic and flexible enough to accommodate more than two electrons per B12 unit. Most of the cuboctahedral borides crystallize in the cubic UB12 structure type, with Fm3m space group where the metal resides in the channels of hexagonal rings (Fig. 28). However, ScB12 is predicted to undergo a D4h distortion into I4/mmm symmetry in its ground state at lower temperatures (< 180 K) as the smaller Sc prefers to cap the square faces of the cuboctahedra.120 The distortion is reflected in the boride lattice as given by the two different intra-cluster bond lengths. Except for Zr, that has a stable þ 2 oxidation state, most other transition metals and rare earths forming this phase prefer higher oxidations states. The frontier bands of these borides comprise of metal d/f orbitals as well as B p orbitals.121,122 The inter-polyhedral 2ce2e bonds are comparatively shorter than those of the other B12 based borides. There is also a report of face sharing cuboctahedral networks in WB4, where the tungsten atom is assumed to partially replace the shared three-membered ring.123

Fig. 27

The crystal structure of Na2B30 in (A) Imma and (B) I212121 space symmetry. The capping boron atom is shown in distinct color.

Bonding in boron rich borides

45

The ring-cap mismatch destabilizes the closo-boron clusters larger than 12 though they are characterized as minima on the potential surface. Initial calculations on the approximate geometry of these supraicosahedral boranes assumed them as strain points and predicted higher stability when they are distributed evenly in the polyhedra surface.124 However, the computed equilibrium geometry reported later125 has shown that such hexagonal caps preferred to be adjacent so that the bond that connects them can be elongated or broken. The broken BeB link creates a distorted square face on the polyhedral surface; yet obey the n þ 1 electron pair rule. This tendency is also observed in some of experimentally reported supraicosahedral carboranes.126 These distorted squares become perfect squares by symmetry for n ¼ 24 snub cube with a chiral O symmetry. Though not known in the molecular world, some of the cubic rare earth borides generally referred to as REB66 have this polyhedral motif as part of a large polyhedral cluster and is the largest ‘closo’ polyhedral unit reported in boron rich borides. The structure of YB66, the first of its kind was characterized long time ago111 and the structure is resolved127to have a cubic unit cell (Fm-3c) containing more than 1600 atoms that also shows large partial occupancies for both the metal and boron sites (Fig. 29). Its structure is usually described as having a super-icosahedral network consisting of a central B12 unit surrounded completely by 12 other B12 units. In addition, it also has B80 cluster with local O symmetry that is riddled with partial occupancies. The BeB distance between some of these boron sites are less than 1 Å making their simultaneous presence an impossibility. At the heart of this B80 cluster is the snub cubic B24 structure, whose dianion is characterized as a minimum by theoretical calculations.125 Despite being surrounded by several other boron atoms with partial occupancies, the square face of the snub cube in the macropolyhedral B80 cluster is open, that implies B24 unit as the primary building block. When some of the boron sites with low occupancy that has abnormally short distances with the high occupancy sites are removed, it gives much simpler O symmetric B48 cluster. The current electron counting rules are difficult to apply for this extremely condensed polyhedra unit due to the uncertainties in determining the number of individual polyhedral units. Despite having established applications as a monochromator for third-generation light sources and holding promise as a thermoelectric material,18 electronic structure and bonding in this material remains a challenge.

3.03.9

Bonding in elemental boron polymorphs

Boron exhibit many allotropes, several of them are continuously refined and several high-pressure modifications are also reported recently. Though ambiguities in their structure and purity exists in several of its allotropes, all these networks typically involve pseudo icosahedral B12 units as the primary motif. The covalent framework exhibited by these allotropes are also frequented with partial occupancies in some of the exopolyhedral capping boron sites and capable of retaining the structure by metal doping. Here we will discuss the chemical bonding in some of the well-established allotropes. The simplest allotrope of boron is the metastable a-rhombohedral boron with its 12-atom unit cell comprising of just one pseudo icosahedral B12 polyhedra with no discernible partial occupancies or interstitials. The symmetry of the B12 unit is reduced to D3d in the lattice that has a layered structure (Fig. 30). The 3ce2e bonds connect the B12 units within the layer and 2ce2e bonds between the layers. The dianionic requirement of B12 polyhedron is achieved by the six 3ce2e bonds that surrounds the B12 unit, as each boron contributes only 2/3 of the electron for each 3ce2e bond. The rest 1/3 electrons from six boron atoms amounts to 2 electrons required by B12.128 The thermodynamically most stable allotrope is b-rhombohedral boron, popularly described by a 105-atom unit cell with partial occupancies and interstitial sites129 (Fig. 31). The covalent framework has four B12 units and a giant D3d symmetric B57 unit that exhibits a complex condensation of six B12 units and two trigonal bipyramidal units that also exhibit vertex sharing, the only example of vertex sharing with boron at the shared site. Assuming the exo-polyhedral bonds are all 2ce2e as they are connected to adjacent B12 units, mno rule predicts (n ¼ 57; m ¼ 8;o ¼ 1) a 3þ charge for this cluster which was confirmed by electronic structure calculations. This idealized structure that ignores partial occupancies and interstitial sites are electron deficient by five electrons and indeed shows a band-gap when five electrons are provided.130 The electron deficiency forces interstitial partially occupied sites to cap the polyhedra from outside, providing

Fig. 28

(A) UB12 type lattice and (B) the dislocated ScB12 lattice.

46

Bonding in boron rich borides

Fig. 29 (A) The structure of the YB66 primitive cell (B) the structure of B80 cluster with all the partially occupied boron atoms. (C) the simplified B48 cluster construed by removing the mutually exclusive atom sites.

the extra electrons needed, which is persistent in this polymorph. This polymorph also hosts an array of metals that correlates with the electronic requirement. Boron also is reported to have two tetragonal polymorphs though their chemical bonding is shrouded with mystery. The a-tetragonal boron, first reported to have 50 atoms in its unit along with some partially occupied interstitial sites has four C2h symmetric B12 units and two tetrahedral boron atoms (Fig. 32). Together, they require 10 electrons by simple electron counting rules that need to be coming from the interstitial boron atoms. After the disputed claims that this structure requires hetero atom impurities like carbon or nitrogen for stability,131 the recent theoretical inquiry with DFT calculations predicted a stoichiometry B52 per unit cell.132There are two interstitial atoms occupying two of the four equivalent bridging sites reducing the symmetry. Surprisingly, bridging of two boron atoms on either interpolyhedral or intra-polyhedral BeB bonds are computed to open a narrow band gap, with the former being more stable. A recent experimental characterization reports the stoichiometry close to these predictions.133 Assuming the two boron atoms provide six electrons, the structure still needs four more electrons, keeping its bonding queries still open. The structure of b-tetragonal boron is isotypic to a-AlB12, in which the aluminium atoms are replaced by interstitial boron atoms satisfying the electronic requirement of the network. A total of eight boron atoms are needed per unit cell to open the band bap as revealed by the DFT calculations.134 The recent revelation is the high-pressure phase of g-boron that has a stoichiometric unit cell of 28 boron atoms comprising of B12 units and B2 chains (Fig. 33), completely devoid of partial occupancies and other disorders.135 The dianionic requirement of the B12 unit is met by the capping of two of the polyhedral bonds by the boron atoms of the B2 unit, more like an interstitial boron atom. Alternative formulation involving 3ce2e bonds between the chain boron atom and two of the B12 unit were originally put forward that makes this structure the first example of a vertex-sharing 3ce2e bond in boron rich borides.

Fig. 30

(A) the covalent network of a-rhombohedral boron and (B) the environment of the B12 unit.

Bonding in boron rich borides

Fig. 31

(A) the unit cell of b-rhombohedral boron and (B) the macropolyhedral B57 unit.

Fig. 32

(A) the unit cell of a-tetragonal boron and (B) the environment of the B12 unit.

Fig. 33

(A) the unit cell of g- boron; (B) and (C) the environment of the B12 unit by two different bonding descriptions.

47

But the unevenness of such a 3ce2e bond with one long BeB distance of 2.089 Å that is longer than the typical 3ce2e bond and two short BeB distances (1.927 and 1.826) that are way too short for 3ce2e bond, along with the question of the electronic requirement of the central triangle that is flanked by the 3ce2e bonds make this proposition unreasonable.

48

Bonding in boron rich borides

The thermodynamically most stable structural motifs of two-dimensional boron nanosheets reported so far are exclusively dominated by planar deltahedral networks. Referred to as borophenes, these are metallic and experimentally characterized only on metal surfaces indicating that the free standing borophenes are extremely reactive.136 There are difficulties in identifying cognizable molecular fragments in these nanosheets and the chemical bonding in these systems are yet to be clearly understood.

3.03.10 Concluding remarks While the nature of covalent motifs underlying the borides depend largely on the number of electrons available to the network, there are a lot of other variables that influence its structural preferences. The array of elemental allotropes exhibited by boron is indicative of the structural diversities. The different bonding motifs observed in molecules, chemical bonding models, developed from frontier orbital interaction theory and electronic structure calculations help decipher the nature of bonding in bulk of the boron rich borides. The chemical viewpoint allows rationalizing the observed geometry by segregating the entire network into a set of chemically meaningful units whose bonding is familiar from its molecular analogs. This helps tremendously in sorting out the origin and nature of disorder that is abundant in boron rich borides and give enormous confidence in exploring the otherwise formidable structures. It also offers enough hints in predicting various electrical, mechanical, thermo-electric properties such as band-gap, hardness, etc. However, there are also covalent motifs found in boron rich borides that are not seen in its molecular world. These can be understood by deconstructing their structure into known and unknown components. The effect of the perturbation relative to the known standard model can then be revealed. In the molecular world, the various non-classical boranes with their strange structures (from the most symmetric to the most chaotic) unheard of in the classical world is primarily due to the frustration of boron not having enough electrons to do its favorite covalent bonding. In boron rich borides, the frustration continues as the preferred closo polyhedral motif still needs two more electrons despite all the non-classical nature. Further, the most stable icosahedral B12 polyhedra being incompatible with the translational symmetry forces a variety of distortions leading to equally strange extended solids. Though the structure of many of the known boron rich borides are understandable from its chemical bonding, there are several others that still pose a challenge, and the list is growing. The perspective from chemical bonding, despite its oddities, prove to be a valuable tool with no alternatives to comprehend the extensive diversities present in boron rich borides.

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Bonding in boron rich borides Jasper, A. L.; Bourgeois, L.; Michiue, Y.; Si, Y.; Tanaka, T. J. Solid State Chem. 2000, 154, 130–136. Jemmis, E. D.; Balakrishnarajan, M. M.; Pancharatna, P. D. J. Am. Chem. Soc. 2001, 123, 4313–4323. Hagenmuller, P.; Naslain, R. C.R. Acad. Sci. 1963, 257, 1294. Albert, B. Angew. Chem. Int. Ed. 1998, 37, 1117. Albert, B.; Kofmann, K. Z. Anorg. Allg. Chem. 1999, 625, 709–713. Morito, H.; Shibano, S.; Yamada, T.; Ikeda, T.; Terauchi, M.; Belosludov, R. V.; Yamane, H. Solid State Sci. 2020, 102, 106166. Jemmis, E. D.; Balakrishnarajan, M. M. J. Am. Chem. Soc. 2000, 122, 7392–7393. Richards, S. M.; Kasper, J. S. Acta Crystallogr. B 1969, 25, 237–239. Jemmis, E. D.; Pancharatna, P. D. Appl. Organomet. Chem. 2003, 17, 480–492. Adasch, V.; Hess, K. U.; Ludwig, T.; Vojteer, N.; Hillebrecht, H. J. Solid State Chem. 2006, 179, 2916–2926. Higashi, I. J. Solid State Chem. 2000, 154, 168–176. Higashi, I. J. Solid State Chem. 1980, 32, 201–212. Higashi, I. J. Solid State Chem. 1983, 32, 333–349. Naslain, R.; Kasper, J. S. J. Solid State Chem. 1970, 1, 150–151. Albert, B.; Hofmann, K.; Fild, C.; Eckert, H.; Schleifer, M.; Gruehn, R. Chem. A Eur. J. 2000, 6, 2531–2536. He, X. L.; Dong, X.; Wu, Q. S.; Zhao, Z.; Zhu, Q. Phys. Rev. B 2018, 97, 0100102-1–0100102-5. Liang, Y.; Zhang, Y.; Jiang, H.; Wu, L.; Zhang, W.; Heckenberger, K.; Hofmann, K.; Reitz, A.; Stober, F. C.; Albert, B. Chem. Mater. 2019, 31, 1075–1083. Korozlu, N.; Surucu, G. Phys. Scr. 2012, 87, 015702-1–015702-6. Ma, T.; Li, H.; Zheng, X.; Wang, S.; Wang, X.; Zhao, H.; Han, S.; Liu, J.; Zhang, R.; Zhu, P.; Long, Y.; Cheng, J.; Ma, Y.; Zhao, Y.; Jin, C.; Yu, X. Adv. Mater. 2017, 18, 1604003. Yeung, M. T.; Mohammadi, R.; Kaner, R. B. Annu. Rev. Mat. Res. 2016, 46, 465–485. Brown, L. D.; Lipscomb, W. N. Inorg. Chem. 1977, 16, 2989–2996. Pancharatna, P. D.; Marutheeswaran, S.; Austeria, M. P.; Balakrishnarajan, M. M. Polyhedron 2013, 63, 55–59. Grimes, R. N. Angew. Chem. Int. Ed. 2003, 42, 1198. Perkins, C. L.; Trenary, M.; Tanaka, T. Phys. Rev. Lett. 1996, 77, 4772–4775. Lipscomb, W. N.; Britton, D. J. Chem. Phys. 1960, 33, 275–280. Hughes, R. E.; Kennard, C. H. L.; Sullenger, D. B.; Weakliem, H. A.; Sands, D. E.; Hoard, J. L. J. Am. Chem. Soc. 1963, 85, 361–362. Bullet, D. W. J. Phys. C 1982, 50, 415. Amberger, E.; Ploog, K. J. Less Common Met. 1971, 23, 21. Waturu, H.; Shigeki, O. J. Solid State Chem. 2010, 22, 1521–1528. Ekimov, E. A.; Lebed, Y. B.; Uemura, N.; Shirai, K.; Shatalova, T. B.; Sirotinkin, V. P. J. Mater. Res. 2016, 31, 2773–2779. Wataru, H. J. Solid State Chem. 2015, 22, 375. Oganov, A. R.; Chen, J.; Gatti, C.; Ma, Y.; Ma, Y.; Glass, C. W.; Liu, Z.; Yu, T.; Kurakevych, O. O.; Solozhenko, V. L. Nature 2009, 457, 863–867. Ranjan, P.; Sahu, T. K.; Bhushan, R.; Yamijala, S.; Late, D. J.; Kumar, P.; Vinu, A. Adv. Mater. 2019, 31, 1900353-1–1900353-8.

3.04

The Zintl-Klemm concept and its broader extensions

A´ngel Vegasa and A´lvaro Lobatob, a Hospital del Rey, Universidad de Burgos, Burgos, Spain; and b MALTA-Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain © 2023 Elsevier Ltd. All rights reserved.

3.04.1 3.04.2 3.04.2.1 3.04.2.2 3.04.2.3 3.04.2.4 3.04.2.5 3.04.3 3.04.4 3.04.4.1 3.04.4.2 3.04.5 3.04.5.1 3.04.5.2 3.04.5.3 3.04.5.3.1 3.04.5.3.2 3.04.5.3.3 3.04.5.3.4 3.04.6 References

Introduction: Structure and bonding in Zintl phases The extended Zintl-Klemm concept Four connected networks Three-connected networks Two-connected networks Branched silicates Discrete oligosilicate anions. Single-connected groups The amphoteric silicon. Si as donor and as acceptor in Ce16 ½Si½6 Si½4 14 O6 N32  The amphoteric germanium. The structure of ðNH4Þ2 Ge½6 ½Ge½4 6 O15  First approach Second approach: The donor Ge[6] atoms Closed packed arrays The EZKC applied to Mg2PN3, Zn2PN3, Li2SiO3 and Ca2PN3 The structure of LiAlSe2 The EZKC in close-packed solids Preferred skeletons in crystals The structures of NiAs and Ni2In The MnP and the Co2Si structures The paradigmatic structure of CsLiSO4 Concluding remarks

52 55 55 57 57 59 60 60 61 61 63 65 65 65 67 67 68 68 70 70 72

Abbreviations AMM Anions in metallic matrices BP Bonding pair BS Building skeleton CP Chemical pressure ELF Electron localization function EZKC Extended Zintl-Klemm concept LP Lone pairs (Lewis pairs) OPC Oxidation pressure concept PS Preferred skeletons VEC Valence electron count ZKC Zintl-Klemm concept

Abstract The Zintl-Klemm concept (ZKC) remains among one of the most important and useful concepts developed in solid state chemistry. Based on the deeply rooted chemical knowledge of electron counting and electronegativity difference, it has allowed to understand crystal structures involving main group elements (aluminates, silicates, phosphates, etc.) as well as structures of alloys (Ni2In, Co2Si, LiAlSe2, CsLiSO4, etc.) which had not been previously explained. Along this chapter we will discuss the principles of the ZKC making emphasis on its extension to complex structures of oxides. Surprisingly, the ZKC holds in the cation arrays of oxides despite of being embedded in an oxygen matrix as it is the case of aluminates, silicates, and germanates among others. Up to now, these compounds had been only classified taxonomically, their structures remaining unexplained. The extended ZKC (EZKC) considers that electrons are transferred from the donor cations to the atoms forming tetrahedral oxopolyanion skeletons. The cation subarrays of these compounds behave as true Zintl phases and the tetrahedral skeletons are true Zintl polyanions. Selected examples are provided to illustrate how Zintl polyanions found in these complex structures resemble the geometry of their parent block-p elements or simple molecules formed with these

Comprehensive Inorganic Chemistry III, Volume 3

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The Zintl-Klemm concept and its broader extensions elements. Specific mention is made to the amphoteric character of atoms like Na, Ni, Si, Al and Ge in order to achieve stable Zintl polyanion configurations. To conclude the chapter, we summarize how the application of the EZKC reveals the presence of preferred skeletons as building blocks of complex structures, making possible to establish a link between chemical composition, structure and atomic octet fulfillment.

3.04.1

Introduction: Structure and bonding in Zintl phases

The Zintl-Klemm concept (hereafter ZKC) was developed by Zintl in the 1930s1,2 to account for the structure and bonding in a new family of compounds formed by the most electropositive metals (Groups 1 and 2 elements) with non-metals and semimetals (elements of the Groups 13–16) collected in Scheme 1. Most Zintl phases obey the formula AaXn (A ¼ donor metal; X ¼ the acceptor electronegative element). The features defining the Zintl phases were suggested by Nesper3 and Miller.4,5 They are semiconductors (gap of about 2.0 eV) or, at least poor conductors, and most of them are diamagnetic. Regarding their structural characteristics, there exists a well-defined relationship between their chemical and electronic structures. The charged Xn- atoms, don’t behave as isolated entities, but are assembled as anionic 1D-, 2D- or 3D-skeletons (X x n polyanions) known as Zintl Polyanions.2 Such a structural arrangement allows the X atoms to complete its octet through the formation of (8-N) directed X-X (2c, 2e) bonds according to the so-called (Pearson’s 8-N rule),6 where N is the total number of valence electrons of the X atoms (their own plus those transferred from the A atoms).3 The charged X x n dskeletons are called Zintl polyanions.2 NaTl was the first Zintl phase reported (1932)1 and despite the long period of time since the ZKC was created, this branch of research continues to be of interest both experimentally and theoretically. On the one hand, the ZKC concept have continuously motivated the search of compounds with new properties and has stimulated new ways of synthesis. Among the different examples which can be cited, it is worthy of mention the synthesis of carbon-free fullerenes of In and Tl7 and the reactions of Zintl phases in non-aqueous solvents7–10 in which the Zintl polyanions remain intact in solution. Likewise, the validity of the ZKC has been often questioned encouraging theoreticians to apply state of the art computational methods with the aim to understand the chemical bonding in these compounds.3–13 For the purpose of this chapter, the most relevant characteristic of these compounds is the tight relationship existing between the electronic and the chemical structures. To illustrate this point, we will describe next the structures of several Zintl phases, beginning with that of NaTl, drawn in Fig. 1. In NaTl, all the valence electrons (3 e of Tl plus one e from Na) are located on the Tl atoms (X), so that each Tl– atom, with 4 e– per atom, can form four Tl-Tl directed (2c, 2e) bonds, according to the (8-N) rule. In this case, the Tl– anions attain their fourfold connectivity forming a diamond-like 3D skeleton (Zintl polyanion) characteristic of the Group 14 elements (tetrels) (see Fig. 1). From this point of view, the atoms forming the Zintl polyanions behave structurally like neutral isoelectronic atoms. This led Klemm14 to give the name of pseudo-atom (Greek symbol Y) to the charged atoms that form the polyanions. Since then, the concept was renamed as the Zintl-Klemm Concept. Zintl phases thus contain to kinds of bonds, i.e., the covalent (2c, 2e) bonds forming the [Tln]ndpolyanion, and the ionic interactions between the 3D-polyanion and the donor Naþ cations. At the first stages, it was believed that the formation of Zintl phases were restricted to the elements at the right of the vertical line separating the 13 and 14 groups shown in Scheme 1 (Zintl border) but further studies showed that some elements at the left of this line, i.e., Ag, Al, In, Tl or Zn, are also capable of forming Zintl polyanions. Zintl polyanions skeletons normally coincide with the structures of elements with the same number of valence electrons, but in other cases, even if they show the expected connectivity, their structure can differ from the one exhibited by the pure elements. This lack of concordance between the Zintl polyanion and elementary structures, led Schäfer et al.15 to propose a redefinition of the ZKC, not limiting it to compounds with known elementary structures but “extending it to skeletons that fit the number of bonds predicted by the (8-N) rule.” We summarize below some examples which emphasize the electron transfers as well as the structure of the Zintl polyanions in CaIn2, CaSi2, NaP, CaP and ScP to show the strength of the ZKC:

Scheme 1

Elements susceptible of forming Zintl polyanions. The thick line separating the groups 13 and 14 represents the so-called Zintl border.

The Zintl-Klemm concept and its broader extensions

53

Fig. 1 Stereopair of the cubic structure (Fd-3m) of NaTl formed by Naþ cations (blue spheres) and the [Tl–]n polyanion (gray spheres) with the diamond-like structure.

CaIn2 : CaSi2 :

Ca2þ þ 2 e– þ 2 In/Ca2þ þ 2 In– hCa2þ ðJ  Sn2 Þ

stuffed  lonsdaleite

Ca2þ þ 2 e– þ 2 Si/Ca2þ þ 2 Si– hCa2þ ðJ  PÞ2 : As  type phosphorous NaP :

Naþ þ 1e– þ P/Naþ þ P– hNaþ ðJ  SÞ helical sulfur

(B) (C)

(A)

(D)

(E)

Fig. 2 The structures of (a) CaIn2 (Ca: blue; In: gray), (b) CaSi2(Ca: green; Si: gray), (c) NaP (Na: blue; P: violet), (d) CaP (Ca: blue; P: ochre) and (e) ScP (Sc: green; P: violet), interpreted according to the ZKC. The Zintl polyanions are: In– h [J-Sn]n, Si– h [J-P]n, P– h [J-S]n, [P2–]2 h [J-Cl]2, P3– h J-Arfcc. Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V.

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The Zintl-Klemm concept and its broader extensions

CaP :


2 Ca2þ þ 4 e– þ 2 P/2 Ca2þ þ 2 P2– h Ca2þ 2 ðJ  Cl2 Þ

ScP : Sc3þ þ 3 e– þ P/Sc3þ þ P3– hðJ  KrÞðJ  ArÞ

Cl2

molecules

2 interpenetrated fcc  arrays

We have selected these five compounds because their J-atoms form Zintl polyanions coincident with the same structures as the elements of Groups 14–18. For instance, In CaIn2, it is assumed that one electron is transferred from Ca to each In atom, formally converting them into J-Sn. Hence, In atoms adopt the structure of a tetrel: the hexagonal diamond or lonsdaleite (see Fig. 2a). In CaSi2 (Fig. 2b) the Ca atoms transfer one electron to each Si atom leading to J-P. In this structure Si– polyanions form threeconnected layers (As-type), also resembling the structure of P at high-pressure. The J-P layers alternate with hcp 36 planar nets of Ca atoms in.ABC. sequence that preserve the Ca-Ca distances of fcc-Ca (see Fig. 2b). Likewise, the structure of NaP (Fig. 2c) can be understood by considering that one-electron is transferred from Na to P, thus the P atoms convert into J-S forming the square helices in which the P atoms are bi-connected as displayed by S and Se elements (sextels). Finally, in CaP (Fig. 2d) the Ca atoms transfer two electrons to the P atoms that convert into J-Cl which, with seven valence electrons, form diatomic groups like Cl2 molecules. We can go further and search for structures such as ScP (Fig. 2e). If the electropositive Sc atoms transfer their three valence electrons to the P atoms, they become Sc3þ (cations, –J-Kr) and P3– individual anions (J-Ar), both adopting the fcc-arrays of Kr and Ar respectively. The result is an ionic NaCl-type structure lacking of any polyanion. The expected structure coincides with that of real ScP.16 Although most Zintl phases fulfill the conditions set out above, it must be noticed that the covalent bonding part has been stressed while any ionic or metallic interactions has been underestimated. As a result, donor atoms have been regarded as a mere electron source to stabilizing the covalent polyanions. We are aware that this almost pure covalent vision does not explain some distortions found in Zintl polyanions when compared with the ideal elemental structures. For example, in the CaIn2 structure (Fig. 2a) the three IneIn bonds, within the horizontal layers, are shorter (2.889 Å) than the vertical bonds (3.171 Å), a fact that differs from the regularity found in the NaTl structure. The fact that some compounds are poor conductors in contrast to the expected semiconducting properties has led to question the validity of the Zintl concept; to this doubts have also contributed the structural behavior of LiTl, NaTl and KTl1 with respect to the electron transfer from the alkali metals to Tl.5 LiTl is CsCl-type; in NaTl, we have shown how the Tl atoms transform into a diamondlike structure (J-tetrel); finally, in KTl, the Tl atoms form octahedral Tl6 clusters. Even if the octahedral clusters are unknown for the Group 14 elements, each Tl atom of the octahedron preserves the fourfold connectivity characteristic of tetrels. Here is to remind the proposal of Schäfer15 to extending the ZKC to skeletons that fit the number of bonds predicted by the (8-N). Finally, LiTl with its CsCl-type structure does not fit the pseudo-atom concept. It has been suggested5 that in this structure, weak ionic and metallic contribution could contribute to the stabilization of the CsCl-type structure in which the Tl atoms form a simple cubic skeleton instead of any structure of tetrels. As a counter-example that reinforces the Zintl concept we can cite the ELF calculations carried out on the butterfly [Si4]6– anion inBa3Si4. Results show that all bonding and non-bonding states are localized on the Si atoms, ensuring the complete electron transfer from Ba to the Si atoms.17 At this point, it seems pertinent to mention the conclusions drawn with respect to the extra electron existing in the Zintl phase Cs3 Bi2 21 : The authors contemplated three possibilities in order to rationalize its stoichiometry, i.e., (a) dimers as radicals Bi2 3– ; (b) the dimers as Bi2 4– with an extra proton included in the compound (A3HBi2); (c) dimers as
 Bi2 2– with an extra delocalized electron 3A þ þ Bi2 2– þ e– : After an exhaustive experimental study, including measurements of the positive paramagnetic molar susceptibility, metallic conductivity and neutron powder diffraction (time-of-flight mode) to check for the presence of any H atoms and also from magnetic measurements, the authors proved “.that this compound can only be rationalized as metal that contain [Bi2]2– dimers, three Csþ cations, and one electron delocalized over the cations and the antibonding orbitals of the dimer.”18 A weakly metallic character also occurs in Na2In8. These conclusions are supported by the fact that the bismuth dumbbells not only exist in the solid state but they can be extracted intact as a double-bonded species when the compound is solved in ethylenediamine. The reader is referred to the review on the chemistry of Zintl polyanions in non-aqueous solvents (NH3, tetramethylethylenediamine, pentamethyldiethylenediamine)9 for an exhaustive discussion on this topic. As it was reported by Vegas, in Chapter 5 of Ref. 19 the ZKC has capabilities not yet fully explored. Its simplicity may be seen, at first glance, devoid of physical sense. However, its power has been nicely highlighted by several authors. For Instance, Hoffmann considers the ZKC20 “. among the most important and useful paradigms in solid state chemistry.” Miller4 also wrote that “. buried within the simple electron counting schemes are complex physical and chemical phenomena. . which affects local electron density distribution.” To these factors, we could add the values of electronegativities, ELF calculations and maybe the chemical pressure, which can become decisive in interpreting the structure formation. The recent work by Lobato et al.21 might help to enlightening some of the challenging aspects just described. Regardless of the disputes about the validity of the ZKC, our aim is to provide new facts that have emerged from the analysis of hundreds (maybe thousands) of crystal structures analyzed in along the last 30 years by our group, to demonstrate the wide-spread applicability of the ZKC. Most of the compounds studied were oxides, but alloys and even some molecular compounds have also considered. The main result can be condensed in the following sentence: The ZKC can be successfully applied to explain the cation arrays of oxides. Thus, the ideal

The Zintl-Klemm concept and its broader extensions

55

electron transfer between cations accounts for the connectivity of the skeletons of thousands of ternary oxides in what has been called the Extended Zintl-Klemm Concept (EZKC). Our intention is to provide the reader with examples of this of the ZKC when “Extended” to oxides and also to some simple alloys. This will evidence that the use of the traditional ionic radii is not any more necessary to explain the structures. They are better understood when regarded as Zintl polyanions with the O atoms located in the vicinity of the bonding pairs or lone pairs of electrons. We will firstly describe some examples of four-, three- and two-connected polyanions, with special emphasis in the compounds where atoms of the same species, say Al, Si, and Ge, show two coordination numbers in the same crystal (CN ¼ 4; CN ¼ 6), a fact that has been interpreted as due to the amphoteric character of such atomic species.12,22,23

3.04.2

The extended Zintl-Klemm concept

In this Section we will show that the ZKC can apply to more complex structures, other than the original Zintl phases. It should be pointed out that the meaning of the term “Extended” used by us in 2003,12,22,24 differs from that given by Papoian and Hoffmann,19 who applied the ZKC to nonclassical electron-rich networks; more precisely to binary compounds of heavy late main group elements such as Sb. These compounds often feature linear nonclassical geometries or two-dimensional square sheets unusual in Zintl polyanions. The differences have been discussed in depth in Chapter 15 of the book by Vegas.19 We used the term extended because the ZKC was broadened, not to new Zintl phases, but to other families of compounds, such as aluminates,12,17,24 silicates17,22,24 sulfides, sulfates and alloys.23 The application of the ZKC to aluminates and silicates could be seen controversial because the Zintl concept was “conceived and restricted” to oxygen-free compounds of elements of the p-block. However, the idea can be better admitted if we know how it came up. During our research we realized that the structure of the parent metal was preserved in many compounds. To illustrate this phenomenon, we will discuss the structure of Ca5(PO4)3(OH) (hydroxyapatite). This structure is of greatest interest because it was involved in what we consider one of the most important structural relationships of crystal chemistry in the 20th century. Wondratschek et al.25 realized that the Ca5P3 substructure, in apatite, was isostructural to the Mn5Si3 alloy (D88-type). Its relevance resides in the fact that the complex structure of an alloy (Mn5Si3) coincided with the cation substructure of another complex oxide (apatite). The authors were aware that this discovery embodied a new, unexpected phenomenon that they tried to define when they wrote: “. It is quite probable that these coincidences are not casual but they should obey a General Principle” (our translation from the original German text). This discovery was ignored up to 1985 when O’Keeffe and Hyde26 uncovered that such similarity affected to many other oxides, leading them to look at oxides as oxygen-stuffed alloys. If we analyze the Ca-substructure, ignoring the P atoms, we find another unexpected phenomenon: The Ca substructure can be seen as formed by two blocks that are fragments of Ca elemental structures. One corresponds to hcp-Ca (named H block) and the other one to the bcc-Ca structure (called C-block). The unit cell constants of the respective phases can be recognized in them17,27 in spite of the two Ca-fragments are welded by sharing atoms. Ca5P3 does not fulfills the valence rules since only 9 of the 10 valence electrons of the Ca atoms are transferred to the 3 P atoms (in this way, each P atom achieve a noble gas configuration). Apparently, there is one extra electron! However, the synthesis of the hydride Ca5PH28 demonstrated the fulfillment of the charge balance. At the same time, this new compound gave full sense to the structure of hydroxyapatiteCa5(PO4)3(OH), where the H atoms occupy precisely a position close to that of the OH group. Thus, hydroxyapatite preserves the structure of its parent alloy Ca5P3H and, in addition, preserves fragments of the Ca structures. The systematic study of the oxide’s structures confirmed that many of them preserved either the complete structure of their parent metal (in some cases, distorted) or fragments of them. Examples are CaO and CaF2, both preserving the structure of fcc-Ca. It is in this context that we arrived to the structures of ternary oxides (aluminates, silicates, phosphates, etc.) in which, new phenomena were discovered as it will be discussed next.

3.04.2.1

Four connected networks

Ternary aluminates such as A[AlO2] (A ¼ Li  Tl) form a stuffed-cristobalite structure, while the compounds A[Al2O4] (A ¼ Sr, Ba, Pb) form a stuffed tridymite-like skeleton. In both cases, the transfer of one electron from the A atoms to each Al atom transforms them into a four-connected skeleton of J-Si, since [AlO2]– h J - SiO2. As an example, we have represented in Fig. 3a the structure of Sr[Al2O4]29 in which the charged Al– atoms form the structure of the lonsdaleite (hexagonal diamond) represented in Fig. 3b. All the O atoms locate near the hypothetical AleAl bonds in order to catch the BP’s. In this way, the O atoms fulfill its octet yielding a tetrahedral coordination around the Al atoms. It should be remarked that under this view, the CN is not a consequence of the radius ratio but the result of the localization of both BP’s and LP’s.12,19,22 The structure of g - LiAlO2 also deserves attention. This phase does not form a stuffed-cristobalite structure as the LiAl alloy does. Instead, all the atoms, say Li, Al and O form part of the skeleton which has a vec ¼ 20 electrons per formula unit with an average of 4 electrons per atom. Accordingly, each atom is bonded to other four (see Fig. 4a), building up the four-connected network represented in Fig. 4a. Interestingly this structure presents a distorted b-BeO-type geometry. The fourfold connection is produced despite of the different number of valence electrons in each species. All the atoms behave as tetrels, forming four bonds to their neighbors.

56

The Zintl-Klemm concept and its broader extensions

Fig. 3 (a) The hexagonal structure of Sr[Al2O4] viewed along [110]. The (Al– h J - Si) atoms form a lonsdaleite-like skeleton. Six AlO4 tetrahedrons are highlighted in the lower right side of (a). (b) Structure of lonsdaleite. Al: blue; O: red; Sr: ochre; C: black. Reprinted from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la UPV.

Fig. 4 (a) Stereopair of the structure of g - LiAlO2 in which, the Li and Al are bonded tetrahedrally to four O atoms and each O atom bonds to two Li and two Al atoms. (b) g - LiAlO2 is a strong distortion of the b-BeO structure drawn in (b). (c) The [Al2Si2] substructure of Ca[Al2Si2O8] (anortite). The Ca atoms are lodged into the octogonal tunnels. Li: blue; Al: gray; O: red; Be: ochre; Ca: dark green; Al: gray; Si: dark blue. Panels (b) and (c): Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018.

The Zintl-Klemm concept and its broader extensions

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The skeleton of g - LiAlO2 can be justified if we consider that one electron from the Al atom is transferred to the Li atom resulting in J-Mg and J-Be, respectively. The compound could be reformulated as [J - Be J - Mg]O2 so that g - LiAlO2 could be considered equivalent to II-VI or III-V compounds that also adopt IV-IV structures like the BeO (wurtzite-type), ZnS or AlP crystals. Similar networks are found in the high temperature phase b-BeO (Fig. 4b), anortite Ca[Al2Si2O8] (Fig. 4c), with a skeleton [Al2Si2]2– h J - Si2Si2 and in the [AlP] skeleton of metavariscite AlPO4 $ 2H2O, all of them with a vec ¼ 4 valence electrons per atom.

3.04.2.2

Three-connected networks

Next, we will discuss the structure of the silicate Na3Y[Si6O15].30 The [Si6O15]6 anion is represented in Fig. 4a where we see that the Si-skeleton forms a trigonal prism where each Si atom is three-connected. The structure of Na3Y[Si6O15] can be justified as follows: The donor cations (Na3 þ Y) can transfer six electrons to each of the six Si atoms, becoming Si– h J - P which, according to the 8-N rule form three SieSi bonds like in pentels. When the O atoms are added, they locate close to the nine SieSi bonding pairs, contributing three O atoms to the tetrahedral coordination of each (SiO4) tetrahedra. The tetrahedral coordination is completed when six additional O atoms situate near the prism’s corners where the existence of six LP attached to each J-P atom. Trigonal prism topology is unknown for pentels although theoretical calculations predict the stability for the N6 molecule drawn in Fig. 5b. In addition, the Si6 skeleton (Fig. 5a) is isoelectronic and isostructural to the molecule of prismane C6H6, shown in Fig. 5c.31 It is possible that the synthesis of N6 can be achieved in the future as for instance it has been synthesized two new  polymeric 3D structure32 and the so-called black-N, isostructural to blackhigh-pressure phases of nitrogen, i.e., the cubic (Ia3) 33 P. Regarding the cubic polymeric structure of nitrogen, the reader is also referred to the article by Vegas et al.34

3.04.2.3

Two-connected networks

In the next examples we will discuss compounds in which the Zintl electron transfer will produce J-elements toward the right side of the Periodic Table like chalcogens and halogens. One of these compounds is Ba16[Al12O36] whose polyanion [Al12O36]36– displays a structure consisting in rings consisted of 12 AlO4 tetrahedrons sharing 2 corners (see Fig. 6a). If the O atoms are neglected, we can see how the Al-substructure coincides with the structure of the S12 molecule, as shown in Fig. 6b. The similarity to S12 had been noticed by Walz et al.,35 but without relating it with the ZKC. Finding physical or chemical explanations of such a similarity is not easy, but our knowledge of the Zintl phases helped to explain the phenomenon. Our finding was simple: if in Ba16[Al12O36], we neglect the O atoms, we obtain the formula Ba18Al12 h Ba3Al2 which is a Zintl phase. Now, if in Ba3Al2 the six valence electrons of Ba3 are transferred to the Al atoms, they would convert into Al3– h [J - S], and Ba16[Al12O36] could be reformulated as [J - Xe]18[J - S12O36]. The same reasoning was successfully applied to the great families of aluminates12,24 and further to the great family of silicates.22,24 The concept also applies to germanates, phosphates, etc. Indeed, similar 12-membered rings are found in the silicate Na16.5Y2.5[Si12O36],36 in the germanate K16Sr4[Ge12O36]37 and in the phosphate V3Cs3[P12036].38 These examples can be easily rationalized considering that a transfer of 24, 24 and 12 electrons respectively, convert Si, Ge and P into J - S12 rings. In other silicates and phosphates, Si and P atoms form cyclic polyanions of general composition J - SO3 as octacycles (J S8O24), hexacycles(J - S6O18), pentacycles (J - S5O15), tetracycles (J - S4O12) and the tricycle (J - S3O9) that is isostructural with the real SO3 molecule (see Fig. 7c and d). The only structural determination of a cyclopentacycle was performed by Jost39 in Na4NH4P5O15 $ 4H2O. In all of them, the Si-skeletons coincide with those of the oxides and also with those of the corresponding Sn molecules. Here it is important to recall that elemental sulfur forms, among others, the S8, S6, S5, S4 and S3 molecules. Moreover, Sulfur also exists as trigonal and square helices like that drawn in Fig. 7g.

Fig. 5 (a)The structure of the [Si6O15]6 anion in Na3Y[Si6O15], showing the trigonal prism formed by the J-P atoms. (b) The structure of the N6 molecule. (c) The molecule of prismane C6H6. Panel (a): Reproduced from Vegas, A. Acta Crystallogr. B 2013, 69, 356–361, With permission of the IUCr. Panels (b) and (c): Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V.

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Fig. 6 (a) Structure of the [Al12O36]36– anion in Ba16[Al12O36] where the Al atoms are converted into J-S. (b) Structure of the S12 molecule. Al: dark gray; O: red; S: yellow. Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V.

These drawings show the feature advanced by O’Keeffe et al.26 when they realized that several binary oxides of the p-block elements maintained the structure of the element itself. For example, in the silica phases, cristobalite maintain the diamond-like structure of Si, tridymite preserves the structure of lonsdaleite and in keatite (also SiO2), the Si atoms maintain the structure of HP-Si, etc. Their list26 was completed in Ref. 19. Structures depicted in Fig. 7 illustrate how the [AgP] and [NaP] substructures of the AgPO3 and NaPO3 ternary oxides behave as the real Zintl phases as the real NaP compound. Considering that the Na (Ag) atoms act as donors, the transfer of one electron from Na / P, transforms the latter into J - Sn, with the same structure as S-II obtained at 55 GPa. The outcome is important because the helical structure of S (Fig. 7f) is reproduced in the Zintl phase NaP (Fig. 2c) and further in the oxide NaPO3 (Fig. 7e). The elemental pattern (sulfur) is reproduced not only in the Zintl phases but also in the ternary oxides with equal stoichiometry, so that the ZKC remains valid when applied to the cationic substructures of ternary oxides of the p-block elements. Another important issue is that O atoms (red spheres in Fig. 7) are always situated near the electron pairs, be BP or LP. So, in the trisilicate anion [Si3O9]6 h J - S3O9 in K6[Si3O9], if we consider that the Si atoms (J - S3) form hypothetical bonds, as they do in the real S3 molecule, three O atoms would approach to the J(SeS) bonds to catch the three BP’s. In addition, each J-S atom has

(A)

(B)

(D)

(C)

(D)

(E)

(F)

(G)

Fig. 7 (a) The polyanion [Si6O18]12 h J - S6O18 in Na4Ca4[Si6O18]. (b) The S6 molecule. (c) The anion [Si3O9]6 h J - S3O9 in K6[Si3O9]. (d) The S3O9 molecule. (e) The polyanion ½PO3 nn h½J  SO3 n in AgPO3 in the form of square helical chains. (f) the square helical chains of P h J Sn in the Zintl phase NaP (see Fig. 2c). The square helices of S-II at 55 GPa. (g) Si: gray/blue; S: yellow; P: purple; O: red. Panels (a) and (b): Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V. Panels (c) and (d): Reproduced from Vegas, A. Acta Cryst. B 2013, 69, 356–361, With permission of the IUCr.

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(A) (B)

(C)

Fig. 8 (a) The structure of the anion [Si2P6O25]12– in In4[Si2P6O25]. The Si2P6 skeleton is equivalent to J - Si2Cl6. (b). The [Sn2P6]6– anion, equivalent to J - Sn2Br6, in Ba6[Sn2P6]. (c) The structure of the Si2Cl6 molecule Si: gray; P: purple; Cl: green. Sn: dark gray. Panels (a) and (c): Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la UPV.

two LP’s arranged tetrahedrally with the BP’s and accordingly, two O atoms can approach to the LP’s to form the SiO4(J - SO4) tetrahedra. The outcome is that the O atoms play the same role as the electron pairs, a phenomenon that was discussed in the AMM theory.40,41

3.04.2.4

Branched silicates

One example of these compounds is the silicophosphate In4[Si2P6O25] whose anion [Si2P6O25]12– is represented in Fig. 8a. Each Si atom is four-connected; one bond is produced to the other Si atom three to P atoms. All of them are tetrahedrally coordinated to four O atoms. The structure fulfills the EZC if we assume that the 12 electrons of the In atoms are transferred to the 6 P atoms resulting in Y-Cl. On the contrary, the Si atoms remain unaltered. The result is four-connected Si atoms whereas the P h J - Cl atoms, with seven valence electrons, have to be single bonded, according to the 8-N rule. As usual, the O atoms, are located near the SieSi and the SieP bonds, as well as near the three LP’s located on each J-Cl atom to complete their tetrahedral coordination. The [Si2P6] skeleton is similar to the isoelectronic anion [Sn2P6]6– in the Zintl phase Ba6Sn2P6, drawn in Fig. 8b. Its skeleton is explained by the transfer of the 12 electrons of the Ba atoms to the 6 P atoms that are converted into J-Cl.42 This compound can be reformulated as (Ba2þ)6[J - Si2Cl6] which is isoelectronic to the Si2Cl6 molecule. Its molecular structure, represented in Fig. 8c, was determined from spectroscopic and electron diffraction data in the vapor phase.43 These compounds provide another example of the structural similarity of a molecule, a Zintl polyanion and an oxide. The compound In4[Si2P6O25] makes rise another important question, i.e., if the tetrahedral coordination of both Si and P atoms, were determined by the radius ratio rule, one could expect that the Si and P atoms might be mixed as [TO4] groups within the polyanion. However, the fact that the Si atom remains four-connected whereas the P atoms behave as J-Cl, means that, according to the EZK, the electrons are transferred to the more electronegative P atoms, that become decisive in determining the structure. The same h i skeleton appears in the compound Si½6 3 Si½4 2 P½4 6 O25 where the hexacoordinated Si[6] atoms act as donors toward the P atoms, producing the same polyanion [Si2P6O25]12–. This compound is also relevant because it contains Si[6] and Si[4] atoms in the same crystal. The former as donor, the latter remains as neutral Si. The same feature occurs in SiP2O7, with Si[6] and P[4]. In the absence of donor atoms, the octahedral coordination of Si was justified by the enhanced basicity of Si with respect to P which makes that “. Si rather than P prefers the octahedral coordination in these compounds.”44 This point is important and will be treated later but it put on the table the question about why an atom like Si, can adopt two different coordination polyhedra and this question shows doubt about whether the so-called ionic radii cause the coordination polyhedron of a given cation.

Fig. 9 (a) The [Si3O10]8– anion in Na4Cd2[Si3O10]. The Na and Cd atoms transfer eight electrons converting the three Si atoms into J - SCl2. (b) The real molecule of SCl2. (c) The [P2O7]4– anion in SiP2O7. (d) The [Si2O7]6– anions in Sc2[Si2O7]. Si: gray; O: red; S: yellow; Cl: green. P: purple. Panels (a–c): Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V. Panel (d): Reproduced from Vegas, A.; Notario, R.; Chamorro, E.; Pérez, P.; Liebman, J. F. Acta Crystallogr. B 2013, 69, 163–175, With permission of the IUCr.

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3.04.2.5

Discrete oligosilicate anions. Single-connected groups

One of them, [Si3O10]8– exists in Na4Cd2[Si3O10].The anion contains three SiO4 tetrahedrons sharing corners, forming the angular T skeleton represented in Fig. 9a. Its geometry fulfills the ZKC since the 8 transferred electrons, convert the three Si atoms into one J-S (central atom) and two J-Cl (terminal atoms). The important outcome is that this J - SCl2 molecule is identical to the real SCl2 molecule represented in Fig. 9b. Skeletons composed of four and even five atoms have also been observed in both aluminates and silicates. The dialuminate, disilicate and diphosphate anions existing in compounds such as SiP2O7 and Sc2[Si2O7] contain two TO4 tetrahedra with a common vertex as it is drawn in Fig. 9c and d. The transfer of four electrons from Si to the two P atoms converts them into J - Cl2 molecules. The same occurs in Sc2[Si2O7], where the two Si atoms also converts into J - Cl2 by the transfer of six electrons from Sc2. The O atoms locate near to or at the center of the SieSi (PeP) bond and near the three LP’s of each J-Cl atom, completing so the TO4 tetrahedra.

3.04.3

i h The amphoteric silicon. Si as donor and as acceptor in Ce16 Si½6 Si½4 14 ðO6 N32 Þ

h i The interest of the compound Ce16 Si½6 Si½4 14 ðO6 N32 Þ (Pa-3; Z ¼ 4) is that it also has both Si[4] and Si[6] atoms coexisting in the structure in spite of being synthesized at ambient pressure.45 The authors described this oxonitridosilicate as a 4  4  4 superstruch i ture of an ideal cubic perovskite of formula A[12]G[6]X3.The unit cell content is Ce64 Si½6 4 Si½4 56 ðO24 N128 Þ .45 Of the 64 octahedral sites G[6] in the cell, 4 are occupied by Si[6], 56 by Si[4] and 4 G positions are empty. On the contrary, all the 64 cuboctahedral positions A[12] are occupied by the Ce3þ cations. Due to the anionic vacancies, the CN of the Ce3þ cation is less than 12. The cationic charge is compensated with 24 O2– and 128 N3– anions, so that 40 of the anionic X positions are also empty. This causes the 56 Si[4] atoms occupying G[6] sites to have to lower their CN from 6 to 4. Taking into account the vacancies the compound can then be h i formulated as Ce64 Si½6 4 Si½4 56 ,4 ðO24 N128 Þ,40 . The vacancies are denoted by the symbol ,. This interpretation was founded on the stability assumed for the topology of the CsCl-type of the cationic substructure of perovskites45 and had been previously applied to account for the structure of some cyclosilicates46 as well as for the Sr3[Al2O6] aluminate.47 As reported by Santamaría-Pérez et al.,22 this description, being true, cannot account for the fact that vacancies, both of anions and cations, are distributed in the factual positions and not in another ones. Next, we will show that the structure can be explained in the frame of EZKC, thus accounting for the tetrahedral Si[4] skeleton, represented in Fig. 10, as well as for the vacancies distribution.22 The structure can be explained according to the electron count shown below, where the Si(1) to Si(4) atoms refers to the four crystallographically independent Si families.22

Fig. 10

h i Perspective of the Si-subnet in Ce16 Si½6 Si½4 14 O6 N32 : The purple spheres represent three-connected J-P atoms, the yellow spheres, bi-

connected J-S atoms. The resulting stoichiometry is (J - P4S3). The donor Si[6] atoms (gray spheres) center the octahedra Si[6]X6(X ¼ O, N). Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V.

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Donor atoms : 64 Ce /64 Ce½8 þ 192 e– 4 Sið4Þ /4 Sið4Þ½6 þ 24 e– 208 e– Acceptor atoms : 128 N þ 128 e– / 128 J-O 24 O unchanged

24 O

–

24 Sið1Þ þ 48 e / 24 J-S 24 Sið2Þ þ 24 e– / 24 J-P 208 e–

8 Sið3Þ þ 8 e– / 8 J-P ½J-P32 S24 O128 O24 h½J-Si56 O152 80–

Assuming that the 64 Ce atoms and the 4 Si[6] atoms act as donors, they can transfer 208 electrons. If 128 e– were captured by the 128 N atoms, they would convert into J-O, unifying so the ideal nature of all the anions. The 24 real O atoms would remain unchanged. The remaining 80 electrons should be transferred to the 56 Si[4] atoms. Of them, 32 Si atoms convert into J-P and 24 into J-S, yielding a Si[4] skeleton of formula J - P32S24(J - P4S3). Fig. 10 shows the Si[4] network consisting of chairconformed rings whose alternate atoms (J-P) bond to three contiguous rings. The J-S atoms remain bi-connected. The X atoms (O and N) located near the midpoint of the Si-Si contacts and also on the lone pairs of both J-P and J-S atoms, forming the [TO4]-skeleton, represented in Fig. 10. This description demonstrate that the number of Si[6] donor cations is that which enables the formation of a chemically sound Si[4]-skeleton. In other words, there is a direct relationship between composition and structure. Similarly, the vacant distribution is not randomly produced but arranges to accomplish a skeleton that fulfills the EZKC. Similarly, the vacant distribution is not at random but so as to accomplish such a skeleton according to the EZKC. To the octahedral coordination sphere of the Si[6] contribute both the N and O atoms. The Si[6]X6 octahedra form a fcc-array, as seen in Fig. 10. h i As we have discussed above, the coexistence of both Si[4] and Si[6] in Ce16 Si½6 Si½4 14 O6 N32 should not be attributed neither to an excess nor to a deficiency of anions. When we described the structure of SiP2O7 (Fig. 9c) we attributed the CN ¼ 6 of the Si[6] atoms to their donor character. The same argument will be extended here. As Al atoms in the aluminates, Si atoms show their amphoteric character, acting as a Lewis base with a CN ¼ 6 and also as a Lewis acid with CN ¼ 4. This outcome evidences that the ionic radius is not the ultimate reason that determines the CN of a conventional cation. Si[6] atoms and the Si[4] atoms are not the same conventional species (Si4þ); the former can be closer to a Si4þ state whereas the latter is closer to a Sin– anion. However, another puzzling feature is that the coordination numbers are opposites to those expected, i.e., the hypothetically smaller Si4þ becomes hexacoordinated while a bigger Sin– anion is tetracoordinated. We should be reminded that this point was treated in Section 3.04.2.4 where we mentioned that in Ref. 44 the CN ¼ 6 for the Si[6] atoms was attributed to their higher basicity in comparison to that of P. Now, it could be asserted that basicity alone should not cause the octahedral coordination. The basicity is materialized in the valence electron transfer from the Si atoms to another ones, say P or Si, as it occurs in the compound under discussion. In other words, the reason why Si adopts de CN ¼ 4 is that the Sin– anions complete their octets when they form Zintl polyanions, so yielding a tetrahedral arrangement of the 4 electron pairs (8-N rule). These four bonds dictate the presence of four O atoms that situate near the BP and/or the LP. It is at this point that the EZKC concept shows its greatest power.

3.04.4

h i The amphoteric germanium. The structure of ðNH4Þ2 Ge½6 Ge½4 6 O15

3.04.4.1

First approach

The structure of (NH4)2Ge7O15 was reported as being a microporous material containing rings in which, GeO6 octahedrons coexist with GeO4 tetrahedrons.48 The presence of both CN’s, already described for aluminates and silicates was less expected in this compound because, at ambient conditions, GeO2 is rutile-type with all Ge atoms as Ge[6]. Only at 1305 K, GeO2 transforms into a quartz-type structure composed of tetrahedral Ge[4] atoms. This fact contrasts with SiO2 that exists as quartz at ambient conditions, undergoing at high pressures the transition quartz-stishovite with an increase in the CN from 4 to 6. The coexistence of both coordination polyhedra in analogous silicates and aluminates, led us to re-examine the structure of (NH4)2Ge7O15 in the light of the Extended Zintl-Klemm Concept.48 Assuming that the Ge[6] atoms together with the two NH4 þ groups act as true cations, they would transfer 6 electrons to the acceptor [Ge6O15] moiety, converting it into the [Ge6O15]6 polyanion, that can be reformulated as 3 (J - As2O5) units. According to the (8-N) rule, the Y-As atoms are threeconnected which, in this compound, adopt the form of condensed ladder-like fragments, forming the skeleton represented in Fig. 11.

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The Zintl-Klemm concept and its broader extensions

(A)

(B)

(C)

(D)

h i Fig. 11 (a) The partial structure of Ge atoms in ðNH4 Þ2Ge½6 Ge½4 6 O15 projected on the ac plane. The skeleton of Ge[4] atoms, with stoichiometry [Ge6O15]6–, is represented by cyan spheres connected by yellow lines. The Ge[6] atoms, as dark blue spheres, bridge the [Ge6O15]6– moieties and are connected to the Ge[4] network by red lines. (b) The Ge-subarray of the tetrahedral [Ge6O15]6 polyanion showing the threefold connectivity of the Ge atoms, converted into J-As atoms according to the Zintl-Klemm concept. The purple spheres here represent the N atoms of the ammonium cations lodged in the tunnels formed by such networks that are visible in (c) and (d). (c) Perspective view of one of the networks showing the tunnels were the Ge[6] and NH4 þ cations are lodged. (d) The network as in (c) in which the O atoms have been added. Reproduced from Vegas, A.; Jenkins, H.D.B. A Revised Interpretation of the Structure of in the Light of the Extended Zintl–Klemm Concept. Acta Crystallogr. B 2017, 73, 94–100. doi:10.1107/S2052520616019181, With permission of the IUCr.

The complete Ge-substructure is represented in Fig. 11a, where the Ge[6] cations, represented as dark blue spheres, serve as bridges between contiguous ladders-like skeletons of the Ge[4] atoms. Two of these ladders appear separated in Fig. 11b by eliminating the donor Ge[6] cations. They are represented as cyan spheres, connected by yellow lines. One of these skeletons of [Gen]n  is separated in Fig. 11c and the complete [Ge6O15]6 polyanion, equivalent to (J - As2O5), is represented in Fig. 11d, in which the GeO4 tetrahedra are highlighted.48 h i h i6– The Ge[4]-skeleton shows similarities with that of sillimanite Al½6 2 Al½4 2 Si½4 2 O10 whose anions Al½4 2 Si½4 2 O10 form rather planar ladders of stoichiometry J - As2O5 (see Fig. 12a) and too with that of the high temperature analog Sb2O3, the mineral valentinite,49 represented in Fig. 12b. In Sb2O3, the Sb atoms, drawn as ochre spheres, are three-connected forming puckered ladder-like chains. The O atoms locate near the center of the (hypothetical) SbeSb bonds, which are drawn with blue lines in Fig. 12b. If in this Figure, we ideally broke some SbeSb bonds, to clearly evidence the pattern of Fig. 12c formed by pairs of rectangles connected by only one bond. These motifs can then rotate with respect to one another so that the two-connected Sb atoms become again three-connected by bonding to one Sb atom of the contiguous ladder. Yielding the J-As skeleton of Fig. 11c. As mentioned above, the central void serves as a cavity into which lodge both the Ge4þ and NH4 þ cations, the latter being normally used as templates in the synthesis of microporous compounds.48 h i The transformation of the ladders, in Sb2O3 (valentinite), into the J - As2O5 partial structure in ðNH4 Þ2Ge½6 Ge½4 6 O15 is probably better understood if, instead of going from the Sb-skeleton in Sb2O3 (Fig. 12b), we start with the J-As skeleton of Fig. 11c. In this case, if we broke the bonds marked by a thick red arrow in Fig. 11c, both the upper and the lower fragments of this skeleton should approach as indicated by the two narrow red arrows, forming new bonds with the atom of the lateral fragments of the skeleton. In this way, we could reconstruct two infinite ladder-like chains similar to that of Sb2O3 in Fig. 12a.49 The [Ge2O5]2– partial structure (J - As2O5) differs from the Sb2O3 structure in their O-content, i.e., in the [Ge2O5] partial structure, the Ge (J-As)

The Zintl-Klemm concept and its broader extensions

63

h i Fig. 12 (a) The ordered [Al2Si2O10]6– anion in sillimanite Al½6 2 Al½4 2 Si½4 2 O10 . Al: blue; Si: black; O: red. (b) The structure of Sb2O3 (Pccn) exhibiting the threefold connectivity of the Sb atoms (in ochre) forming ladder-like chains from which the Ge[4]-skeleton of Fig. 2b can be derived. The O atoms (represented by small red spheres) are near the centers of the Sb-Sb contacts, mimicking the hypothetical bonding pairs. The positions close to the lone pairs (LP) remain empty. Note that in the Ge[4]-network of Fig. 11d these positions are occupied so completing the GeO4 tetrahedra. (c) The same structure without the O atoms and where some SbeSb bonds have been eliminated to leave pairs of rectangles like those forming the h i network of Fig. 2c. (d) Projection of the ðNH4 Þ2Ge½6 Ge½4 6 O15 structure on the ac plane, similar to that represented in Fig. 2a. Here, some Ge[4]  Ge[4] contacts have been eliminated giving rise to a set of isolated blocks formed by Ge½6 Ge½4 6 . Big dark blue spheres represent Ge[6] atoms. N atoms are represented as purple spheres. Red lines connect the Ge[6] atom to the six closest Ge[4] (J-As) atoms. Yellow lines connect Ge[4] (JAs) atoms. Reproduced from Vegas, A.; Jenkins, H.D.B. A Revised Interpretation of the Structure of in the Light of the Extended Zintl–Klemm Concept. Acta Crystallogr. B 2017, 73, 94–100. doi:10.1107/S2052520616019181, With permission of the IUCr.

atoms are tetrahedrally coordinated, whereas in Sb2O3 the lower O-content leads to a CN ¼ 3, so creating vacancies at the expected LP’s positions (Fig. 12b).

3.04.4.2

Second approach: The donor Ge[6] atoms

So far, we have explained the tetrahedral Ge-skeleton (J-As), according to the EZKC and have rationalized its connectivity, a feature that could not be accomplished if both kinds of polyhedra are taken together, as it was done in its first description.50 Now, we will focus on the donor Ge[6] atom. As pointed out by Miller et al.5 the donor atoms, once converted into cations, are ideally ignored or treated as mere spectators of the Zintl polyanion. This aspect was already discussed when we reported the new description of h i ðNH4 Þ2Ge½6 Ge½4 6 O15 ,48 as well as in other studies concerning double sulfates.23 In fact, the donor Ca atoms were also considered in the structures of CaIn2 and CaSi2 (Fig. 2a and b) where they form hcp-layers with Ca-Ca distances slightly longer than in fcc-Ca. h i Under this view, the structure of ðNH4 Þ2Ge½6 Ge½4 6 O15 leads us back to Figs. 11a and 12c and will be focused on the donor Ge[6] atoms (dark blue spheres) in Figs. 12d and 13a. Fig. 12d shows that each Ge[6] atom is connected to six Ge[4] atoms (cyan spheres) at distances Ge[6] Ge[4] of 6  3.19 Å. These groups of seven Ge atoms form blocks that become more visible when the connections between Ge[4] atoms (yellow lines in Fig. 12a) are omitted. The result is the pattern of Fig. 12d where the N atoms have now been drawn as purple spheres. These clusters are separated in Fig. 12d and one of them is isolated in Fig. 13a. The relevance of this fragment is that it reproduces, in some extent, the rutile-like structure of GeO2. In fact, the fragment of Fig. 13a corresponds to two cells of GeO2 (rutile-like) in which two Ge atoms are missing at opposite corners. This structure can be easily identified when compared with the real structure of GeO2, represented in Fig. 13b and c. In Fig. 13d the rutile-like fragment is projected onto the hypothetical (001) plane. The central Ge atom (dark blue) is octahedrally coordinated, sharing the six corners with six GeO4 tetrahedra as shown in Fig. 13d, and in turn, mimicking the corner-sharing of the octahedral arrangement found in the real rutile-like structure drawn in Fig. 13e.48 h i Regarding dimensions, the rutile-like unit cell fragments in NH4 Ge½6 Ge½4 6 O15 , denoted as a and c in Fig. 13a, have Ge-Ge distances of 4.23  4.28  3.02 Å, with six Ge-Ge distances of 3.19 Å (red lines in Fig. 13a). These values are comparable to those

64

The Zintl-Klemm concept and its broader extensions

Fig. 13 (a) Perspective view of the octahedral Ge[6] atom (dark blue) connected to the six tetrahedral Ge[4] atoms (cyan) in a fashion that reproduces partially the structure of the rutile-type GeO2 represented in (b). (b) Drawing of two unit-cells of the rutile-type structure of GeO2. The unit cell edges have been omitted but the fragment reproduced in (a) has been highlighted with yellow lines, together with the central GeO6 octahedron. (c) New representation of the two unit-cells of the rutile-type structure of GeO2. The bonds drawn in ochre color are those forming the corresponding fragment in (b) and in (d). (d) The fragment represented in (a) but projected to show the connectivity of the central octahedron to the six tetrahedral. (e) A projection of the rutile-like structure of GeO2 to be compared with the fragment in (c) and with the defect structure in (d). Reproduced from Vegas, A.; Jenkins, H.D.B. A Revised Interpretation of the Structure of in the Light of the Extended Zintl–Klemm Concept. Acta Crystallogr. B 2017, 73, 94–100. doi:10.1107/S2052520616019181, With permission of the IUCr.

of the unit cell of the rutile-like GeO2 (Fig. 13e), with a ¼ b ¼ 4.39; c ¼ 2.90; d ¼ 8  3.42 Å. These parameters are close to the  distances labeled as a and c for the germanate fragment in Fig. 13a and b. The Ge-Ge-Ge angles (86 ) deviate from the ideal values  of 90 in the rutile structure (compare Fig. 13d and e). Coming again to Fig. 12c we see that if the N atoms (purple spheres) were removed the “rutile-like” Ge7-blocks would collapse, reconstructing the GeO2 structure. From the above structural description, two aspects are evident. Firstly, that the Extended Zintl-Klemm Concept, previously applied to aluminates and silicates12,22,24 works equally well for the germanate, despite the lower electronegativity of Ge with respect to Si. This compound provides additional evidence for the possibility of charge transfer between nominal cationsdeven if they are of the same speciesdas has been illustrated for the compounds CsLiSO4 and Na2SO4.23 Further application to ternary oxides and chalcogenides have emphasized the validity of this concept.19,34,51,52 Of special interest is the application of these novel principles to the interpretation of the stuffed-bixbyites structures as reported in Ref. 34. From the data described above, it seems that the achievement of these cation networks, intimately related to the structures of p-block elements, is a fundamental factor of both the underlying structural and the thermodynamic stability. If the stoichiometry does not allow for the formation of such a network, then some of the atoms act as true cations in order to transform the other similar atoms into the Zintl-polyanions like that of [Ge6O10]6 as shown in Fig. 2b. The second aspect to be outlined in this chapter refers to the behavior of the Ge[6] atom. Its role is of crucial importance. At first glance, one might think that its true cationic character would cause it to behave as an individual entity. However, as it was seen above (Fig. 13), its octahedral coordination provokes the extension of a distorted, modified, structure of the rutile-like GeO2. After having dissected this crystal structure, each of the two opposite characters exhibited by the Ge atoms (Ge4þ and Ge–) yield two opposite partial structures, i.e., rutile and the Zintl polyanion respectively which must necessarily become compatible.

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65

The existence of a unique GeO6 octahedron forces it to connect to six GeO4 tetrahedra to simulate the partial structure of rutile and consequently, the Ge-Ge distances almost reproduce the unit cell dimensions of the rutile-like GeO2. At the same time, the ladder-type skeleton of J-As is spatially altered but maintaining its threefold connectivity. The simultaneous existence of several substructures within a compound has been described in previous articles. We can mention the dissections of the stuffedantibixbyites Li3 AlN4 34 and the olivine-like FeLiPO4.51,52 In the case of the germanate (NH4)2Ge7O15 it is surprising how the [Ge6O10]6 anion, in the form of a (J - As6O10) Zintl pseudo-oxide, is “devised,” under the appropriate conditions, to allow the reconstruction of the partial structure of the rutilelike GeO2 moiety.48 The TGA and DTA experiments, reported by Cascales et al.,50 showed an endothermic process corresponding to the loss of one H2O and two NH3 molecules. These seem to be the necessary conditions for such reconstruction, i.e., the loss of  the moiety (NH4)2O. When this loss occurs, the structure of Fig. 11a is maintained after heating to 312 C.Then, the structure  collapses, at 450 C, to yield GeO2, in agreement with the collapse proposed by us in Fig. 12d, where if the N atoms (purple spheres) were removed the Ge7-blocks would collapse, reconstructing the GeO2 structure.

3.04.5

Closed packed arrays

3.04.5.1

The EZKC applied to Mg2PN3, Zn2PN3, Li2SiO3 and Ca2PN3 

The catena-polynitridophosphate Zn2PN3 was synthesized under high pressure (8 GPa) and high temperature conditions (1200 C) by Sedlmaier et al.53 This compound is isostructural to Mg2PN3,54 and to Li2SiO355 and is related but not equal to Ca2PN3.54 The structures of Zn2PN3 and Mg2PN3 have been described as ordered superstructures of wurtzite. Thus, since the sets of atoms [Mg2P], [Zn2P] and [Li2Si] equal the three N/O atoms, these compounds can be formulated as pseudo-XY compounds in which the [Mg2P] nominal cations occupy one half of the tetrahedral voids of a hcp-array of N/O atoms. The vec ¼ 4 electrons per atom, common to the three compounds justifies their wurtzite-type structures, characteristic of tetrels. The structure is drawn in Fig. 14a where the horizontal axis is the [110] direction and the vertical axis is [001]. On the other hand, the groups PN3 and SiO3 have the stoichiometry of the catena-phosphates and catena-silicates. These coincidences find explanation by applying the EZKC. Thus, if we admit that in Li2SiO3 the two Li atoms transfer their valence electrons to the Si atom, the latter converts into J-S and the group SiO3 becomes J - SO3. Similarly, if in Mg2PN3 and Zn2PN3, both the Mg and Zn atoms transfer their 4 valence electrons, the catena-polynitridophosphates (PN3 groups) is transformed into J - SO3. The chains of (PN3)4 J - SO3 are represented in Fig. 14b. In the related Ca2PN3 the different nature of the Ca atoms in comparison with Mg makes that the PN4 tetrahedra adopt a different conformation54 (Fig. 14d) which, also differs from the real SO3. Compare Fig. 14b–d. Each tetrahedron share two corners with the adjacent ones in such a way that the P (J-S) atoms, when connected, form an infinite zigzag chain, like the S atoms do in real SO3, represented in Fig. 14c. Both zigzag chains, that of P (Fig. 14b), and that of S (SO3) in Fig. 14c, are identical to that drawn in Fig. 14d that is a fragment separated from the structures of either diamond or lonsdaleite (wurtzite-like). The process by which the elemental structures of tetrels decompose progressively to yield those of pentels, sextels and halogens, was mentioned in the Introduction and discussed in depth in Chapter 9 of Ref. 19. A question, often overlooked when describing crystal structures, is why the atoms occupy the concrete positions they have into the unit cell. In our case, why both Mg2 and P atoms distribute in the voids so as to produce a wurtzite-type structure. The above description can provide an explanation, i.e., the Mg atoms are located in the voids that yield the PN3(J - SO3) chains (Fig. 14b) At that positions, both moieties satisfy the structural requirements of the EZKC. If the cations Mg2þ and P5þ (two Liþ plus one Si4þ in Li2SiO3) would locate randomly in the tetrahedral voids, the EZKC should not be accomplished because the (J - SO3) chains would not be built; under these conditions, no law could be claimed to account for that alternative structure. The only available recourse to justify the structure would be the non-sense radius ratio rule. A final feature deserves discussion: if in the (PN3)4– polyanion of Fig. 14b, instead of considering the planar chains of P atoms that are connected with ochre lines, we consider the P atoms and the bridging N atoms, we obtain a wavy chain of composition PN (J-SO). This square wavy chain resembles the chain of J-S that can be separated either from the black-P or from the lonsdaleite structure. Such fragments are drawn in Fig. 14f and g. The chain can also be formulated as J-SO and the PN3 tetrahedra are completed with the two terminal N atoms so that the PN chain should be considered as J-SO and the (PN3)4– as J - SO3. Thus, the EZKC justify he existence of both J-SO and J - SO3 in the superstructure of the wurtzite-type Mg2PN3.

3.04.5.2

The structure of LiAlSe2

Like in the compounds described in Section 3.04.5, LiAlSe2 also has a vec ¼ 4 electrons per atom [1 þ 3 þ (2  6) ¼ 16 electrons] implying that, even if composed of species so different as Li, Al and Se, each atom behaves as a tetrel adopting a wurtzite-type structure.56 Unlike Zn2PN3 where the N atoms formed an hcp-array. Here, the majority component (Se), as well as Li and Al, are allocated in an apparent random distribution as it can be noted in Fig. 15a. Like in Zn2PN3, discussed above, here, both Li and Al atoms, are ordered forming a superstructure of the wurtzite structure, as if the four valence electrons of Li and Al (two e– per atom), would form with the Se atoms (six valence electrons) a II-VI (ZnSe-like) compound capable of building an IV-IV skeleton. The ordering of both Li and Al can be explained, however, by applying the EZKC. Assuming that Li transfers its valence electron to the Al atom, it converts into J-Si and the compound can be reformulated as

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Fig. 14 (a) The structure of Mg2PN3(Cmc21) projected on (110). The N atoms (blue spheres) form an irregular hcp-array. The two Mg and one P atoms occupy one half of the tetrahedral voids, resulting in an irregular wurtzite-like structure. Mg (ochre); P (purple). (b) One infinite chain of (PN3)4– tetrahedra, sharing corners. The P– atoms (J-S), linked by ochre lines, form a zigzag chain similar to that of (J-S) in real SO3 drawn in (c) in which the identity of the whole pattern (PN3)4– h J - SO3 is made visible. S: yellow; O: red. (d) The zigzag chain of PN4 tetrahedra in the related compound Ca2PN3. (e) A bi-connected chain of (J-S) separated from the diamond-like structure of silicon resembles both the P (J-S) in Mg2PN3 and the S chains in SO3. (f) A wavy bi-connected chain of P (J-S) atoms separated from the structure of black-P. (g) A similar chain of Si (J-S) separated from both the lonsdaleite and the diamond-like structures. Panels (c and (e–g)): Reproduced from Vegas, A. Structural Models of Inorganic Crystals. From the Elements to the Compounds; Valencia: Editorial de la U.P.V., 2018, With permission of the Editorial de la U.P.V.

Liþ[AlSe2]– h Liþ[J - SiSe2]. According to the 8-N rule, such a skeleton should have tetra-connected Si atoms and bi-connected Se atoms, a interpretation that agrees with the features observed in Fig. 15b where the Li atoms have been omitted. Because the charged Al– atoms behave as J-Si, they should form a four-connected skeleton as is shown in Fig. 15b. To achieve it, the Li atoms, and the Al atoms too, must be located in the positions allowing them to contribute to the IV-IV network of LiAlSe2 at

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Fig. 15 (a) The distorted wurtzite-like structure of LiAlSe2. The Li and Al atoms are tetrahedrally coordinated. Li: yellow; Al: blue; Se: red. (b) The Al-subarray (J-Si) showing the fourfold connectivity of a distorted diamond-like structure. In the lower part are drawn the J - SiSe4 tetrahedra of a chair-like hexagon as a part of the J - SiSe2 network. Al: blue; Se: red.

the same time that the induced [Al]– h [J - Si] polyanion is four-connected. Some J - SiSe4 tetrahedra are drawn in the lower part of Fig. 15b.

3.04.5.3

The EZKC in close-packed solids

Along the chapter we have seen that the application of the EZKC to oxides such as silicates/aluminates lead to somewhat surprising results that indicated that, in these compounds, the cations substructures behaved as real Zintl phases. The aluminates/silicates/germanates polyanions are real Zintl polyanions in which the O atoms locate in the vicinity of BP and LP. In fact, if in Li2SiO3, for example, the O atoms were neglected, we obtain the formula of the Zintl phase Li2Si. In this line of work, the dissection of the structure of the germanate (NH4)2Ge7O1548 described in the Section 3.04.4.1 has also shown the power of the EZKC by unveiling the tendency of the donor cations (in that case Ge[6]) to building up its own octahedral i6– h GeO2 structural (rutile-like structure) when trying to conciliate it with that of the Ge½4 6 O15 polyanion. In this section, we intend to give a step forward in the generalization of the ZKC, to demonstrate that the ZKC provides the key to understand more complex structures, provided that a formal charge transfer between cations takes place in the crystal. We have shown the amphoteric behavior of Al, Si and Ge in solids, and the Li / Al transfer in LiAlSe2. In this section, we will show that such an unexpected electron transfer can also accounts for structures like alkali sulfides and sulfates and also explain structures so simples as NiAs, MnP, etc. as if the ZKC concept might be more universal than previously thought.

3.04.5.3.1

Preferred skeletons in crystals

After analyzing several hundred compounds, we found that a number of preferred skeletons (PS) appeared in a recurrent fashion either in the structure itself (in binary and ternary alloys) or in the cation subarrays (in ternary and quaternary oxides).23 The structure of LiAlSe2 discussed above illustrate this finding. We also found that the number of such PS was limited to those of the p-block elements, specifically those of Group 14 and that the cationic substructures of oxides could be rationalized in terms of the PS exhibited by binary alloys in the light of the ZKC. We think pertinent to call in mind the thought reported by Vegas and García-Baonza23:, “.if we assume that a generalization of the ZKC holds, a universal principle seems to work as follows: In any oxide, a pair of cations tends to produce skeletons characteristic of the p-block elements in such a way that the vec is satisfied and the 8-N rule is maintained. As already noted above, it appears that the PS are those of Group 14, as if a more or less homogeneous charge distribution would be a stability factor for a given structure. The analysis of the relative occurrence of a given PS reveals that those derived from (IV)–(IV)-type compounds (diamond, blende, SrAl2, etc.) are certainly quite common.” Vegas and García-Baonza summarized the idea in the following points23: (1) If the number of valence electrons shared by the structure completes an octet, the compound tends to acquire a PS of an element of Group 14. Therefore, it appears that some sort of “IV-IV rule” holds in the formation of a given compound. (2) Whenever possible, a pair of cations tends to acquire a full-octet configuration, to form a (IV)-(IV)-like structure. If the octet is not completed, then, these cations tend to acquire electrons from others present in the cell whatever type it is, even if they are of the same kind, to fulfill the octet configuration. (3) To achieve this, the necessary formal electron transfer between cations should be produced to form what we have named the building skeleton [BS]. Within this view, the BS is considered to be formed by J-atoms in the original Klemm’s sense.

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(4) A given structure should be universally formulated as Dn[BS]Xm, where D is the donor cation, the [BS] cations can be regarded as cations ex-officio57,58 and, if present, X are the anions. The charge can be transferred, in Zintl’s sense, to the [BS] as a whole. To illustrate this idea, we shall consider the Na3P structure,59,60 where two Na atoms can formally donate charge to the entity (NaP)2– h [AlP], a III-V compound that forms graphene-like layers (Group 14 elements), intercalated with layers of Naþ cations. Accordingly, Na3P can be reformulated as Na2[NaP]. We will describe next the binary structures NiAs and MnP, and their stuffed variants Ni2In and Co2Si.

3.04.5.3.2

The structures of NiAs and Ni2In

To demonstrate the wide-spread applicability of the generalized ZKC (EZKC), let us analyze the NiAs structure and that of its related Ni2In alloy.23 Both are hexagonal (P63/mmc, Z ¼ 2). NiAs consists of a hcp-array of As atoms, with all the octahedra filled by the Ni atoms. It is represented in Fig. 16a. The structure of Ni2In (Fig. 16b) is closely related to NiAs. The In atoms, at 2c (1/3, 2/3, ¼), form a hcp-array, like As does in NiAs. The Ni atoms occupy two crystallographically independent sites: Ni(1) at 2a (0,0,0), green spheres in Fig. 16b, center the In6 octahedra; Ni(2) as yellow spheres are at 2d (1/3, 2/3, ¾), centering trigonal bipyramids of the hcp-In substructure (see Fig. 16b). With this in mind, we suggest the following re-interpretation of the Ni2In structure. The Ni(2) atoms (yellow spheres in Fig. 16b) transfer two electrons to the BS, the final formula being Ni(2)[Ni(1)In], with the BS [Ni(1)J-Sb] having the NiAs-type structure. It is stunning that this structure coincides with that of the real NiSb alloy (also NiAs-type) drawn in Fig. 16a. Ni2In is then a Ni-stuffed NiSb structure or, said in other words, the Ni2In alloy contains a Ni2In-type structure formed by the [Ni(1)In] pair. In Fig. 16b, the In atoms (J-Sb) (gray) are connected with blue lines to remark the hcp-array of J-Sb with all the octahedra filled with the green Ni(1) atoms. The important outcome is that, according to the EZKC, the Ni(2) atoms induce the formation of the NiAs structure. To our knowledge, this is the first time that a simple and rational explanation was given for this compound.19,23 Ni2In can also be described as a simple hexagonal array of Ni(1) atoms in which, the J-Sb and the Ni(2) atoms occupy alternate trigonal prisms (Fig. 16c). Such filling process leads to a structure in which hcp-layers of Ni2þ cations (36 planar nets) alternate along the c axis with graphite-like layers formed by [Ni(2)In]2– entities (Fig. 16c). This interpretation was already considered by Zheng and Hoffmann61 as a superstructure of the AlB2-type.

3.04.5.3.3

The MnP and the Co2Si structures

A similar reasoning can explain the MnP structure and its relationship with that of Co2Si, represented in Fig. 17. Co2Si is also tightly related to the cotunnite (PbCl2)-type structure. As seen in Fig. 17b, MnP is a distortion of the NiAs-type; the P atoms (purple spheres) form a very distorted hcp-array with all the octahedral voids occupied by the Mn atoms. The great irregularity of the Parray is visible when comparing Fig. 17b with Fig. 16a corresponding to NiAs. An alternative view is drawn in Fig. 17a where we have drawn the Mn-Mn distances instead of the P-octahedra in Fig. 17b. The connected Mn atoms (green spheres) in Fig. 17a form vertical broken lines that facilitate its comparison with the structure of Co2Si, represented in Fig. 17c where such lines are formed by the Co(1) atoms. This comparison is pertinent because in the same manner that Ni2In contains implicit a NiAs-type structure (see Fig. 16), so the Co2Si-structure (and hence, all cotunnite-type structures) contain implicit a MnP-type structure. In other words, the MnP structure is the parent skeleton of the Co2Si structure. To illustrate this point, in MnP (Fig. 17a), we have inserted an extra atom of Mn (yellow sphere), to show how this extra atom yields the Co2Si (cotunnite-like) structure of Fig. 17c.19

Fig. 16 (a) The structure of NiAs (P63/mmc) viewed along [110]. The As atoms, connected with red lines, form a hcp-array with all the octahedra centered by the Ni atoms. As: gray; Ni: green. (b) The structure of Ni2In (also P63/mmc) highlighting the hcp-array of the In atoms (gray) with the Ni(1) atoms (green) centering the In6 octahedra and the Ni(2) atoms (yellow) centering the trigonal bipyramids. (c) Alternative description of the Ni2In structure based on graphene-like layers of Ni(2)In intercalated with 36 layers of Ni(1) atoms. Colors as in (b). Panels (a) and (c): Reproduced from Vegas, Á.; García-Baonza, V. Pseudoatoms and Preferred Skeletons in Crystals. Acta Cryst. B 2007, 63, 339–345. doi:10.1107/ S0108768107019167, With permission of the IUCr.

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Fig. 17 (a) The structure of MnP where the Mn atoms are connected as zigzag lines like those formed by Co(1) atoms in Co2Si in (c). An ideal yellow atom has been added to show that its role in converting MnP into Co2Si. The yellow atom in (a) mimics the Co(2) atom in (c). (b) The same MnP structure in which the P atoms are connected to show the irregular hcp-array of P and the distorted P-octahedra occupied by the Mn atoms. (c) The Co2Si structure (cotunnite-type, Pnma) showing the zigzag chains of Co6Si trigonal prisms. All atoms at z ¼ 1/4, 3/4. Adjacent chains are displaced c/2 respect to each other. Panels (a) and (c): Reproduced from Vegas, Á.; García-Baonza, V. Pseudoatoms and Preferred Skeletons in Crystals. Acta Cryst. B 2007, 63, 339–345. doi:10.1107/S0108768107019167, With permission of the IUCr.

When the yellow atom is bonded to two Mn atoms (green spheres) as has been done in Fig. 17a, one forms one trigonal prism, centered by the P atom, just like the Si atoms center the equivalent prisms in Co2Si (compare Fig. 17a and c). The Co2Si structure can then be reasoned in the light of the EZKC. Thus, if Co(1) atom transfers one electron to the Si atom, Co(1) converts into J-Fe and Si into J-P, so that Co2Si can be reformulated as Co(2)[J-FeP]. Since FeP is MnP-type, we can assert that the MnP structure is included into Co2Si. Summarizing, the phase relations quoted below hold23: Compound

Structure type

D þ BS

BS

BS structure

Ni2In Mn2Ge Co2Si

Ni2In Ni2In Cotunnite

Ni þ NiAs Mn þ MnGe Co þ CoSi

NiJ-Sb MnJ-Se CoJ-P

NiAs NiAs MnP

It is noteworthy that the ternary compounds MnNiSi and CoVSi are also Co2Si, confirm the underlying transformations expressed in the above diagram. Thus, if in MnNiSi the Ni atom transfers one electron to the Si atoms the formula becomes JMnCoP and the important issue is that [CoP] is also MnP-type. Similarly, if in CoVSi the transfer of two electrons from Co to the V and Si atoms (one electron each) the formula becomes J-MnCrP where both the [MnP] and [CrP] substructures are of the MnP type.

Fig. 18 (a) The structure of CsLiSO4 (Pnma) projected on the ac plane. The Cs and Li atoms form zigzag chains of trigonal prisms, marked with red lines, centered by the SO4 groups, producing a cotunnite-type structure, similar to that of Co2Si (Fig. 17c). After the EZKC, the one-electron transfer from Cs to Li, converts Li into J-Be forming with the S atoms the four-connected BS J-[BeS] (II-VI) drawn with blue lines in (a). The Cs atoms are lodged into the octogonal tunnels. (b) An identical skeleton formed by the Al atoms transformed into J-Si by the electron transfer from Sr / Al. The Sr atoms are lodged into the octogonal tunnels. Atomic species are labeled in the drawings. Reproduced from Vegas, Á.; García-Baonza, V. Pseudoatoms and Preferred Skeletons in Crystals. Acta Cryst. B 2007, 63, 339–345. doi:10.1107/S0108768107019167, With permission of the IUCr.

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3.04.5.3.4

The paradigmatic structure of CsLiSO4

The structure of CsLiSO4, represented in Fig. 18a, is of greatest interest. It is a variant of the cotunnite structure and Co2Si-type, drawn in Fig. 17c. It is also formed by many ternary oxides, such as silicates, phosphates, sulfates, selenates, etc. As examples, we can mention b-Ca2SiO4, CsZnPO4, KBaPO4, b - K2SO4, Cs2SO4, KNaSO4, Cs2S, Ag2Se, among others. As it was pointed out,26 in all these compounds their cation substructures [Ca2Si], [KZnP], [Cs2S], [CsLiS] or [K2S] are cotunnitetype, just as they are many of their parent binary alloys (Ca2Si, Cs2S), or the high-pressure phase of K2S (see the Co2Si structure in Fig. 17c). The structure of CsLiSO4 is apparently complex. However, its explanation is straightforward. It is represented in Fig. 18a in which we have highlighted two separate frameworks: The first one, corresponds to the [CsLiS] substructure and is marked with red lines. It corresponds to a cotunnite-type (Co2Si) structure (compare Fig. 18a with Fig. 17c). Both the Cs and Li atoms form the zigzag chains of trigonal prisms centered by the S atoms (more properly, the SO4 2– nominal anions), comparable with the Co2Si structure of Fig. 17c. According to the EZKC, the structure is explained by assuming that the two alkali atoms transfer two electrons to the S atoms converting them into J-Ar whose four electron pairs, disposed tetrahedrally, attract the four O atoms to form the SO4 tetrahedral groups. From an electrostatic point of view, the structure could be explained by assuming the formula Liþ Csþ(SO4)2–. However, two questions arise: the first one is why in many of these compounds the parent cationic substructure remain in the oxide. This is the case of the alloys Ca2Si, K2S, Cs2S, etc. The second question is why the Li and the Cs atoms occupy the positions they have in CsLiSO4 and not the opposite ones. The first question has been widely treated in the frame of the AMM model.12,19,22,40,41 Thus, the fcc-Ca structure is maintained in CaO and CaF2 or the alloy BaSn is preserved in BaSnO3.21,62 The second question is less easy but it can be responded if we apply the EZCK. Thus, assuming that the Cs atom transfers one electron to the more electronegative Li atom, Li would convert into Be and the compound could be reformulated as Csþ[J - BeSO4]. According to the preferred skeletons idea, the [J-BeS] substructure form a fourfold connected framework that is marked in blue in Fig. 18a. Although this network differs from those of BeSO4 and BeSO4-type (both cristobalite-type) and also from BeS (sphalerite-type), the [BS] [J-BeS] keeps the predicted fourfold connectivity. Its structure is identical to the Al-network in the Zintl phase SrAl2, represented in Fig. 18b. In SrAl2, the transfer of two electrons from Sr / 2 Al atoms, converts them into a four-connected [BS] of J-Si, comparable to the blue skeleton in Fig. 18a. In both compounds, the octogonal tunnels, lodge the donor atoms (Cs and Sr) that induce the four-connected BS in CsLiSO4 and SrAl2. The same pattern is obtained in Ag2Se in which the disproportionation of the Ag atoms gives rise to Agþ[AgSe]d equivalent to Agþ[J - CdSe] It should be recalled that real CdSe forms the sphalerite and the wurtzite-type structures (IV-IV compounds).52 Now, we can understand why the [BS] is [LiS] but not [CsS]. The actual framework assumes a Cs / Li transfer yielding the JBeS-type [BS]. However, the exchange between the Li and Cs positions would produce a [BS] of the type [CsS]d h J - BaS. This would imply the unexpected electron transfer Li / Cs. In support of the first interpretation we can mention the compound Cs [LiCr]O4 in which, the positions of Cs and Li coincide with those of the sulfate. Assuming that Cs donates its valence electron to Li, the [BS] of the type J-BeCr is formed, in which Cr is chemically equivalent to S, as it is well-known. We can extend this model to the high-pressure phase of Na2S (cotunnite-type)63 in which, some sort of disproportion between the two Na atoms occurs. Thus, one Na atom donates its electron to the [BS] [NaS]– forming a J-MgS (II-VI) array, equivalent to a IV-IV compound. This is the first time that the cotunnite structure has been considered a stuffed (IV-IV)-like structure. However, such interpretation appears naturally within the EZKC by assuming a charge transfer between a pair of atoms of the same species to stabilize octets, yielding the (IV-IV) skeletons.

3.04.6

Concluding remarks

Our final comments will include some ideas that although are not strictly related to the ZKC, have served to develop the concepts we have exposed along the chapter. The analysis of hundreds of structures since the of O’Keeffe and Hyde’s pioneering work26 in 1985 up to now leads to the following conclusions: (1) The cationic substructures of many oxides preserve the structure of the parent alloy. We can mention as examples the pairs of compounds: apatite Ca5(PO4)3OH/Ca5P3H, SiO2 (cristobalite)/Si (diamond like), b - Ca2SiO4/Ca2Si (cotunnite-like), or the molecular species of the oxides P4Ox (x ¼ 6–10) and the P4 molecule of white-P. In many instances, the cationic array does not adopt the structure of the alloy stable at ambient conditions but another one that is stable at high pressure or high temperature. An illustrative example is provided by the alloy BaSn (CrB-type). When it is oxidized, BaSnO3 forms the perovskite-type structure whose [BaSn] substructure is CsCl-type. The important issue is that at high pressures BaSn undergoes the transition CrB / CsCl, indicating that the oxidation can produce the same effect than the application of an external pressure64 (see Ref. 21). (2) The coincidences existing between oxides and alloys evidences the important role played by the cations in determining the structure. The preponderance of the cation arrays may lead to consider26 oxides as oxygen-stuffed alloys. When the high-pressure exerted by the O atoms was discovered and the high-pressure phases of alloys were considered, oxides were renamed64 as real oxygen-stuffed alloys. As an example, we can mention the trigonal structure of quartz whose Si-array is of the HgS-type (cinnabar),

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a phase not found so far in the elements of the Group 14 and that might be a metastable phase of Si, stabilized by the O atoms.   Therefore, at 870 C and 1470 C, in the double transition quartz / tridymite / cristobalite the oxygen-pressure is compensated by temperature, recovering in cristobalite, the Si substructure (diamond-like) stable at ambient pressure. For a complete study on these structural coincidences, the reader is referred to Refs. 19,22,23,26,63,65. More recently21 the pressure exerted by the O atoms on the structure has been quantified and explained by means of a generalized stress-redox equivalence by which cations can be either oxidized or reduced, depending on the electronegativity of the anion. Thus, in CaO the fcc-Ca structure is compressed around 37% in volume, with respect to that at ambient conditions, as due to the strong oxidation of Ca. However, in both CaS and CaSe, the fcc-Ca substructure is expanded because S and Se are not so electronegative and need less electron transfer from the Ca atoms producing the chemical reduction of the Ca-subarray by expanding its fcc-structure (see Ref. 21). More recently21 the pressure exerted by the O atoms on the structure has been quantified and explained by means of a generalized stress-redox equivalence by which cations can be either oxidized or reduced, depending on the electronegativity of the anion. Thus, in CaO the fcc-Ca structure is compressed around 37% in volume, with respect to that at ambient conditions, as due to the strong oxidation of Ca. However, in both CaS and CaSe, the fcc-Ca substructure is expanded because S and Se are not so electronegative and need less electron transfer from the Ca atoms producing the chemical reduction of the Ca-subarray by expanding its fcc-structure (see Ref. 21). (3) The important role played by cations made Santamaría-Pérez et al.12,22 feel that cations in aluminates, silicates and phosphates might behave as Zintl phases in spite of being embedded in an oxygen matrix. The examples given along this chapter demonstrate that really the Zintl-Klemm Concept holds in these oxides (see Figs. 6–8). This extension of the ZKC to oxides has two important consequences: (a) This great family of polyoxoanions had been previously classified in a taxonomic way as oligo-aluminates, chain-silicates, tectosilicates, ring silicates, phyllosilicates, etc., but their structures remained unexplained. Now, the EZKC permits to put this large family of compounds on a common basis, as well as to rationalize their skeletons in terms of the charge transfer between cations. A first attempt to classifying the structures with anionic tetrahedron complexes using valence electron criteria was due to Parthé et al.57,58 However, even if they claimed for the Zintl concept to predict the connectivity of tetrahedra, they failed in offering an intuitive explanation of all the structure types. They also failed in relating the Al/Si skeletons with pseudo-elements or pseudo-molecules (see Figs. 6–8). Nevertheless, the term coined by them to denote the T atoms (Al/Si) as “cations ex-officio” implies the admission that the T atoms had lost some of their cationic character. (b) A second consequence of the ZKC Extension to these compounds is that, for the first time, the tetrahedral coordination number of Al and Si is explained without the need of using “ionic radii” criteria. Since in the Zintl polyanions, each atom completes its octet forming 8-N bonds, each Al/Si/P atom has four electron pairs (bonding and lone pairs) that arrange tetrahedrally. Now, if we admit that each O atom locates near any electron pair, then, a tetrahedral coordination is produced around the central atom. Therefore, the CN is a consequence of the electron pairs distribution and not of their ionic radii, as predicted by the “ionic bond” model. The EZKC also accounts for the coexistence in the same crystal of Si and Ge atoms, with CN 4 and 6, as it was shown in Sections h h i i 3.04.3 and 3.04.4 when we described the compounds Ce16 Si½6 Si14 ½4 ðO6 N32  and ðNH4 Þ2Ge½6 Ge½4 6 O15 : The amphoterism exhibited by Al, Si and Ge in some oxides, proves that electron transfer between cations of the same species is feasible when needed to form stable Zintl polyanions. Recall that such transfer also explains the structures of Ni2In, Co2Si, Na2S and Mn2Ge, listed in Section 3.04.5.3.3. We wish to make a final comment on a matter that is normally overlooked in textbooks, i.e., the tight relationship between the structures of the elements of the p-block. This matter has been treated in Chapter 9 of Ref. 19 entitled as “The progressive simplification of the structures of the p-Block.” It demonstrates that, if we start from all the structures of the Group 14 and if we ideally break (not capriciously) some selected bonds, we can obtain all the structures of the Group 15. In the same way, the latter yield the structures of the Group 16 and from them, the dumbbells of the X2 molecules of halogens can be derived. The final step is obtaining the single noble gases atoms. The process just described is ideal because we cannot break the bonds with a scissor but the process can be achieved if in every of the steps described above, we add one electron to each atom of the previous group. This process that is not feasible so far by physical methods, is fortunately achieved chemically as it has been described in the Zintl phases and condensed in the EZKC. Thus, “the stage where such progressive simplification of the elemental structures takes place are the Zintl phases”; more precisely, in those chemical compounds in which the EZKC applies. The reader can get an image of such process by comparing the Zintl anions of Fig. 2. The As-type structure of CaSi2 derives from that of Si (diamond-like) or from that of lonsdaleite in Fig. 2a. The square helices of S-II (Fig. 7g) are implicit in the black-P structure and too in the diamond or silicon structure. The S6-chairs are fragments of the Y-P structure in Ca2Si (Fig. 2b) and so on. We believe that the structures are better understood in this way. Nowadays, chemists spend many efforts to structures prediction. Even if predicting is not lacking of interest, our aim was always to understand the reasons why the structures are as they are. At this point we are in the line of thought of Thom66: “To predict is not to explain,” a thought already expressed in Chapter 18 of Ref. 19.

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M., Ed.; Chemistry, Structure and Bonding of Zintl Phases and Ions, VCH: Weinheim, 1996. 12. Santamaría-Pérez, D.; Vegas, A. The Zintl-Klemm Concept Applied to Cations in Oxides. I. The Structures of Ternary Aluminates. Acta Cryst. B 2003, 59, 305–323. https:// doi.org/10.1107/S0108768103005615. 13. Savin, A.; Nesper, R.; Wengert, S.; Fässler, T. E. ELF: The Electron Localization Function. Angew. Chem. Int. Ed. 1997, 36, 1808–1833. https://doi.org/10.1002/ anie.199718081. 14. Klemm, W.; Busmann, E. Volumeninkremente und radien einiger einfach negativ geladener ionen. Z. Anorg. Allg. Chem. 1963, 319, 297–309. https://doi.org/10.1002/ zaac.19633190511. 15. Schäfer, H.; Eisenmann, B.; Müller, W. Zintl Phases: Transitions between Metallic and Ionic Bonding. Angew. Chem. Int. Ed. Engl. 1973, 12, 694–712. https://doi.org/ 10.1002/anie.197306941. 16. Parthé, E.; Parthé, E. Note on the Structure of ScP and YP. Acta Cryst. 1963, 16, 71. https://doi.org/10.1107/S0365110X63000141. 17. Nesper, R. Structural and Electronic Systematics in Zintl Phases of the Tetrels. In Silicon Chemistry; Jutzi, P., Schubert, U., Eds., Wiley-VCH: Weinheim, 2003; p 171. https:// doi.org/10.1002/9783527610761.ch13. 18. Gascoin, F.; Sevov, S. C. Synthesis and Characterization of A3Bi2 (A ¼ K, Rb, Cs) with Isolated Diatomic Dianion of Bismuth, [Bi2]2-, and an Extra Delocalized Electron. J. Am. Chem. Soc. 2000, 122, 10251–10252. https://doi.org/10.1021/ja002606t. 19. Vegas, A. Structural Models of Inorganic Crystals, Editorial de la Universitat Politècnica de València: Valencia, 2018. 20. Papoian, G. A.; Hoffmann, R. Hypervalent Bonding in One, Two and Three-Dimensions: Extending the Zintl-Klemm Concept to Nonclassical Electron-Rich Networks. Angew. Chem. Int. Ed. 2000, 39, 2408–2448. https://doi.org/10.1002/15213773(20000717)39:143.0.CO;2-U. 21. Lobato, Á.; Hussien, H. O.; Salvadó, M. A.; Pertierra, P.; Vegas, Á.; Baonza, V. G.; Recio, J. M. Generalized Stress-Redox Equivalence: A Chemical Link between Pressure and Electronegativity in Inorganic Crystals. Inorg. Chem. 2020, 59, 5281–5291. https://doi.org/10.1021/acs.inorgchem9b01470. 22. Santamaría-Pérez, D.; Vegas, Á.; Liebau, F. The Zintl-Klemm Concept Applied to Cations in Oxides. II. The Structures of Silicates. Struct. Bond. 2005, 118, 121–177. https:// doi.org/10.1007/b137470. 23. Vegas, Á.; García-Baonza, V. Pseudoatoms and Preferred Skeletons in Crystals. Acta Cryst. B 2007, 63, 339–345. https://doi.org/10.1107/S0108768107019167. 24. Santamaría-Pérez, D. Reinterpretacio´n de las estructuras de aluminatos y silicatos. Equivalencia entre oxidacio´n y presio´n, Universidad Carlos III: Leganés, 2005. Ph.D. Tesis. 25. Wondratschek, H.; Merker, L.; Schubert, K. Beziehungen zwischen der Apatit-Struktur und der Struktur der Verbindungen vom Mn5Si3 - (D88-)Typ. Z. Kristallogr. 1964, 120, 393–395. Erratum correction: ibid 120, 478. https://doi.org/10.1524/zkri.1964.120.4-5.393. 26. O’Keeffe, M.; Hyde, B. G. An Alternative Approach to Non-molecular Crystal Structures. With Emphasis on the Arrangement of Cations. Struct. Bond. 1985, 61, 77–144. https://doi.org/10.1007/BF0111193. 27. Vegas, Á.; Romero, A.; Martínez-Ripoll, M. A New Approach to Describing Non-molecular Crystal Structures. Acta Cryst. B 1991, 47, 17–23. https://doi.org/10.1107/ S010876819001062X. 28. Xie, L. S.; Schoop, L. M.; Seibel, E. M.; Gibson, Q. D.; Xie, W.; Cava, R. J. Potential Ring of Dirac Nodes in a New Polymorph of Ca3P2. APL Mat. 2015, 3, 083602. https:// doi.org/10.1063/1.4926545. 29. Avdeev, M.; Yakovlev, S.; Yaremchenko, A. A.; Kharton, V. V. Transitions between P21, P63 and P6322 Modifications of SrAl2O4 by In Situ High-Temperature X-Ray and Neutron Diffraction. J. Solid State Chem. 2007, 180, 3535–3544. https://doi.org/10.1016/j.jssc.2007.10.021. 30. Haile, S. M.; Maier, J.; Wuensch, B. J.; Laudise, R. A. Structure of Na3YSi6O15- a Unique Silicate Based on Discrete Si6O15 Units, and a Possible Fast-Ion Conductor. Acta Crystallogr. B 1995, 51, 673–680. https://doi.org/10.1107/S0108768194014096. 31. Katz, T. J.; Acton, N. Synthesis of Prismane. J. Am. Chem. Soc. 1973, 95, 2738–2739. https://doi.org/10.1021/ja00789a084. 32. Eremets, M. I.; Gavriliuk, A. G.; Trojan, I. A.; Dzivenko, D. A.; Boehler, R. Nat. Mater. 2004, 3, 558–563. https://doi.org/10.1038/nmat1146. 33. Laniel, D.; Winkler, B.; Fedotenko, T.; Pakhomova, A.; Chariton, S.; Milman, V.; Prakapenka, V.; Dubrovinsky, L.; Dubrovinskaia, N. High-Pressure Polymeric Nitrogen Allotrope with the Black Phosphorus Structure. Phys. Rev. Lett. 2020, 124 (7), 216001. https://link.aps.org/doi/10.1103/PhysRevLett.124.216001. 34. Vegas, A.; Martin, R. L.; Bevan, D. J. M. Compounds with a Stuffed’ Anti-Bixbyite-Type Structure, Analysed in Terms of the Zintl–Klemm and Coordination-Defect Concepts. Acta Cryst. B 2009, 65, 11–21. https://doi.org/10.1107/S010876810803423X. 35. Walz, L.; Heinau, M.; Nick, B.; Curda, J. Die kristallstrukturen der erdalkalialuminate Ba3Al2O6 und Ba2.33Ca0.67Al2O6(Ba7Ca2Al6O18). J. Alloys Comp. 1994, 216, 105–112. https://doi.org/10.1016/0925-8388(94)91050-2. 36. Többens, D. M.; Kahlenberg, V.; Kaindl, R.; Sartory, B.; Konzett, J. Na8.25Y1.25Si6O18 and its Family of zwo¨lferring Silicates. Z. Kristallogr. 2008, 223, 389–398. https://doi.org/ 10.1524/zkri.2008.0039. 37. Baumgartner, O.; Völlenkle, H. Die Kristallstruktur der Verbindung K4SrGe3O9, ein Cyclogermanat mit Zwölferringen. Z. Kristallogr. 1977, 146, 261–268. https://doi.org/ 10.1524/zkri.1978.146.16.261. 38. Lavrov, A. V.; Voitenko, M. Y.; Tselebrovskaya Izvet. Akad. Nauk SSSR, Neorg. Mater. 1981, 17, 99–103. 39. Jost, K. H. Struktur Eines Cyclopentaphosphate. Acta Crystallogr. B 1972, 28, 732–738. https://doi.org/10.1107/S0567740872003103. 40. Vegas, A.; Santamaría-Pérez, D.; Marqués, M.; Flórez, M.; García-Baonza, V.; Recio, J. M. Anions in Metallic Matrices Model: Application to the Aluminium Crystal Chemistry. Acta Crystallogr. B 2006, 62, 220–227. https://doi.org/10.1107/S0108768105039303. 41. Marqués, M.; Flórez, M.; Recio, J. M.; Santamaría, D.; Vegas, A.; García Baonza, V. Structure, Metastability, and Electron Density of Al Lattices in Light of the Model of Anions in Metallic Matrices. J. Phys. Chem. B 2006, 110, 18609–18618. https://doi.org/10.1021/jp063883a.

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42. Eisenmann, B.; Jordan, H.; Schäfer, H. Darstellung und Struktur von Ba6Sn2P6. Z. Naturforsch. B 1983, 38, 404–406. https://doi.org/10.1515/znb-1983-0323. 43. Swick, D. A.; Karle, I. L. Structure and Internal Motion of C2F6 and Si2Cl6 Vapors. J. Chem Phys. 1955, 23, 1499–1504. https://doi.org/10.1063/1.1742337. 44. Königstein, K.; Jansen, M. Ein einfacher Weg zu Silicium in oktaedrischer Sauerstoffkoordination. Chem. Ber. 1994, 127, 1213–1218. https://doi.org/10.1002/ cber.19941270706. 45. Köllisch, K.; Schnick, W. Ce16Si15O6N32: An Oxonitridosilicate with Silicon Octahedrally Coordinated by Nitrogen. Angew. Chem. Int. Ed. 1999, 38, 357–359. https://doi.org/ 10.1002/(SICI)1521-3773(19990201)38:33.0.CO;2-D. 46. Liebau, F. Silicates and Perovskites: Two Themes with Variation. Angew. Chem. Int. Ed. 1999, 38, 1733–1737. https://doi.org/10.1002/(SICI)1521-3773(19990614) 38:123.0.CO;2-5. 47. Alonso, J. A.; Rasines, I.; Soubeyroux, J. L. Tristrontium Dialuminum Hexaoxide: An Intricate Superstructure of Perovskite. Inorg. Chem. 1990, 29, 4768–4771. https://doi.org/ 10.1021/ic00348a035. 48. Vegas, A.; Jenkins, H. D. B. (NH4)2Ge7O15. Acta Crystallogr. B 2017, 73, 94–100. https://doi.org/10.1107/S2052520616019181. 49. Svensson, C. The Crystal Structure of Orthorhombic Antimony Trioxide, Sb2O3. Acta Crystallogr. B 1974, 30, 458–461. https://doi.org/10.1107/S0567740874002986. 50. Cascales, C.; Gutiérrez-Puebla, E.; Monge, M. A.; Ruiz-Valero, C. (NH4)2Ge7O15: A Microporous Material Containing GeO4 and GeO6 Polyhedra in Nine-Rings. Angew. Chem. Int. Ed. 1998, 37, 129–131. https://doi.org/10.1002/(SICI)1521-3773(19980202)37:1/23.0.CO;2-T. 51. Vegas, A. FeLi[PO4]: Dissection of a Crystal Structure. The Parts and the Whole. Struct. Bond. 2011, 138, 67–91. https://doi.org/10.1007/430_2010_35. 52. Vegas, A. On the Charge Transfer between Conventional Cations: The Structures of Ternary Oxides and Chalcogenides of Alkali Metals. Acta Crystallogr. B 2012, 68, 364–377. https://doi.org/10.1107/S0108768112021234. 53. Sedlmaier, S. J.; Eberspecher, M.; Schnick, W. High-Pressure Synthesis, Crystal Structure, and Characterization of Zn2PN3dA New catena-Polynitridophosphate. Z. Anorg. Allg. Chem. 2011, 637, 362–367. https://doi.org/10.1002/zaac.201000403. 54. Schultz-Coulon, V.; Schnick, W. Mg2PN3 und Ca2PN3dPhosphor(V)-nitride mit eindimensional unendlichen Ketten eckenverknüpfter PN4-Tetraeder. Z. Anorg. Allg. Chem. 1997, 623, 69–74. https://doi.org/10.1002/zaac.19976230112. 55. Völlenkle, H. Verfeinerung der Kristallstrukturen von Li2SiO3 und Li2GeO3. Z. Kristallogr. 1981, 154, 77–81. https://doi.org/10.1524/zkri.1981.154.1-2.77. 56. Kim, J.; Hughbanks, T. Synthesis and Structures of New Ternary Aluminium Chalcogenides: LiAlSe2, Alpha-LiAlTe2, and Beta-LiAlTe2. Inorg. Chem. 2000, 39, 3092–3097. https://doi.org/10.1021/ic000210c. 57. Parthé, E.; Engel, N. Relation between Tetrahedron Connections and Composition for Structures with Tetrahedral Anion Complexes. Acta Cryst. B 1986, 42, 538–544. https:// doi.org/10.1107/S0108768186097732. 58. Parthé, E.; Chabot, B. Classification of Structures with Anionic Tetrahedron Complexes Using Valence Electron Criteria. Acta Cryst. B 1990, 46, 7–23. https://doi.org/10.1107/ S0108768189008074. 59. Dong, Y.; DiSalvo, F. J. Reinvestigation of Na3P Based on Single-Crystal Data. Acta Crystallogr. E 2005, 61, i223–i224. https://doi.org/10.1107/S1600536805031168. 60. Yu, X.-f.; Giorgi, G.; Ushiyama, H.; Yamasita, K. First-Principles Study of Fast Na Diffusion in Na3P. Chem. Phys. Lett. 2014, 612, 129–133. https://doi.org/10.1016/ j.cplett.2014.08.010. 61. Zheng, C.; Hoffmann, R. Conjugation in the 3-Connected Net: The Aluminium Diboride and Thorium Disilicide Structures and their Transition-Metal Derivatives. Inorg. Chem. 1989, 28, 1074–1080. https://doi.org/10.1021/ic00305a015. 62. Vegas, A.; Mattesini, M. Towards a Generalized Vision of Oxides: Disclosing the Role of Cations and Anions in Determining the Unit Cell Dimensions. Acta Crystallogr. B 2010, 66, 338–344. https://doi.org/10.1107/S0108768110013200. 63. Vegas, A.; Grzechnik, A.; Syassen, K.; Loa, I.; Jansen, M. Reversible Phase Transitions in Na2S Under Pressure: A Comparison with the Cation Array in Na2SO4. Acta Crystallogr. B 2001, 57, 151–156. https://doi.org/10.1107/S0108768100016621. 64. Martínez-Cruz, L. A.; Ramos-Gallardo, A.; Vegas, A. MSn and MSnO3 (M ¼ Ca, Sr, Ba): New Examples of Oxygen-Stuffed Alloys. J. Solid State Chem. 1994, 110, 397–398. https://doi.org/10.1006/jssc.1994.1186. 65. Vegas, A.; Jansen, M. Structural Relationships between Cations and Alloys. An Equivalence Between Oxidation and Pressure. Acta Crystallogr. B 2002, 58, 38–51. https:// doi.org/10.1107/S0108768101019310. 66. Thom, R.; Noel, E. Pre´dire n’est pas expliquer, 2ème; Éditions Eshel: Paris, 1991.

3.05

An introduction to the theory of inorganic solid surfaces

E´milie Gaudry, University of Lorraine, CNRS, Institut Jean Lamour, Nancy, France © 2023 Elsevier Ltd. All rights reserved.

3.05.1 3.05.1.1 3.05.1.1.1 3.05.1.1.2 3.05.1.1.3 3.05.1.2 3.05.1.3 3.05.1.4 3.05.2 3.05.2.1 3.05.2.2 3.05.2.3 3.05.2.4 3.05.2.4.1 3.05.2.4.2 3.05.2.5 3.05.2.5.1 3.05.2.5.2 3.05.2.5.3 3.05.2.6 3.05.3 3.05.3.1 3.05.3.1.1 3.05.3.1.2 3.05.3.1.3 3.05.3.1.4 3.05.3.2 3.05.3.2.1 3.05.3.2.2 3.05.3.3 3.05.3.3.1 3.05.3.3.2 3.05.3.3.3 3.05.3.4 3.05.4 3.05.4.1 3.05.4.1.1 3.05.4.1.2 3.05.4.1.3 3.05.4.1.4 3.05.4.2 3.05.5 3.05.6 Acknowledgments References Relevant websites

Basics of surface science Surface energy Amount of reversible work to create a surface Thermodynamic approach Surface energy anisotropy and Wulff construction Surface symmetries Beyond bulk-truncated surfaces Summary First principles modeling of surfaces Many-body Schrödinger equation Density Functional Theory Impact of the functional in surface computations Structural models for surfaces within DFT Convergence issues Supercell slab generation Surface stability from DFT and atomistic thermodynamics Surface energies of clean surfaces Temperature effects Surface energies modified by adsorption Summary Phenomenological models Early concepts The free electron model extended to the surface The broken bond model at metal surfaces Can the broken bond model apply at intermetallic compound surfaces? Pauling rules extended to the surfaces of ionic compounds The concept of dangling bonds Dangling bonds at elemental semiconductor surfaces Dangling bonds at intermetallic and oxide surfaces Electron counting rules Electron counting rules at sp semiconductor surfaces Electron counting rules involving d electrons Electron counting rules at oxide surfaces Summary Morphologies and electronic structures of inorganic compound surfaces Surface morphologies Elemental boron Al-based complex intermetallics Cage compounds Oxide surfaces Electronic states at the surface Conclusion Further reading

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Abstract The purpose of this chapter is to provide a general introduction to the surfaces of inorganic solids. The goal is to set the stage of this field by providing an overview of the present understanding of the chemical and physical concepts involved. A theoretical point of view is adopted, based on Density Functional Theory (DFT). After a short reminder of basics in surface

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science and a brief review of DFT methods, a few phenomenological approaches are examined and illustrated. In metals, bulk-truncated, flat and dense terminations are observed in most cases, in agreement with the broken bond model. Numerous reconstructions occur at the surface of covalent materials, driven by the minimization of the dangling bonds. In ionic solids, stable surfaces are those that can compensate the surface charges, in agreement with the electron counting rules. Most intriguing morphologies likely arise at the surface of inorganic compounds which combine several types of bonding interactions, like perovskites, cage compounds or Heusler alloys. The deep understanding of the driving forces leading to specific surface structures will provide the necessary knowledge for an efficient development of surfaces for technological applications.

The term surface is intrinsically multi-disciplinary. In the scientific context, it consist of the upper boundary of something. It can be as thick as several kilometers, like at the surface of the earth, or as evanescent as in mathematical objects, as defined by a twodimensional locus of points. In this chapter, surfaces are considered in the field of inorganic chemistry. They are defined as the exterior part of a solid. Depending on the experimental technique used to probe them, their thickness is assumed to correspond to a few atomic layers. This thickness is tiny, regarding the dimensions of a macroscopic solid. This corresponds to a negligible number of atoms with respect to the bulk onesda cubic solid of 1 cm3 contains about 1024 atoms but only 6
1016 atoms at its surface. This is why in most cases, surfaces do not contribute to the bulk properties. Therefore, surfaces of inorganic solids are often ignored. There are however a few situations that requires the consideration of the crystal’s surface. When the solid has a very small volume, like in nanoparticles, the proportion of surface atoms increases significantly, thus amplifying the impact of the surface on the properties. For instance, decreasing the size of a cubic particle from 1024 atoms to 1000 atoms leads to an increase of this ratio from 6
10 8 to 0.6. It accounts for the specific and size-dependent properties of nanoparticles, mainly stem from surface and quantum size effects, which makes them very attractive in numerous fieldsdfor drug delivery in medicine, for energy conversion, as gas sensing and capturing in depollution technologies, etc. Surfaces cannot be ignored either when the physical or chemical properties of interest actually take place at the surface, or involve surfaces. This applies to a wide number of phenomena, such as electronic emission for metals, atomic transport through the surface, surface passivation, heterogeneous catalysis, wetting or crystal growth. These examples highlights the significance and wealth of inorganic surface science, lying in its interdisciplinary naturedinvolving physics, chemistry, material science, crystallography, etc.das well as in the diversity of questions it raises, both on a fundamental and applied level. The knowledge of the atomic arrangements at the surface is a prerequisite to further studies aiming to understand and predict surface properties. The situation would be rather simple if all surface structures resulted from a bulk truncation. However, the reality is that the observed surface structures are the lowest free energy structures kinetically accessible under the preparation conditions. As such, they sensitively depend on cleavage, annealing, and growth conditions, which can generate different observations, for the same surface, using different experimental techniques (see the NIST Surface Structure Database1). This makes the determination of surface structures complex, but also challenging. The surfaces investigated here are crystalline surfaces. Thus, they present long-range ordered atomic arrangements. They are formed using a large number of crystal types, from elemental solids like dense-packed metals and semiconductors, to binary or ternary compoundsdoxides, polar semiconductors, alloys and inter-metallics. This correspond to different types of interactions in the bulk,2 that are distinguished through the way valence electrons are shared between neighboring atoms. They can effectively be sorted using a two-dimensional van Arkel-Ketelaar diagram3 (Fig. 1). Within this scheme, the A-B bonds are characterized using their average electronegativities hci and the electronegativity difference Dc between species A and B. The three main types of solids, i.e. ionic, covalent and metallic crystals, are located at the three vertices of the triangle. More precisely, the upper corner identifies the compound with the highest electronegativity difference, where electrons can be fully transferred from one atom to its neighbors, leading to bonds with a strong ionic character. At the base of the triangle, Dc is weak, thus avoiding any large interatomic charge transfer. In this region, the electron localization increases from left to right. As a consequence, the lower right region is the one of covalent solids, with electrons shared between nearest neighbors, while the lower left region is the one of metals, with electrons shared in the whole crystal. Such a rationalization of the bonding in bulk systems, combined with several electron counting rules (Hume-Rothery, Wade, 18-n, etc.) has been proven to be very useful to account for the myriad of bulk structures in inorganic chemistry.4–6 In the following, insights to rationalize the surface structures observed so far are suggested, in connection with the bonding network in the corresponding bulk compounds. In this vast context, the whole field of the surface inorganic chemistry cannot be covered in a single chapter. The objective here is to offer a general introduction, as well as an overview of the current understanding of the physical and chemical processes involved. The purpose aims to indicate tracks for future research, rather than to present a final picture. A theoretical point of view is adopted, focusing on basic concepts, illustrated by recent examples in surface inorganic chemistry. The extension to the surface of fundamental aspects, well-established for bulk solids, especially those related to the chemical bonding, is discussed throughout this chapter. Quantitative arguments are provided by calculations based on Density Functional Theory. The latter is introduced, as a reliable and precise method to investigate the surface structures and properties, in combination to experimental studies if applicable. A few phenomenological approaches, such as the minimization of broken or dangling bonds, as well as electron counting rules, are reviewed using recent examples in different classes of solids. The wide diversity of surface morphologies and electronic structures are illustrated, before a short conclusion and a few perspectives are given.

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Fig. 1 Schematic view of a Van Arkel-Ketelaar triangle (left). Major bonding types are labeled. Van Arkel-Ketelaar triangle map (right) built for 28 equiatomic binaries (AB) and seven elemental crystals (period 3 of the Periodic Table) using the Allen electronegativity data (c). The average B and electronegativity difference is Dc ¼ | cA  cB |. Three families are considered: equiatomic Zintl compounds AX electronegativity is hci ¼ cA þc 2 (A ¼ Li and Na and X ¼ Al, Ga, In, and Tl) (•) and Al-, Zn-and Cd-TM compounds (•) and P-based compounds (•). Redrawn from Ref. Mizutani, U.; Sato, H. The Physics of the Hume-Rothery Electron Concentration Rule. Crystals 2017, 7, 9 under the Creative Common CC BY license.

3.05.1

Basics of surface science

Thermodynamics is at the heart of many first discoveries in materials science. The concept of surface energy is a prime example. Indeed, it originates from the discovery of capillary phenomena by Leonardo da Vinci. 7 It has been later on formalized by Young and Laplace8,9 in the beginning of the 19th century. It is detailed below.

3.05.1.1 3.05.1.1.1

Surface energy Amount of reversible work to create a surface

Surface energy and surface tension are both related to the amount of work to create a surface. For a liquid, the surface tension and the surface energy have the same numerical value. But in solids, the surface tension is generally not equal to the surface energy (Fig. 2). Surface tension measures the mechanical work required to stretch the surface. It depends on the surface stress, sij, a second-order tensor in two dimensions. A surface can also be created by cleavage, i.e., by splitting a crystalline materials along definite crystallographic structural planes, with no deformation. The surface energy g is then defined as the amount of reversible work dWcre needed to create new area dA: dW cre ¼ g dA

(1)

Fig. 2 surface created by stretchingdno broken bondsdor by cleavageda few bonds are broken. In comparison, the sublimation breaks all bonds that previously held a crystal together.

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As a first approximation, and considering elemental metals, the energetic cost related to the cleavage can be estimated from the energy required to cut the bonds which ensure the crystal cohesion, in connection to the heat of sublimation. It depends on both the number and the strength of the broken bonds resulting from the cleavage process. This is the basis of the broken bond model, a phenomenological approach useful to analyze surface energies and their anisotropy (see Section 3.05.3.1). The surface energy and the surface stress are not independent. They are connected through the Shuttleworth equation.10 Recently, the Shuttleworth equation has been revisited in the framework of statistical mechanics.11 Thanks to molecular dynamics simulations performed with a simple one-component system of particles interacting via the Lennard-Jones potential, the consistency of the two approachesdclassical thermodynamic and statistical mechanicsdhas been validated within statistical uncertainty. In the following, the surface stress will not be discussed, but we will focus on the surface energy.12

3.05.1.1.2

Thermodynamic approach

According to thermodynamics, using the Gibbs dividing plane approach,10,13–15 the surface energy is defined as a function of excess quantities g¼

Usurf ¼ usurf A

(2)

where A is the total surface area and Usurf is the excess energy of the grand potential, also known as the Landau free energy. It can be expressed as the difference i Usurf 1 h surf F  Gsurf ¼ f surf  g surf ¼ A A

(3)

between the excess free energy and the excess Gibbs free energy, where X X surf surf F surf ¼ U surf  TSsurf and Gsurf ¼ mi Ni ¼ A mi Gi i

(4)

i

In the previous equation, mi and Gi surf are the chemical potential and the excess atomic density of species i at the surface, respectively. Eqs. (5–7) show without any ambiguity that the surface energy cannot be confused with the surface free energy density N surf

¼ Ai ¼ (fsurf s g), except in the case of a pure body (i ¼ 1) where the Gibbs dividing plane can be chosen in such a way that Gi surf 0. The values of the thermodynamic potentials c (c ˛ {F, G.}) depend on the selection of the Gibbs dividing plane, but the equilibrium surface energy is a physical property that only depends on thermodynamic variables such as chemical potentials and temperature. The chemical potentials of species i are intrinsic properties of inorganic compounds. They are defined as the derivative of the Gibbs free enthalpy G for a given phase with respect to the number of particles i and fixed numbers of other particles {Nj} apart   vG . For elemental solids, the chemical potentials are then connected to the cohefrom Ni. More precisely, it writes mi ¼ vN i surf

P;T;Nj

i sive energies mi bulk ¼  Ecoh , cohesive energies being positive within this definition. In the case of inorganic compounds, the chemical potentials are given by the Gibbs phase rule (equilibrium conditions). It means that the total energy of the bulk can be expressed P as a weighted sum of the chemical potentials: Ecoh ¼  iNimi.

3.05.1.1.3

Surface energy anisotropy and Wulff construction

Due to the crystalline anisotropy, the surface energy depends on the surface orientation. The latter is expressed by three (hk‘) indices, and four for hexagonal structures, i.e., (hkp‘) with h þ k þ p ¼ 0. They are called the Miller indices. They denotes a plane that inter! ! ! ! ! cepts the three points A, B and C with OA ¼ ! a =h, OB ¼ b =k and OC ¼ ! c =‘, where ! a , b , and ! c are the lattice vectors of the periodic crystal cell. Surface anisotropy implies that two crystallographic surfaces with different Miller indices (hk‘) and not equivalent by symmetry, present different surface energies, denoted by g(hk‘) in the following. Surface energies are not straightforward to deduce from cleavage experiments. Measurements may be subject to errors if the samples do not present clean surfaces, since adsorbates can result in a decrease of the surface energy.16 In addition, all plane orientations are not necessarily cleavage planes, and thus cannot be easily formed. Methods based on the investigation of a crack propagation in a crystal require the knowledge of the crystal elastic constants.17 In metals, the extrapolation of liquid surface tensions has given reliable estimates of surface energies.18 This method can be extended in alloys in a few cases.19,20 Unfortunately, this approach is hardly applicable to all inorganic compounds. Most popular derivations of surface energies rely on the analysis of the equilibrium shape of small crystallites. Within this framework, the crystal is assumed to be large enough so that edge and apex effects can be neglected. The method is based on the fact that R R ! the particle shape results from the minimization of its excess grand potential Usurf ¼ g n dA, with the volume of the particle kept constant. In the previous equation, g is the surface energy and ! n is a unit vector perpendicular to the surface element dA. In the isotropic case, where the surface energy is orientation-independent and hence just a positive constant, the shape is obviously spherical (if one neglects gravity), but in the general case it is less symmetric. It result from the Wulff construction, in which the length of a vector drawn normal to a crystal facet is assumed to be proportional to its surface energy. The resulting g-plot, i.e., the polar plot of

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surface energy as a function of orientation,21–23 is used to graphically determine which crystal facets are present at the crystallite surface, as illustrated in Fig. 3 in two-dimensions. A precise and general statement, as well as a rigorous proof of Wulff’s theorem have been given by the American mathematician Jean Taylor.24 The Wulff shape must be consistent with the crystallographic point group symmetry of the underlying crystal. In the next section, the symmetries of surfaces resulting from bulk truncations are summarized.

3.05.1.2

Surface symmetries

With the exception of aperiodic solids, a perfect crystal is an ordered phase built from an elementary cell reproduced identically to itself in the three directions of space. It is characterized by a long range order and a specific symmetry, strongly related to the compound anisotropy and properties. According to crystallography, periodic structures are classified on the basis of their symmetries in 230 space groups, 14 Bravais lattices and 7 crystal systems. According to computational25–29 and experimental30–32 bulk structure databases, the distribution of the samples within the crystal systems is not uniform.33 There are far more structures experimentally revealed in cubic (3404) than triclinic systems (352), e.g., 26% cubic vs. 3% triclinic. On the other hand, computations on low-symmetry structures are over-represented relative to their experimental counterparts (e.g., 35% were performed in triclinic or monoclinic systems, with respect to 16% observed experimentally). Surface structures result from the minimization of the surface energy. Thus, an efficient approach to theoretically determine surface structures is based on the generation of surface models, for which the surface energy is calculated. Because the number of candidates can be high, the surface models must be generated strategically so that the screening space is reduced without overlooking potential models. In most cases, the search efficiency is improved by implementing symmetry groups. Three-dimensional (3D) crystallography can be used as a starting point to infer surface symmetries. Here, one consider ideal surfaces built by bulk truncation. All atoms located in an half-space above the considered surface plane are removed, thus leading to the loss of all symmetry operations along the direction perpendicular to the surface plane. Although the surface is a three-dimensional object, with a given thickness, all symmetries of the surface are two-dimensional, i.e. that the surface is periodic only in two directions. As a consequence, the number of translational and point symmetry operations is much lower in 2D than in 3D. Four planar crystal systemsdsquare, rectangular, hexagonal and obliquedgenerate 5 two-dimensional Bravais lattices (four primitive and a centered rectangular lattice, Fig. 4a–e), and 10 point groups (1, 2m, 2mm, 3, 3m, 4, 4mm, 6, 6mm). The combination of the translational and point group symmetries then gives the 17 planar space groups (Fig. 4). According to the previous classification, the low-index surfaces of the closed-packed metals present several different symmetries (Fig. 5). The symmetries of the fourfold fcc(100) and bcc(100) surfaces are those of the p4m space group. The threefold fcc(111) and bcc(111) surfaces belong to the p3m1 space group. The dense bcc(110) and the fcc(110) surfaces show the symmetries of the cmm and pm space groups, respectively. Lower surface symmetries can be achieved by the selection of the highest surface indices.34 A huge diversity of surface structure symmetries can also emerge from the wide diversity of inorganic crystal structures, spread in the full range of the 3D space groups. Several of them, with specific surface symmetries, show intrinsic surface chirality.35

3.05.1.3

Beyond bulk-truncated surfaces

The bulk truncated surfaces considered in the previous section as an ideal picture to derive surface symmetries, actually undergo several structural and compositional changes. Surface relaxation corresponds to a change in the interlayer spacing of the outermost surface layers, with respect to the bulk layers. It is defined by the averaged atomic displacements perpendicular to the surface plane, according to Meyer and Vanderbilt.36 It may affect several surface layers and decays gradually into the bulk. For most surface structures, an inward relaxation of the

(A)

(B)

(C)

Fig. 3 Schematic Wulff construction for a two-dimensional material. (A) The Wulff construction begins with the g-plot (in red). (B) A radius vector is then drawn in each direction (in brown) and a perpendicular plane is plotted where this vector hits the Wulff plot (in dark blue). (C) The interior envelope of the family of “Wulff planes” thus formed, forms the crystal shape (in blue). Adapted from a drawing by Michael Schmid under the GFDL and Creative Commons BY 2.5.

An introduction to the theory of inorganic solid surfaces

Fig. 4

79

Two dimensional space groups (left) and Bravais lattices (right).

topmost layer is observed, corresponding to a contraction of the interlayer spacing, attributed to the reduced coordination number of surface atoms. A few surfaces however show an outward relaxation, for instance Al(111) and Pt(111).37 In the case of multicomponent solids, surface relaxation is accompanied by intralayer corrugation, buckling or rumpling, i.e. the vertical displacement of anions and cations away from in-plane positions. Because of the existence of forces of different strength acting on the positively and negatively charged species, the displacement of anion and cations generally differs. All surfaces undergo multilayer relaxations whatever their orientation and whatever the material. This mechanism is expected to decrease the thermodynamic cost related to the surface creation, but does not necessarily alone lead to the most stable surface configuration. The latter can be reached through a more drastic modification of the surface structure. Surface reconstructions involve, in most cases, a lateral displacement of atoms and/or a modification of the surface atomic density. The symmetry of the surface changes with respect to the bulk. It can be specified by a matrix notation proposed by Park and Madden.38 It consists in giving the relationship between the set of 2D basis vectors of the non-reconstructed and reconstructed surfaces, denoted as     ! ! ! a ; b and ! a S ; b S , respectively:

(111)

(100)

(110)

fcc

bcc

Fig. 5 Structures of the bulk-truncated (unreconstructed) low-index surfaces of fcc and bcc metals. The surface unit cell is shown in each case. The surface atomic density na varies in the following way for fcc and bcc metals: na fccð111Þ >na fccð100Þ >na fccð110Þ and na bccð110Þ >na bccð100Þ >na bccð111Þ .

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An introduction to the theory of inorganic solid surfaces 0

1 " u11 a S @! A ¼ bS u21

u12 u22

#

! a ! b

(5)

    ! ! Most of the time, the ! a S ; b S and ! a ; b vectors are collinear and/or can be deduced from each other by a simple rotation. It leads to the reduced notation, proposed by Wood,39 which is frequently used in the literature, but impractical for largest unit cells. pffiffiffi  pffiffiffi pffiffiffi For example, the 2 2
2 R45 reconstruction of Al2Cu(001) indicates that the lattice parameters are multiplied by 2 2 and pffiffiffi 2 along the a and b directions, and that the surface cell is rotated by 45 with respect to the underlying layers.40 Surface segregation is commonly encountered in metallurgy, but also in the field of nanoalloys. It refers to variations in the composition near the surface at thermodynamic equilibrium.41 It can have a great impact on the macroscopic properties. It is assumed to result from a competition between three main driving forces,42 involving the component surface energies, the compound enthalpy of mixing and the elastic stress induced by the component atomic sizes. More precisely, the species with the lowest surface energy tend to diffuse to the surface to minimize the total surface energy. Positive mixing enthalpies, i.e., tendency to demixion, favors the segregation. Finally, the size effect tend to trigger the larger atoms at the surface.

3.05.1.4

Summary

The equilibrium surface structures of inorganic compounds can be successfully derived from the minimization of the excess grand potential at constant chemical potentials and other thermodynamic state variables:   X with usurf ¼ f surf  mi Gi (6) geq ¼ min usurf T;mi

i

Thus, the computation of the surface energy would enable to make a selection, at least based on energetic arguments, among a priori models, inspired by human intuition, suggested by experimental observations or generated by systematic methods. In the next section, Density Functional Theory is introduced, as a successful computational quantum mechanical modeling method to investigate surface structures.

3.05.2

First principles modeling of surfaces

3.05.2.1

Many-body Schrödinger equation

A quantitative theoretical investigation of solids and their surfaces starts with the many-body Schrödinger equation43 HJ ¼ EJ

(7)

involving the Hamiltonian (H) of the system made of N electrons and M ions and described by its energy E and its wave function J. ! Using m, ! r i and e for the electron mass, positions and charge, respectively, and MI, R i and ZI for the nuclei mass, positions and atomic number, respectively, the time-independent Schrödinger equation can be expressed as H ¼ T e þ T n þ V nn þ V en þ V ee e

(8)

n

nn

ee

where T and T are the kinetic energies of the electrons and nuclei, respectively, V and V are the Coulomb repulsion between pairs of nuclei and electrons, respectively, and Ven is the Coulomb attraction between electrons and nuclei. These contributions can be written as H¼

N X

M X X e2 Z2 Z2 2 1 X e2 ZI ZJ ZI 1 X e2 1  þ  þ    V2i þ  VI þ       ! 2 2 2m 2M 4pε 4pε 4pε ! ! ! I 0  0 ! 0 r i! r j  i;I  IsJ I¼1 isj i¼1 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}  R I  R J  r i  R I  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Tn Te |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} V ee V nn

(9)

V en

where Vi 2 and VI 2 are the Laplacian operators acting on each of the i electrons and I nuclei. Owing to the mass differences between electrons and ions, the previous equation can be separated into two equations, one for the electrons and one for the nuclei. For this purpose, the total wave-function is written as a product of an electron-only wavefunction, which parametrically depends on the set of nuclear coordinates (fR), and a nuclear-only wave-function (c), i.e., J ¼ fRc with e

(10) T þ V ee þ VRen fR ¼ EeR fR ER fR c þ ½T n þ V nn fR c ¼ Etot fR c

(11)

where the subscript R indicates that it depends on all the nuclear coordinates. Solving the previous equations can lead to many equilibrium properties of solids and surfaces, the latter being related to energies and eigenstates at their ground states. However, it is

An introduction to the theory of inorganic solid surfaces

81

impossible to solve for anything but the simplest model systems. This is because of the many-body physics of the electronic structure problem.

3.05.2.2

Density Functional Theory

A large range of strategies do exist to obtain accurate results from the numerically intractable many-body Schrödinger equation. The most recent ones benefit from machine learning methods.44 Among standard methods, DFT is currently the most widely used,45 owing to its reliability and transferability. DFT typically yields very accurate results, allowing not only direct comparisons with experimental measurements but also providing reliable predictions of materials properties. This modeling technique can be applied to different classes of materials, thus providing a universal character to this approach. The introduction of standardized and possibly open-source software additionally boosted its development. A comprehensive overview of DFT is out of the scope of this chapterdinterested readers can refer to excellent books on this topic.46–48 Here, we briefly introduce the basics required to understand the potential and limitations of DFT approaches. The fundamental concept of DFT is that the energy of an electronic system can be expressed in terms of its electronic density n, i.e. a quantity which only depends on three variables. It follows the original idea of Thomas and Fermi49,50 and was laid down by the two theorems of Kohn and Sham.51,52 Within this scheme, the ground-state electronic wave function J0 is a unique functional of the ground state electron density n0. As a consequence, all observables are also functionals of the ground-state density. The groundstate density unambiguously determines the many-body Hamiltonian, from which the equilibrium properties of the system can be specified, including the ground-state electronic energy E0e for a given configuration of the ions: e ee en Ee0 ¼ Ee ½n0  ¼ min ! hJ½njT þ V þ V jJ½ni n r

(12)

A critical problem is the evaluation of the electronic functional, since there is only one term (Ven) for which the dependence on the density is explicit. The kinetic and electron-electron Coulomb repulsions only implicitly depend on the density. To overcome this fact, the idea of Kohn and Sham was to map this system of interacting electrons onto a system with non-interacting electrons in a fictitious potential that has the same total energy and the same electronic density51: Z     r þ EH ½n þ Exc ½n d! rn ! r Vn ! (13) Ee ½n ¼ Tse ½n þ where Ts e[n] is the kinetic energy functional of non-interacting electrons of density n, Ven is expressed explicitly in terms of the density and EH[n] is the Hartree energy describing the (classical) electron-electron Coulomb repulsion given by   ! Z Z n ! r n r0 1 0 H  d!  r d! r E ½n ¼ (14) ! !0  2 r  r  The term Exc contains everything that is left out and is called the exchange and correlation energy. The single-particle Schödinger equation is deduced from Eq. (16):   0            Z n ! r 1 2 dExc ½n 0 ! ef f ! i ! i i ! ef f !   d! r þ   r f0 r ¼ ε0 f0 r with V r ¼V r þ (15)  V þV 0   2 dn ! r r ! r  ! which yields the orbitals that reproduce the density of the original many-body system. N   2   X  i !  n ! r ¼ f0 r 

(16)

i¼1

Solving the previous equations is achievable through a self-consistent method. A relevant choice of the basis functions with which to expand the Kohn-Sham orbitals is of critical importance to efficientlydwith a minimal computational costdsolve the Kohn-Sham equations. For solid state calculations, planes-wave sets are currently the most popular basis functions. They are frequently combined with a projector-augmented wave (PAW) potential53 to describe the electron-ion interactions. The basis-set size is usually defined through an energy cutoff (Ecut), i.e., that all plane waves with a kinetic energy smaller than Ecut are included in the basis set. Uniform grids over the Brillouin zone are built in order to calculate the total electronic energy by numerically integrating the occupied electronic bands. Thus, accurate bulk calculations require careful studies of the energy convergence as a function of the k-point sampling and Ecut.

3.05.2.3

Impact of the functional in surface computations

According to DFT, there is a functional which gives the exact ground state energy and density. However, its expression is not known. Approximations are required to build a functional which tend to provide an excellent cost/accuracy ratio. Great efforts have been

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An introduction to the theory of inorganic solid surfaces

done in this direction these last years,54,55 to go beyond the traditional Local Density Approximation (LDA) and the Generalized Gradient Approximation (GGA). To date, more than 200 functionals have been developed for the description of exchangecorrelation interactions.55 But most studies are still performed within the GGA scheme,56 using the popular PBE and PW91 functionals.57,58 In surface computational modeling, LDA however outperforms GGA, due to a favorable cancellation of errors in the exchange-correlation contribution to the surface energy. Indeed, LDA over-estimates the exchange contribution to the surface energy, but underestimates the correlation contribution, whereas PBE underestimates both quantities.59,60 Most semi-local density functionals predict metal surfaces to be more stable than they are experimentally.61 The PBE-GGA has been shown to underestimate both their surface energies and work functions. Using a database containing aluminum and seven d-electron metals,62 the over-evaluation is about 24% and 4%, respectively. Van der Waals effects have also be pointed as unexpectedly important for the surface energy of elemental metals. Using the previous database, the integration of long range van der Waals contributions within the SCANþ rVV10 functional63,64 increases the surface energies by about 10%, and increases the work functions by about 3%.62 High-level computations, using the Random Phase Approximation scheme65–67 give better results, but at a substantial computational cost.61 The modification of the surface energies caused by the choice of the functional, does not necessarily alter the qualitative picture drawn from the comparison of several structural models for an oriented surface of a given compound. In PbS, PbSe,68 Ba8Au5.25Ge40.75,69 Al13Co4,70 the relative stabilities of model surfaces are not drastically modified when dispersion corrections are considered. This conclusion is not general. For instance, the surface phase diagram of graphene films on SiC undergo large changes when van der Waals interactions are included, due to the over-stabilization of graphite.71 Beside the qualitative ranking of model stabilities, the under-estimation of surface energies may impact the nano-crystal shapes when achieving Wulff constructions. Indeed, with the assumption that van der Waals contributions result in a given offset for all considered surface energies, the most stable surface models and orientations may be the most influenced by the choice of the functional.68 Surface energies are also frequently underestimated in the case of oxide surfaces.72 In these systems, the interactions between strongly correlated electronic states (usually d- or f-orbitals) are poorly described by LDA and GGA. Standard functionals can be extended to the DFT þ U form by adding parameterized terms, which provide corrections. In the Dudarev’s formulation, only one effective U parameter is needed, thus making this approach very attractive. The original drawback of the method, associated with no general rule for choosing the values of U, has been resolved in subsequent years, at least to a certain extend.73 Besides its affordability, the DFT þ U method suffer from local structure effects74 and cannot simultaneously correct surface energies and adsorbate binding energies.72 Hybrid functionals that incorporate a portion of exact exchange from Hartree-Fock theory, are generally more efficient but at higher computational costs. To summarize, a huge number of exchange-correlation functionals have been developed so far. Given the diversity of bonding types in inorganic compounds, resulting from ionic, covalent, metallic interactions, and where van der Waals and hydrogen bonding cannot systematically be excluded, the selection of the appropriate functional for a given compound is not trivial, and must be guided by the optimization of the compromise between the computational cost and the accuracy.

3.05.2.4

Structural models for surfaces within DFT

The numerical implementation of DFT for solids and surfaces can benefit from the periodic nature of the systems. Infinite solids can be treated by considering a three-dimensional simulation box, thanks to the Bloch’s theorem. The threedimensional periodicity is however a drawback when considering surface modeling, because of the loss of translational symmetry in the direction perpendicular to the surface. A standard approach is based on super-cell slabs, which is the structural model used in DFT simulations of surfaces. Slabs are infinite two-dimensional thin films oriented to expose the facet of interest, separated from periodic images by a vacuum thickness. In this approach, the three-dimensional periodicity of the simulation box is kept.

3.05.2.4.1

Convergence issues

When dealing with surfaces, the convergence of surface properties as a function of the slab and void thicknesses is an important point. Slabs should be thick enough to avoid interactions between opposite surfaces through the bulk, and the vacuum thickness between slabs should be large enough to avoid interactions between periodic images. Several papers report the non-convergence of surface energies as the thickness of the slab increases.75–77 Divergences arise when the well-converged bulk energy is different from the bulk energy of the surface slab, which is equal, for simple metals, to the incremental increase per layer of the slab energy (Eslab tot (N þ 1)  Eslab tot (N) where N is the slab thickness). An efficient strategy to avoid such issues is based on surface-oriented bulk simulation boxes and surface slabs constructed from them, with consistent convergence parameters for both systems (Ecut, k-point grid, etc.).78 Thus, the two systems present the same two-dimensional surface Brillouin zone, which avoids numerical inconsistencies in Brillouin zone integration between bulks and surfaces. This results in a rapid convergence of surface energy with respect to slab thickness. Depending on the targeted properties, the minimum slab thickness to consider may vary. While the surface energies of metals can converge with 10-layer thick slabs,37 electronic properties can require larger thicknesses. In case of the Si(100) surface, the dispersion features of the surface bands, such as the band shape and width, converge when the slab thickness is larger

An introduction to the theory of inorganic solid surfaces

83

than 30 layers. Complete convergence of both the surface and bulk bands in the slab is only achieved when the slab thickness is greater than 60 layer.79 With respect to metal surfaces, the convergence of properties at oxide surfaces can be extremely delicate, because of the surface polarity. To introduce this concept, let’s consider an oxide surface modeled by a stacking of infinite alternating anionic and cationic layers, along a direction perpendicular to the surface, with charge densities  s. It is equivalent to an assembly of parallel-plate capacitors in series. When the number of capacitors increases, the electrostatic potential difference between the two end sides become infinite, leading to a divergence of the surface energy. Within such a rigid charge model, polar surfaces are thus unstable.80,81 Several mechanisms can experimentally compensate the polarity at surfaces, such as structural reconstructions, changes in the surface morphology, or chemical doping.82–84 In computations, the polar instability can be treated through several approaches, such as the introduction of a planar dipole layer in the middle of the slab vacuum region85 or the addition of a compensating ramp-shaped potential in the vacuum region that cancels the artificial field together with an energy correction term.86

3.05.2.4.2

Supercell slab generation

The automatic generation of supercells containing slabs with a given surface orientation is an important step in the context of highthroughput calculations. The three Miller indices uniquely defines the ideal surface structure of simple metals resulting from bulk truncation. This is not the case for more complex inorganic materials. Because they present, in most cases, bulk structures described by a stacking of non-equivalent planes along the surface normal, several terminations are conceivable at inorganic compounds surfaces. Thus, at least, an additional parameter, besides the three Miller indices, is required to unambiguously define their topmost layer. The shift relative to a given origin frequently enables the identification of the surface termination. In more complex cases, the knowledge of the surface orientation and the shift are not enough to unambivalently describe the surface terminations. It happens when terminations are incomplete surface planes, such as topmost planes altered by (ordered) surface vacancies, or highly corrugated surface structures derived from the presence, at the surface, of intact bulk clusters. The parameters mentioned beforedMiller indices, shift relative to a given origin, slab and void thicknessdare used as input of numerous algorithms developed these last years to design slabs.78,87 The latter have therefore allowed the automated treatment of inorganic compounds surfaces. The implementation of additional conditions can help to reduce the number of models. By assuming that low-energy surfaces are planes with high atomic density, a fast screen for low-energy surface orientations has been made available.88 With the objective to avoid polar instability issues, methods to identify whether a slab with a given surface orientation would be intrinsically polar, have been recently designed.89 Within this framework, and depending on the considered crystal, a clever cleavage of the bulk can be used to build non polar slabs. The method described above enables the automatic generation of slabs guided by human intuition or experimental observations. More systematic approaches can be achieved through genetic algorithms, generating realistic systems by relying on biologically inspired operators such as mutation, crossover and selection, beside random approaches. This has already been successfully applied.90,91

3.05.2.5

Surface stability from DFT and atomistic thermodynamics

The equilibrium surface structure of a given compound results from the minimization of the surface energy as defined in Eq. (9). With the exception of elemental solid surfaces, for which the Gibbs dividing plane can be chosen such as the free energy density equals the surface energy (fsurf ¼ g), the surface energies of inorganic compounds depends on the chemical potentials of the species i, which are strongly influenced by the material growth conditions. In the following, we describe how the surface free energy can be derived. Surfaces are first considered under ultra-high vacuum without any adsorbates, and the temperature effects are discussed. In a second part, the focus is made on modifications of the surface energy induced by adsorption.

3.05.2.5.1

Surface energies of clean surfaces

According to Eqs. (5–7), the surface energy of a symmetric slab model with (hk‘) orientation and termination s is given by. # " X 1 gshk‘ ¼ (17) mi Nis Eshk‘  TSshk‘  2A i In Eq. (20), 2A is the surface area of the symmetric slab, the factor of 2 in the denominator accounting for the two identical s s and S(hk‘) are the total electronic energy and entropy of the slab with termination s, respectively. surfaces. The quantities E(hk‘) s The integers Ni are the numbers of species i in the slab, and mi are their chemical potentials. Eq. (20) can be applied to single element or multi-component systems. For simple metals, at T ¼ 0, it can be rewritten as i 1 h slab E ðN Þ  N Ebulk (18) gmetal hk‘ ¼ hk‘ 2A hk‘ slab bulk where Ehk‘ (N) is the total energy of a slab built with N atoms, and Ehk‘ ¼ Ecoh is the energy per atom of the bulk built using an ! ! oriented unit cell where the a and b lattice vectors are parallel to the (hk‘) plane. This approach has been used to build several databases for surface energies.92–94 For multi-component systems, at T ¼ 0, the surface energy is given by

84

An introduction to the theory of inorganic solid surfaces

gsðhk‘Þ

" # X 1 s s ¼  mi Ni E 2S slabðhk‘Þ i

(19)

which involves the chemical potentials since the stoichiometry of the slab generally differs from the one of the bulk. In the previous equation, the numerator can be understood as the difference between the total energy of the slab and the energy of the corresponding “bulk” with the same stoichiometry. Although the chemical potentials of species can theoretically take any positive and negative values, boundary conditions can be set. They assume that species i do not segregate at the surface. Their chemical potentials are then lower than the ones in the stable component phase. It gives mi  mi bulk, c i where mi bulk are the chemical potential of species i in the elemental i stable crystal. Combined with the Gibbs rule, the previous inequalities implies that the surface energy can be plotted in a space of p  1 dimensions, p being the number of distinct chemical species involved (p  2 for a multi-component system). An example is given for a ternary intermetallic compound in Fig. 6. The horizontal and vertical axis represents the range of variations for the chemical potentials of two speciesdthe third one is indirectly considered through the Gibbs phase rule. The color code reflects the values of the surface energies. If applicable, additional conditions can be written when p  3, to account for the absence of the formation of compounds made of p 1 species. An example is given in Ref. 95 where additional conditions have been considered to avoid the formation of the GeTe and Sb2Te3 at the surfaces of Ge1Sb2Te4(0001), Ge2Sb2Te5(0001) and Ge1Sb4Te7(0001). Surface phase diagrams have been built from the method described above, for almost all types of inorganic surfaces, like in metal carbides,97 III–V semiconductors,98,99 Al-based intermetallics,40,100–104 cage compounds,69,96,105 Tellurides,106–108 Chalcogenides,109 Heusler compounds.110,111 A wide variety of surface structures have been identified, from bulk truncated flat surfaces, to more corrugated ones, due to surface reconstructions involving surface vacancies or to the emergence of bulk clusters at the surface. Temperature effects and gas phase environments can modify the previous picture. This is detailed in the following.

3.05.2.5.2

Temperature effects

In most studies, the effect of the temperature is disregarded. Standard DFT calculations are indeed performed at T ¼ 0 K, and zero point vibrations EZP are neglected. Is it really relevant to neglect the temperature effect in the previous equations? The latter contains at least two contributions, i.e. the vibrational and configurational ones, which are not straightforward to quantitatively assess. The free energy is written as F ¼ U  TS ¼ Ee  TSe þ Evib þ EZP  TSvib  TSconf |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Fe

(20)

F vib

where a division into parts makes a clear distinction between the electronic free energy (Fe), the vibrational free energy (Fvib) and the contribution from configurational entropy (TSconf). The electronic free energy contains the total electronic energy of the system

Fig. 6 surface energy diagrams for Ba8Au5.25Ge40.75(110). Re-plotted from Ref. Anand, K.; Allio, C.; Krellner, C.; Nguyen, H. D.; Baitinger, M.; Grin, Y.; Ledieu, J.; Fournée, V.; Gaudry, E. Charge Balance Controls the (100) Surface Structure of the Ba8Au5.25Ge40.75 Clathrate. J. Phys. Chem. C 2018, 122, 2215–2220.

An introduction to the theory of inorganic solid surfaces

85

(Ee) and the electronic entropy term (TSe). Besides the vibrational energy of the ions (Evib) and the vibrational entropy term (TSvib), the vibrational free energy contains the zero point energy (EZP). The calculation of Fvib requires the calculation of the full phonon spectra:      Z N Zu Zu du (21) F vib ¼ þ g ðuÞ kB Tln 1  exp  kB T 2 0 where g(u) is the vibrational density of states. The calculation of vibrational surface properties are computationally demanding and usually requires thick slabs, as the 60-layer thick slab built to investigate the low-energy surface-phonons on a-quartz(0001).112 Overall, the vibrational contributions have a stabilizing effect, whose magnitude is of the order of a few tens of percent of the surface energyd10% (x 5 meV Å 2, T ˛ {773, 973} K) for the low-index surfaces of GaAs and InAs,113 x 30% (0.6 eV, melting point) for the low-index surfaces of iron.114 It does not drastically change the particle shapes, as in the case of GaAs and InAs, but can impact the anisotropy of the g surface energy, as shown for iron surface, where a slight decrease of the anisotropy gð001Þ is mentioned when the temperature 110

increases. The contribution arising from the configurational entropy strongly depends on the investigated system. It is generally evaluated by statistical physics.115 A basic example is the one of a weakly disordered surface, induced by the presence of a small number of surface defects. By considering nd defects arranged in Nsite surface sites, each with an area Asite, the configurational contribution to the surface energy is      TSconf kB T n n Nsite ¼ ln 1 þ d þ d ln 1 þ (22) Nsite Asite Asite Nsite Nsite nd d Using Nnsite ¼ 0:1 and T ¼ 1000 K, it is smaller than 3 meV Å 2. Non-negligible contribution to the surface energy can however

occur. At the GaSb(111) surface, the configurational contribution to the surface energy has been shown to stabilize specific surface reconstructions.116 A stabilization mechanism based on surface disorder has also been highlighted on CeO2 surfaces,117 showing the importance of configurational entropy in stabilizing polar surfaces in general. When the dimension of the configurational space becomes huge, Monte Carlo simulations have proven to be very helpful. This method has been extensively used to investigate the surface segregation in alloys,118,119 such as Pd-Ni,120 Cu-Ni,121 Co-Pt,122 using parameterized energetic models (tight-binding or embedded atom models). More recently, DFT coupled with Monte Carlo and machine learning methods, provided full explorations of surface phase diagrams and automated the discovery of realistic surfaces.123,124

3.05.2.5.3

Surface energies modified by adsorption

The focus is made here on possible modifications of the surface energy induced by adsorption. The theoretical approach suitable for such an investigation is well-known. Ab initio atomistic thermodynamics125,126 describes the thermodynamic stability of surfaces in contact with different gaseous environment, at various pressure and temperature conditions. It has been successfully applied in several metal oxides,127–133 as well as in surfaces covered by other types of adsorbatesdP,134 Cl,135–138 CO,139,140 to cite a few. A few theoretical databases gather the energies of simple adsorbates on metallic surfaces.94,141 Within this framework, the surface is assumed to be in thermodynamic equilibrium with the surrounding gas phase, which is treated as a reservoir. If one consider that the adsorbed species are different from the one involved in the slab, then the surface energy covered with adsorbates (gcover(T, P)) writes as the sum of the surface energy without adsorbates (gclean), and gads(P,T), the contribution of adsorption to the surface energy: gcover(T,P) ¼ gclean þ gads(P,T). The term gads(P,T) depends on the temperature and pressure, though " # X  1 ads PX slab nX EX  kB TlnZX þ kB Tln E E  (23) gads ðP; T Þ ¼ A kB T X In the previous equation, Eads is the total energy of the slab covered with the species, Eslab is the total energy of the clean slab, EX is the gas phase energies of the adsorbate X and nX the number of adsorbed species X. The main contributions to the partition function ZX come from adsorbates reduced translations (z*Xtrans), rotations zrotX and vibraX X , all gathered in the total partition function ZX ¼ z*XtranszrotXzvib . They can be evaluated through statistical thermodynamics, tions zvib within the ideal gas model, in the gas phase:   2pmkB T 3 2 with m the mass of X zX trans ¼ (24) 2 h 8p2 IkB T if X is linear ðI; moment of inertiaÞ h2

(25)

pffiffiffi  2  1 p 8p kB T 3 2 ðI I I Þ2 if X is not linearðI ; moment of inertia and s symmetry numberÞ a A B C h2 s

(26)

zXrot ¼ zXrot ¼

86

An introduction to the theory of inorganic solid surfaces 

2 zvib ¼

Y6 6 4

exp

hvi 2kB T

 3

7  7 5where vi are the vibrational modes i 1  exp hv kB T

(27)

All translation degrees of freedom of adsorbates are generally considered to be lost when adsorbed, and the vibrational and configurational entropies are assumed to be negligible. A few examples nevertheless show that it is not necessarily correct. Reconstructions at polar ZnO(0001) surfaces in a humid environment have indeed been found to be driven by vibrational entropy.142

3.05.2.6

Summary

First-principles simulations are extensively used for surface studies. First, they can generate large data sets and benchmarks for ideal systems, extremely useful for quantities hardly reachable through experiments, such as surface energies. Although the computational effort required is hugedeach bulk material typically has dozens of distinct low-index crystalline surfaces and terminationsdsuch approaches have been undertaken these last years, to complete databases like the Materials Project.26 The latter now contains the work functions, surface and interfacial energies of a wide range of solids.92,143,144 Second, DFT calculation can help to reveal new surface structures and properties. Early milestones in this direction include the determination of the 7
7 reconstruction of Si(111),145,146 or the explanation of anomalous thermal contraction of the Al(110) surface.147 More recent examples will be discussed in the next section. Finally, DFT simulations can be genuinely predictive. This is especially relevant to investigate the surface structures of oxides or intermetallics, since their experimental investigations are not straightforward. Besides DFT, phenomenological models are also relevant approaches to derive a good understanding of surface structures on the basis of physical and chemical laws. This is reviewed in the following.

3.05.3

Phenomenological models

As mentioned in Section 3.05.1.1.1, the surface energy can be estimated from the energy required to cut the bonds which ensure the crystal cohesion. Overall, phenomenological approaches are based on this observation, and have different expressions depending on the type of bonds in the bulk material. One of the simplest models is the broken bond model, very popular for metal surfaces. When the electron localization increases in the system, the concept of dangling bonds is generally more suitable to rationalize the surface structures. When charge transfer occurs between the atomic constituents, electron counting rules can be very relevant. The three approaches are discussed in the following.

3.05.3.1 3.05.3.1.1

Early concepts The free electron model extended to the surface

In metals, electrons are shared among all atoms. Cohesion is ensured by metallic bonds, which are largely non directional. Apart from a few exceptions (Mn, Ga, etc.), such interatomic interactions lead to elemental solids crystallizing in structures arising from optimal packing, i.e. closed packed structures. One of the first scheme to describe these systems is the free electron model.148 Within this framework, a metal is an ensemble of non-interacting electrons where ions play almost no role. The Fermi energy EFdthe energy of the highest electronic state at zero temperaturedand the Fermi wave vector kF can be expressed as a function of the elec 1 1 1 Z2 k2 tronic density n ¼ 4 pr 3 , where rs/a0 is the ratio of the Wigner-Seitz and Bohr radii: kF ¼ 3p2 n 3 ¼ 3:63
A and EF ¼ F ¼ 3:63
A . 3

rs =a0

s

2m

rs =a0

Despite its simplicity, this model successfully predicted many properties of the free electron gas, thus explaining many properties of bulk metals. How can it be extended to qualitatively describe metal surfaces? This is derived below. Let us assume a surface created by cleaving the bulk perpendicularly to one direction, for instance z. The broken solid is of infinite dimensions in the directions parallel to the surface (x, y) but of finite dimension in the z direction, perpendicular to the surface. The boundary conditions are then the periodic Born-von-Karman boundary conditions in the x and y directions supplemented by a condition for canceling the wave function in the surface planes (z ¼ 0). When the surface is created, the quantum states characterized by the wave-vector kz ¼ 0 of the three-dimensional solid, do not exist anymore. The electrons, which occupy these states, in the reciprocal space of the non-split bulk, will therefore have to be moved to unoccupied states, i.e. just above the Fermi level, in the k2

reciprocal space of the split solid. The number of electrons involved in this process, is 2pF A where A is the area of the considered surface. The excess energy, i.e. the energy difference between the initial and final situations, per surface unit, defines the surface E k2

F F energy. It can be written as a function of the Fermi energy and wave-vector: g ¼ 16p . Within this approach, the only knowledge of the electronic density, is required to evaluate the surface energy. The values of the surface energies, calculated for several metals in the framework of the free electron model (FEM), are reported in Table 1 and compared to experimental and DFT values. While the order of magnitude for surface energies, calculated within the FEM, is reasonable for noble metals, like Ag and Au, it appears that the model fails for light alkaline metals like Na and Li. Indeed, it neglects, at least, the interactions with the ions, the surface anisotropy and the electronic correlations. Density Functional Theory

An introduction to the theory of inorganic solid surfaces Table 1

87

Comparison of several experimental and theoretical surface energies (J/m2) calculated within the free electron model (FEM) and DFT (PBE functional57).

Metal

FEM

DFT 92

Exp. 18

Exp. 150

Cu(111) Ag(111) Au(111) Li(110) Na(100) K(110) Rb(110) Cs(110)

4.13 2.52 2.58 1.90 0.87 0.38 0.29 0.21

1.34 0.76 0.71 0.50 0.22 0.11 0.08 0.06

1.790 1.246 1.506 0.522 0.260 0.145 0.117 0.095

1.825 1.250 1.500 0.525 0.261 0.130 0.110 0.095

thus gives much better results, as first noticed by Lang and Kohn149 within the Local Density Approximation (LDA) and further confirmed by extensive DFT calculations.92 Within the previous scheme, the electronic density n(z) can be derived from the integration of the wavefunction module in the kk2

space. It is an oscillating function at the surface which tends to nN ¼ 3pF in the bulk. The typical thickness where the electronic density is distorted by the surface, is calculated to correspond to one or two atomic planes. An improvement of this model can be achieved by considering a finite barrier at the surface (height of W0), instead of an infinite well. This does not modify n(z), except for a shift towards the vacuum: " # cosð2kF uÞ nðzÞxnN 1  3 (28) ð2kF uÞ2 Z with u ¼ z þ pffiffiffiffiffiffiffiffiffiffi . An excess of electron charge is then observed outside the solid (z < 0) and an excess of positive charge in the bulk, 2mW 0

close to the surface (z > 0), thus giving rise to a surface dipole. Because it can interact with the electrons, the existence of the surface dipole explains why a minimum thermodynamic workdcalled the work functiondis required to get an electron out of the solid. The spreading of the electron charges, also called the Smoluchowski smoothing,151 is often invoked to explain the interlayer relaxation phenomena occurring at metal surfaces, through an electrostatic attraction of the top layer ions with the underlying crystal. This model gives insight into the magnitude of the surface relaxations, which are generally smaller at more dense metal surfaces than at open surfaces. Indeed, electronic corrugations are rather flat for closed-packed surfaces, and larger at more open surfaces.

3.05.3.1.2

The broken bond model at metal surfaces

Within the broken bond model, the work required to split a crystalline metal may be calculated by counting the bonds broken during the cleavage process (Fig. 2). The surface energy g(hk‘), can thus be written as gðhk‘Þ ¼

ðhk‘Þ

Ncut f 2Aðhk‘Þ

(29)

(hk‘) In the previous equation, only the first-neighbor interactions are considered, through the number of broken bonds Ncut and their strength f. The surface area is implemented by A(hk‘). Let’s take an example. The Pt metal crystallizes in the fcc closed-packed structure. Therefore, each atom in the bulk is surrounded coh . by 12 first neighbors. The energy of the Pt-Pt interactions (EPt-Pt) can be evaluated from the cohesive energy with EPt-Pt ¼ E12 Throughout the cleavage process, 12  Nsurf bonds per surface atom are cut, where Nsurf is the number of remaining neighbor atoms (111) ¼9 per surface atom. Thus, the low-index surface energies can be written as gðhk‘Þ ¼ 32EPt-Pt and g(100) ¼ 2EPt-Pt, given that Nsurf (100) 1 37 1 37 and Nsurf ¼ 8. According to DFT, cohesive energies for bulk Pt are 7.16 eV$at. (LDA) and 5.59 eV$at. (GGA). The resulting bond strengths are thus calculated to be 0.298 eV$at. 1 (LDA) and 0.233 eV$at. 1 (GGA). This leads to surface energies, calculated with the broken bond model, in good agreement with the ones evaluated by more costly DFT calculations (Table 2). Limitations of the broken bond model are however revealed when moving to a sp metal like Al, where energy differences with DFT can reach 30% (Table 2). Indeed, several contributions have not been considered in the previous approach, such as surface relaxations, interactions beyond the first neighbors and variations of the bond strength with the coordination number.152,153 For dense metal surfaces, the surface energy differences calculated with relaxed and non-relaxed slabs are tiny, in agreement with small surface relaxation effects, as suggested by the Smoluchowski smoothing argument.154 Surface energy predictions based on the broken bond model are therefore rather accurate in these systems, as shown for several orientations of noble metal surfaces.155 More recently, an almost-perfect linear correlation between experimental and theoretical surface energies given by the broken bond model has been derived for transition metal surfaces based on DFT calculations.156 Surface energy predictions are however hardly extendable to more open surfaces of metals, for which non-negligible deviations between DFT and the broken bond model have been revealed.157

88

An introduction to the theory of inorganic solid surfaces Table 2

surface energies of Pt(111), Pt(100) and Al(111) within the broken bond model.

surface

surf. energy (eV$at. 1)

Ref.

Pt(111)

0.90 0.70 0.91 0.71 0.65 1.19 0.93 0.90 0.51 0.45 0.39 0.33

Broken bond model (LDA) Broken bond model (GGA) LDA37 GGA37 GGA154 Broken bond model (LDA) Broken bond model (GGA) GGA154 Broken bond model (LDA) Broken bond model (GGA) LDA37 GGA37

Pt(100) Al(111)

Overall, the broken bond model is reasonably suited for closed-packed metal energy surfaces. It nicely correlates the surface atomic density with the surface stabilitydmore dense surfaces are more stabledand gives reasonable quantitative values. The consideration of interatomic interactions beyond first-neighbors may improve the model, especially for sp-bonded metals. However, a few metal surface structures cannot be predicted by the broken bond model, especially reconstructions. The latter are rather unusual at metal surfaces, except for 5d metals.158 They can occur at constant surface atomic density, like in displacive reconstruction involving intralayer lateral relaxations, lowering the surface layer symmetry. A typical example is the c(2
2) reconstruction of W(100),159 attributed to a coupling between specific surface states and surface phonon modes.160 Reconstructions connected with changes in the surface atomic density are more frequent. In most cases, the surface atomic density is reduced, like in the “hex” phases of Ir(100), Pt(100) and Au(100)161–163 or the missing row reconstructions of Ir(110), Pt(110) and Au(110)164,165 arising from a competition between surface-substrate mismatch and stress-related energy gain.166 The opposite occurs in the famous herring bone reconstruction of Au(111), which is driven by an increased packing density in the reconstructed termination167,168 related to a uni-axial contraction at the surface.

3.05.3.1.3

Can the broken bond model apply at intermetallic compound surfaces?

Intermetallic compounds are ordered solids containing two or more metallic elements at a well-defined chemical compositions.169 They include Hume-Rothery compounds, but also Laves phases, Frank-Kasper phases, Nowotny phases, Zintl phases, to cite a few. Most of them present a metallic character. However, the bonding network in these solids is very specific, caused by the unique combination of covalent and ionic interactions as well as the presence of conducting electrons. Thus, although intermetallic compounds are made from metallic elements, their properties may be quite different from those of metals. It is especially true for Al-based quasicrystalline surfaces. The latter present original properties, as compared to the pure Al ones, such as a non-wetting behavior,170,171 oxidation resistance,172 or promising catalytic properties.173–175 The understanding of these properties at an atomic level requires the design of relevant surface models. This is not straightforward, since the absence of translational symmetries in these systems prevents the use of standard DFT approaches based on periodic boundary conditions. The broken bond model can however propose reasonable insights in these systems176 Stable topmost layers are assumed to be dense planes, i.e., terminations that can minimize the number of broken bonds. In addition, bulk truncation is generally predicted to occur between planes separated by large interatomic distances, i.e., weakly bonded planes. In the case of the fivefold quasicrystalline surface of i-AlPdMn, these two rules defines two main families of possible structures.176 An additional rule, related to the chemical composition, states that termination planes rich in the element with the lowest surface energy, stabilizes the structure. In fact, Albased quasicrystals or Al-TM related approximants (TM ¼ transition metal), are generally terminated with an Al-rich plane.177 More detailed analyses based on the broken bond model are however not straightforward. Surface energies and surface atomic densities are not necessarily related to the number of broken bonds at intermetallics surfaces. This has been investigated at the Al5Co2(001) bulk-truncated surface.178 Two different terminationsdF-type and P-typedcan be obtained by a cleavage, perpendicularly to the [001] direction, in-between dense atomic layers (7 at./surf. cell). Both terminations present the same surface atomic density, but the number of broken bonds is larger at the P-type model, because of the different atomic environments around each surface atomdfive inequivalent Wyckoff positions are occupied in this compound. The number of broken bonds, i.e. missing first neighbors, is indeed 30 at the P-type termination, and 27 at the F-type one. Despite its larger number of broken bonds, the Ptype termination is more stable than the F-type one.101,179 This clearly demonstrates the non-applicability of the simple broken bond model here. The broken bond model is also not necessarily valid at alloy surfaces. This is especially true when surface reconstructions are observed. For instance, the Ag/Cu(111) system shows a n
n superstructure accommodating the size mismatch and with a strong corrugation of the Ag ad-layer.180 Reconstructions at Pt-M, Au-M and Ir-M (M ¼ Fe, Ni, Co) alloy surfaces are also expected, since they contain a metal (Pt, Au, Ir) for which the (100) and (110) surfaces reconstruct, whereas the same surfaces do not reconstruct for

An introduction to the theory of inorganic solid surfaces

89

the M-constituent. Indeed, at the (111) and (100) surfaces of Cox-Pt1-x, several bi-dimensional structures are predicted, especially pffiffiffi pffiffiffi the 3
3 R30 for the (111) and the (110) surfaces, respectively, with no equivalent ordering in bulk.181 In the two previous examples, the driving force for the observed reconstructions is related to elastic forces, induced by the different atomic radii of the multi-component system.  pffiffiffi pffiffiffi Reconstructions can also be observed at intermetallic surfaces. An interesting example is the 2 2
2 R45 reconstruction of Al2Cu(001), which is attributed to the ordering of vacancies at the surface, in the form of missing rows of surface Al atoms. The structure of this intermetallic compound, which crystallizes with the lattice parameters a ¼ b ¼ 6.04 Å and c ¼ 4.86 Å (I4/mcm space group), can be described by interpenetrating graphite-like nets of Al atoms bound by covalent-like interactions, with interlayer pffiffi  directions. Each copper atom is located in the tetragonal distances for the graphite-like planes equal to a 2 along the [110] and ½110 2

anti-prismatic cavity of this network and is bound to the aluminum environment by 8 three-center bonds.182 At the surface, the pffiffi pffiffi reconstruction arises from the ordering of vacancies in rows with specific directions and width: 3
a 2 2x13
A, 4
a 2 2x17
A and 5
pffiffi 40 a 2
 2 x21 A along the [110] and ½110 directions, in relation to the graphite-like Al nets. It is driven by the low formation energy of  directions.40 vacancy pairs separated by 2.71 Å along the [110] and ½110

3.05.3.1.4

Pauling rules extended to the surfaces of ionic compounds

In ionic solids, a few electrons are fully transferred from one atom to the other, in such a way, that, the outer electronic shells of these ions are either completely filled or empty. Cohesion is ensured by electrostatic forces termed ionic bonding. A theoretical treatment of ionic crystal structures has early been proposed by Born.183–185 Using a ionic solid containing N formula units and Z anioncation bonds per formula unit, and assuming ionic charges to be  Q and þ nQ (e.g., TiO2 with Q ¼ 2 and n ¼ 2), the total energy E of a binary compound is written in a Lennard Jones form  2  Q a ZA þ n (30) E ¼ EM þ Erep ¼  N R R In this equation, the total energy E is assumed to be the sum of the (attractive) Madelung energy EM, where a is the Madelung constant (a  Z186), and the short-range repulsion energy Erep between first neighbors, containing two parameters which are functions of the interacting ions, A and n with 5 < n < 12. The distance R is a function of the smallest anion-cation distance (R ¼ rþ þ r) or only of r, depending on the size of the cations. Thus, ionic crystal structures generally offer optimal packing of differently sized ions. The equilibrium distance R0 between two ions of opposite charge is an increasing function of the coordi  1 nation number Z : R0 f Za n1 . The reduced coordinance of surface atoms thus leads to relaxations at the surface. They are the weakest at dense surfaces which have the largest Madelung constant. The conclusion concerning the interlayer relaxations at the surface is thus similar in ionic compounds and in metals, despite the difference in the cohesive forces. In agreement with the energetic approach described above, the structures of ionic crystals have been rationalized using the Pauling’s rules.187,188 Within this scheme, cations are assumed to be surrounded by an anion polyhedron, the shortest inter-atomic distance being equal to the sum of the ionic radii. The coordination is set by the radius ratio rrþ (rule 1), while preserving the local electronegativity (rule 2). Coordination polyhedra are generally linked by vertices, all the more so if the charge of the cation is high and its coordination number is small. Indeed, the structure stability decreases when edges or faces are shared (rule 3). In addition, in a crystal with several types of cations, the polyhedra around those which have the highest charges or the lowest coordinations, tend to avoid each other (rule 4). Finally, a cation prefers to occupy the same local environment in a given crystal structure, thus limiting the number of different environments for a given cation in a given structure (rule 5). A recent statistical assessment of the performances of these five empirical rules led to the conclusion that only 13% of the oxides simultaneously satisfy the last four rules, indicating a much lower predictive power than expected.189 In a few studies, Pauling rules have been extended to the surface to rationalize the structures of several oxide terminations. For example, at quartz-type GeO2 surfaces, the surface energy, computed for several terminations, increases with the density of shared edge SiO2 tetrahedra (occurrence per area).190 In addition, many perovskite surface structuresdSrTiO3(100), SrTiO3(110), SrTiO3(111), LaAlO3(110), BaTiO3(100)dcan be understood as a result of the minimization of the type of building block units and the shared edge polyhedra.191 The broken bond model has been considered as a valid model to qualitatively rationalize the surface energies of several lowindex oxides. It is the case at the low-index surfaces of rutile192 where the relaxed surface energies were found to increase with the surface densities of unsaturated sites (considering Ti and O simultaneously). More precisely, the surface energy follows the trend of coordinationdthe lower the surface atomic coordination, the higher the surface energy. Thus, TiO2(001), where Ti and O are poorly coordinated, shows a rather high surface energy (1.30 J/m 2), whereas TiO2(110), where the atoms are more coordinated, is more stable (0.55 J/m2,193 0.57 J/m2,194 0.47 J/m2,195). It suggest that the surface atomic density can be used to identify candidate low-energy facets, a priori. Within this approach, systematic investigations of a large number of possible surface orientations88 have 2g  and f1014g  surfaces of hematite a-Fe2O3, the {311} surfaces of successfully predicted the observed low-energy, high-index f101 cuprite Cu2O, and the {112} surfaces of anatase TiO2. Unfortunately, the number of broken bonds at the oxide surfaces cannot explain the trends in surface energies for all oxides. A systematic investigation of M2O3 surfaces with M ˛ {Ti, V, Cr, Fe} shows that surface energies in transition metal oxides do not always have a direct relationship with surface coordination, and that the chemistry of the redox-active element can influence the

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relative facet stability.88 Besides the number of broken bonds, the bonding strength can have a non-negligible influence on surface energies. It is expected that the more cohesive a solid, the higher its corresponding surface energies. This trend has indeed been 2g,  f1011g  and {0001} non-relaxed surfaces of the corundum structure of Al2O3, Ga2O3 and In2O3,88 but observed for the f101 was found to be altered upon atomic relaxations. Moreover, the opposite trends have been calculated for MO2(110) surfaces.196 This conclusion has been drawn from the counter-intuitive correlation observed between the formation energy DEf of 15 different rutile MO2(110) surfaces and the bulk heat of formation of MO2 rutile systems (DGf). Indeed, more stable compounds, which are expected to have stronger bonds, show lower surface formation energies. This demonstrates the limitations of the broken bond model in the case of oxides. The broken bond model is a good starting point to investigate surface structures. Of course, its applicability is not systematic at oxide and intermetallic surfaces, where the iono-covalent character of the bonding network can modify the picture initially built for systems where electrons are delocalized. In addition, this model cannot explain surface reconstructions, especially if they imply a lower surface atomic density. To go beyond the broken bond model, we first consider systems with a covalent-like network of bonds. In this case, the concept of dangling bonds, i.e. unsatisfied valence on surface atoms, can be more suited to rationalize the surface structures.

3.05.3.2

The concept of dangling bonds

The chemical bonding pattern at the surfaces of solids can be investigated through the electron localization function. This method reveals the rearrangements in the electron distribution due to the surface formation and to the changes in the local coordination. At the surface of aluminum, the most packed surface, Al(111), can be regarded as a jellium surface, while the least packed surface, Al(110), shows a free-atom electron distribution.197 Dangling bonds are revealed at the surface of solids characterized by a higher electron localization, like in silicon.198 The minimization of their number drives the myriad of reconstructions at covalent solid surfaces. This is detailed in the following.

3.05.3.2.1

Dangling bonds at elemental semiconductor surfaces

In covalent solids, atoms form bonds with their close neighbors in order to gain enough electrons to fill their valence shells. Electrons are shared between adjacent bonding partners through the strong overlap of the wave functions on the adjacent atoms. Covalent bonds are therefore highly directional. As a result, structures of covalent solids tend to optimize the orbital hybridization. Tetrahedral atomic arrangements are therefore preferred, enabling sp3 hybridization, such as in the diamond and zinc blende structures. Examples of covalent solids include the well-known elemental C, Si, Ge, and a-Sn crystals and the III–V semiconductors (GaAs, BN, etc.). At the surface, atoms cannot fulfill their optimized environment. In covalent solids, it leads to the emergence of dangling bonds, i.e., sp3 orbitals facing an atom vacancy. Dangling bonds are obviously energetically very expensive.14,158,199,200 In most cases, surface atoms rearrange to minimize their number, by the formation of new bonds and/or re-hybridization. It generally leads to surface reconstructions. A typical example is the carbon diamond (111) surface, that can form one or three dangling bonds per surface atom depending on where the cleavage is performed. As expected, the (111) termination with three dangling bonds per surface atom is less stable than the (111) cleavage plane with one dangling bond per surface atom.201 At the (001) surface, the number of dangling bonds is two per surface atom, whatever the position of the cleavage plane (Fig. 7). This number can be decreased by a pairing mechanism, i.e. by the formation of dimers (Fig. 7), as first proposed by Schlier and Farnsworth.202 The two bridging p-orbitals between the topmost atoms form a s bond, and the two p-orbitals perpendicular to the surface form a p interaction. Each of the s and p bonding states are occupied with two electrons, thus passivating the surface. Interestingly, the CeC bond length of the dimer at the C(100)-2
1 surface is very close to the one of the C2H4 molecule. The mechanism described previously is very general, and dimers have been found to be very useful as basic building blocks to describe the surface reconstructions of numerous covalent compounds. Symmetric dimers have been observed at other (100)

1× 1

2× 1 Symmetric Dimer

2× 1 Asymmetric Dimer

Fig. 7 Structures of the (001) surface of the diamond lattice (side view). (left) Bulk-truncated structure. (middle) The (2
1) reconstruction of the C(001) surface. (right) The (2
1) reconstruction of the Si(001) surface. Structures are plotted with the VESTA software. From data offered by the Materials Project Tran, R.; Xu, Z.; Radhakrishnan, B.; Winston, D.; Sun, W.; Persson, K.; Ping-Ong, S. Surface Energies of Elemental Crystals. Scientific Data 2016, 3, 160080.

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surfaces of semiconductors, such as SiC (2
1 reconstruction203). More recently, a reconstruction involving a SneSn dimer has been observed at the (001) surface of the semiconducting half-Heusler NieTieSn alloy.204 Dimers also form at the Si(100) and Ge(100) surfaces (Fig. 7). However, in these cases, the surface dimer is distorted. Indeed, an additional energy gain is achieved through a Jahn–Teller distortion (Fig. 7). This makes the surface semiconducting rather than metallic. The reconstructions at semiconductor surfaces strongly depend on the surface preparation conditions. Therefore, several types of surface reconstructions are generally observed for a given system. To rationalize the observed surface structures, general principles have been proposed a few years ago by Duke.199 They states that 1. A surface tends to minimize the number of dangling bonds by the formation of new bonds. The remaining dangling bonds tend to be saturated (principle 1). 2. A surface tends to compensate charges (principle 2). 3. A semiconductor surface tends to be insulating or semiconducting (principle 3). Principle 1 is indeed obeyed in the previous examples through the formation of dimers. The driving force for the additional atomic rearrangement in the form of buckled dimers at Si(100) and Ge(100) surfaces, is related to Principle 3.205 The principle 2 is strongly related to the electron counting rules and applies when charge transfer occurs. It is discussed below. Before that, the concept of dangling bonds is examined at ionic and intermetallic compound surfaces.

3.05.3.2.2

Dangling bonds at intermetallic and oxide surfaces

Several oxide and intermetallic compounds contain bonds with a covalent-like character. Thus, the question arises about the existence of dangling bonds at their surfaces, and their role to drive the observed surface structures. The best example to start with is probably the one of titanium oxide surfaces, especially TiO2, where the covalent-like effect is assumed to be the strongest.206 Arguments related to the minimization of dangling bonds are then expected to be valid to explain  direcsurface reconstructions, like the one observed for TiO2(011) in the form of row-like (n
1) structures running along the ½011 tion.207 Surface energies calculations, performed on the (110), (001) and (011) low-index TiO2 surfaces, result in a single value for the surface energy per Ti dangling bond (DBTi), whatever the surface orientation (x 0.8 eV/DBTi). Since the TiO2 (011)-(2
1) and (4
1) reconstructions present a density of dangling bonds equal to one half and one quarter, respectively, with respect to the bulk terminated TiO2 (011)-(1
1) structure, they are energetically favored and are actually observed. The observation of a single value for the surface energy per dangling bond, whatever is the surface orientation, is not limited to TiO2 low-index surfaces. It has also  been revealed, for instance, at the (110), (111) and ð111Þ low-index surfaces of Ge, GaAs and ZnSe.208 Intermetallic clathrates, a class of cage compounds, also contain covalent-like bonds. Type I clathrates present a cubic crystal structure defined by the composition A8E46 where A are guest atoms (alkaline-earth or rare-earth elements) and E denote group 14 elements (Si, Ge, Sn) that form the cages hosting the A-type atoms. They show a metallic or a semiconducting character, controlled by the Zintl-Klemm rule. For instance, K8Ge46 has an excess of 8 electrons ([(4b)Ge0]46[Kþ]8
8e, where (4b) represent 4-bounded atoms), thus suggesting a metallic character. Vacancies enable the modification of the electronic structure, like in K8Ge44,2 ([(4b)Ge0]36[(3b)Ge 1]8[Kþ]8
0e, where (3b) represents 3-bounded atoms and , is the vacancy). The charge compensation in the bulk thus leads to a compound with a semiconducting character. In addition, the calculations of the electron localization function, performed on several Ge-based type I clathrates209 show that the network of bonds within the cages has a covalent character, i.e., is very similar to the one in the Ge-diamond crystals. As a consequence, the two crystals are found to share similar properties, especially the semiconducting character. The band gap is much larger in the Ge-clathrate than in the Gediamond210 and the bonding strength is evaluated to be weaker as a result of the larger atomic volumes in the Ge-clathrate than in the diamond phase. Since intermetallic clathrates show common features with elemental semiconductors, their surface structure is expected to be driven by the minimization of dangling bonds. It is indeed the case at the low-index surfaces of the Ba8Au40.67Ge5.33 clathrate69,96 ([(4b)Ge0]40.67[(4b)Au 3]5.33 [Baþ 2]8
0e) The crystal structure consists in a rigid network made of cages built with Au and Ge atoms, The guest atoms (Ba) lie at the center of the cages. The analysis of the chemical bonding identifies a covalent character for the GeeGe and GeeAu bonds within the rigid framework. It also highlights a covalent-like (dative) interaction between the Ba guest atom and the Au atoms of the cages.209 A combination of surface science studies and DFT calculations show that the rigid cages are kept intact at the (100) and (110) low-index surfaces of Ba8Au40.67Ge5.33, leading to highly corrugated surface morphologies. The ordered arrangement of Ba topmost atoms, located at the center of the cage left by the bulk truncation process, contributes to the stability. This occurs through electron charge transfer, from protruding Ba to surface Ge and Au atoms, saturating the dangling bonds.69,96 As a consequence, topmost atoms recover an electronic environment similar to that of the bulk phase, even if the weakly metallic character of the Ba8Au40.67Ge5.33 (110) surface, is not similar to the semiconducting character of the bulk.96 The concept of dangling bonds is very useful to explain the reconstructions at semiconductor surfaces, as well as at inorganic compound surfaces described by a network of bonds with a covalent character. As suggested by the previous examples, the surface charge compensation and the surface polarity are also important factors to consider to evaluate the surface stabilities. This is discussed below.

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3.05.3.3

Electron counting rules

The minimization of dangling bonds at the surface drives the reconstruction of most elemental semiconductor surfaces. The same concept can be applied to polar semiconductors, as illustrated above. However, in this case, charge transfer also plays a role, since it can contribute to saturate the existing dangling bonds. At the surface, charge compensation can be rationalized by electron counting rules. They assume that bonding and non-bonding surface states that lie below the Fermi level at the surface must be filled, whereas the non-bonding and antibonding states which lie above the Fermi energy must be empty. For semiconductors, this criterion is in agreement with the criterion of the insulating (semiconducting) behavior of the surface, i.e., Principle 3. The applicability of electron counting rules has a more general character and can be extended to ionic insulators. A few examples are given below.

3.05.3.3.1

Electron counting rules at sp semiconductor surfaces

The typical example to illustrate the electron counting rule at polar semiconductor surfaces is the one of zinc-blende AC(100) surfaces,14,211 where A and C are the most electronegative and electropositive elements, respectively. Upon cleavage, surface atoms end up with dangling bonds. Polarity implies that the dangling bonds on the cations would be empty, and would be filled on the anions. In a simple approach, only homo-dimers form at the surface, thus causing a 2
m reconstruction (Fig. 8), where the m periodicity arises from missing surface dimers, leaving D dimers per surface cell (D  N). The integers D and m are not independent but constrained by the electron rule. More precisely, since the total number of electrons on the dimerized surface should be equal to that of the bulk truncated surface with a missing anion pair, then 6D ¼ 4fAD þ 4fC (m  D). The fraction of electrons in cation and anion dangling bonds are fc ¼ NC/4 and fA ¼ NA/4 where NA and NC are the number of valence electrons of the atoms in the AC C compounds (NA þ NC ¼ 8). As a consequence, the number of dimers is D ¼ 12 NNC 1 m. The previous condition can be fulfilled with NC ¼ 2 (D ¼ m) or NC ¼ 3 (D ¼ 3 m/4). This is consistent with the 2
4 reconstructions observed on III–V (100) semiconductor surfaces such as GaAs(100),212 and with the 2
4 reconstructions observed on II-VI (100) semiconductor surfaces such as ZnSe(100).211 In contrast, missing dimer reconstructions are not observed on elemental semiconductor surfaces (NC ¼ 4), i.e., C(100), Si(100) or Ge(100) neither on I–VII ones (NC ¼ 1) like CuBr(100).213 The success of the electron counting rule in explaining a wide range reconstructions at semiconductor surfaces214 is manifest. A few exceptions have also been identified. For instance, the GaSb(100) surface reconstruction is driven by elastic deformation.215 Obviously, the factors controlling the surface structures cannot be restricted to a single rule but should include all processes occurring at the surface. The electron counting rules can also be applied in the case of adatoms deposited on a semiconductor surface. A relevant example is the one of atomic adsorbates deposited on the surface of a binary semiconductor. The adatom filled and unfilled states can contribute to the saturation of the substrate dangling bonds and should be considered in the electron counting. It applies for Cs adsorbed on the (110) surface of GaAs, a cubic zinc-blende compound.216,217 The GaAs(110)- 1
1 surface forms zig-zag chains of alternating As and Ga atoms parallel to the [001] direction, each with three nearest neighbors and one broken bond. The cation and anion dangling bonds are partially filled with 3/4 and 5/4 electrons, respectively. A deformation of the surface, consisting in a combined bond-rotation relaxation and a bond contraction, leads to a modification of the local atomic arrangement around cations and anions. Ga atoms are involved in a sp2 bonding with the three neighboring As atoms, resulting in a tendency for a local planar geometry with bond angles close to 120 , while As atoms hybridize with the three neighboring cations (p3 bonding), resulting in a local pyramidal geometry with bond angles tending to 90 . Upon adsorption of two Cs atoms, each transferring one electron to the substrate, one nearby Ga atom is lifted vertically, due to the filling of the Ga dangling bond, causing it to evolve from sp2 bonding to p3 bonding.

3.05.3.3.2

Electron counting rules involving d electrons

In the previous examples, only sp interactions are considered. Nevertheless, a large number of semiconductors involve pd interactions as well. It is typically the case of the XYZ half Heusler compounds, such as TiNiSn or TiCoSb, where X and Y are transition metals and Z is a p-element. Their crystal structure consists in three interpenetrating face-centered cubic (fcc) sub-lattices with

Fig. 8

surface cell (top view) of the 2
m reconstruction observed at AC(100) surfaces. Here, m ¼ 3 and D ¼ 2 (see text for explanations).

An introduction to the theory of inorganic solid surfaces

93

transitions metals (X, Y) and p-element (Z), lying at (¼, ¼, ¼), (½, 0, 0) and (0,0,0), respectively. It can be described as a zinc-blende XZ sub-lattice with Y at the octahedral sites or as a rocksalt YZ sub-lattice with X at the tetrahedral sites (Fig. 9A). The interatomic XZ and YZ interactions are covalent and ionic, respectively.218,219 Stable and semiconductor compounds, with filled bonding and unfilled anti-bonding states, are found among the alloys with a total valence electron count equal to 18. The surface of these compounds have attracted the interest of the research community, due to the large tunability of their properties, such as topological states,220–222 half-metallic ferromagnetism,223 and novel superconductivity.224 In (001) orientation, the structure consists of alternating atomic planes of X and YZ. Thus, a surface obtained by bulk truncation can be either X- or YZ-terminated, depending on the surface plane selection (Fig. 9B). Focusing on the CoTiSb(001) surface, it corresponds to alternating planes with charges equal to þ 1 and  1, using the formal atomic charges of the bulk compounddCo1 (d10), Sb3 (s2p6), and Ti4þ (d0). In addition, the topmost Sb atoms are left with one dangling bond each. Unreconstructed surfaces are thus not likely, since they break the general principles established so far for semiconductor surfaces. The surface can lower its energy by hybridizing half of the dangling bonds into SbeSb dimers (Fig. 9C), in the same way as on the C(100) surface. The resulting (2
1) reconstructed surface cell contains two Co atoms involved in four CoeSb bonds, one SbeSb dimer and 2nTi Ti atoms (0 < nTi < 1). Assuming that electrons from Co are shared with the two nearest TiSb layers, it leads to a total number of electrons equal to 2(Sb dimer) þ 8(CoSb back) þ 4(Sb dangling) þ 10(Co d10), i.e., 24 electrons per (2
1) surface cell. In the previous equation, Sb dangling bonds are fulfilled with 2 electrons per bond. By considering half-filled bonds, the total number of electrons per surface cell decreases to 22. On the other hand, the number of electrons available at the surface is 2
92 ðCoÞ þ 2
5ðSbÞ þ 2
4
nTi ðTiÞ, i.e., 19 þ 8nTi. It thus leads to a partial occupancy of topmost Ti atoms with nTi ¼ 3/8, which form metallic surface states within the bulk band-gap. This model is demonstrated to be quantitatively accurate, as benchmarked against experiments and DFT.110

3.05.3.3.3

Electron counting rules at oxide surfaces

The electron counting rules are relevant when differences in electronegativities between involved atoms are noticeable, i.e., when the ionic character of the compound is not negligible. At oxide surfaces, the surface polarity is well known to have a significant impact on the surface structure. The surface polarity has been classified by Tasker a few years ago225 (Fig. 10). Surfaces belonging to type 1 are described by a stacking of identical and neutral planes, they are thus non-polar. Examples include the {100} and {110} faces of rocksalt crystals (Fig. 10A), in which the outer layers contain as many anions as cations, and the {110} faces of the cubic fluorite structure. In most oxide surfaces, however, the atomic layers are not neutral. Surfaces belonging to type 2 are described by a stacking of repeat units, each of them being non polar, but made from a stacking of polar atomic layers. Overall, they do not present any polarity. For instance, the rutile {110} faces, and the oxygen-terminated {111} surfaces of fluorite structures belong to this family (Fig. 10B). Polar surfaces are therefore gathered in type 3 of the Tasker’s classification (Fig. 10C). They are described by a staking of non-neutral layers and the repeat unit bears a non-zero dipole moment. Strong polar divergence may arise when modeling these systems by DFT based on the finite slab method. Within the slab approach, the simplest model of a polar surface is a semi-infinite stacking of alternating anionic and cationic infinite layers, with inter-plane distances R1 and R2, and charge densities  s, as shown in Fig. 10A. It behaves electrostatically as an assembly of N capacitors in series, with N / N. The electric potential varies monotonically across one capacitor, with V1 ¼ sR1. It reaches the value VN ¼ NsR1 between the two sides of the slab, thus diverging when N / N. Within this framework, polar surfaces are then unstable. Compensating charges Ds, located on the two topmost surfaces, can help to stabilize the system, through the related potential, which can be written as Vcomp ¼ NDs(R1 þ R2), that compensates VN if Ds ¼ s2. Let’s take an example. In bulk SrTiO3, each strontium atom has 12 oxygen neighbors, and each titanium atom is surrounded by six oxygen atoms (Fig. 11B). The charge transfers can be written as QTibulk ¼ 4  6x, QSrbulk ¼ 2  12y, and QObulk ¼  2 þ 2x þ 4y, where x and y are the electron transfer per TieO and SreO bond. Along the [001] direction, SrO and TiO planes alternate, with charges (in the

Fig. 9 (A) Bulk structure of the CoTiSb Heusler compound. (B) Unreconstructed (001) surface. The two topmost layers are shown. (C) 2
1 reconstruction of CoTiSb(001) highlighting the SbeSb dimers and the partial occupancy of surface To atoms. Color code: blue ¼ Co, green ¼ Ti, brown ¼ Sb. Surface Ti vacancies are pointed by semitransparent green circles.

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Fig. 10 (A) Schematic polar surface built by a stacking of charged planes (surface charge density s). It behaves electrostatically as an assembly of capacitors in series (condensator thickness R1), separated by a distance R2. (B) Categorization of surfaces according to the Tasker’s approach. The surface normal (black line) is vertical. Tasker type 1 surfaces have charge neutrality based on formal charge in every layer. Tasker type 2 surfaces present with charged planes but no net dipole moment. Tasker type 3 surfaces are polar. Color code: cation ¼ blue, anion ¼ green. bulk bulk bulk) equal to QSrO ¼ 2x  8y and QTiO ¼ 2  4x þ 4y. Thus, the bulk-truncated (001) surface can be terminated by either a SrO or a TiO plane. Due to the reduced coordinance, the charge transfers are modified at the surface. More precisely, the number of oxygen neighbors is reduced to eight for Sr (SrO termination) and to five for Ti (TiO termination). Thus, the charges are surf ¼ x0  4y0 at the SrO termination QTisurf ¼ 4  5x0 and QSrsurf ¼ 2  8y0 and QOsurf ¼  2 þ x0 þ 4y0 , i.e., QSrO surf surf 0 0 0 0 QO ¼  2 þ 2x þ 2y , i.e., QTiO ¼ 2  3x þ 2y at the TiO termination. Since it corresponds to half of the charges in the surf bulk surf bulk bulk, i.e., QSrO ¼ QSrO /2 (SrO termination) and QTiO ¼ QTiO /2 (TiO termination), the surface stabilization can be fulfilled 0 0 without noticeable changes in the bond character (x ¼ x and y ¼ y). Charge compensation can involve more drastic modification in the electronic structure. For instance, in bulk MgO, the coordination of Mg and O atoms is equal to six (Fig. 11A). Thus QMg ¼ 2– 6x and QO ¼  2 þ 6x, with x the electron transfer per MgeO bond. At the (111) surface, assuming an insulating behavior, the surf threefold coordinated atoms carry charges equal to QMg ¼ 2  3x0 and QO ¼  2 þ 3
0 . The condition Ds ¼ s2 requires that x0 ¼ 1 x þ 3. This is a very large charge variation and before it is achieved, the surface bands overlap leading to metallization of the surface.226 The previous electron counting rules are the basis of the more general bond valence method, also called the bond valence sum, derived from the Pauling’s rules.227 This concept has recently been applied to rationalize a homologous series of structures on the SrTiO3(110) surface.228 Along the [110] direction SrTiO3 is described by a stacking of alternating PSrTiO and PO layers, composed of SrTiO4þ and O24 motifs, respectively (Fig. 11C). Thus, an unbalanced macroscopic dipole results from the uncompensated þ 4/ 4 charges per surface cell on alternating layers. The stabilization of the SrTiO3(110) surface is expected to occur through a nominal excess surface valence of either þ 2 or  2. Experimentally, several n
1 surface reconstructions have been observed, including 3
1, 4
1, 5
1 and 6
1 reconstructions. These reconstructions consist in the arrangement of rings of TiO4 tetrahedra, the size of the ring increasing with n. The bond-valence sums (BVS) are found very relevant to rationalize these crystal structures.   P ‘ ‘ The BVS is defined as the sum of the bond valences around the considered ion i, i.e., j exp 0 b ij where ‘ij is the interatomic

distance between the i and j atoms, ‘0 is a bond valence parameter empirically determined using experimental roomtemperature structure data and b is a parameter concerning bond softness.229 By this metric, the coordination for the stable

(A)

(B)

(C)

Fig. 11 (A) Bulk structure of MgO. Color code: blue ¼ Mg and red ¼ O. (B) Bulk structure of the SrTiO3 perovskite. Color code: oxygen (red), strontium (green), Ti (blue). (C) Bulk structure of the SrTiO3 perovskite, using a different orientation.

An introduction to the theory of inorganic solid surfaces

95

reconstructions (n ˛ {2, 3, 4, 5, 6}) is comparable to that of bulk SrTiO3, whereas it is not the case for the less stable surface structure derived from n ¼ N.

3.05.3.4

Summary

The phenomenological approaches reviewed above are strongly connected to the electronic structure of the underlying bulk. In metals, where electrons are delocalized, the minimization of the broken bonds is relevant to rationalize surface structures. In most cases, it leads to flat and dense terminations. In covalent materials, the low coordination of surface atoms leads to the formation of dangling bonds, which are reminiscent of the strong electron localization occurring in the bulk. In ionic solids, stable surfaces are those that can compensate the surface charges. Thus, electron counting rules are relevant to discriminate the surface models that are charge-neutral. Since most inorganic compounds present bonds that are not purely covalent, ionic or metallic, this gives rise to a countless variety of surface atomic arrangements, morphologies and electronic structures. Several examples illustrating original surface morphologies and surface electronic states are discussed in the next section.

3.05.4

Morphologies and electronic structures of inorganic compound surfaces

3.05.4.1

Surface morphologies

The closed-packed structure of metals leads to surface models that are typically flat planes, with atomic steps separating terraces, even if this picture can be altered by restructuring phenomena like reconstruction, segregation or roughening.161,230 Similarly, flat terminations are also widely observed at the surface of inorganic compounds. The previous picture can be altered by faceting. It corresponds to the favored formation of a larger surface area of low energy terminations over the formation of a single layer of a high-energy plane. Many high-index planes are known to facet at equilibrium,231 such as vicinal surfaces of Si(111), GaAs(100) or Pt(100). Faceting also occurs at low-index surfaces of more complex compounds, like in twofold surfaces of quasicrystals232 or related phases, like Al13Co4(010).104 Because of the large diversity of their bulk structures, a multiplicity of surface morphologies are expected at the surface of inorganic compounds. Such an assumption also relies on the observation that among the 38,000 intermetallic structures contained in the ICSD233 and Pearson databases,30 less than 1/3 are described by a hexagonal lattice, which corresponds to the closest packing of spheres on a plane.234 A more relevant description is based on a stacking of building units, using most frequently 12-vertex polyhedra (33% of the studied structures), but also multi-shell nano-clusters.235 Corrugated surface structures may then arise, in the form of highly cohesive clusters emerging from the bulk lattice. The examples detailed in the following include surfaces of elemental solids, intermetallics and oxides.

3.05.4.1.1

Elemental boron

Because of the unique bonding character of boron, ranging from covalent 2-electron 2-center (2e2c) bonds to poly-centered, metallic-like236 as well as ionic bonding,237,238 the phase diagram of elemental boron is rather complex.239 In the bulk, multicenter bonds dominates, through the formation of icosahedral shell complexes that stabilize 13 strong multi-center intraicosahedral bonds (requiring 26 electrons, Wade’s rule). Are these clusters kept intact at the surface? The answer to this question is not straightforward. Indeed, the choice of the surface orientation and termination can lead to an infinite number of surface models. Systematic computations of the surface energies have been performed for this system, selecting low-index surfaces, and using surface models built by minimizing the number of broken bonds.240 Improved methods such as the minima hopping method or ab initio evolutionary structure prediction have revealed several low energy surface reconstructions.241,242 Among them, the most stable one satisfies the electron counting rules.243 The corresponding surface morphology deviates from a flat termination. Icosahedral clusters are kept intact at the surface but the atomic density of the topmost layer is increased by the presence of additional interstitial boron atoms, which stabilizes this unexpected surface model.

3.05.4.1.2

Al-based complex intermetallics

Cluster-based structures are also rather common in quasicrystals and related phases. They can be simple cages, like the ones of the Al13Fe4 quasicrystalline approximant, or more complex multi-shell clusters, like the ones in g-Al4Cu9, made of concentric inner tetrahedral, outer tetrahedral, octahedral and cubo-octahedral polyhedra. The dense low-index surfaces of these two compounds show various structures, attributed to the different type of bonding network occurring in the bulk. The Al13Fe4 quasicrystalline approximant contains pentagonal bi-pyramid clusters. They are kept intact at the Al13Fe4(010) surface, which can be described as incomplete puckered planes made of adjacent pentagons of Al atoms, each centered by a protruding Fe atom. The surface is nanostructured (roughness x2 Å). A similar corrugated surface structure is predicted for Al13Co4(100), but not necessarily under ultra-high vacuum conditions.244 These morphologies are driven by the strong covalent like bond strengths identified in the TM-Al-TM atomic linear group,245,246 thanks to quantum chemical calculations. Resulting surfaces structures avoid the breaking of such strong bonds. This principle is rather general and can be applied to other surface orientations,104,247 or to related structures.70,101,102

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In contrast, clusters are not kept intact at the surface of g-Al4Cu9(110).248 More precisely, the surface is characterized by dense flat terminations, consisting in two possible terminations. Their surface energies are very similar (within 0.06 J/m2). The corresponding simulated STM images are in reasonable agreement with the experimental ones and mirror the experimental voltage dependence. The two terminations differ by their surface composition, and by the fact that only one of the two cuts the inner tetrahedra. Such a surface morphology leads to a rather small surface corrugation (0.6–0.7 Å), in agreement with the general trends observed for related quasicrystalline phases. In this case, the role of the three-dimensional cages on the surface morphology is negligible. The bonding picture in g-Al4Cu9 is dominated by its metallic character, in agreement with its specific composition-averaged valence satisfying the Hume-Rothery electron concentration rule.249 Thus, dense and flat surfaces are favored, like in metals.

3.05.4.1.3

Cage compounds

Cages compounds are a class of materials formed by a three-dimensional network of cages as building blocks. They can be metallic, like the Ce3Pd20Si6 compound identified as a heavy-fermion system,250 or semiconducting like the Ba8Au5.25Ge40.75 Zintl clathrate. Different mechanisms are at play to stabilize their surface structures, as detailed in the following. Bulk Ge-based clathrates contains two pentagonal dodecahedra and six tetrakaidecahedra per cell. Their surface morphologies are rather corrugated (x2 Å), and controlled by the saturation of the dangling bonds located on sp2 Ge surface atoms, thanks to a charge transfer from Ba adatoms to the surface, thus achieving surface charge compensation.69,96 The nanostructuration at the surface can be tuned through the selection of the type of intact clusters at the surface, i.e., by the surface orientation. In contrast, Ce3Pd20Si6 is metallic. Its structure consists in two interpenetrating sub-lattices: one face-centered cubic sub-lattice of Ce-filled Pd12Si6 cages and one simple cubic sub-lattice of Ce-filled Pd16 cages.251 The (100) surface forms at specific terminations of the bulk structure, identified as layers of Pd12Si6 cages which are further stabilized by additional Pd atoms, rendering the surface termination more compact.105 It is consistent with the broken bond model, and leads to a termination with a rather high surface atomic density, and a small corrugation, as in metals.

3.05.4.1.4

Oxide surfaces

Large surface corrugations can be observed at oxide surfaces as well. For instance, the corrugation at the KTaO3(001) surface is about 0.4 nm, i.e., the KTaO3 lattice constant.252 According to STM, the entire surface area is covered by alternating KO and TaO2 terraces separated by half-unit-cell steps. The KO planes are always on top, whereas TaO2 is in the valley region, suggesting that the KO plane fractures more easily during cleavage. The driving force for this non trivial surface structure is the polarity compensation. A full KO or TaO2 termination plane would indeed lead to polar instability, since the KO and TaO2 terminations have a formal charge of  1e and þ 1e per unit cell, respectively. The observed structure successfully compensates the polarity at a long-range scale. Theoretical calculations suggest that a competition between a cation-exchange mechanism253 and a striped structure occurs at the KTaO3 (001) surface,254 the surface energies between the two resulting surface structures being only slightly different (0.07 J m 2). The fact that striped phases are actually observed in experiment may be thus an indication of a high kinetic barrier for the cation-exchange reconstruction on this surface. In summary, numerous surface morphologies emerges at the surfaces of inorganic solids, driven by the high variety of bonding characteristics in these compounds. Compared to elemental surfaces, original electronic structures can also be observed, as detailed in the following.

3.05.4.2

Electronic states at the surface

At the surface of sp metals, the jellium model is very useful to describe several properties, like the work function, or the rumpling resulting from the Smoluchowski smoothing. Tight-binding arguments are more appropriate when focusing on transition metals. Within this framework, several hopping integrals are canceled at the surface, because of the lower coordination of surface atoms. This reduces the average width of the d-states at a surface atom relative to the bulk, as illustrated in Fig. 12 for the Al2Cu(001) an Al45Cr7(010) intermetallic surfaces. This bands narrowing is accompanied by a shift of the center of gravity of the bands, in order to avoid (or to at least reduce) possible unrealistic lack or surplus of electronic charge on the surface atoms depending on the filling of the band. The magnitude and direction of the shift S, with respect to the zero energy, taken at the center of the bulk d-band, can be ! qffiffiffiffiffiffiffi Csurf approximated with the rectangular d-band model S ¼ EF 1  Cbulk where Cbulk and Csurf are the coordination numbers of atoms in the bulk and at the surface, respectively. The shift S is negative for less-than-half-filled bands, positive for more-than-half-filled bands, and zero for an exactly half-filled band.158 Because S increases with the number of broken bonds at the surface, it is larger for more open surfaces. The narrowing of the d-band when atoms are isolated is supposed to lead to improved chemical properties. This is the key point in the field of Single Atom Catalysts. These materials consist of an inactive matrix with active metal atoms dispersed across the surface, such that no bonds between neighboring active sites form. Using the example of AgCu alloys that contain dilute Cu concentrations, projected density of states calculations, performed on bulk systems, show that the solute’s d-states resemble those of a free atom, whereby the d-electron orbitals are nearly degenerate.255 A series of computational experiments reveals that this phenomenon arises when solutes in dilute concentrations exhibit a weak interaction with the electronic states of the matrix element. Isolated atoms dispersed in a metallic matrix are not limited to solid solution alloys, but can occur in intermetallic compounds as well,

An introduction to the theory of inorganic solid surfaces

80

97

10

Al 2Cu(001)

Al 45Cr7(010) 8

surface d-states bulk d-states

DOS (states/eV)

60

6

40 4

20 2

0

-6

-4

-2

0

2

-6

-4

-2

0

2

0

Energy (eV) Fig. 12

Projected density of states showing the d-states contributions of surface and bulk Cu atoms.40,103

although in these cases the interaction of the isolated site may be stronger with its neighbors. An interesting example is the one of the Al13Co4 quasicrystalline approximant.245 The projected density of states of transition metal (TM) atom in the bulk is much narrower than the ones in elemental TM bulk. This is suggested to give rise to improved catalytic properties.256 Besides the narrowing of bands at the surface, more drastic differences between the bulk and the surface are likely. They are called “Surface States”. In a bulk metal, the Bloch waves describe the eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential. When the periodicity is removed in the direction perpendicular to the surface, some modifications of the electronic structure can be observed. Two cases are considered in the following. First, bulk states can terminate in an exponentially decaying tail to the vacuum, with a possible resonance at the surface. The second type of states decays exponentially both into the vacuum and the bulk crystal. They correspond to surface states with wave functions localized close to the crystal surface. They are often referred to as Shockley states and Tamm states.257,258 They are not rare at metal surfaces. They can also emerge at the surface of semiconductors, like Si(111).259 In inorganic compounds, surface states can differ from the one in elemental crystals. Indeed, they can emerge for topological reasons,260,261 and thus can be much more robust against disorder. While the energy gap in trivial insulators is not sensitive to the change in boundary conditions, the edges or surfaces of a finite sample can see topologically protected states emerge, within the bulk energy gap. The topological character makes the states very robust against disorder. Spin-orbit coupling can favor these topologically protected states. This has been shown at the surface of Bi1  xSbx. While a pure Bi crystal is topologically trivial, topologically non trivial phases can be achieved by reducing the strength of the spin orbit coupling through Sb doping.262 In these compounds, the surface states are influenced by finite size effects, as shown for Bi2Se3 and HgTe.263

3.05.5

Conclusion

The surfaces of inorganic compounds show a myriad of structures. In some cases, the surface atoms drastically rearrange, in other cases, they just slightly relax from their bulk positions. This behavior is driven not only by the intrinsic properties of the underlying crystal, but also by the experimental conditions. The determination of surface structures by theoretical approaches only is then challenging. Several phenomenological rules can guide the development of such structural models. Overall, they are related to driving forces arising from elastic, electronic, or configurational contributions. In pure metals, the elastic contributions are scarce, except for a few 5d metals. In multi-component systems, the different atomic sizes of the constituents can imply complex surface structures, especially in alloys where they also influence the surface segregation. Besides the minimization of the broken and dangling bonds, electronic effects are crucial factors to elaborate realistic surface models. They are accounted for through the concept of charge compensation, leading to electron counting rules. At least, the configurational entropy can have a great impact on surface structures, especially those containing disorder, i.e. surface vacancies, adatoms or adsorbates arranged with no long-range order.

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The original morphologies observed at inorganic compound surfaces can lead to intriguing surface electronic structures, for instance at topological insulator surfaces. The surface state topology is uniquely determined by the bulk state topology. Surface band dispersion changes as the specific surface condition is varied.264 The surface structures developed in this chapter have been revealed through “traditional methods”, i.e. in most case the combination of experimental and DFT-based methods. Due to their low computational cost and short development cycle, machine learning methods coupled with powerful data processing are expected to strongly impact the field of inorganic solid surfaces in the coming years. Unlocking the surface structure-property relationships in this class of materials, through a “universal phase diagram” seems then to be unreachable right now. But several factors are very promising for the future. First, the large number of investigations achieved so far, in most cases through a bottom-up investigations of how the arrangement of atoms and interatomic bonding in a material determine its macroscopic behavior, can be very useful as databases for further analysis. Second, the combination of statistical approaches and the increased power of computers, are expected to lead to major breakthroughs in surface science. A few examples already do exist for bulk materials. For instance, the thermodynamic scale of metastability has been quantified for 29,902 observed inorganic crystalline phases using a large-scale data-mining approach. Another example deals with the unraveling of the complete phase stability network of all inorganic materials, through a densely connected complex network of 21,000 thermodynamically stable compounds (nodes) interlinked by 41 million tie lines (edges) defining their twophase equilibria.265 With concerted progress in theoretical methods, computing power, as well as in crystal synthesis and experimental probe techniques, the design of surfaces with specific structures and properties would become a reality, propelling advances in technological applications.

3.05.6

Further reading

Aside from the many original articles, review papers, and books cited already, the interested reader may consult the following books for complementary information. Books are ranked according to the publication date.

• • • • • •

Physics at Surfaces by A. Zangwill (1988)266: This book is a graduate-level introduction to the physics of solid surfaces, divided in two parts. The first one deals with clean surfaces while the second one focuses on adsorbates. Handbook of Surface Science, Volume 2 edited by K. Horn and M. Scheffler (2000)267: It contains several detailed chapters on the surface electronic structure of metals and semiconductors. Principles of Surface Physics by F. Bechstedt (2003)14: The book is based on lectures given at the Humboldt-Universität zu Berlin and the Friedrich  Schiller  Universitt€ Jena and on student seminars. It is a comprehensive overview of surface physics with a particular emphasis on semiconductor surfaces. Physics and Chemistry at Oxide Surfaces by C. Noguera186: This book provides a comprehensive overview of surfaces in the field of oxides. Theoretical models are proposed to synthesize the experimental or numerical results obtained on given systems. Theoretical Surface Science; A Microscopic Perspective by A Gross (2009)268: a comprehensive overview, with the emphasis on fundamental concepts that govern the atomic and electronic structures as well as the processes occurring on the surfaces. surface and Interface Science edited by K. Wandelt (2012)269: This series of two books provides and a comprehensive review of surface science. An excellent introduction to the theory of elemental solids and their surfaces is proposed in the first volume (Introduction to the theory of metal surfaces158).

Acknowledgments I would like to thank all my collaborators for fruitful discussions, especially my colleagues at Institut Jean Lamour and L. Piccolo at IRCELYON. Financial support was provided by the COMETE project (COnception in silico de Matériaux pour l’EnvironnemenT et l’Énergie) co-funded by the European Union under the program FEDER-FSE Lorraine et Massif des Vosges 2014–2020. This work was granted access to the HPC resources of TGCC, CINES and IDRIS under the allocation 99642 attributed by GENCI (Grand Equipement National de Calcul Intensif). High Performance Computing resources were also partially provided by the EXPLOR center hosted by the University de Lorraine (project 2017M4XXX0108).

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Soc. 2007, 129, 10474–10478. 208. Zhang, S. B.; Wei, S.-H. surface Energy and the Common Dangling Bond Rule for Semiconductors. Phys. Rev. Lett. 2004, 92, 086102. 209. Zhang, H.; Borrmann, H.; Oeschler, N.; Candolfi, C.; Schnelle, W.; Schmidt, M.; Burkhardt, U.; Baitinger, M.; Zhao, J.; Grin, Y. Atomic Interactions in the P-Type Clathrate I Ba8Au5.3Ge40.7. Inorg. Chem. 2011, 50, 1250. 210. Herman, Z. S. The Nature of the Chemical Bond in Metals, Alloys, and Intermetallic Compounds According to Linus Pauling. Theor. Comput. Chem. 1999, 6, 701–746. 211. Pashley, M. D. Electron Counting Model and Its Application to Island Structures on Molecular-Beam Epitaxy Grown GaAs(001) and ZnSe(QQ1). Phys. Rev. B 1989, 40, 1081. 212. Xue, Q.-K.; Hashizume, T.; Sakurai, T. Scanning Tunneling Microscopy of III-V Compound Semiconductor (001) Surfaces. Prog. Surf. Sci. 1997, 56, 1. 213. Dassanayake, U. M.; Chen, W.; Kahn, A. Atomic Arrangement at the CuBr(100) Surface and CuBr/GaAs(100) Interface: Application of the electron Counting Method. J. Vac. Sci. Technol. B 1993, 11, 1467–1471. 214. Srivastava, G. The electron Counting Rule and Passivation of Compound Semiconductor Surfaces. Appl. Surf. Sci. 2006, 252, 7600–7607. 215. Whitman, L.; Thibado, P.; Erwin, S.; Bennett, B. R.; Shanabrook, B. V. Metallic III-V (001) Surfaces: Violations of the Electron Counting Model. Phys. Rev. Lett. 1997, 79, 693–696. 216. Yang, S.; Zhang, L.; Chen, H.; Wang, E.; Zhang, Z. Generic Guiding Principle for the Prediction of Metal-Induced Reconstructions of Compound Semiconductor Surfaces. Phys. Rev. B 2008, 78, 075305.

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217. Zhang, L.; Wang, E. G.; Xue, Q. K.; Zhang, S. B.; Zhang, Z. Generalized Electron Counting in Determination of Metal-Induced Reconstruction of Compound Semiconductor Surfaces. Phys. Rev. Lett. 2006, 97, 126103. 218. Gautier, R.; Zhang, X.; Hu, L.; Yu, L.; Lin, Y. Y.; Sunde, T.; Chon, D.; Poeppelmeier, K.; Zunger, A. Prediction and Accelerated Laboratory Discovery of Previously Unknown 18electron ABX Compounds. Nat. Chem. 2015, 7, 308–316. 219. Anand, S.; Xia, K.; Hegde, V. I.; Aydemir, U.; Kocevski, V.; Zhu, T.; Wolverton, C.; Snyder, G. J. A Valence Balanced Rule for Discovery of 18-electron Half-Heuslers with Defects. Energ. Environ. Sci. 2018, 11, 1480–1488. 220. Logan, J. A.; Patel, S. J.; Harrington, S. D.; Polley, C. M.; Schultz, B. D.; Balasubramanian, T.; Janotti, A.; Mikkelsen, A.; Palmstrom, C. J. Observation of a Topologically Nontrivial Surface State in Half-Heusler PtLuSb (001) Thin Films. Nat. Commun. 2016, 7, 11993. 221. Lin, H.; Wray, L. A.; Xia, Y.; Xu, S.; Jia, S.; Cava, R. J.; Bansil, A.; Hasan, M. Z. Half-Heusler Ternary Compounds as New Multifunctional Experimental Platforms for Topological Quantum Phenomena. Nat. Mater. 2010, 9, 546–549. 222. Chadov, S.; Qi, X.; Kübler, J.; Fecher, G. H.; Felser, C.; Zhang, S. C. Tunable Multifunctional Topological Insulators in Ternary Heusler Compounds. Nat. Mater. 2010, 9, 541–545. 223. de Groot, R. A.; Mueller, F. M.; van Engen, P. G.; Buschow, K. H. J. New Class of Materials: Half-Metallic Ferromagnets. Phys. Rev. Lett. 1983, 50, 2024. 224. Nakajima, Y.; Hu, R.; Kirshenbaum, K.; Hughes, A.; Syers, P.; Wang, X.; Wang, K.; Wang, R.; Saha, S. R.; Pratt, D.; Lynn, J. W.; Paglione, J. Topological RPdBi Half-Heusler Semimetals: A New Family of Noncentrosymmetric Magnetic Superconductors. Sci. Adv. 2015, 1, e1500242. 225. Tasker, P. W. The Stability of Ionic Crystal Surfaces. J. Phys. C: Solid State Phys. 1979, 12, 4977. 226. Pojani, A.; Finocchi, F.; Goniakowski, J.; Noguera, C. A Theoretical Study of the Stability and Electronic Structure of the Polar (111) Face of MgO. surf. Sci. 1997, 387, 354–370. 227. Brown, I. D. The Chemical Bond in Inorganic Chemistry, Oxford University Press: Oxford, 2002. 228. Enterkin, J. A.; Subramanian, A. K.; Russell, B. C.; Castell, M. R.; Poeppelmeier, K. R.; Marks, L. D. A Homologous Series of Structures on the Surface of SrTiO3(110). Nat. Mater. 2010, 9, 245. 229. Brese, N.; Okeeffe, M. Bond-valence parameters for solids. Acta Crystallogr. Sect. B Struct. Sci. Cryst. Eng. Mater. 1991, 47, 192–197. 230. Somorjai, G.; Hove, M. V. Adsorbate-Induced Restructuring of Surfaces. Prog. Surf. Sci. 1989, 30, 201–231. 231. Shchukin, V. A.; Bimberg, D. Spontaneous Ordering of Nanostructures on Crystal Surfaces. Rev. Mod. Phys. 1999, 71, 1125. 232. Garn, A.; Levine, D. Faceting and Roughening in Quasicrystals. Phys. Rev. Lett. 1987, 59, 1683. 233. Belsky, A.; Hellenbrandt, M.; Karen, V. L.; Luksch, P. New Developments in the Inorganic Crystal Structure Database (ICSD): Accessibility in Support of Materials Research and Design. Acta Crystallogr., Sect. B: Struct. Sci. 2002, 58, 364–369. 234. Akhmetshina, T. G.; Blatov, V. A.; Proserpio, D. M.; Shevchenko, A. P. Topology of Intermetallic Structures: From Statistics to Rational Design. Acc. Chem. Res. 2018, 51, 21–30. 235. Shevchenko, V.; Medrish, I.; Ilyushin, G.; Ilyushin, G. D.; Blatov, V. A. From Clusters to Crystals: Scale Chemistry of Intermetallics. Struct Chem. 2019, 30, 2015–2027. 236. Albert, B.; Hillebrecht, H. Boron: Elementary Challenge for Experimenters and Theoreticians. Angew. Chem. Int. Ed. 2009, 48, 8640. 237. Oganov, A. R.; Chen, J.; Gatti, C.; Ma, Y.; Ma, Y.; Glass, C. W.; Liu, Z.; Yu, T.; Kurakevych, O. O.; Solozhenko, L. V. Ionic High-Pressure Form of Elemental Boron. Nature 2009, 457, 863. 238. Mondal, S.; van Smaalen, S.; Schönleber, A.; Filinchuk, Y.; Chernyshov, D.; Simak, S. I.; Mikhay-lushkin, A. S.; Abrikosov, A. I.; Zarechnaya, E.; Dubrovinsky, L.; Dubrovinskaia, N. Electron-Deficient and Polycenter Bonds in the High-Pressure g-B28 Phase of Boron. Phys. Rev. Lett. 2011, 106, 215502. 239. Shirai, K. Phase Diagram of Boron Crystals. Jpn. J. Appl. Phys. 2017, 56, 05FA06. 240. Hayami, W.; Otani, S. surface Energy and Growth Mechanism of b-Tetragonal Boron Crystal. J. Phys. Chem. C 2007, 111, 10394–10397. 241. Amsler, M.; Botti, S.; Marques, M. A. L.; Goedecker, S. Conducting Boron Sheets Formed by the Reconstruction of the a-Boron (111) Surface. Phys. Rev. Lett. 2013, 111, 136101. 242. Zhang, H.; Shang, S.; Wang, W.; Wang, Y.; Hui, X.; Chen, L.; Liu, Z. Structure and Energetics of Ni from Ab Initio Molecular Dynamics Calculations. Comput. Mater. Sci. 2014, 89, 242–246. 243. Zhou, X.-F.; Oganov, A. R.; Shao, X.; Zhu, Q.; Wang, H.-T. Unexpected Reconstruction of the a-Boron (111) Surface. Phys. Rev. Lett. 2014, 113, 176101. 244. Gaudry, E.; Chatelier, C.; Loffreda, D.; Kandaskalov, D.; Coati, A.; Piccolo, L. Catalytic Activation of a Non-noble Intermetallic Surface through Nanostructuration under Hydrogenation Conditions Revealed by Atomistic Thermodynamics. J. Mater. Chem. A 2020, 8, 7422–7431. 245. Scheid, P.; Chatelier, C.; Ledieu, J.; Fournée, V.; Gaudry, E. Bonding Network and Stability of Clusters: The Case Study of the Al13TM4 Pseudo-10fold Surfaces. Acta Crystallogr. A 2019, 75, 314–324. 246. Jeglic, P.; Vrtnik, S.; Bobnar, M.; Klanjsek, M.; Bauer, B.; Gille, P.; Grin, Y.; Haarmann, F.; Dolin-sek, J. M-Al-M Groups Trapped in Cages of Al13M4 (M¼ co, Fe, Ni, Ru) Complex Intermetallic Phases as Seen Via NMR. Phys. Rev. B 2010, 82, 104201. 247. Gaudry, E.; Chatelier, C.; McGuirk, G.; Loli, L. S.; DeWeerd, M.-C.; Ledieu, J.; Fournée, V.; Felici, R.; Drnec, J.; Beutier, G.; de Boissieu, M. Structure of the Al13Co4(100) Surface: Combination of Surface X-Ray Diffraction and Ab Initio Calculations. Phys. Rev. B 2016, 94, 165406. 248. Gaudry, E.; Shukla, A. K.; Duguet, T.; Ledieu, J.; deWeerd, M.-C.; Dubois, J.-M.; Fournée, V. Structural investigation of the (110) surface of g-Al4Cu9. Physical Review B 2010, 82, 085411. 249. Asahi, R.; Sato, H.; Takeuchi, T.; Mizutani, U. Verification of Hume-Rothery Electron Concentration Rule in Cu5Zn8 and Al4Cu9 g Brasses by Ab Initio FLAPW Band Calculations. Phys. Rev. B 2005, 71, 165103, 1–8. 250. Takeda, N.; Kitagawa, J.; Ishikawa, M. New Heavy-electron System Ce3Pd20Si6. J. Physical Soc. Japan 1995, 64, 387–390. 251. Gribanov, A. V.; Seropegin, Y.; Bodak, O. Crystal Structure of the Compounds Ce3Pd20Ge6 and Ce3Pd20Si6. J. Alloys Compd. 1994, 204, 9–11. 252. Setvin, M.; Reticcioli, M.; Poelzleitner, F.; Hulva, J.; Schmid, M.; Boatner, L. A.; Franchini, C.; Diebold, U. Polarity Compensation Mechanisms on the Perovskite Surface KTaO3(001). Science 2018, 359, 572–575. 253. Deacon-Smith, D. E. E.; Scanlon, D. O.; Catlow, C. R. A.; Sokol, A. A. A.; Woodley, S. M. Interlayer Cation Exchange Stabilizes Polar Perovskite Surfaces. Adv. Mater. 2014, 26, 7252–7256. 254. Zhao, Z.-J.; Liu, S.; Zha, S.; Cheng, D.; Studt, F.; Henkelman, G.; Gong, J. Theory-Guided Design of Catalytic Materials Using Scaling Relationships and Reactivity Descriptors. Nat. Rev. Mater. 2019, 4, 792–804. 255. Greiner, M.; Jones, T.; Beeg, S.; Zwiener, L.; Scherzer, M.; Girgsdies, F.; Piccinin, S.; Armbrüster, M.; Knop-Gericke, A.; Schlögl, R. Free-Atom-like D-States in Single-Atom Alloy Catalysts. Nat. Chem. 2018, 10, 1008–1015. 256. Piccolo, L.; Chatelier, C.; de Weerd, M.-C.; Morfin, F.; Ledieu, J.; Fournée, V.; Gille, P.; Gaudry, E. Catalytic Properties of Al13TM4 Complex Intermetallics: Influence of the Transition Metal and the Surface Orientation on Butadiene Hydrogenation. Sci. Tech. Adv. Mater. 2019, 20, 557–567. 257. Shockley, W. On the Surface States Associated with a Periodic Potential. Phys. Ther. Rev. 1939, 56, 317. 258. Tamm, I. On the Possible Bound States of Electrons on a Crystal Surface. Phys. Z. Sowjetunion 1932, 1, 733. 259. Lapena, L.; Müller, P.; Quentel, G.; Guesmi, H.; Tréglia, G. Kinetics Study of Antimony Adsorption on Si(111). Appl. Surf. Sci. 2003, 212-213, 715–723. 260. Bradlyn, B.; Elcoro, L.; Cano, J.; Vergniory, M. G.; Wang, Z.; Felser, C.; Aroyo, M. I.; Bernevig, B. A. Topological Quantum Chemistry. Nature 2017, 547, 298–305. 261. Vergniory, M. G.; Elcoro, L.; Felser, C.; Regnault, N.; Bernevig, B. A.; Wang, Z. A Complete Catalogue of High-Quality Topological Materials. Nature 2019, 566, 480–485. 262. Zhang, H.-J.; Liu, C.-X.; Qi, X.-L.; Deng, X.-Y.; Dai, X.; Zhang, S.-C.; Fang, Z. Electronic Structures and Surface States of the Topological Insulator Bi1–xSbx. Phys. Rev. B 2009, 80, 085307.

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An introduction to the theory of inorganic solid surfaces Linder, J.; Yokoyama, T.; Sudbo, A. Anomalous Finite Size Effects on Surface States in the Topological Insulator Bi2Se3. Phys. Rev. B 2009, 80, 205401. Xiao, J.; Yan, B. First-Principles Calculations for Topological Quantum Materials. Nat. Rev. Phys. 2021, 3, 283–297. Hegde, I. V.; Aykol, M.; Krklin, S.; Wolverton, C. The Phase Stability Network of All Inorganic Materials. Sci. Adv. 2020, 6, eaay5606. Zangwill, A. Physics at Surfaces, Cambridge University Press: Cambridge, 1988. Horn, K. M. S. Electronic Structure. In Handbook of Surface Science, 2nd ed.;; vol. 2; Springer: North Holland, 2000. Gross, A. Theoretical Surface Science, A Microscopic Perspective, Springer, 2009. Wandelt, K., Ed.; surface and Interface Science, Wiley, 2012.

Relevant websites A few theoretical databases for bulk solids have been cited in this chapter: https://materialsproject.org/dThe Materials Project.25,26 http://oqmd.org/dThe Open Quantum Materials Database.27 http://www.aflow.org/dAutomatic Flow for Materials Discovery.28 https://www.topologicalquantumchemistry.com/#/dTopological Materials Database.260,261 https://cmr.fysik.dtu.dk/dThe Computational Materials Repository29 Large theoretical databases do exist for surfaces as well. They are listed below: http://crystalium.materialsvirtuallab.org/dSurfaces and Wulff shapes of elemental solids.92 https://cmrdb.fysik.dtu.dk/surfaces/dSurface energies.94 https://cmrdb.fysik.dtu.dk/adsorption/dAdsorption energies94 https://www.catalysis-hub.org/energiesdElectronic structure database for surface reactions.141

3.06

Bond activation and formation on inorganic surfaces

Yuta Tsuji, Faculty of Engineering Sciences, Kyushu University, Kasuga, Fukuoka, Japan © 2023 Elsevier Ltd. All rights reserved.

3.06.1 3.06.2 3.06.2.1 3.06.2.2 3.06.3 3.06.4 3.06.4.1 3.06.4.2 3.06.4.3 3.06.5 3.06.5.1 3.06.5.2 3.06.5.3 3.06.6 References

Introduction Bond activation on metal surfaces Dissociative activation of molecules Non-dissociative activation of molecules Bond formation on metal surfaces CeH bond activation on late transition metal oxide surfaces Overview of the CeH activation of methane on the IrO2 surface Orbital interaction between methane and the IrO2 surface Bond dissociation viewed as a bond formation NeH bond formation on early transition metal hydride surfaces Overview of ammonia synthesis Ammonia synthesis on ionic compound surfaces Electronic origin of catalytic activity of hydride catalysts for ammonia synthesis Conclusion

105 106 107 109 114 119 120 122 128 129 129 130 131 136 137

Abstract The activation and formation of bonds on inorganic surfaces are important elementary processes in heterogeneous catalysis. The purpose of this chapter is to provide one way to clarify these processes in terms of the electronic structure of solid surfaces. In particular, the reader will learn how to rationalize the origin of the activation energy for bond breaking and formation on surfaces, focusing on the orbital interactions between the surface and the bond. There are very useful theoretical chemistry tools such as crystal orbital overlap population and crystal orbital Hamilton population that can be used to effectively retrieve information on bonds that are obscured in a bunch of surface bands. Local information on bonding on the surface is obtained. Information on the occupancy of electrons in the bonding and antibonding orbitals of the bond will prove to be very useful. Using such tools, one prescription is offered for the question of how to understand why the activation energy of bond breaking and formation on inorganic surfaces is high or low. What is covered in this chapter begins with simple metal surfaces and ends with more complex oxide and hydride surfaces. The interaction of various inorganic surfaces with bonds will be overviewed using examples from important topics in catalytic chemistry such as CO activation, FischerTropsch synthesis, methane activation, and ammonia synthesis.

3.06.1

Introduction

A catalyst is generally defined as a substance that speeds up the reaction rate of a particular chemical reaction and itself does not change before and after the reaction.1 Today, many types of catalysts have been developed according to the type of reaction targeted. Catalysts are frequently used in the modern chemical industry. In fact, it would not be an exaggeration to say that chemical industries without catalysts are rare. There are many examples of how the discovery of a new catalyst has revolutionized the chemical industry. Many readers may be familiar with the establishment of a method for producing ammonia by reacting hydrogen and nitrogen at high temperature and high pressure over an iron-based catalyst (Haber-Bosch process),2 and the development of a method for producing high-density polyethylene at low pressure using a catalyst system based on titanium halides and organoaluminum (Ziegler-Natta catalyst).3 Recently, catalysts have been used not only in chemical plants but also in various consumer products and air pollution control devices.4,5 Thus, the scope of application of catalysts is expanding. There are a wide variety of catalytic reaction phases, catalyst materials, and catalyst morphology. By and large, catalysts are classified into two. When the catalyst is in the same phase as the reactants, for example, when both catalyst and reactants are dissolved in the same solvent, the catalyst is called a homogeneous catalyst.6 In the case of a reaction where the catalyst and reactant are in different phases, for example, where the catalyst is solid and the reactant is gas or liquid, the catalyst is called a heterogeneous catalyst.7 A typical example of the homogeneous catalyst is a molecular catalyst such as a metal complex, whereas that of the heterogeneous catalyst is a solid surface. When developing homogeneous catalysts, it is possible to design catalysts at the molecular level and high selectivity can be achieved.8 However, since the reaction temperature is limited by the boiling point of the solvent, the catalytic activity is not set to be high.9 On the other hand, as for heterogeneous catalysts, the catalyst surface is not necessarily homogeneous, and it may

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be difficult to design and control the catalyst at the atomic level; there may also be a chance that the surface is poisoned.10,11 However, solid catalysts are robust, capable of catalytic reactions at high temperatures, and have high catalytic activity.12 In addition, separation of products and recovery of catalysts are easy, making them suitable for large-scale chemical processes.13 Compared to homogeneous catalysts, there is still much we do not know about heterogeneous catalysts. One of the reasons for this is that it is very difficult to identify the reaction intermediates in heterogeneous catalytic reactions.14 There is a need for a better understanding of heterogeneous catalytic reactions. In this chapter, we will focus on heterogeneous catalysts and discuss chemical processes on solid surfaces. Catalytic reactions on solid surfaces involve several important steps.15 In the first half of the reaction, the adsorption of the reactant onto the catalytically active site and the subsequent activation of the bond will be important. In the second half of the reaction, the formation of the product by bond formation and its desorption from the surface should not be overlooked. Surface diffusion of the adsorbate is also included during these stages. Unfortunately, it is not possible to describe all these processes in detail one by one in this chapter. This is not only due to the limitation of space, but also due to the limitation of our knowledge. Since we believe that bond activation and bond formation are key steps in catalytic reactions on solid surfaces, we would like to consider these two processes. The technology to analyze solid catalysts has advanced dramatically. For example, scanning tunneling microscopy and electron microscopy have made it possible to observe catalyst surfaces at the atomic or molecular level.16–18 Photoemission spectroscopy uses the photoelectric effect due to ionizing radiation, providing information on the chemical composition of the catalyst surface and the bonding nature of adsorbed materials.19,20 Low-energy electron diffraction can be used to determine the atomic arrangement near the catalyst surface.21,22 Recently, these surface science experiments have been combined with first-principles calculations, and a comparative study of them has enabled us to better understand the chemical properties of surfaces, including catalysis.23,24 First-principles calculations can be used to determine the energies of intermediates and transition states in catalytic reactions, and by comparing these with experimentally obtained information, it is possible to infer the mechanism of catalytic reactions.25,26 In this process, it is also possible to obtain information on the electronic state of the surface, which can be used to obtain an essential understanding of the activation energy of catalysis. A detailed analysis of such energies does not necessarily have to be deduced from the first principles. It is also possible to apply the tight-binding approximation to the surface structure optimized with the firstprinciples calculation to approach the essence of the energetics from the viewpoint of orbital interaction.27 Nowadays, the performance of computers has improved dramatically, and even calculations with a relatively high computational load, such as surface reactions, can be completed in an acceptable amount of time. Not only high-throughput experiments, but also high-throughput calculations can lead to new catalyst candidates.28,29 It is a joy to find a good catalyst. But it would be even more gratifying if we could understand why the catalyst is good. It may be too early to ask high-throughput computation and artificial intelligence to do that. It is still a matter to be clarified by humanity’s steady theoretical research. It is important to gain an understanding of the elementary processes in catalytic reactions on solid surfaces. It is also important to know how to obtain such an understanding. To do so, we must first define what we mean by understanding. If we can trace the problem back to the molecular orbitals (MOs) of the fragments that make up the system, as Roald Hoffmann did in his book on solids and surfaces,30 then we may as well say that we understand it. In this chapter, the orbital aspects of bond activation and formation on solid surfaces are discussed with some examples. Here, the choice of examples is very arbitrary, but we have chosen them so that we can maximize both what we can tell and what the reader can get out of it. We begin with an overview of the orbital theoretical framework for understanding bond activation and bond formation, using metal catalysts as an example. Based on the concepts presented, we then discuss the issues of CeH bond activation and NeH bond formation. The former is an important step in methane conversion,31 and the latter in nitrogen fixation.32 These processes on metal surfaces do not need to be described in detail by us, for much has been written about them by many authors.33–36 Therefore, we will focus here on the processes on inorganic compounds such as metal oxides and metal hydrides, rather than on metal surfaces. Such inorganic compounds are generally used to support metal nanoparticles or metal clusters that serve as catalysts, rather than functioning as catalysts themselves. Catalyst supports are used to disperse metallic catalysts on the surface.37,38 In addition, for the cases of some types of supports, the interaction between the metal and the support can change the electronic state of the catalyst so that the catalytic activity can be enhanced.39 Alternatively, the interface between the metal and the support may become the catalytic active site.40 Usually, such inorganic compounds play a supporting role in catalysis. That is why it would be interesting to see examples of them acting as catalysts themselves, and there should be much we can learn from them.

3.06.2

Bond activation on metal surfaces

The activation of molecules on metal surfaces can be divided into two categories: one is activation due to dissociative adsorption of a molecule, while the other is activation due to chemisorption (coordination) of a molecule. Activation by coordination and that by dissociation may be inseparable, as molecules are usually activated by coordination followed by bond dissociation. However, for the sake of convenience, we will distinguish between them in this discussion.

Bond activation and formation on inorganic surfaces 3.06.2.1

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Dissociative activation of molecules

Many transition metal surfaces have the ability to dissociate molecular hydrogen.33,41 Group 10 elements, namely, Ni, Pd, and Pt, are typical metals that can dissociate molecular hydrogen at room temperature.42,43 The driving force behind this dissociation process may be the chemical affinity of the metal surface for hydrogen atoms. In order to understand the degree of affinity of various metal elements for hydrogen atoms, it is a good idea to look at the binding energy of a hydrogen atom to various metal surfaces. Miyazaki used the Pauling-Eley equation to evaluate the binding energy of a hydrogen atom to the metal atom of various metal surfaces based on the energy of the metal-metal bond, the bond dissociation energy of molecular hydrogen, and the electronegativity of the metal and hydrogen atoms.44 He summarized the results in a table. Here is a graph created based on those figures, shown in Fig. 1A. Similarly, the binding energies of O and N atoms to the metal surface have been calculated, and they are shown in Fig. 1B and C, respectively. The bond order between the metal atom and the H atom is assumed to be 1, whereas that is 2 and 3 for the O and N atoms, respectively. When interpreting these figures, it should be noted that the vertical axis scales are different in the three panels. Nevertheless, it is possible to find features common to all three. As one moves from left to right on the periodic table, the bonding energy basically decreases: the group 10 metals have the lowest binding energies for H, O, and N atoms in each period. The trend that can be read from these figures is seemingly contradictory to what was explained in one previous paragraph. Note that the bond dissociation energies of H2, O2, and N2 are 103, 118, and 226 kcal/mol, respectively.44 They are dissociated on the surface to produce two H atoms, two O atoms, and two N atoms. If the strength of the interaction between the metal and H atom is 51.5 kcal/mol or more, that between the metal and O atom is 59 kcal/mol or more, and that between the metal and N atom is 113 kcal/mol or more, the state after the bond dissociation is expected to be more stable. The strength of the NieH, PdeH, and PteH bonds falls within the range of about 60–65 kcal/mol. This satisfies the requirement that the affinity for hydrogen must be greater than 51.5 kcal/mol, as discussed above. The affinity of these metals for hydrogen is on the weaker side among the transition metals, but is sufficient to cause HeH bond dissociation. Since the affinity of early transition metals for H, O, and N atoms is very high, bond dissociation on these surfaces is expected to occur very easily. On the surface of such transition metals, the main group elements would be very strongly bound to the surface. Even if a molecule came near these atoms, they would not be able to react with it. It can be said that H, O, and N atoms on the surface of the late transition metals are more activated than those on the surface of the early transition metals. What is the origin of the very strong bond formation on early transition metal surfaces? Or, what is the origin of the moderately strong bond formation on late transition metal surfaces? Answering these questions is very important in understanding the differences in catalytic activity of these surfaces. There are two very useful models for achieving this goal: one proposed by Hoffmann and co-workers,45 and the other by Nørskov and co-workers.46 The former model is based on the electron transfer between a surface and an adsorbed atom, and is suitable for understanding the formation of ionic bonding. On the other hand, the latter model is based on orbital interactions between the surface and the adsorbed atom, and is suitable for understanding the formation of covalent bonding. In particular, the latter is well known as the d-band center theory. Fig. 2 shows a conceptual diagram of each model. These figures are taken from the supplementary material of our previous publication.47 The adsorbed atomic species, such as H, O, and N atoms, which are produced by the dissociative adsorption of molecules on transition metal surfaces, are themselves radicals. The H atom has one unpaired electron, the O atom has two unpaired electrons, and the N atom has three unpaired electrons. This is the valence bond theory view. From the MO theory point of view, those atoms can be said to have singly occupied molecular orbitals (SOMOs). Since these are atoms, not molecules, to be precise, they might as well be called singly occupied atomic orbitals (SOAOs). However, in general, when considering the

Fig. 1 Bonding energies of (A) H, (B) O, and (C) N atoms to the metal surfaces plotted as a function of the group of the metal. Data were taken from the literature.44 Note that the scale of the vertical axis is different among the three panels.

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Fig. 2 (A) Schematic representation of the interaction of the band of an early transition metal surface with the SOMO of an adsorbate upon the formation of an ionic bond between them. (B) Schematic representation of covalent bonding formation between the s and d bands of a late transition metal surface and the SOMO of an adsorbate. Reproduced from Ref. Hori, M.; Tsuji, Y.; Yoshizawa, K. Phys. Chem. Chem. Phys. 2021, 23, 14004– 14015 with permission from the Royal Society of Chemistry.

interaction between a surface and an adsorbed species, the adsorbed species is not necessarily an atom. It is quite possible to consider fragments of molecules. Therefore, we will use the term “SOMO” here. The SOMO might as well be called the Highest Occupied Molecular Orbital (HOMO). This is because the levels with higher energy than the SOMO of the adsorbate are empty, while the levels with lower energy are occupied. We can think of the number of SOMOs as being the same as the number of unpaired electrons. For simplicity, in the case of Hatom adsorption, the orbital that interacts with the surface is the SOMO (SOAO) of the H atom, which is nothing but the 1 s orbital of the H atom. That is why the SOMO level of the H atom interacting with the surface band is drawn in Fig. 2. The shaded area in Fig. 2A represents the energy band of the metal occupied by electrons. Usually, the Fermi level (EF) of a metal is energetically higher than the SOMO (HOMO) of the adsorbate. This is evident by comparing the work function of the metal with the ionization energy of the adsorbed molecules and atoms. The work functions of various metals fall within the range of about 3– 6 eV.48–50 The ionization energies of typical elements, on the other hand, fall within the range of about 6 to 20 eV.51 If the vacuum level is taken as the energy reference, the work function is the energy difference between the vacuum level and the Fermi level, while the ionization energy is the energy difference between the vacuum level and the HOMO level.52 Therefore, the work function and ionization energy are useful indices to estimate how energetically deep the Fermi and HOMO levels are from the vacuum level, respectively. Let us see how the work function of a metal and the Fermi level determined from it change when the type of metal changes. The work function data shown below was obtained from the Materials Project database.53 This database contains the work functions calculated for each metal facet existing in the Wulff shape. Here, we present data for the weighted work function, which is a work function obtained by weighting the work function for unique facets in the Wulff shape by the area of each facet. Fig. 3A shows how the work function of each transition metal changes as one moves from left to right in the periodic table. Fig. 3B shows a graph

Fig. 3 (A) Weighted work functions of transition metals were taken from the Materials Project database53 and plotted as a function of the group of the metal. (B) The graph of the work function shown in (A) is converted to the graph of the Fermi level.

Bond activation and formation on inorganic surfaces

109

in which the change in work function is converted to a change in Fermi level. The general trend that can be read from this is that the Fermi level goes down as one moves from left to right in the periodic table. Due to the relationship between the Fermi level of the metal and the SOMO level of the adsorbate, as shown in Fig. 2A, electron transfer at the adsorption interface usually occurs in the direction from the metal to the adsorbate. The adsorbate becomes anionic while the metal atoms on the surface become cationic. Since the Fermi level is very high in early transition metals, the energy of stabilization of electrons transferred from the surface to the adsorbate is very high when the surface interacts with main group elements (see Fig. 2A). The larger the energy difference between the Fermi level and the SOMO, the greater the stabilization energy of the electrons associated with charge transfer. This is a reason why the early transition metals form very strong bonds with the main group elements, as seen in Fig. 1. In the late transition metals, the Fermi level is lower so it gets closer to the SOMO level of the adsorbate. To understand what happens in this situation, it is more useful to look at the phenomenon from the perspective of covalent bond formation based on orbital interactions, as shown in Fig. 2B, rather than from the perspective of ionic bond formation based on electron transfer, as shown in Fig. 2A. In Fig. 2B, the bonding and antibonding orbitals (states) formed as a result of the interaction of the SOMO with the d band after its coupling with the s band are shown. It can be seen that the electron occupancy of the d band affects that of the antibonding orbitals formed between the metal and the adsorbed species. As one moves from left to right in the periodic table, the number of d electrons increases, so the occupancy of the d band increases. Correspondingly, the number of electrons in the antibonding orbitals (states) produced by the interaction of the SOMO of the adsorbate with the d band increases. This means that the bond between the surface and the adsorbate weakens. This is the reason why the main group elements form weak bonds with the surface of late transition metals, as shown in Fig. 1. Such a weak bond is just right for the catalysis. Now that we have a good understanding of the interactions of main group elements such as hydrogen, oxygen, and nitrogen with transition metal surfaces, we are ready to look at some typical catalytic reactions on those surfaces. The hydrogen atoms produced by the dissociative adsorption of a hydrogen molecule on the late transition metals are more reactive than the hydrogen molecule itself. This can be viewed as the state in which molecular hydrogen is activated. When an olefin is adsorbed near a hydrogen atom adsorbed on the metals, the hydrogen atom readily adds to the olefin to form an alkyl group. If another hydrogen atom is added, an alkane is formed. In this way, the hydrogenation of olefins proceeds.54,55 When such hydrogen atoms are added to carbon monoxide adsorbed on the surface, formyl and formaldehyde are formed.56 Oxygen molecules can be dissociated on the surface of Pt.57,58 The resulting oxygen atoms on the surface are highly reactive. They can oxidize hydrocarbons and alcohols,59–61 seen as an activated state of oxygen. As we saw above, changing the type of catalyst metal changes the strength of the interaction between the surface and the oxygen atom. Therefore, it may be possible to modulate the reactivity of the oxygen atoms by selecting an appropriate metal. This would allow more flexibility in catalyst design, such as letting the catalyst perform selective oxidation or complete oxidation. Nitrogen molecules can be dissociated on the surface of such metals as Fe, Ru, and Os with a relatively low activation barrier.62 When hydrogen atoms approach the nitrogen atoms produced as a result of the dissociation, NH, then NH2, and finally NH3 are formed. This is the essence of the Haber-Bosch process.63 The details of this reaction have been investigated experimentally using surface observation techniques by Ertl and co-workers.64,65 Nørskov and co-workers investigated it theoretically using ab initio calculations.66 It should be noted, however, that metal type alone does not tell us everything about catalytic activity. Somorjai and co-workers studied the rate of ammonia synthesis on Fe crystal surfaces with different surface facets. As a result, it became clear that even if the same metal, Fe, is used, different surface structures of the catalyst can cause very large differences in catalytic activity: the reaction rate on the most active (111) surface is more than 100 times faster than that on the least active (110) surface.67

3.06.2.2

Non-dissociative activation of molecules

In the previous section, we have seen the activated states of atomic H, O, and N produced by the dissociation of molecules adsorbed on the transition metal surfaces. As a preliminary step to dissociative adsorption, adsorption in the molecular state can be observed. Adsorption can be broadly classified into physisorption and chemisorption. In physisorption, the attractive interaction between the molecule and the solid is due to dispersion forces.68 On the other hand, in chemisorption, it is due to electronic interaction.69,70 Therefore, in chemisorption, the molecule can be said to “react” with the solid surface. Compared to physisorption, the structure of the adsorbed molecule can change significantly in chemisorption. This is a sign that the molecule has been activated on the solid surface. In this section, we will specifically look at what kind of electronic interaction between the adsorbed molecule and the surface leads to the activation of that molecule. We have seen in the previous section the interaction of atoms with transition metal surfaces. In the following, we will look at the interaction of molecules with the transition metal surface. In the previous section, we saw that the affinity of a metal for main group element atoms changes as the type of metal changes. A similar trend is known for the chemisorption of molecules. Trapnell investigated the chemisorption of molecules on clean metal surfaces generated on evaporated metal films, summarizing the trends in chemisorption of various molecules in a table.71 Table 1 reproduces it. The metals investigated were classified into six categories, A-F, in order of decreasing activity. This table has since been augmented with data by other researchers. Depending on the source, group A may include Ru and Os.72 N2 is chemisorbed only on the metals of group A. On the other hand, O2 and C2H2 are chemisorbed on most of the metals. This implies that the N2 molecule is very inert. It can be seen that only a limited number of metals can activate the N2 molecule. Most of

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Bond activation and formation on inorganic surfaces Table 1

Activity of metals in chemisorption. Gasses

Group

Metals

N2

H2

CO

C2H4

C2H2

O2

A

W, Ta, Mo, Ti, Zr, Fe, Ca, Ba

þ

þ

þ

þ

þ

þ

B

Ni, Pd, Rh, Pt



þ

þ

þ

þ

þ

C

Cu, Al





þ

þ

þ

þ þ

D

K









þ

E

Zn, Cd, In, Sn, Pb, Ag











þ

F

Au





þ

þ

þ



þ indicates that the gas is chemisorbed;  indicates that the gas is not chemisorbed.71

the metals in group A are early transition metals. Therefore, the very high adsorption capacity of the metals belonging to group A seems to be related to the strength of the bond between the metal and the main group element seen in Fig. 1. Table 1 is useful to know which metals are more active and which are less active. At the same time, the table also tells us which gas molecules are more active and which are less active. Among these gas molecules, CO is one of the molecules whose adsorption on metal surfaces has long been investigated. In 1954, Eischens, Pliskin, and Francis reported the first observation of chemisorbed CO on the surface of transition metals by infrared spectroscopy.73 Subsequently, the adsorption states of various adsorbates have been investigated using this technique.74 The infrared spectra provide information on the adsorption sites (adsorption geometry) and the extent to which the CO bond is weakened by adsorption. With the advancement of techniques such as high resolution electron energy loss spectroscopy (HREELS), low-energy electron diffraction (LEED), and thermal desorption spectroscopy (TDS), more detailed information on CO adsorption has become available.75,76 The mixture of CO and H2 is called syngas; it is one of the basic feedstocks in C1 chemistry.77 For example, in the Fischer-Tropsch process, liquid hydrocarbons are synthesized from syngas using a catalytic reaction. This is also one of the reasons why the study of CO adsorption on catalyst surfaces is of interest. Therefore, in this section, we will take the CO molecule as an example and review the basics of its activation on the catalyst surface. In 1985, Sung and Hoffmann published a very educational paper on the adsorption of CO on metal surfaces.78 The essence of that work was further discussed in Hoffmann’s series of works on solids and surfaces.30,79 The interaction between the metal surface and CO is understood in terms of MO theory using the so-called Blyholder model.80 The validity of this model has been supported by extended Hückel calculations by Hoffmann et al.78 and density functional theory (DFT) calculations by Nørskov et al.81,82 We revisit the importance of orbital interactions in CO adsorption by reproducing what was done in the landmark 1985 paper by Hoffmann et al.78 Ti was chosen as a representative of early transition metals and Ni as a representative of late transition metals. The (0001) plane was examined for Ti, which has a hexagonal-closest-packed (hcp) structure, and the (111) plane was examined for Ni, which has a face-centered cubic (fcc) lattice. Fig. 4A and B show the density of states (DOS) calculated for the Ti and Ni surfaces. The major contributors to these DOS peaks are the 3d orbitals of Ti and Ni. Comparing these DOS plots, one can notice some important features: 1) The DOS peak of Ti is located at higher energy than that of Ni. 2) The electron occupation of the Ti band is about one third, while the Ni band is almost occupied. 3) The DOS peak of Ti is broader than that of Ni. Feature 1) can be understood from the difference in the energy levels of the 3d orbitals of Ti and Ni.84 Feature 2) can be understood from the difference in the number of d electrons between Ti and Ni. Feature 3) is a very interesting point. This can be understood from the difference in the difuseness of the 3d orbitals of Ti and Ni.78 In general, more electropositive elements have diffuse AOs than the AOs associated with a more electronegative atom.85 Fig. 4C shows the DOS plot of the CO molecule alone. Each peak in this plot can be assigned by comparing the energies of the peak positions with the energy levels of the MOs. Since the p orbitals are doubly degenerate, their peak heights are twice those of the s orbitals. Looking at the wave functions for the 2s, 3s, and 2p orbitals, one can see that the amplitudes on the C side are larger than those on the O side. Only in the wavefunction of the 1p orbital is the opposite trend observed. Therefore, it is expected that the C atom interacts more strongly with the metal surface than the O atom. The 3s and 2p orbitals will play an important role in the adsorption of CO. The 2s orbital is energetically distant from the dband of the metal, so its contribution to the adsorption is expected to be small. Let us see how the 3s and 2p orbitals interact with the surface orbitals. There are tools that can be of great help for this analysis: The Crystal Orbital Overlap Population (COOP) and the Crystal Orbital Hamilton Population (COHP).30,79,86–88 These indices are excellent for determining whether the orbital interaction between an atom A and another atom B is bonding or antibonding at a given energy level. COOP and COHP are calculated by using the following equations:

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111

Fig. 4 DOS profiles calculated for slab models of (A) Ti(0001) and (B) Ni(111) surfaces. The structure of the slab model used is shown in the inset. (C) DOS profile calculated for a CO molecule placed in a large unit cell as shown in the inset. The DOS peaks of CO are attributed to the corresponding MOs, and the distributions of the MOs are shown near the peaks with typical orbital notation. The dashed line shows the Fermi level (EF). These were calculated at the extended Hückel level of theory. The surface structures were drawn using VESTA.83 We use this software for visualization here and there in this chapter.

COOPA;B ðEÞ ¼

X X

          !  ! ! ! k Ci;n k Cj;n k d 3 n k  E ;

(1)

X X

          !  ! ! ! k Ci;n k Cj;n k d 3 n k  E :

(2)

Sij i˛A;j˛B ! n; k

and COHPA;B ðEÞ ¼

Hij i˛A;j˛B ! n; k

COOP and COHP are essentially equivalent, the only difference being whether to use the overlap integral element Sij between AO i     ! ! centered on atom A and AO j centered on atom B, or the corresponding Hamiltonian matrix element Hij. Ci;n k and 3 n k are ! the expansion coefficient of AO i in the nth wave function at k-point k and its eigenvalue, respectively. In general, there is a relationship between the Hamiltonian and the overlap integral as follows: Hij f  Sij.89 Therefore, when trying to understand COOP and COHP, one has to pay attention to the signs. COOP is positive for the bonding interaction and negative for the antibonding, while COHP is negative for the bonding and positive for the antibonding. The COOP profiles calculated for the structures with CO adsorbed on the Ti(0001) and Ni(111) surfaces are shown in Fig. 5. The COOP curves are drawn focusing on the CeO and metal-C bonds. Two characteristic peaks can be seen at around E ¼  15 eV. In both peaks, there is a contribution of bonding interactions to the CeO and metal-C bonds. Comparison of the energy levels of the carbon monoxide MOs shown in Fig. 4C with the positions of these COOP peaks shows that they originate from the 1p and 3s orbitals. Why do we reject the possibility that the two peaks at around E ¼  15 eV originate from 2s and 2p orbitals? The fact that the COOP values calculated for the CeO bond are positive in that energy region does not suggest that these peaks originate from the 2p orbital. In the 2s orbital, the CeO bond has a bonding character, but there is no possibility that these peaks originate from this orbital: the energy of these peaks is much higher than that of the 2s orbital. The antibonding interaction between the 2s orbital and the surface orbitals could push up the energy of the resulting orbitals, but the COOP values calculated for the metal-C bond in that energy region suggest that the orbital interaction for that bond is bonding. So which of the two peaks, the higher or lower energy one, originates from the 1p orbital and which from the 3s orbital? To answer that question, we need to decompose the COOP curve into contributions from each atomic orbital. Let us do it using the interaction of the Ti surface with CO as an example. Fig. 6 shows the decomposed COOP curves. They were drawn by focusing on the anisotropy of the 2p orbitals of the carbon atom. We define the z-axis as the direction perpendicular to the surface. Thus, the interaction of the 2pz orbital of the carbon atom with the surface orbitals exhibits s symmetry, while that of the 2px and 2py orbitals with the surface orbitals exhibits p symmetry. In Fig. 6, one can see that the peak corresponding to the orbital interaction with p-symmetry is located at a lower energy position than that with s-symmetry. Thus, the lower peak is from the 1p orbitals and the higher peak is from the 3s orbital. The d orbitals on

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Bond activation and formation on inorganic surfaces

Fig. 5 COOP profiles calculated for structures with CO adsorbed on-top sites on (A) Ti(0001) and (B) Ni(111) surfaces. The COOP curves were calculated for the CeO and metal-C bonds as indicated respectively by the solid and dotted lines. The dashed line shows the Fermi level (EF).

Fig. 6 COOP profiles calculated for structures with CO adsorbed at an on-top site on the Ti(0001) surface. The COOP curve is shown separately for the interaction of the 2pz orbital of the carbon atom with the d orbitals of the Ti atom and for the interaction of the 2px and 2py orbitals of the carbon atom with the d orbitals of the Ti. The former interaction is indicated as s (solid line) and the latter as p (dotted line). Note that we assume that the 2px and 2py orbitals are oriented parallel to the surface, whereas the 2pz orbital is oriented perpendicular to the surface. The orbital interaction corresponding to each peak is shown schematically.

the surface that can interact with the 1p orbitals are the dxz and dyz. One of the resultant orbital interactions is shown schematically in Fig. 6. On the other hand, the d orbitals on the surface that can interact with the 3s orbital is the dz2. The resultant orbital interaction is shown schematically in Fig. 6. This orbital interaction is the so-called s donation. This is called “donation” because it represents the charge transfer interaction from the ligand CO to the metal.

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113

Fig. 4C shows that the 3s orbital is the HOMO of the CO molecule. Therefore, this orbital is occupied by two electrons. This orbital cannot accept any more electrons. On the other hand, is the dz2 orbital occupied by electrons? In the adsorption mode shown in Fig. 5A, there is only one Ti atom that interacts with the CO molecule. There is of course only one dz2 orbital in that atom. However, the dz2 orbital interacts with the orbitals of the surrounding atoms to form a band. This is illustrated in Fig. 7. It shows the partial density of states (PDOS) calculated for the dz2 orbital of a Ti atom located in the topmost surface layer of the Ti(0001) slab before CO is adsorbed. The dz2 orbital can be seen to have a broad DOS in a wide energy range from  14.5 to  5.5 eV. The Fermi level is located at  11.5 eV, indicating that only about a third of the dz2 band is occupied by electrons. The dz2 band has room for electrons, and electrons from the 3s orbital flow into such an energy state. Let us move on to the vicinity of the Fermi level in Fig. 6. The symmetry of the orbital interaction in that energy region is p. Looking at the same energy region in Fig. 5A, one can see that the orbital interaction between the C and O atoms is antibonding. Therefore, one can judge that the orbital interaction in that energy region originates from the 2p orbital as schematically depicted in Fig. 6. This is the so-called p back donation. The reason why it is called “back donation” is because it corresponds to the charge transfer from the metal to the ligand. As can be seen from Fig. 4C, the 2p orbitals are the LUMOs of the CO molecule, so before CO adsorption, those orbitals do not have any electrons. Therefore, upon adsorption, they just accept electrons. In Figs. 5A and 6, most of the COOP peaks corresponding to the p back donation are located below the Fermi level. In other words, they are occupied by electrons. Since it is the dxz and dyz orbitals that interact with the 2p orbitals, the PDOS for the dxz and dyz orbitals on the Ti(0001) surface before CO adsorption is shown in Fig. 7. Similar to that for the dz2 orbital, about a third of it is occupied by electrons; the electrons injected into the 2p orbitals come from there. The s donation leads to the outflow of electrons to the metal involved in the bonding orbital interaction between the C and O atoms in the 3s orbital. On the other hand, the p back donation leads to the flow of electrons from the metal into the 2p orbitals, providing electrons involved in the antibonding orbital interaction between the C and O. This creates quantum mechanical resonance states between the orbitals on the surface and the 3s and 2p orbitals of the CO molecule, and so the surface and molecule interact strongly. Such surface orbital interactions have been found not only in catalytic processes but also in various fields of surface science such as adhesion phenomena.69,90 The s donation and p back donation reduce the electron density involved in the bonding interaction and increase the electron density involved in the antibonding interaction between the C and O atoms, thus making the CeO bond easier to break. As such, the CO molecule adsorbed on the Ti surface is highly activated through these orbital interactions. COOP curves similar to the ones shown in Fig. 6 were also calculated for the adsorption of the CO molecule on the Ni surface. The resultant COOP curves are drawn in Fig. 8. The peak corresponding to the s donation is found around E ¼  15 eV. However, the charge transfer from the 3s orbital of the CO molecule to Ni may not be very effective. This is because, as shown in Fig. 4B, most of the Ni d-band is occupied by electrons, and there may not be room for more electrons.

Fig. 7

Plot of the PDOS of the dz2, dxz, and dyz orbitals of a Ti atom located at the topmost surface layer in the Ti(0001) slab without CO.

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Bond activation and formation on inorganic surfaces

Fig. 8 COOP profiles calculated for structures with CO adsorbed at an on-top site on the Ni(111) surface. The COOP curve is shown separately for the interaction of the 2pz orbital of the carbon atom with the d orbitals of the Ni atom and for the interaction of the 2px and 2py orbitals of the carbon atom with the d orbitals of the Ni. The former interaction is indicated as s (solid line) and the latter as p (dotted line). Note that we assume that the 2px and 2py orbitals are oriented parallel to the surface, whereas the 2pz orbital is oriented perpendicular to the surface.

The peak corresponding to the p back donation is found around E ¼  14 eV. This peak is much smaller than that of the interaction between the Ti surface and the CO molecule. This is because the 2p orbitals and the Ni d band are energetically distant from each other, as shown in Fig. 4B and C, so the interaction between them is smaller. These results indicate that the CO molecule adsorbed on the Ni surface is not as activated as that adsorbed on the Ti surface. On the surface of the early transition metals, CO will dissociate easily. The reasons for this can be summarized as follows: 1) the Fermi level and d band of the early transition metals are located high in energy, 2) their d bands are wide, and 3) they have room in the d-band to accept electrons. The first two are important for promoting the p back donation, and the last one is important for promoting the s donation. What has been discussed in this section is very useful not only for understanding the activation of the CO molecule but also for discussing the activation of N2, O2, H2, etc.33,91,92

3.06.3

Bond formation on metal surfaces

In the previous section, we looked at the orbital interactions that lead to the activation of bonds on the surface of transition metal catalysts, focusing on the adsorption of CO. We then obtained a framework for understanding which metals are favorable for the dissociation of the CeO bond. The role of the catalyst is not only to break bonds. It also plays an important role in the formation of bonds. Now that we have looked at CO activation, in this section we will continue to consider the fate of activated CO on the catalyst surface. First, let us look at a very important chemical process involving CO. “Gas to liquids” is a technology to produce liquid fuels or chemicals (hydrocarbons, alcohols, etc.) from natural gas via synthesis gas (mixture of CO and H2).93 The core of this technology is the Fischer-Tropsch (FT) synthesis, first reported by Fischer and Tropsch in 1923.94 It is known that the type of catalyst and the conditions under which it is used can affect the molecular weight and structure of the main product obtained. The most typical catalysts used for the FT synthesis are Fe and Co.95 Fig. 9 shows what kind of products can be obtained when these catalysts are used. The typical reaction conditions under which the FT synthesis is performed are in the temperature range of 200–350  C and pressure range of 15–40 atm.95 The FT reaction is much more complex than the simple diagram in Fig. 9 suggests. However, there is general agreement on its initial process. As shown in Fig. 10A, the CO bond of the CO molecule activated on the catalyst surface is cleaved to form a surface carbide and surface oxide. They are hydrogenated to produce methine (CH), methylene (CH2), methyl (CH3), and H2O.95 There is a lot we do not know about what happens after that. One promising proposal is that CH3 serves as the initiator and the polymerization of CH2 occurs (carbide mechanism), as shown in Fig. 10B.95,97 Another one suggests the involvement of vinyl species

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115

Fig. 9 Overview of the Fischer-Tropsch reaction. Note that the products are different depending on whether Fe or Co is used as the catalyst. This figure was generated based on Ref. 96.

Fig. 10 (A) Initial step of the FT reaction: the CeO bond of the activated CO molecule is cleaved, and the C and O generated on the surface are hydrogenated. (B) Progress of carbon chain growth by carbide mechanism and (C) alkenyl mechanism. (D) One model for the termination step of the FT reaction.

(HC]CH2), which is what is called the alkenyl mechanism.96,98 As shown in Fig. 10C, the reaction of CH with CH2 produces CH]CH2. It reacts further with another CH2. This is followed by isomerization, which is not shown in the figure. How does the FT reaction stop? This is an important issue that concerns the factors that determine the length of the alkyl chains produced by FT synthesis. Take the carbide mechanism, for example. In this mechanism, an alkyl adsorbed species is produced as shown in Fig. 10B. The coupling of adsorbed H or CH3 species with the alkyl chain leads to the formation of alkanes.99 The alkyl species adsorbed on the surface can ultimately be modeled by the CH3 adsorbed species. Therefore, the ultimately simplified model of the termination step of the FT reaction would be the coupling of two CH3 species on the catalyst surface shown in Fig. 10D.45 In this section, we use this coupling reaction as an example to explain the orbital view of bond formation on transition metal surfaces. Recently, we have simulated the coupling reaction of two CH3 species adsorbed on the Pt(111) surface using DFT calculations.100 The activation energy for this reaction was estimated to be 63 kcal/mol. This activation energy would generally be judged to be relatively high. A fairly high temperature environment would be required to trigger this reaction. It is generally thought that the coupling reaction to produce ethane from methane molecules is dominated by coupling between methyl radicals in the gas phase rather than CeC bond formation on the catalyst surface.101,102 Nevertheless, be the activation energy high or low, we can learn a lot from this model reaction. Fig. 11 shows the potential energy curve calculated using the Nudged Elastic Band (NEB) method103 for the coupling reaction of CH3 species on the Pt surface. Up to this point, we have analyzed the static state of molecules and atoms adsorbed on a surface. The

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Bond activation and formation on inorganic surfaces

Fig. 11 Potential energy diagram of the coupling reaction between two CH3 species on the Pt(111) surface. Structures 1, 3, and 6 correspond to the initial, transition, and final states, respectively.

calculation of the potential energy surface is very useful to understand the dynamic behavior of adsorbed atoms and molecules on a surface. In addition to the initial and final states of the reaction, the structures of the intermediate steps are also shown in Fig. 11. The reaction looks asymmetric: one can see one of the CH3 species leaving the surface and jumping onto the other CH3 species. What is the origin of the very large activation energy seen in Fig. 11? In order to answer this question, we need to take a closer look at the process of bond formation, and one way is to look at the COOP curve calculated for the interaction between the two carbon atoms.45 Fig. 12A shows the COOP curve calculated for the interaction between the two carbon atoms of the CH3 species in structure 1 (initial state of the reaction). Since the interaction between the two carbon atoms is very weak, the height of the peak in the COOP curve is low. Nevertheless, one bonding peak and one antibonding peak can be seen. The former is labeled s and the latter s*. Since the interaction is very weak, the splitting between the s and s* orbitals is very small. Let us look at Fig. 12B. There one can see the COOP curve calculated for structure 2 in the same way as for structure 1. One can see that the heights of the s and s* peaks are slightly higher than those in Fig. 12A. Furthermore, one can see that the energy difference between the s and s* orbitals has increased. These facts imply that the interaction between the C atoms has become stronger. Let us take a look at the interaction between the carbon atoms in the transition state (Fig. 12C). One can see that the height of the s and s* peaks is even higher, and the energy difference between them has also increased. It is suggested that the CeC bond has become even stronger. In addition, new states have emerged between the s and s* peaks. The peaks for these states are more broad, suggesting that the states are delocalized. This suggests that the CeC bond interacts with the orbitals on the surface.30 There is a crucial difference between the COOP curves shown in Fig. 12A and B and Fig. 12C. That is, in Fig. 12C, the s* peak is above the Fermi level, whereas in Fig. 12A and B, it is below the Fermi level. This means that in structures 1 and 2, there are electrons in the antibonding orbital of the CeC bond, but in structure 3, no electrons there. Using a simple diagram, let us try to explain again what has been explained above. A simple illustration of how the s and s* peaks in the COOP curves shown in Fig. 12 arise from orbital interactions is shown in Fig. 13. It shows how s and s* orbitals are formed by the interaction of two CH3 orbitals. In structures 1 and 2, both s and s* orbitals are located at energies below the Fermi level, so both are occupied by two electrons, resulting in the two-orbital-fourelectron situation. This situation leads to destabilization.85 This is because the orbital interaction stabilizes two electrons but destabilizes the other two, and the degree of destabilization is usually greater than the degree of stabilization. This can be understood in the following way. Fig. 14 shows an example of the two-orbital-four-electron orbital interaction. Since two degenerate orbitals j1 and j2 interact with each other, this is a kind of what is called degenerate orbital interaction. The energy of j1 and j2 is 3 . Let us take this as the

Bond activation and formation on inorganic surfaces

117

Fig. 12 COOP curves calculated for the interaction between the two carbon atoms in structures (A) 1, (B) 2, and (C) 3 shown in Fig. 11. The dashed line shows the Fermi level (EF).

Fig. 13

Diagram to understand the change in the COOP curve shown in Fig. 12 by reducing it to a simple orbital interaction.

origin of the energy axis. The energies of the bonding orbital jþ and antibonding orbital j resulting from the orbital interaction are 85 H H 3þ ¼ 1þS and 3  ¼  1S, respectively. These are easily obtained by solving the secular determinant. Here, using the Hamiltonian b 1 D ¼ Cj2 j Hj b 2 D and S ¼ Cj1 | j2D. Note that H < 0 and S > 0. It b the following relations hold: H ¼ Cj1 j Hj operator of the system, H, can be seen that the fact that the degree of destabilization of the antibonding orbital is greater than that of the bonding orbital is due to the overlap integral S. 1 ¼ 1  S þ S2  / and 1 ¼ Let us simplify the mathematical expressions for 3 þ and 3  by using the following identities: 1þS 1S 1 þ S þ S2 þ /. Now we have 3þ

and

¼

    H zH 1  S þ S2 ¼ k  S þ S2  S3 ; 1þS

(3)

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Bond activation and formation on inorganic surfaces

Fig. 14 Degenerate orbital interaction: orbitals j1 and j2 with energy 3 interact with each other to produce a bonding orbital jþ with energy 3 þ and an antibonding orbital j with energy 3 .

3

¼ 

    H z  H 1 þ S þ S2 ¼ k S þ S2 þ S3 : 1S

(4)

Here, we set H ¼ -kS (k > 0) in order to use the relation H f  S shown in Section 3.06.2.2. Before the orbital interaction, four electrons occupied the energy levels with energy 3 , so the energy of the system was 43 in the first approximation. This is the so-called Hückel approximation.104 We set 3 ¼ 0, so the energy of the system before the orbital interaction was 0. After the orbital interaction, two electrons occupy the orbital with energy 3 þ and the other two occupy the orbital with energy 3 . Therefore, the energy of the system as the first approximation is 23 þ þ 23 . Here, using Eqs. (3), (4), one can see that only the second term survives: 23 þ þ 23  ¼ 4kS2. As can be seen in Figs. 11 and 12, as the reaction proceeds, the distance between the carbon atoms shortens, and the magnitude of the overlap integral in between increases. The discussion in the previous paragraph implies that as S gets larger, the energy of the system gets higher in proportion to the square of S. So, will the energy of the system continue to rise as the CeC bond length shortens? In the case of isolated molecules, the answer is yes. However, in the case of extended systems, the answer is no. Look at the orbital interaction diagram on the right side of Fig. 13. As the bond length shortens, the magnitude of the overlap integral increases, and the energy of the antibonding orbital (s* orbital) rises, exceeding the Fermi level. Therefore, the electrons in the s* orbital flow out to the empty level just above the Fermi level. Then, the energy of the system is no longer 4kS2. It can be expressed as 2k( S þ S2  S3) þ 2EF using the energy levels of the bonding orbitals and the energy of the Fermi level. Since f(S) ¼  S þ S2  S3 is a monotonically decreasing function, the energy of the system starts to decrease as S gets larger. While the very simplified energy argument that we have discussed so far is very useful, we also need to have a good grasp of the shortcomings of this approximation: an insufficient account of the atomic repulsive forces.105 As the distance between the carbon atoms becomes shorter and shorter, the energy of the antibonding orbital exceeds the Fermi level, and the energy of the system begins to decrease, the decrease in energy stops somewhere because the repulsive energy between the carbon atoms also starts to work. Based on the understanding of the origin of the activation energy, which we have discussed at length, the strategy for reducing the activation energy of this reaction is obvious. The key is how early one can turn the energy of the system to stabilization. To do so, the energy level of the s* orbital must reach the Fermi level as soon as possible. Ultimately, this problem comes down to the question of how to reduce the difference between the Fermi level and the energy of the s* peak in Fig. 12A. We are presented with the choice of pushing down the Fermi level or pushing up the s* orbital. A low Fermi level means a high work function. Therefore, if a surface with a high work function is used as a catalyst, the activation energy of the CeC coupling is expected to be lower. It should be added that the work function of Pt is the largest among the transition metals shown in Fig. 3A. Furthermore, it should be recognized that a low work function of the catalyst surface may be detrimental to bond activation, the initial step of catalysis, as we saw in Section 3.06.2.2. Let us consider making the energy of the s* orbital higher. In Fig. 12A, the s* orbital is located near  15.5 eV. From Fig. 13, one can see that the positions of the energy levels of the s and s* orbitals are determined by the positions of the energy levels of the CH3 orbitals. The main contributor to the CH3 orbital is the 2p orbitals of the carbon atom, also shown in Fig. 13. Fig. 15 shows the PDOS plot for the 2p orbitals of the carbon atoms in structure 1. One can notice that the energy position of the highest peak in Fig. 15 is almost the same as the energy position where the s* orbital is located in Fig. 12A.

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Fig. 15

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PDOS plot for the 2p orbitals of the carbon atoms in structure 1.

The ionization energy of the carbon atom is 11.26 eV.106 If we take the vacuum level as the reference of energy, one can say that the energy level of the 2p orbitals of the carbon atom is located at  11.26 eV. The position of the main peak of the 2p orbitals of the carbon atoms shown in Fig. 15 is located lower than that. Note that the 2p orbitals interact with the orbitals of the hydrogen and platinum atoms nearby. Nevertheless, the ionization energy of the carbon atoms is useful for estimating the energy levels from which the s and s* orbitals begin to grow. The energy of the 2p orbital of an isolated carbon atom is  11.26 eV, but it is pushed down to  15.5 eV when it interacts with the surface (and the hydrogen atoms). If this interaction is weakened, the energy level of the 2p orbitals of the carbon atom will exist at a much shallower energy position. Thus, the energy cost of pushing the s* orbital to the Fermi level will be small. This sounds a roundabout discourse, but it says the obvious: the weaker the binding of the CH3 species to the surface, the lower the barrier to their coupling will be. In addition to the above, another alternative method can be proposed. Through antibonding orbital interactions between the CH3 species and other adsorbed species nearby, it would be possible to push up the energy level of the CH3, which in turn would push up the energy level where the s* orbital starts to grow. This explanation is abstract. However, we will see a more concrete example in a later section.

3.06.4

CeH bond activation on late transition metal oxide surfaces

Natural gas, one of the three major fossil energy sources, has abundant reserves and is relatively inexpensive compared to coal and oil.102 Methane is one of the main components of natural gas, attracting much attention as a future source of hydrocarbons.107 The increased availability of shale gas through hydraulic fracturing has also led to increased interest in designing catalysts for the partial oxidation of methane.108 The direct conversion of methane into liquid chemicals such as methanol suitable for transportation and storage is highly desirable due to its industrial importance.109 The conversion process of methane begins with the adsorption of methane on the catalyst surface and the activation of its CeH bond.110 Because methane is a highly symmetric, nonpolar molecule, it is very difficult to trap it on the catalyst surface. On top of that, since methane has the strongest CeH bond among hydrocarbons, its activation is also very difficult.111,112 Note that the bond dissociation energy of the CeH bond of methane is as high as 104 kcal/mol. In order for the CeH bond cleavage reaction to proceed at a practical rate, a high temperature environment is required. However, because of the need to operate the reaction at high temperatures, it would be difficult to control the subsequent reaction steps and selectively produce value-added products.113 If methane could be activated under mild conditions, selective chemical transformations would be possible. Specifically, if the barrier of the initial CeH cleavage is lower than that of the subsequent reaction steps, it would be possible to generate methyl groups on the catalyst surface under conditions where complete oxidation occurs only slowly. Avoiding complete oxidation after

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the formation of methyl intermediates provides an opportunity to develop chemical pathways for selective conversion of methane to valuable products such as methanol and ethylene.113 Recently, much attention has been paid to the fact that the surface of some late transition metal oxides is very effective in methane activation at low temperatures.114–116 Experiments using temperature-programmed reaction spectra (TPRS) under ultrahigh vacuum have shown that the magnitude of the adsorption energy of methane on the surface of the late transition metal oxides is very large and the activation energy required for the cleavage of the CeH bond of methane is very low. IrO2, RuO2, and PdO are typical examples of such transition metal oxides.114,115 In particular, the activity of the (110) surface of IrO2 is remarkable because the absolute value of the adsorption energy of methane is large, about 10 kcal/mol, and the cleavage of the CeH bond occurs at an extremely low temperature of 150 K.117 In addition, a first-principles materials screening of methane activation catalysts was performed by Nørskov et al., reporting IrO2 at the top of the computational volcano plot.118 In this section, we would like to try to understand why the IrO2 surface has such high CeH bond activation capacity in terms of orbital interactions.

3.06.4.1

Overview of the CeH activation of methane on the IrO2 surface

Theoretical analysis of the CeH bond cleavage reaction of methane adsorbed on the IrO2(110) surface has been performed by several groups (including us) using DFT calculations.117,119–125 In all these studies, the same initial, transition, and final state structures of the CeH bond cleavage reaction were identified. The adsorption energy of methane and the activation energy of the CeH bond cleavage were calculated. The calculated energies are consistent with what was experimentally observed by Weaver and coworkers.117 Let us base our discussion here on the results of our previous calculations.121 IrO2 crystalizes in a tetragonal rutile structure as shown in Fig. 16A. In this structure, the Ir cation is coordinated to six oxides and is in an octahedral ligand field. Each oxide is bound to three Ir cations and has a trigonal planar structure. The typical crystal plane for this crystal type is the (110) plane, whose structure is shown in Fig. 16B.126 On the surface, one can see rows of 2-coordinated O, 3-coordinated O, and 5-coordinated Ir. The coordination numbers of O and Ir in the bulk are 3 and 6, respectively, so the 2coordinated O and 5-coordinated Ir are coordinatively unsaturated (cus). It is known that the presence of cus metal and cus oxygen pairs on the surface is important for methane activation.113 The cus oxygen is the acceptor of a hydrogen atom and the cus metal forms a bond with CH3. Fig. 17 shows how the CeH bond of methane is cleaved on the IrO2(110) surface. One can see the stable adsorption structure of methane, the initial state of the reaction. The methane molecule is located directly above an Ircus atom: two hydrogen atoms facing the surface side, while the other two hydrogen atoms face the opposite side. The distance between the Ircus and the C atoms is approximately 2.5 Å. In Fig. 17, how the potential energy changes in this reaction is also shown. The activation energy for this reaction is estimated to be about 10 kcal/mol. Considering that the activation energy for the CeH bond cleavage of methane on late transition metal surfaces is approximately 20 to 30 kcal/mol,127 the activation energy of 10 kcal/mol can be said to be small.

Fig. 16 (A) Crystal structure of IrO2. (B) The (110) surface structure of IrO2: oblique (left) and top (right) views are shown. Coordinatively unsaturated O and Ir atoms (denoted respectively as Ocus and Ircus) are indicated.

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Fig. 17 Top: Initial, transition, and final state structures of the CeH bond dissociation reaction of methane on the IrO2 (110) surface. Bottom: Potential energy diagram for the CeH bond cleavage reaction on the IrO2 surface. Energy values are taken from Ref. 121, where Edes and Ea are the energy barriers for the desorption of methane from the surface and the activation energy for the CeH bond dissociation, respectively. Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2020, 124, 17058–17072. Copyright 2020 American Chemical Society.

As shown in Fig. 16, this reaction is highly exothermic. The magnitude of the heat of reaction is 26 kcal/mol. Thus, the initial state is energetically closer to the transition state than the final state. Such a transition state is called early transition state.128 According to Hammond’s postulate,129 the structure of the early transition state is closer to that of the initial state than that of the final state, and its properties are also considered to be similar to those of the initial state. Therefore, if one wants to better understand the origin of the low activation energy of this reaction, one should focus on the initial state, i.e., the adsorption state of methane. Fig. 18 shows the adsorption structure of methane in more detail. The HeCeH bond angle of methane in the gas phase is 109.5 , while the surface-facing H1eCeH2 bond angle of methane adsorbed on the IrO2 surface is 122.6 . The H3eCeH4 angle on the opposite side of the surface is also greatly widened. It was suggested that adsorption significantly distorts the structure of methane. Note that the deformation of methane has been recognized as an important factor in the activation of its CeH bond since the vigorous studies by Shestakov and Shilov.130 The direct indicator of bond activation is none other than the bond length. Keeping in mind that the CeH bond length of the methane molecule in the gas phase is 1.09 Å,121 let us look at the CeH bond length of the methane molecule shown in Fig. 18. One

Fig. 18 Side views of the structure of methane adsorbed on the IrO2 (110) surface taken from two different angles. Selected bond distances are shown in Å.

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Fig. 19

Two coordination modes of a methane molecule to the metal center M.

of the two CeH bonds oriented toward the surface has a bond length of 1.15 Å. On the other hand, the bond length of the other one is 1.10 Å. Thus, only the longer CeH bond is implied to be non-dissociatively activated. The coordination mode of methane to the Ir atom on the IrO2 surface can be viewed as an intermediate between what is called h2eH,H and h2eC,H131 using the nomenclature of coordination chemistry. Readers unfamiliar with this nomenclature will immediately understand what it means when they see Fig. 19.

3.06.4.2

Orbital interaction between methane and the IrO2 surface

Let us see what orbital interactions activate the CeH bond of methane on the IrO2 surface. To do this, we first need to know about the MOs of methane. Visualization of MOs is easy if molecular calculations are performed. However, what we are going to consider is the surface, which is not the subject of molecular calculations. So, let us dare to understand the MOs of methane using tools for solid and surface calculations. We made full use of COOP in Section 3.06.2.2. to understand the interaction between metal surfaces and CO. The same could be done here using COOP. However, since we used COHP in our previous work, we will reuse that data here. Therefore, the discussion proceeds using COHP. Fig. 20A shows the COHP curve calculated for the CeH bond of methane in the gas phase. In this plot the COHP value is multiplied by  1 so that positive -COHP values indicate a bonding interaction. The interpretation of the -COHP curve is the same as that for the COOP. In this graph, the energy reference point is set at the Fermi level of the IrO2 surface. The well-known MO diagram of methane111 is shown in Fig. 20B. One a1 orbital and three degenerate t2 orbitals are occupied; one empty a*1 orbital and three degenerate t*2 orbitals can also be seen. The a1 and t2 orbitals have a bonding character while the a*1 and t2* orbitals have an antibonding character. Two bonding peaks and two antibonding peaks can be seen in Fig. 20A, so they can

Fig. 20 (A) COHP curve for the CeH bond of methane optimized in the gas phase, where the interaction between methane and the surface is negligible. The Fermi level of the IrO2 surface is set to E ¼ 0 in this fig. A one-to-one correspondence exists between the peaks of the COHP curve for the CeH bond of methane and (B) the MO levels of methane. Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2018, 122, 15359–15381.

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easily be traced back to the MOs. The correspondence between the COHP peaks and the MO levels is indicated by using symmetry species. Let us see how the COHP spectrum for the CeH bond changes when methane is adsorbed on the IrO2 surface. In Fig. 21A, the COHP spectra before and after adsorption are compared. Several effects of adsorption on the shape of the COHP spectrum can be noted: 1) a shift in peak position can be seen and 2) some peaks are broadened. These effects are particularly pronounced in the peak for the t2 orbitals. Before adsorption, the peak was located around E ¼  3 eV. After adsorption, the peak was broadened, dropping to around E ¼  6.5 eV. However, this is not the only important change regarding this peak. After adsorption, one can witness the appearance of a new, albeit small, peak in the vicinity of E ¼ 1 eV. Since the -COHP value for such a newly emerged peak is positive, it cannot be attributed to the a*1 or t*2 orbitals. Therefore, it is reasonable to assume that it is derived from the t2 orbitals. Thus, the MOs that interact most strongly with the IrO2 surface are the t2 orbitals. We notice that the CeH bond highlighted in the inset of Fig. 21A interacts with the Ircus atom. However, we cannot directly visualize the COHP between the atom and the bond. To explore orbital interactions between the surface and the adsorbate, it is a good idea to look at COHP profiles calculated for a pair of atoms in the surface and the adsorbate. In Fig. 21B, COHP spectra calculated for the Ircus-C and Ircus-H interactions are shown. No essential difference is found between these two COHP curves. Since we assume that the t2 orbitals mainly interact with the surface, let us analyze the Ircus-C and Ircus-H COHP profiles by focusing on the energy regions where the COHP peaks derived from the t2 orbitals are identified, namely around E ¼  6.5 eV and E ¼ 1 eV. Interestingly, bonding peaks are observed near E ¼  6.5 eV and an antibonding peak near E ¼ 1 eV. From the above, we can say that the t2 orbitals split into two states due to bonding and antibonding interactions with the Ircus atom. This could actually be confirmed by visualizing the wave function in the relevant energy region. Fig. 21C illustrates it. Note that it shows the square of the wavefunction, i.e., the charge density, rather than the wave function itself. The charge density is observable. It does not contain information about the phase of the wave function. The direct distinction between bonding and antibonding interactions is the phase of the wave function, but it is possible to infer the phase from the charge density as demonstrated below. Fig. 22 shows the wave functions and charge densities for the bonding and antibonding orbitals of a H2 molecule as an example. As far as the bonding orbital is concerned, there is little difference in the shape of the wave function and charge density. As for the antibonding orbital, there is a difference between the wave function and the charge density. The charge density profile for the bonding orbital and that for the antibonding orbital are similar: Both have peaks corresponding to the position of the H nucleus. However, there is a difference, too: the charge density profile for the antibonding orbital takes zero at the midpoint of the HeH bond. This point corresponds to the node of the wave function. Therefore, when looking at the charge density distribution, the presence or absence of nodes can be used to determine whether the orbital interactions are bonding or antibonding.

Fig. 21 (A) How the shape of the COHP curve for the CeH bond of methane changes as it interacts with the IrO2 surface. The blue line shows the COHP profile before the interaction and the red line shows that after the interaction. The CeH bond that is the subject of the calculation is circled by the red dashed line in the inset. The COHP profile indicated by the blue line is essentially the same as that already shown in Fig. 20A. (B) COHP curves calculated for the IreH (pink) and IreC (green) interactions in the structure of methane adsorbed on the IrO2 surface. The IreH and IreC interactions are indicated by the pink and green dashed lines, respectively, in the inset. In these figures, the Fermi level of the IrO2 surface is set to E ¼ 0. (C) Partial charge density maps on the (100) plane calculated using wavefunctions in the energy ranges of 1.0 eV  E  2.0 eV (top) and  8.0 eV  E  5.0 eV (bottom). Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2018, 122, 15359–15381. Copyright 2018 American Chemical Society.

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Fig. 22 Diagram comparing wave function (j) and charge density (|j |2) using (A) the bonding orbital (s orbital) and (B) antibonding orbital (s* orbital) of a H2 molecule as an example.

Returning to Fig. 21C, now let us look at the charge density corresponding to the energy range 1.0 eV  E  2.0 eV, where there is a blue region between the CeH bond and the Ircus atom. This region is the region of zero electron density, corresponding to the node of the wave function. On the other hand, there is no such region between the C and H atoms in the CeH bond. Therefore, we can say that the interaction between the CeH bond and the Ircus atom is antibonding, while the interaction between the C atom and the H atom is bonding. This is entirely consistent with what the COHP curves shown in Fig. 21A and B imply. The reader may similarly recover information about the phase of the wave function from the charge density in the energy region  8.0 eV  E   5.0 eV. From the charge density map corresponding to the energy range of 1.0 eV  E  2.0 eV shown in Fig. 21C, it is inferred that it is the dz2 orbital of the Ircus atom that is interacting with the CeH bond. To be sure of this, we need to be familiar with the electronic structure of the surface. Knowledge of the ligand field of the Ircus cation is required. As illustrated using Fig. 16, the Ir cation exists in a 6-coordination environment of oxide ligands in the bulk, but once the surface is exposed to the vacuum, it transitions to a 5-coordination environment. This change in the coordination environment is schematically depicted in Fig. 23. In the six-coordination environment in the bulk state, the well-known octahedral ligand field causes the d orbitals to split into sets of doubly degenerate eg orbitals and triply degenerate t2g orbitals. When the surface is formed, one of the axial oxide ligands is removed. The dz2 orbital is stabilized because the antibonding orbital interaction between the Ircus cation and

Fig. 23 How the ligand field of the Ir cation changes when bulk IrO2 is cleaved to form the IrO2 surface. Only the metal d orbitals and s-type ligand orbitals are depicted.

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the oxide ligand is eliminated. However, the other axial oxide ligand is still present, so an antibonding orbital interaction there. As such, the dz2 orbital is still located higher in energy than the t2g set. The valence of Ir in IrO2 is þ 4, so formally Ir has five d electrons. Thus, in Fig. 23, it is depicted as if there is one half occupied level in the t2g set. The Fermi level of IrO2 is inferred to cross the upper part of the t2g band. The correctness of this inference can be justified by looking at the PDOS plots calculated for the d orbitals of the Ircus atom as shown in Fig. 24. Note here that the PDOS peak of the dz2 orbital is located just above the Fermi level, left unoccupied. Recall that the t2 orbital of the methane molecule is located just below the Fermi level, as seen in Fig. 20A. The dz2 and t2 orbitals are in close energetic proximity, resulting in a significant orbital interaction. The orbital interaction between the dz2 orbital and one t2 orbital is schematically shown in Fig. 25. The methane molecule is deformed on the surface and its symmetry is lowered, so the degeneracy of the t2 orbitals is lifted. The distribution of the orbital changes slightly accordingly, but is essentially the same as that shown in Fig. 25. See the literature for details.121 The orbital interaction shown in this figure is the so-called charge transfer interaction, namely the interaction between an occupied orbital and an empty orbital. This orbital interaction suggests a charge transfer from the t2 orbital to the dz2 orbital. The orbital interaction shown in Fig. 25 is very important for the CeH bond activation of methane. The t2 orbital has a CeH bonding character, and the orbital interaction weakens it because a portion of the electrons in that orbital flows out to the empty dz2 orbital. If this orbital interaction could be made stronger, more electrons would flow out of the t2 orbital, which would be expected to weaken the CeH bond and lower the activation barrier for the CeH bond breaking. Let us address this issue below. S2

When orbitals with energies 3 1 and 3 2 interact with each other, the strength of the orbital interaction is expressed as j3 1 123 2 j, where S12 is the overlap integral of these orbitals.85 If the energy difference between the t2 and dz2 orbitals can be reduced, the interaction between these orbitals will be strengthened and the CeH bond will be weakened more. To achieve this goal, we can either push the t2 orbital up or push the dz2 orbital down, or both. The energy level of the t2 orbital is determined by the nature of methane, and it is extremely difficult to influence its energy level. So let us consider pushing down the energy level of the dz2 orbital. We have proposed two strategies to push down the energy level of the dz2 orbital: (1) one is to use the surface of an oxide of a metal more electronegative than Ir121; (2) the other is to change the axial anion to a more electronegative one instead of oxide.124 Let us start with strategy (1). It is known that the greater the electronegativity, the lower the energy level of the AO of the atom.85 This is basically true for the levels of the d orbitals of transition elements (although there are some exceptions)84: it is certain that the d orbital energy of Pt is lower than that of Ir. Therefore, we would like to consider platinum oxides. Known platinum oxides are PtO, Pt3O4, and PtO2.132 Several structures have been identified for PtO2, including a-PtO2 (hexagonal CdI2-type structure), b-PtO2 (orthorhombic CaCl2-type structure), and b’-PtO2 (tetragonal rutile-type structure).133 Prior theoretical studies using DFT on the

Fig. 24 Plot of the PDOS of the d orbitals of the Ircus atom in the IrO2(110) surface, which has no interaction with any adsorbate. Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2018, 122, 15359–15381. Copyright 2018 American Chemical Society.

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Fig. 25

Schematic depiction of the orbital interaction between the dz2 orbital of the Ircus atom and one of the t2 orbitals of methane.

electronic energy for these three phases confirmed that the total energy calculated at 0 K increases in the order of b-PtO2 z a-PtO2 < b’-PtO2.134–136 Unfortunately, rutile-type PtO2 (b’ phase) is not the most stable. Fortunately, however, the structure of b-PtO2 is quite close to the rutile-type structure. The structures of a-PtO2 and b-PtO2 are compared in Fig. 26. At first glance, it is very difficult to find the difference between the structure of b-PtO2 and that of the rutile type. In fact, the structure of b-PtO2 can be regarded as a distorted rutile-type structure. Our free energy calculations show that b-PtO2 becomes more stable than a-PtO2 above 500 K, though the energy difference is at most 0.004 eV/f.u. Thus, there is a good chance that the b-phase exists and can be used as a catalyst. Fig. 27A shows the initial, transition, and final state structures calculated for the first CeH bond cleavage reaction of methane. The side view of the methane adsorption structure is similar to that on the IrO2 surface shown in Fig. 18. In IrO2, the IreOeIr bond angle is 180 , but in b-PtO2, the PteOePt bond angle is smaller than 180 (ca. 160 ), which indicates the effect of distortion. In Fig. 27B, energy diagrams of the CeH bond cleavage reaction on the IrO2 and b-PtO2 surfaces are compared. The activation energy of the CeH bond cleavage reaction on the b-PtO2 surface is found to be only 1 kcal/mol. The enhanced activity as expected confirms the validity of strategy (1). It has been reported that Pt doping on the (110) surface of rutile-type TiO2 also results in an improved CeH bond activation ability.137 In the following, we will explore measures to control the activity of the IrO2 surface according to strategy (2). As shown in Fig. 23, the IrO5 cluster is useful as the ultimately simplified model for considering the electronic structure of the Ircus atom on the IrO2 surface. Given that the valences of Ir and O are þ 4 and  2, respectively, the charge of the cluster would be  6. Fig. 28 shows how the interaction between the orbitals of the square planar [IrO4]4 cluster and those of the axial oxide ligand gives rise to the MOs of the [IrO5]6 cluster. Around E ¼  10 eV one can find an MO whose main contributor is the dz2 orbital of the Ir atom, the a1 orbital, which is what we have been focusing on all along. It is important to note that the dz2 orbital is not the only orbital that contributes to this MO. There is also mixing of the 2pz orbital of the axial oxide ligand into this MO.

Fig. 26

Crystal structures of (A) a-PtO2 and (B) b-PtO2.

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Fig. 27 (A) Initial, transition, and final state structures of the first CeH cleavage reaction of methane on the (110) surface of b-PtO2. Selected bond lengths are shown in Å. (B) Energy diagram of the CeH bond cleavage reaction on the b-PtO2 surface is compared with that on the IrO2 surface. Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2018, 122, 15359–15381. Copyright 2018 American Chemical Society.

Because of the antibonding orbital interaction between the dz2 orbital and the pz orbital, the energy level of the dz2 orbital in the IrO5 cluster is higher than that of the dz2 orbital in the IrO4 cluster. We would like to push down the energy level of the dz2 orbital in the IrO5 cluster. The first choice would be to lower the energy level of the dz2 orbital of the metal. It was shown above that this can be accomplished by changing the type of metal. Let us consider the other contributor to the a1 orbital of the IrO5 cluster we are focusing on, namely the pz orbital. As can be seen from the mode of orbital interaction in Fig. 28, if the energy level of the pz orbital can be

Fig. 28 Orbital interaction diagram for the formation of the [IrO5]6 cluster. The MOs of the cluster can be constructed from the square planar [IrO4]4 cluster and the axial O2 ligand. For clarity, the symmetry species in the C4v and D4h point groups are used to label the orbitals of [IrO5]6 and [IrO4]4, respectively. The distribution of the a1 (dz2) orbital is shown as the inset. Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2020, 124, 17058–17072. Copyright 2020 American Chemical Society.

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pushed down, the energy level of the a1 orbital will also go down. To achieve this, the O atom should be exchanged with a more electronegative element, say F. A structure in which the axial O ligand coordinating to the Ircus atom in the IrO2 surface is replaced by an F atom is hereafter referred to as FeIrO2. A material in which multiple anions coexist in a single phase is called a mixed anion compound.138,139 Compared to single anion compounds such as oxides and nitrides, mixed anion compounds are expected to exhibit rich catalytic chemistry due to their unique coordination and electronic structures.140,141 Fig. 29A shows the initial, transition, and final state structures calculated for the first CeH bond cleavage reaction of methane on the FeIrO2 surface. Compare the adsorption structure of methane in this figure with that shown in Fig. 18. The HeCeH bond angle facing the surface is wide open, and the CeH bond to be broken is already elongated. This indicates that the CeH bond of methane is fully activated. In Fig. 29B, energy diagrams of the CeH bond cleavage reaction on the IrO2 and FeIrO2 surfaces are compared. The activation energy for the CeH bond cleavage on the FeIrO2 surface is approximately 2 kcal/mol, suggesting that the introduction of F significantly increases the activity of the IrO2 surface as expected. This result supports the validity of strategy (2). Moreover, we have proposed a catalyst that can catalyze the synthesis of methanol from methane by bringing the concept of mixed anion compounds to the platinum oxide.142

3.06.4.3

Bond dissociation viewed as a bond formation

The title of this section would sound a bit odd. But its implication will soon become clear. All this time we have been considering the activation or cleavage of CeH bonds in methane. As can be seen in Figs. 17, 27, and 29, beside the cleavage of the CeH bond, OeH and C-metal bonds are being formed. The question we have considered at length so far can be viewed as a question of how well a surface can activate the CeH bond, as well as how easily a surface can bond to hydrogen and carbon. The problem of activation of the CeH bond of methane on the IrO2 surface can be reduced to the question of how easy it is to form a bond between the H atom and the Ocus atom on the surface. In Section 3.06.3, we have described a view of bond formation on a transition metal surface using the example of CeC bond formation. Here we apply that argument to the discussion of the OeH bond formation on the IrO2 surface. First, we have to focus on the pair of atoms between which a bond is being formed. Fig. 30A shows the COOP curves calculated for the interaction between the Ocus atom and the H atom abstracted by it (indicated by the dashed line in the inset). These COOP calculations were performed using the adsorption structure of methane. The fact that the b-PtO2 surface has a higher methane

Fig. 29 (A) Initial, transition, and final state structures of the first CeH cleavage reaction of methane on the (110) surface of FeIrO2. Selected bond lengths are shown in Å. (B) Energy diagram of the CeH bond cleavage reaction on the FeIrO2 surface is compared with that on the undoped IrO2 surface. Adapted with permission from ref. Tsuji, Y.; Yoshizawa, K. J. Phys. Chem. C 2020, 124, 17058–17072. Copyright 2020 American Chemical Society.

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Fig. 30 (A) Plots of COOP curves calculated for the interaction between the H1 atom and the Ocus atom on the IrO2 (black) and b-PtO2 (gray) surface. The energy origin corresponds to the Fermi level of the surface. (B) Schematic representation of how orbital interactions between the Ocus and H atoms change with the formation of the OeH bond.

activation capacity means that its surface has a higher hydrogen abstraction capacity. To understand this, we are checking the COOP for OeH. There are two main peaks in each OeH COOP profile, which may be attributed to the s and s* orbitals of OeH judging from the sign and energy ordering of them as shown in this figure. It can also be seen that the b-PtO2 surface has a higher peak height due to the shorter OeH distance. In these COOP plots, the energy origin is set at the Fermi level of each surface. The s and s* orbitals are both located below the Fermi level and would correspond to the orbital interaction shown on the left side of Fig. 30B. Not only the bonding orbital but also the antibonding orbital contains electrons. Such an electron occupancy is unfavorable for bond formation, resulting in a repulsive interaction. This is thought to be one of the origins of the activation energy for this reaction. As seen in Fig. 13, the energy of the s* orbital gets higher as the bond gets formed, so at some point it will be above the Fermi level. Then the s* orbital can dump its electrons at the Fermi level, instability removed, and the system turns to stabilization. The key to hydrogen abstraction by the Ocus atom in the surface is how early the s* orbital can come above the Fermi level. For this reason, it is preferable that the Fermi level of the surface be as low as possible. Our extended Hückel calculations show that the Fermi level of the IrO2 surface is  11.37 eV with respect to the vacuum level, and the b-PtO2 surface  11.51 eV. In fact, the COOP in Fig. 30A shows that the s* orbital peak is closer to the Fermi level for the case of the b-PtO2 surface than the IrO2. Thus, the s* orbital can reach the Fermi level with less energy cost on the b-PtO2 surface. This may be one reason for the lower activation energy of the CeH bond cleavage on this surface.

3.06.5

NeH bond formation on early transition metal hydride surfaces

3.06.5.1

Overview of ammonia synthesis

Ammonia is a substance used in various applications such as fertilizers,143 raw materials for chemicals,144 and refrigerants,145 an important basic chemical that is produced at about 170 million tons per year.146 It also has the advantage of high mass and volume hydrogen densities, making it a candidate for hydrogen storage and transportation media in the hydrogen-using society that is currently attracting much attention.147–149 Global demand for ammonia is increasing year by year against the backdrop of economic development in emerging countries, and this demand is expected to increase further in the future.150,151 Ammonia is industrially synthesized directly by passing a mixture of nitrogen and hydrogen gases through a catalyst layer at high temperature (around 500  C) and high pressure (approximately 150–200 atm).152 In 1908, Haber established the theoretical basis of the technology,153,154 and Bosch and Mitasch devised high-pressure equipment and catalysts, respectively, and the first industrialization took place in 1913.155 This is what is known as the so-called Haber-Bosch process. This method made it possible to supply large quantities of nitrogen fertilizers at low cost, and led to a dramatic development in agricultural production.153 The chemical equation for ammonia synthesis by the Haber-Bosch process is written as follows: 3H2 þ N2 ➔2NH3 DH ¼  46 kJ=mol 156

(5)

lower temperatures are more favorable for ammonia production. To achieve ammonia Since this reaction is exothermic, synthesis, nitrogen and hydrogen molecules have to be activated and nitrogen has to be hydrogenated (see Fig. 31). In general, the rate-limiting step is known to be the dissociative adsorption process of N2 (Fig. 31A).12,96 Catalysts play a central role in the activation of such small molecules at low temperatures. In the current Haber-Bosch process, metallic iron is used as a catalyst in the presence of small amounts of K2O and SiO2 or Al2O3 promoters.96,157 K2O acts as an electronic promoter to enhance catalytic

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Fig. 31 Schematic description of three important elementary steps in the Haber-Bosch process: (A) N2 activation, (B) H2 activation, and (C) NeH bond formation.

activity, because it contributes to an increase in the electron-donating capacity of the surface; SiO2 or Al2O3 acts as a structural promoter to stabilize the catalyst structure. Metals other than Fe, such as Ru and Os, can also catalyze ammonia synthesis, showing high catalytic activity.144 Since the ratelimiting step is the dissociation of N2, the activation energy for this step becomes higher when a late transition metal is used as a catalyst, resulting in a lower reaction rate.96 On the other hand, early transition metals, such as Mo and Re, can effectively adsorb N2 dissociatively, but because of the strong metal-N interaction (see Fig. 1C), the N atoms will remain retained on the surface, blocking the reaction sites. That is why, when the catalytic activity of each transition metal is plotted against the number of d electrons, the plot looks like a mountain shape, so it is called the volcano plot.158 Metals with neither too many nor too few d electrons will be highly catalytically active.

3.06.5.2

Ammonia synthesis on ionic compound surfaces

Solid catalysts usually consist of a support, which is usually an ionic compound, and metal particles supported on it. The metal is the catalytically active site. The support may have an electronic effect on the supported metal particles, or it may simply serve to physically hold the particles in place and enhance their stability and dispersiveness.159 The metal particles are the main players in catalytic reactions, on which many researchers focus their attention. However, solid catalysts that do not support metal particles have been attracting attention in recent years. Recently, it was found experimentally that various early transition metal hydrides and oxyhydrides, such as TiH2, VH0.39, NbH0.6, and BaTiO2.5H0.5, catalyze ammonia synthesis in the absence of metal particle catalysts.140,160 Under 1–5 MPa at the temperature range 325–400  C, the catalytic activity of these hydrides was reported to be in the order VH0.39 > TiH2 > NbH0.6,160 with VH0.39 and TiH2 showing higher or comparable activity than the well-known Cs  Ru/MgO catalyst.161,162 In contrast, when nitrides or oxides of these transition metals (TiN, VN, TiO2, Ti2O3, etc.)140,160 are used as catalysts, very little product is detected, suggesting that hydrides are an essential catalytic component.163 Meanwhile, the effectiveness of these hydrides as supports has of course been vigorously studied.164 A very straightforward way to understand this reaction would be that the hydride acts as a hydrogenation catalyst just as the oxide acts as an oxidation catalyst.165,166 Interestingly, partially-dehydrogenated titanium hydrides have recently been reported to be active for dehydrogenation.167 Detailed surface scientific observations of hydrogen defect formation on TiH2 surfaces have also been conducted.168 These catalytic hydrides are characterized by the fact that the metals contained are all early transition metals. The active site for the reaction is expected to be Ti, V, and Nb sites in their surfaces.139 These metals would be excellent for activating N2 because their affinity for the N atom is very high (see Fig. 1C); there is concern that the subsequent reaction would not proceed. In reality, ammonia is produced. Therefore, the mechanism of catalytic reactions with these hydrides is of great interest. Let us summarize what we know experimentally about the kinetics of the ammonia synthesis reaction using these hydride catalysts. The reaction rate r of ammonia formation is given by the following equation169: r ¼ kpaN2 $pbH2 $pgNH3 ;

(6)

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where k is the reaction rate constant and pN2, pH2, and pNH3 are the partial pressures of nitrogen, hydrogen, and ammonia, respectively. a, b, and g are the corresponding reaction orders. For catalytic reactions using conventional catalysts, a is close to 1,169–171 which is related to the fact that the cleavage reaction of the adsorbed N2 species is the rate-limiting step.140 When TiH2 or VH0.39 is used as a catalyst, a is considerably reduced to less than 0.5.140,160 For the BaTiO2.5H0.5 catalyst, a is smaller than 1, although not as small as the values for these TieH and VeH systems. An a value as low as 0.5 was observed in an experiment by Kitano et al. where ruthenium-loaded 12CaO$7Al2O3 electride (Ru/C12A7:e) was used as a catalyst.172 The low reaction order supports their conclusion that the rate-determining step of ammonia synthesis on Ru/C12A7:e is not the N2 cleavage but the NeH bond formation. Wang et al.173 reported on catalytic reactions combining various transition metals with LiH, where a low N2 order (a z 0.5) was observed. It follows from this that the dissociation of N2 is likely to be fast, and hydrogen addition to the adsorbed N species is considered to be the rate-limiting step. Based on the above, the rate-limiting step in ammonia synthesis using TiH2 or VH0.39 as a catalyst is not N2 dissociation but NeH bond formation. The reaction order b for H2 is of practical importance because it is a good indicator of the degree of hydrogen poisoning of precious metal catalysts. b becomes negative for most of the Ru-based catalysts, falling in the range from  1 to 0,169,170,174 meaning that the reaction rate is reduced under high pressure due to excessive hydrogen adsorption on the surface. When Ru/C12A7:e or transition-metal-loaded LiH is used as a catalyst, b is reported to be positive, close to 1.172,173 These catalysts do not suffer from hydrogen poisoning and maintain high catalytic performance under high pressure. Ammonia production activity is expected to increase in proportion to total pressure.170 b values nearly equal to 1 have also been reported for the TiH2 and VH0.39 catalysts.140,160 For the BaTiO2.5H0.5 catalyst, b is positive but does not reach 1, being about 0.5.140 Although these hydride catalysts have much in common with metalsupported electrides and LiH, it should be noted that the stoichiometry of H is known to be variable for these hydrides.175–180 This suggests that sufficient spillover capacity/pathway is provided for the active site to avoid hydrogen poisoning.140 As for the reaction order of NH3, g, negative values are identified for many catalysts, including those presented above.140,160,172,173 Their g values range from  1.9 to  1.0. Such negative values indicate that the catalysts exhibit high catalytic performance for ammonia decomposition and suppress NH3 formation. To prevent loss of catalytic activity due to inhibition by NH3, the generated ammonia must be efficiently removed from the catalyst bed. As described above, the experimental kinetic analysis revealed the following notable properties of the hydride catalysts. 1) N2 cleavage does not appear to be the rate-limiting step. 2) The rate-limiting step is probably NeH bond formation. 3) Hydrogen poisoning does not occur so the catalytic activity is not reduced. In the next section, we will explore the origin of hydride catalyst activity by analyzing orbital interactions on the hydride surfaces.

3.06.5.3

Electronic origin of catalytic activity of hydride catalysts for ammonia synthesis

In the previous section we have described the specificity of such hydride catalysts as TiH2, VH0.39, NbH0.6, and BaTiO2.5H0.5 for ammonia synthesis based on experimental results. Here, we will focus on TiH2, whose surface structure is well known, to deepen our understanding of its catalytic activity. It would be a good idea to compare this with metallic Ti. According to the scaling rule,181 ammonia is not synthesized on Ti. Then why does the Ti hydride come to function as a catalyst for ammonia synthesis? It is the purpose of this section to answer this question. Let us start with a comparison of the structure of TiH2 and Ti. Fig. 32A and B show the crystal structures of TiH2 and Ti, respectively. TiH2 has a fluorite-type crystal structure, which can be described as a face-centered cubic (fcc) network of Ti2þ ions with all tetrahedral holes occupied by H ions. Metallic Ti has a hexagonal close-packed (hcp) structure. The most densely packed plane for the fcc Ti lattice is the (111) plane, while that for the hcp Ti is the (0001).182 In fact, the most stable surface known for TiH2 is the (111) (see Fig. 32C),183 and the Ti (0001) surface (see Fig. 32D) is one of the most stable surfaces of Ti.53 Whether these surfaces actually provide active sites for catalytic reactions remains to be seen, but it would be a good idea to start with these basic surface structures as a first step in research. Comparing Fig. 32C and D, notice that the arrangement of the Ti atoms is the same. In other words, the TiH2 (111) surface can be regarded as a structure in which H atoms have penetrated into the gaps between the Ti atoms on the Ti(0001) surface. Due to the penetration of the H atoms, the TieTi distance on the TiH2 surface is longer than that on the Ti surface. This similarity in structure makes it easier to compare the catalytic activity of these surfaces. We have seen in Section 3.06.2.1 that information on the Fermi level and work function of a surface is very important in gaining insight into dissociative bond activation. Thus, here we want to know the work functions of the TiH2 and Ti surfaces. In Table 2, the work function values evaluated for the TiH2 (111) and Ti (0001) surfaces are summarized. For reference, that for the Ru (0001) surface is shown. Experimentally measured work function values for these materials are also shown. By and large, one can see a good correspondence between the calculations and experiments. From this table, one can see that the Fermi level of Ti is high because the work function of Ti is small. This suggests that Ti has high electron emission capacity and can easily inject electrons into the antibonding orbitals of N2. Therefore, dissociation of N2 on the Ti surface will proceed with a small activation energy. This is evident from the calculation of the potential energy diagram for N2 dissociation on the Ti surface (see Fig. 33). The adsorption energy of N2 is  3.5 eV, and its absolute value is very large, suggesting a strong interaction between the Ti surface and N2. The Bader charge187 per N atom is shown. It can be seen that the charge transfer from Ti to N2 seems to be effective. The NeN bond length of N2 in the gas phase is 1.10 Å,188 but that of the adsorbed on the Ti surface is 1.33 Å, indicating that the NeN bond is considerably activated at the stage of adsorption.

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Fig. 32 Crystal structures of (A) TiH2 and (B) Ti. (C) The (111) surface of TiH2 and (D) the (0001) surface of Ti. Selected atom-atom distance is indicated. Table 2

Work functions calculated for the TiH2 (111), Ti (0001), and Ru (0001) surface slab models are compared with those measured in experiments.

Materials

Work function (calc.) [eV]

Work function (exp.) [eV]

TiH2 Ti Ru

4.1a 4.4a 5.0a

4.0b 4.6c 5.4d

a

Taken from ref. 91. Taken from ref. 184. c Taken from ref. 185. d Taken from ref. 186. b

From Fig. 33, it can be seen that NeN bond that is substantially activated due to adsorption so it is expected to be cleaved easily. Note that the energy of the transition state for N2 cleavage is well below that of the N2 molecule in the gas phase. This means that the apparent activation energy is negative.117 Compared to the N2 molecule in the gas phase, the dissociated state into two N atoms on the Ti surface is 5.3 eV more stable. Since the N atoms on the Ti surface are so stable, it would be difficult to hydrogenate the N atoms and remove them from the surface. Table 2 shows that the work function of Ru is larger than that of Ti, so the Fermi level of Ru is lower than that of Ti. Therefore, the injection of electrons from the Ru surface into N2 would not be so effective compared to that on the Ti surface. Indeed, N2 dissociation on the Ru (0001) surface has been well documented in the literature189,190 and its activation energy is reported to be approximately 2.0 eV. Actually, Ru is a very good catalyst for ammonia synthesis, but it is known that the catalytic reaction occurs on the stepped surface, not on the (0001) surface.191 What about TiH2? Interestingly, its work function is smaller than that of Ti. That is, the Fermi level of TiH2 is higher than that of Ti. Thus, charge injection into N2 on the TiH2 surface is expected to occur as effectively as on the Ti surface, and dissociation of N2 is expected to occur as readily as on the Ti surface. This is in good agreement with the very small reaction order for N2 observed when using the TiH2 catalyst, as described in the previous section. Next, let us look at the formation reactions of N-Hn species on the Ti and TiH2 surfaces. We begin with the reactions on the Ti (0001) surface. Fig. 34 shows the initial, transition, and final state structures and the activation energies and reaction heats for the NHn (n ¼ 1, 2, 3) formation reactions on the Ti surface. As n increases, both activation energy and reaction heat are found to increase.

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Fig. 33 Calculated potential energy diagram of N2 dissociation on the Ti (0001) surface. Initial, transition, and final state structures are shown; the NeN distance (d(NeN)) and the Bader charge per N atom (q(N)) are shown below each structure. Adapted with permission from ref. Tsuji, Y.; Okazawa, K.; Kobayashi, Y.; Kageyama, H.; Yoshizawa, K. J. Phys. Chem. C 2021, 125, 3948–3960. Copyright 2021 American Chemical Society.

Fig. 34 Top views of the initial, transition, and final state structures for the hydrogenation reactions on the Ti (0001) surface: (A) N* þ H* ➔ NH*, (B) NH* þ H* ➔ NH2  , and (C) NH2  þ H* ➔ NH3  . The activation energy (Ea) and heat of reaction (DH) for each elementary reaction are shown on the left side of the structures. Adapted with permission from ref. Tsuji, Y.; Okazawa, K.; Kobayashi, Y.; Kageyama, H.; Yoshizawa, K. J. Phys. Chem. C 2021, 125, 3948–3960. Copyright 2021 American Chemical Society.

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Fig. 35 Top views of the initial, transition, and final state structures for the hydrogenation reactions on the TiH2 (111) surface: (A) N* þ H* ➔ NH*, (B) NH* þ H* ➔ NH2 , and (C) NH2 þ H* ➔ NH3 . The activation energy (Ea) and heat of reaction (DH) for each elementary reaction are shown on the left side of the structures. Adapted with permission from ref. Tsuji, Y.; Okazawa, K.; Kobayashi, Y.; Kageyama, H.; Yoshizawa, K. J. Phys. Chem. C 2021, 125, 3948–3960. Copyright 2021 American Chemical Society.

By comparing Figs. 33 and 34, one can see that the rate-limiting step for ammonia synthesis on the Ti (0001) surface is not N2 dissociation but N-H bond formation, in particular, the formation of NH3 from NH2 . The formation reactions of N-Hn species on the TiH2 surface are shown in Fig. 35. It is possible to find similarities with what was observed on the Ti surface in that as n increases, the activation energy and heat of reaction increase. The change in activation energy for a change in n is summarized in Fig. 36. However, we must scrupulously note the difference in absolute values. Above all, the activation energy of the process of NH*3 formation through hydrogenation of NH2 is very large on the Ti surface, 2.1 eV, while on the TiH2 surface it is very small, 1.0 eV. This difference is what causes TiH2 to be a good catalyst for ammonia synthesis while Ti is not. We would like to clarify the origin of this difference based on orbital interactions. It would be good to focus on the reaction NH2 þ H* ➔ NH3 to see the difference in catalytic activity between Ti and TiH2. Nevertheless, as can be seen in Fig. 36, there seems to be an essential difference between Ti and TiH2, regardless of the reaction

Fig. 36

How the activation energy for the reaction of NHn-1* þ H* ➔ NHn on the Ti (blue) or TiH2 (orange) surface varies depending on n.

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Fig. 37 How the COHP plot for the NeH bond formed in the reaction N* þ H* ➔ NH* on the Ti (0001) surface changes as the reaction proceeds. The COHP was calculated for the (A) initial, (B) transition, and (C) final state structures of the reaction. The dashed line denotes the Fermi level (EF) of the surface. The number in parentheses indicates the energy relative to the initial state. The NeH distances (d(NeH)) are shown below the structures. Adapted with permission from ref. Tsuji, Y.; Okazawa, K.; Kobayashi, Y.; Kageyama, H.; Yoshizawa, K. J. Phys. Chem. C 2021, 125, 3948–3960. Copyright 2021 American Chemical Society.

step. Then we may as well focus on the N* þ H* ➔ NH* reaction. This reaction is very simple, just the coupling of two atoms on the surface. Thus, we will be able to extract from it an intrinsically important insight. If one wants to get some clues about bond formation, one may want to look at COOP or COHP between two atoms that form a bond, as we have done in Section 3.06.3. COHP plots for the interaction between N* and H* in each of the initial, transition, and final state structures of the N* þ H* ➔ NH* reaction on the Ti surface are depicted in Fig. 37. Let us begin with Fig. 37A, the initial state. Bonding peaks around E ¼  5.2 eV and antibonding peaks around E ¼  3.4 eV can be found. The former and latter can be attributed to sN  H and sN* H orbitals, respectively. These peaks have energies below the Fermi level of the surface, and their corresponding energy states are all occupied by electrons. Such a situation corresponds to the two-orbital-four-electron orbital interaction case, as we saw in Fig. 14, resulting in destabilization. It would be reasonable to consider this as the origin of the activation energy. From Fig. 37, it can be seen that as the reaction proceeds, the energy of the s* orbital rises and the energy of the s orbital falls. In the transition state, the peak of the s* orbital is located just near the Fermi level. As the s* orbital is destabilized beyond the Fermi level, the electrons in that orbital begin to flow to other levels near the Fermi level, and the s* orbital becomes empty. When this happens, the orbital no longer contributes to the destabilization of the system, and the system turns to stabilization. Thus, as we saw in Section 3.06.3, how close the antibonding peak is to the Fermi level in the initial state is an important factor in determining the magnitude of the activation energy. The closer the peak is to the Fermi level, the smaller the activation energy, since the antibonding orbital can be emptied at a small energy cost. Let us look at COHP for the NeH interaction on the TiH2 surface. As mentioned above, it is sufficient to look only at the initial state of the NeH bond formation reaction. As shown in Fig. 38A, the peak for the NeH antibonding interaction on the TiH2 surface

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Fig. 38 (A) COHP plot calculated for the NeH1 interaction in the initial state of the NeH bond formation reaction on the TiH2 (111) surface shown in Fig. 35A (see top right for atom numbering). (B) Combined COHP plots calculated for the H1-H2 and H1-H3 interactions on the same surface. The black dashed lines indicate the Fermi level (EF) of the surface. The partial charge densities for the wave functions with eigenvalues corresponding to the three peaks indicated by the blue dashed lines are projected onto the plane through the N, H1, H2, and H3 atoms (right side of the figure). Adapted with permission from ref. Tsuji, Y.; Okazawa, K.; Kobayashi, Y.; Kageyama, H.; Yoshizawa, K. J. Phys. Chem. C 2021, 125, 3948– 3960. Copyright 2021 American Chemical Society.

is found around E ¼  2.4 eV; it is located 1 eV above that on the Ti surface. Thus, the energy required to exclude electrons from this orbital is less than that required to exclude them from it on Ti. This is why NeH bonds are more likely to form on the TiH2 surface than on the Ti surface. The bond is formed between the N and the H1 atoms. Around them are hydrogen atoms labeled H2 and H3. Since such atoms are not found on the Ti surface, it is natural to assume that these may hold some key. Finally, we conclude this section by answering the question of why the NeH antibonding orbital on the TiH2 surface is energetically higher than that on the Ti. A look at the structure will give us an important clue. The local structure around the NeH bond is shown in the upper right corner of Fig. 38. Let us look at the interaction of these atoms with the NeH bond. We computed COHP plots for the H1-H2 and H1-H3 interactions, and their summing up is shown in Fig. 38B. One can find H1-H2 and H1-H3 antibonding interactions in the energy region around  2.4 eV, which we have been focusing on. To gain a better understanding of this peak, a charge density map of the wave function with an eigenvalue corresponding to the energy of this peak was calculated, which is also shown in Fig. 38B. The  2.4 eV eigenstate clearly shows the NeH1 antibonding character. At the same time, the H1 atom interacts in an antibonding fashion with the surrounding H2 and H3 atoms. The presence of the antibonding interactions between the hydrides on the TiH2 surface and the NeH bond that is being formed on that surface results in pushing up of the sNH * orbital on the TiH2 surface. Fig. 38B suggests that the interaction between the H1 atom and the H2 and H3 atoms is repulsive. A naïve interpretation of this would be to think that the NeH1 bond is likely to form simply because the H2 and H3 atoms are pushing the H1 atom toward the N atom.

3.06.6

Conclusion

It may not be an overstatement to say that bond activation and bond formation on inorganic solid surfaces is one of the cornerstones of the modern chemical industry, since it is an important elementary process in heterogeneous catalysis. While there may be several ways to approach these processes microscopically, this chapter provided an example of how to understand them from

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137

a quantum mechanical perspective, based on orbital interactions. Orbital interactions are a method for describing the relationships between electronic states among discrete chemical entities such as atoms, molecules, and molecular fragments. Hoffmann and coworkers extended the orbital interaction concept to solid and surface problems. At the surface of a solid, the wave function of a bond hybridizes with the surface wavefunctions and becomes broad, making the analysis of the density of states alone elusive. Theoretical tools such as COOP and COHP make it possible to successfully retrieve the wave function of the bond buried in the bands. They used these useful tools primarily to address the problem of bond activation and formation on metal surfaces. This chapter has reproduced just a few of those vast efforts. Through the question of how to interpret COOP and COHP spectra and how to project them onto a picture of molecular orbital interactions, a picture of the electronic structure of surfaces that appeals to the chemist’s intuition has been provided. The key is how much electrons can flow into the antibonding orbital of the bond to be activated and how much electrons can flow out of the bonding orbital of that bond. In terms of bond formation, in the initial stage of a reaction, there are often electrons in both the bonding and antibonding orbitals of the bond that is being formed, resulting in a two-orbital four-electron situation. Therefore, it is important how effectively electrons are drained from the antibonding orbital of the bond to stabilize the system. This perspective is not unique to metal surfaces. It is applicable to a variety of inorganic surfaces, including ionic compound surfaces. The second half of this chapter has illustrated how such concepts can advance our understanding of catalytic activity, using methane activation and ammonia synthesis on oxide and hydride surfaces as examples, respectively.

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Bond activation and formation on inorganic surfaces DeKock, R. L.; Gray, H. B. Chemical Structure and Bonding, University Science Books: Sausalito, CA, 1989. Diekhöner, L.; Mortensen, H.; Baurichter, A.; Huntz, A. C. J. Vac. Sci. Technol., A 2000, 18, 1509–1513. Dahl, S.; Logadottir, A.; Egeberg, R. C.; Larsen, J. H.; Chorkendorff, I.; Törnqvist, E.; Nørskov, J. K. Phys. Rev. Lett. 1999, 83, 1814–1817. Jacobsen, C. J. H.; Dahl, S.; Hansen, P. L.; Törnqvist, E.; Jensen, L.; Topsøe, H.; Prip, D. V.; Møenshaug, P. B.; Chorkendorff, I. J. Mol. Catal. A: Chem. 2000, 163, 19– 26.

3.07

Chemical bonding with plane waves

Ryky Nelson, Christina Ertural, Peter C. Mu¨ller, and Richard Dronskowski, Institute of Inorganic Chemistry, RWTH Aachen University, Aachen, Germany © 2023 Elsevier Ltd. All rights reserved.

3.07.1 3.07.1.1 3.07.1.2 3.07.1.3 3.07.1.4 3.07.1.5 3.07.1.6 3.07.1.7 3.07.2 3.07.2.1 3.07.2.2 3.07.2.3 3.07.2.4 3.07.2.5 3.07.2.6 3.07.2.7 3.07.2.8 3.07.2.9 3.07.2.10 3.07.2.10.1 3.07.2.10.2 3.07.3 3.07.3.1 3.07.3.2 3.07.3.3 3.07.3.4 3.07.3.5 3.07.3.6 3.07.3.7 3.07.3.8 3.07.4 Acknowledgments References

Part I: Foundations Introduction Chemical bonding for the simplest molecule: H2 Schrödinger’s and Fock’s equation, Kohn-Sham equations: DFT More complicated molecules, LCAO in general Plane Waves and Bloch’s theorem One-dimensional systems Concept of charges and bonding Indicators (COOP and COHP) Part II: Methods Basis functions Pseudopotentials in general Projector augmented wave method Plane-wave-based quantum chemistry programs Projection method Atomic basis sets in LOBSTER The projected DOS, COOP, COHP, fatband, and k-dependent COHP and time-reversal symmetry Mulliken and Löwdin population analysis Crystal orbital bond index Averaged bonding descriptors Covalent bonding indicators: Averaged COHP and effective interaction number Ionic bonding indicators: Ionicity and Madelung energy Part III: Applications Molecular systems Bonding between molecules (H and X bonding), bonding on surfaces Semiconductors such as GaAs and Phase-change materials Spin polarization and magnetic systems Rechargeable battery electrode materials Thermoelectrics and Zintl phases Materials mapping Interplay of bonding and structure Computational details

142 142 143 144 147 148 149 151 155 155 157 158 159 160 162 163 166 168 171 171 173 176 176 178 180 184 187 189 191 194 196 196 196

Abstract Two kinds of basis functions, plane waves and local orbitals, have been dominantly used in quantum chemistry due to their specific advantages. The former excels because of its simple computational implementation, whereas the latter is often chosen because of its convenience of extracting and analyzing chemical aspects of materials. In this review, we thoroughly describe a projection method bridging these two kinds of basis functions in order to harvest the benefits of their properties. Although the projection technique can in principle be applied to any quantum-chemical method, a particular implementation has been realized in a computer program called Local Orbital Basis Suite Towards Electronic-Structure Reconstruction (LOBSTER) processing data from density-functional theory (DFT) calculations. The reconstruction, furthermore, facilitates chemical bonding analysis through various bonding indicators and extraction of chemical information. Prior to the description of the projection method, the concept of chemical bonding is introduced to emphasize its importance in understanding chemical systems. Afterward, formulations of various quantum-chemical methods helping to understand bonding problems in many-electron systems are also described. Finally, we present diverse interesting material studies, including molecular systems, semiconductors, phase-change materials, electrode materials, etc. in great detail to demonstrate the advantages of exploiting both kinds of basis functions simultaneously.

Comprehensive Inorganic Chemistry III, Volume 3

https://doi.org/10.1016/B978-0-12-823144-9.00120-5

141

142

Chemical bonding with plane waves

3.07.1

Part I: Foundations

3.07.1.1

Introduction

Strange, isn’t it, that chemists can talk for hours about the concept of chemical bonding (let alone writing review articles about that very subject) although, strictly speaking, such phenomenon dubbed “chemical bonding” does not exist at all if we hide in the incredibly tiny box labeled “physics reductionism”. As will be clarified somewhat later and as provocatively formulated by Richard Feynman quite early, all theoretical chemistry is really physics, and all theoretical chemists know it. But seriously and for the moment being, let us predict what we are going to witness in the coming sections: to solve those quantum-mechanical equations envisioned by Schrödinger or by Kohn and Sham serving us well when dealing with atoms, molecules, and solids, the quantum-chemical (there we go!) interpretations but very often also starting assumptions make us aware, puzzlingly enough, that even a quantum-mechanical model of a system dubbed “molecule” does not relate to atoms and bonds (the chemist’s language) but to nuclei and electrons (the physicist’s language). Even worse, the more sophisticated the calculations, the more estranged we may become from chemistry such that, eventually, all the chemical understanding (and the chemical bonding, too) may vanish into thin air. So where do we start? The good ol’ triangle by van Arkel1 and Ketelaar,2 depicted in Fig. 1, will serve us because it already comprises three (out of five) flavors of chemical bonding, at least qualitatively. If we agree on the concept of electronegativity EN,3 namely the tendency of an atom to pull electron density towards itself inside a molecule (or whatever chemical entity), irrespective of whether electronegativity is defined empirically or even quantum-mechanically,4 the stage is set for things to come. There are “ionic” compounds to begin with, seen in the upper red corner, which result from chemical reactions between those atoms that strongly differ in their electronegativities, in other words: electronegative nonmetals and not-so-electronegative (lab jargon: “electropositive”) metals. Here, the EN difference is so large that an electron eventually hops from one atom (metal which turns into a cation) to the other atom (nonmetal becoming an anion), and the amount of electron localization at the anion is so tremendous (the electron now fully belonging to the nonmetal atom) that this very electron may be nicely described by a wave function exclusively belonging to that atom it eventually resides on. Quantum-chemically speaking, it is a kind of “one-center” bonding because just one atom is involved to hold the electron. As a result, there will be a Coulomb force between cation and anion, and in a threedimensional solid there results a Madelung field which lends itself to classical bonding theory typically found in textbooks dealing with ionic solids such as CsCl or MgO. The right blue corner could not be more different. If two highly electronegative atoms (the nonmetals) lacking an EN difference come together, the outer electrons of those atoms will experience the nuclear charge of both atoms, so both atoms and their wave functions will engage in holding those electrons such that a phenomenon called “covalent” bonding must result. Although there are quantum-mechanical principles which somewhat complicate the entire storydthe electrons cannot be distinguished and experience so-called electronic exchange due to their Fermionic naturedit is safe to assume that this kind of bonding involves two partners, so it is a “two-center” bonding. This is the type of bonding for which valence-bond and molecular-orbital theory were developed, eventually clarifying the meaning of Lewis’ formulation of chemical bonding by drawing a stick for two “paired” electrons, say, in the H2 molecule. We are surrounded by and composed of such molecules featuring covalent bonds between nonmetal atoms so the importance of that two-center bonding type cannot be overstated. And yet, there is another corner on the left for which there is also no EN difference but with much smaller electronegativities, so this is all about the interaction of metal atoms, say, in metallic Cs and even in metallic Fe, that is, “metallic” bonding. In metallic systems, there are plenty of metal atoms whichdbeing metal atomsdare always short of a sufficient number of electrons, so some

Fig. 1 The van Arkel-Ketelaar triangle of chemical bonding, grouping all chemical species by their EN differences and the EN size of their involved atoms. Reprinted from Deringer, V. L.; Dronskowski, R., Computational Methods for Solids. In: Comprehensive Inorganic Chemistry II: From Elements to Applications, Reedijk, J.; Poeppelmeier, K., Eds. Elsevier: Amsterdam, 2013; vol. 9, pp 59–87 with permission from Elsevier.287

Chemical bonding with plane waves

143

kind of electronic socialism sets in to lower the individual burden for all atoms by making an individual electron collectively belonging to plenty of atoms at the same time, a kind of “many-center” bonding. Technically speaking, dealing with two or more electrons is not a big difference, and we safely regard “metallic” bonding as a special case of “covalent” bonding. Before we describe how to proceed in terms of real computations, let us also comment on two more “bonding” types which do not show up in the magic triangle. Even if nonmetal atoms have no electrons to share, such as between light noble-gas elements like He, there is a tendency to come together and lower their energies by some weak “dispersion” forces as originally dubbed by London5 or much earlier, in a classical picture, by van der Waals.6,7 The reason is the fluctuation of the vacuum (or Heisenberg’s uncertainty principle, a matter of taste) which induces momentary atomic dipoles inducing other atomic dipoles, so there results a force from, well, nothing which scales with the polarizability of the atoms but without electrons in-between and without electron transfer. It is a small effect (which may become significant for, say, molecular iodine), usually incorporated by perturbation theory or empirical corrections. And there is also the biochemist’s delight, so-called hydrogen bonding, exclusively found between hydrogen atoms and neighboring atoms with a large EN such as oxygen. In fact, this bonding type is not an independent bonding type but a complicated mélange of the other ones. Because H bonding is so prominent and important (say, for storing genetic information), however, it counts as bonding type no. 5 for sheer reasons of convenience.8,9 So how do we proceed, at least in terms of the first three bonding types? Let us first quantum-chemically characterize two-center covalent bonding between nonmetals because this is what quantum chemistry was made for, successfully so. As a hole-in-one, essentially the same (delocalized) equations but now for metal atoms will make us understand and quantitatively calculate many-center metallic bonding without too much trouble. As another hole-in-one, the same equations will show, once in a while, that some of the electrons are less shared than others, so some atoms will end up with slightly lower/higher electron counts than in their neutral state, thereby arriving at slightly (or even strongly) charged atoms. That is to say that by solving the puzzle of covalent (metallic) bonding given that the atoms do show a little EN difference, we will also, indirectly so, successfully model ionic onecenter bonding.

3.07.1.2

Chemical bonding for the simplest molecule: H2

One of the most iconic and also simplest example of the covalent bond type is given by the H2 molecule. Even simpler is the hydrogen atom with a single electron, so there is no correlation such that analytical (nonrelativistic) solutions to Schrödinger’s equation are exactly known. These hydrogen-like solutions are often used as approximations for other systems.10–12 Likewise, the hydrogen molecule is often used as the archetypical case to introduce molecular orbital (MO) theory. In the MO framework, the hydrogen molecule allows for covalent bonding by the formation of the simplest type of molecular orbital, a s orbital, from two 1s orbitals, the simplest type of atomic orbital. At the same time, Pauli’s exclusion principle forbids that two electrons have the same quantum numbers. In Fig. 2, the formation of a bonding s (positive overlap of orbitals because of non-orthogonality) and an antibonding s* (negative overlap) state for H2 is shown. Here, the favorable bonding level is occupied and stabilizes the system, unlike the antibonding level. The molecular orbitals are linear combinations of atomic orbitals (LCAO), which, for the H2 molecule, read

Jj ðrÞ ¼ cA jA ðrÞ þ cB jB ðrÞ;

(1)

and cA and cB are the LCAO expansion coefficients for the two H atoms in H2, both having 1 s wave functions jA and jB which are identical for both hydrogen atoms. The LCAO coefficients cA and cB are crucial because the symmetric combination (cA ¼ cB ¼ þ1) yields the bonding MO s orbital with an even (gerade) parity sg and the antisymmetric combination (cA ¼ þ1, cB ¼ 1) leads to the antibonding s* orbital with odd (ungerade) parity su.10–14 A very different approach to depict the H2 molecule is given within valence-bond theory3,11,12,15,16 which will only be discussed very briefly. For example, the so-called Heitler-London-Pauling-Slater wave function3

Fig. 2

MO scheme for the hydrogen molecule with its bonding s and antibonding s* orbitals.

144

Chemical bonding with plane waves

JVB ð1; 2Þ ¼ ð1  hÞJcov ð1; 2Þ þ hJion ð1; 2Þ

(2)

with Jcov(1, 2) ¼ jA(1)jB(2) þ jB(1)jA(2) and Jion(1, 2) ¼ jA(1)jA(2) þ jB(1)jB(2) for H2 is a representation of its total wave function that includes both the covalent and ionic structures and allowing for a variational parameter h between the covalent and ionic parts. In contrast, the MO picture, an “uncorrelated” description, puts both contributions on equal footing.

3.07.1.3

Schrödinger’s and Fock’s equation, Kohn-Sham equations: DFT

To more comprehensively understand chemical bonding, some mathematics is needed to describe it quantitatively. For obvious reasons, quantum mechanics must be employed. Due to the wave nature of electrons (and other atomic entities), position and momentum are no longer deterministic. Instead, the fundamental property of a given quantum system is called a (quantum) state that can provide a probability distribution of a particular observable, such as position, momentum, energy, etc. One of the formulations of quantum mechanics was proposed by Schrödinger in his 1926 paper through an equation17 named after him:
 Z2 2 vJðr; t Þ (3) V þ V ðr Þ Jðr; t Þ ¼ iZ  : 2m vt This equation describes the evolution of the state, J(r, t), of a quantum particle moving in a potential V(r). Moreover, J(r, t) is also called a wave function, and hence the formulation is called wave mechanics, in contrast to matrix mechanics, which is another formulation by Heisenberg.18 The term inside the square bracket of the left part of Eq. (3) is called the Hamiltonian, H, whereas its first term is called the kinetic energy operator: Z2 T ¼  V2 ; 2m

(4)

v þ v þ v being the Laplacian. For systems whose potential is time-independent, J(r, t) can be constructed as with V2 ¼ vx 2 vy2 vy2 iEt a product of a function which only depends on r and e =Z : 2

2

2

Jðr; t Þ ¼ Jðr Þe

iEt= Z:

After inserting (5) into (3) and applying the partial derivative over time, we will get:
 Z2 2  V þ V ðr Þ Jðr Þ ¼ E Jðr Þ; 2m

(5)

(6)

iEt

where e =Z has been taken out because it no longer affects any further calculation. Eq. (6) is called the time-independent Schrödinger equation, or often just called the Schrödinger equation. Moreover, J(r) is called an eigenfunction associated with an eigenvalue (eigen energy), E. Right after its formulation, the equation was used to study the hydrogen spectra and the results agreed with the experimental results.19 Further details, such as fine and hyperfine structure of hydrogen, however, require considering relativistic effects which are out of the scope of our discussion. Furthermore, a system with one electron, like hydrogen, is called a one-particle system and has a special place in quantum chemistry because its eigenfunctions can be used to construct the solutions for manyelectron systems, as discussed later. Eq. (6) can be extended for many-electron systems, such as molecular and solid-state materials, by including all the contributions of each electron and atom into the Hamiltonian: 2 3 N M N X M N X N 2 M X M X X Z2 2 X Z2 2 1 4X ZA e2 X e ZA ZB e2 5 : (7)   V  V  H¼  2m i 2M A 4p3 0 i¼1 A riA r RAB i¼1 A¼1 i¼1 j>i ij A¼1 B>A The first and the second terms, respectively, describe the kinetic energy of electrons and nuclei, whereas the third one is the attractive Coulomb interaction between an electron and a nucleus separated by a distance riA. The fourth and the fifth terms are, respectively, the repulsive Coulomb interaction between two electrons separated by a distance rij and a pair of nuclei separated by a distance RAB. Despite its transparent formulation, the Schrödinger equation is practically insolvable without making further approximations. The first one is called the Born-Oppenheimer approximation20 whereby we assume that the electronic and the nuclear parts do not affect each other and can be solved independently. This assumption is based on the argument that the velocities of nuclei and electrons differ drastically. Furthermore, we drop the Coulomb potential between the atoms. Henceforth, we use the Hamiltonian to mean the electronic part only: H¼

N N X M N X N 2 X Z2 2 1 X ZA e2 1 X e Vi   : 2m 30 r 30 r 4p 4p iA i¼1 i¼1 A i¼1 j>i ij

(8)

Before going further, it is important to remember that electrons are quantum particles. Hence, electrons are indistinguishable, so if two electrons are swapped in a system, the system is the same as before. Second, here is the exclusion principle named after Pauli:

Chemical bonding with plane waves

145

two electrons or more cannot enter the same quantum state.21 As a consequence, the total wave function of an electronic system has to be antisymmetric. Back to Eq. (8)! Although the nuclear parts have been taken out, the problem is still far from being solvable due to the third term correlating all electrons with each other, so that the states of all electrons need to be determined simultaneously to get the total wave function. In other words, Eq. (8) forms a set of coupled partial-differential equations of 3N variables, where N is the number of electrons with two previous rules as the boundary conditions. Keeping in mind that N can be as large as 1023, this problem is impossible to tackle, even with any current supercomputers. The first prominent and simplest approximation to solve the Schrödinger equation for many-electron systems is the one proposed by Hartree22 in 1928. Here one assumes that the total wave function is equal to the product of those of individual electrons, which is later called the Hartree product:

Jðr Þ ¼ ja ðr 1 Þjb ðr 2 Þjc ðr 3 Þ/;

(9)

where ja(r1), jb(r2), jc(r3), / describe one-electron orbitals. Upon inserting this approximated wave function into Schrödinger’s equation and introducing a Lagrange multiplier, 3 i, to enforce orthonormalization, the minimization of the Lagrange functional with respect to the one-electron orbitals follows the variation principle, namely "Z # X Z v  2 dr drjj ¼ 0: (10) J ð r Þ H J ð r Þ  3 ð r Þj i i vjj i This leads to a set of one-electron equations


 Z2 2 V þ Veff ðr Þ ji ðr Þ ¼ 3 i ji ðr Þ; 2m

(11)

in which the original potential has been replaced by an effective one, Veff(r), which consists of the potential by the nuclei and the potential of the average interaction felt by an electron in the vicinity of other electrons: Z M 1 X ZA e2 e2 rðr 0 Þ (12) dr 0 Veff ðr Þ ¼  þ ; 4p3 0 4p3 0 A riA jr  r 0 j where rðr 0 Þ ¼

2 P   jj ðr 0 Þ is the density of (N1) electrons. The obvious advantage of the Hartree approximation is that the partial

jsi

differential equations have been decoupled and can be solved independently. However, despite offering decisive progress in dealing with many-electron systems, it ignores the anti-symmetry principle of electronic wave functions. To cure this problem, in 1929 Slater23 decided to construct the wave function as a (Slater) determinant of so-called one-electron spin orbitals:      j ðr Þ j ðr Þ / jN ðr 1 Þ   1 1 2 1       j ðr Þ j ðr Þ / j ðr Þ   2 2 2 1 2 N 1  : (13) Jðr Þ ¼ pffiffiffiffiffiffi   N!   « « 1 «        j1 ðr N Þ j2 ðr N Þ / jN ðr N Þ    A spin orbital, ji, is a function of the space and spin coordinates. Unless described otherwise, spin orbitals will be used interchangeably with one-electron wave functions in future texts. Furthermore, later in 1930 Fock24 reduced the many-electron problem into a set of linear one-electron equations: Fji ðr Þ ¼ 3 i ji ðr Þ;

(14)

which is later known as the Hartree-Fock (HF) equations. Here, F is the Fock operator which plays a role of a one-electron energy operator. Furthermore, the Fock operator consists of three terms: F¼hþ

N  X

 Jj  Kj ;

(15)

j¼1

where h¼

M Z2 2 1 X ZA e2 ; V  4p3 0 A riA 2m

(16)

146

Chemical bonding with plane waves

Jj ¼

e2 4p3 0

Kj j i ðr Þ ¼

e2 4p3 0

Z

Z

dr 0

jj ðr 0 Þjj ðr 0 Þ

dr 0

jr  r 0 j

;

jj ðr 0 Þji ðr 0 Þ jr  r 0 j

jj ðr Þ:

(17)

(18)

The operator h describes an electron’s kinetic energy and its potential energy due to its interactions with all M nuclei surrounding PN j¼1 Jj , is called the Coulomb term and represents the potential energy of an electron due to an average field P created by all N electrons in the system. The third term, N j¼1 Kj , is called the exchange term, and its existence mirrors the antisym-

it. The second term,

metry of the wave function. Note that when j ¼ i, an electron will have non-physical self-interactions from the Coulomb and exchange terms, but these two terms are conversely equal and cancel each other. Furthermore, as detailed in Eq. (18), a single operator Kj acting on a spin orbital ji will “swap” the electrons in ji and jj. The exchange term is a purely quantum mechanical effect and has no classical analogy. To solve Eq. (14) several sophisticated algorithms have been developed but will not be described in detail here. Nevertheless, the integro-differential HF equations have to be solved self-consistently, so the solving process starts with a set of trial spin orbitals for calculating the Fock operator, and then the HF equations are solved for a new set of spin orbitals that repeat the solving process recursively until all spin orbitals converge. Furthermore, Roothaan and Hall in 1951 proposed how to solve the HF equations by employing more general, non-orthogonal basis functions to construct spin-orbitals and transforming the original HF integrodifferential equations into algebraic equations,25 which is computationally more practical to implement. Due to the use of the Slater determinant, the HF method has been a prominent and reliable tool in quantum chemistry since the time it was formulated. However, since the elements of the exchange term matrix are obtained from two-electron integrals of basis functions, the HF method scales as N4, where N is the number of basis functions. By doubling the number of basis functions, the computational cost, i.e., time, memory, etc., will roughly increase by a factor of 24 ¼ 16. This restricts the HF method to modestly sized molecular systems. Still, electronic correlation is missing in the HF theory. Regardless of the spin states, electrons repel each other through the Coulomb force of individual electrons, not their average density. The missing electronic correlation is more clearly seen by looking at the Hartree products that construct the Slater determinant. A Hartree product implies that the probability of simultaneously finding all electrons at their given positions is the products of probabilities of individual electrons. Although electronic correlation, or sometimes called Coulomb correlation, is easy to comprehend, it is very hard to formulate and then to solve. The simplest mathematical definition can be written as the difference between the exact energy and the HF energy: Ecorr ¼ Eexact  EHF :

(19)

Due to this missing electronic correlation, the results from the HF method often deviate significantly from the experimental data. When applied to solids, the HF method may behave even worse because it wrongly predicts that simple metals, which are modeled using free electrons, are insulators.26 Luckily, in most of the molecular systems made up by light atoms the effect of the electronic correlation is not that strong.27 Therefore, the HF method works quite well for this kind of systems, especially organic molecular compounds.26 More sophisticated methods, however, have been developed based on the HF method to include the electronic correlation, such Møller-Plesset perturbation theory,28 configuration interaction,27,29 etc., but will not be discussed further here. Another approach commonly used in quantum chemistry and physics is density-functional theory (DFT). Unlike the two previous approaches, DFT utilizes the electron density instead of the wave function and bases its formulation on two surprisingly simple theorems proposed by Hohenberg and Kohn.30 First, the external potential, and hence the total energy, of any interacting system is a unique functional of the ground state electron density. So, by knowing the ground state density of the system of interest, the Hamiltonian is fully determined and the many-electron wave functions can be obtained. Second, the total energy functional can be defined in terms of the density and only be minimized globally by the true ground-state density. The proofs of both theorems can be found in many textbooks.29,31 Despite their robustness, the two theorems do not provide any practical implementation on how to calculate the exact density and the corresponding exact energy functional. To make DFT practicable as a quantum chemistry tool, Kohn and Sham proposed a further approximation by utilizing an auxiliary system of non-interacting electrons32: Z2 Haux ¼  V2 þ V ðr Þ: 2m

(20)

Up to this point we do not make any assumption on how V(r) should look like. The associated density of this auxiliary system is given by: rðr Þ ¼

N X

jji ðr Þj2 :

(21)

i¼1

Here, ji represent one-electron orbitals, known as Kohn-Sham (KS) orbitals, in terms of which the kinetic energy of noninteracting electrons, Ts, can be written as

Chemical bonding with plane waves Z Ts ½r ¼

 Z2 2 dr ji ðr Þ  V ji ðr Þ: 2m

147

(22)

Furthermore, the KS idea is to use the density of the auxiliary system to represent the density of the many-body system, so the Coulomb interaction energy between electrons, i.e., the Hartree energy (EH), can be represented as: Z Z e2 rðr Þ rðr 0 Þ (23) drdr 0 ; EH ½r ¼ 8p3 0 jr  r 0 j and total energy functional of the given interacting-electron system as: Z E½r ¼ Ts ½r þ dr Vext ðr Þ rðr Þ þ EH ½r þ EXC ½r;

(24)

where Vext is the potential energy due to all nuclei in the system and EXC is the so-called exchange-correlation energy functional. The latter is the most enigmatic term in DFT and various methods have been developed to approximate EXC. Unlike the HF method, the KS approach also cleverly anticipates EXC and separates it from the rest to allow its systematic improvement. Using the variational principle and the Lagrange multiplier to enforce the orthogonality of the KS orbitals, one derives a set of linear one-electron equations, known as the Kohn-Sham (KS) equations: HKS ji ðr Þ ¼ 3 i ji ðr Þ; (25) where HKS is the same Hamiltonian given by Eq. (20) and the potential is now given by dEH dEXC V ðr Þ ¼ Vext ðr Þ þ þ ¼ Vext ðr Þ þ VH ðr Þ þ VXC ðr Þ: drðr Þ drðr Þ

(26)

The simplest approximation to EXC, which was also firstly introduced by Kohn and Sham,32 is called the local density approximations (LDA). Here, the exchange-correlation energy of a given system is approximated with the one of the homogenous electron gas with the same density. This means EXC only depends on the (local) density at each point in the system:
 Z r ð r Þ (27) ELDA ½ r  ¼ dr r ð r Þ ε XC XC εXC is usually separated into two terms, namely the exchange and correlation terms, i.e., εXC(r) ¼ εX(r) þ εC(r). In this case the exchange term is represented analytically by the Dirac functional33:  3 3 1 3 rðr Þ: εX ðrÞ ¼  (28) 4 p εC(r), however, does not have an analytical form but has been accurately determined by quantum Monte Carlo calculations and then parameterized.34 Despite its primitive assumptions, namely slowly varying densities, LDA works surprisingly very well with many systems, even for atoms and molecules where densities do not vary slowly, the reason being some error cancellation from the exchange and correlation terms.35–38 A better approximation to EXC than LDA is the so-called generalized gradient approximation (GGA) for which the functional also depends on the gradient of density: Z dr rðr Þ εXC ½rðr Þ; Vrðr Þ: (29) EGGA XC ½r ¼ More advanced approximations to EXC have also been developed with their advantages and disadvantages. For more information on this topic, readers may to refer to, for instance, Ref. 31. The reader will recall that, in order to solve the electronic structure of many-body systems, all of the three methods presented above aim for downsizing many-body problems into one-electron problems, such that one-electron wave functions and their eigen energies are the products. In the next sections, we will show further, mostly within the DFT framework, how this simplification to one-electron problems has greatly advanced the exploration of the electronic structure of many materials. It goes without saying that, in order to solve one-electron equations for their wave functions, basis functions are needed. In general, there are two kinds of basis functions, namely extended and localized functions. Plane-waves (PWs) belong to the former kind, whereas any type of atomic orbitals (AOs) to the latter. The combination of the former and the latter, through mixing or augmentation, is also possible as basis functions. These different basis sets have their own advantages and disadvantages and depending on the nature of the system, one type can be more beneficial than the other. In the coming sections, we will focus on plane-waves due to their simple application, and atomic orbitals, which are the natural basis functions for chemical bonding analysis and for extracting various kinds of chemical information.

3.07.1.4

More complicated molecules, LCAO in general

When going from a simple molecule such as H2 to more complex ones, things naturally get more complicated. With more orbitals to optimize, more electrons and their interaction among each other, more nuclei to include into the calculation, more approximations are needed, as illustrated in this and the following sections. The size of the system that can be examined within the quantum-mechanical framework in reasonable time is naturally limited due to computational resources. Still, the study of small molecules can give us new insights into much larger problems, for example,

148

Chemical bonding with plane waves

the enzyme family of nitrogenases which are metalloproteins that convert nitrogen and hydrogen into ammonia,39–41 an essential step in the biosynthesis of amino acids and, therefore, for plants and animals. The industrial Haber-Bosch process of the ammonia synthesis requires temperatures up to 500  C and pressures up to 350 bar for that reaction.41 In contrast, the bio-catalyzed reaction occurs at standard conditions due to the special structure of nitrogenases containing iron sulfide units acting at the active sites.40 Let us look at the quantum-chemical study of a diatomic sub-unit of FeS to improve our general understanding of proteins like nitrogenases. We will only concentrate on the so-called MO scheme and a qualitative representation of the molecular orbitals, modeled from results of an ab initio Hartree-Fock calculation,42,43 for didactic purposes. As seen from Fig. 3, the MO scheme for the 5D experimental state44 (it is not 5S as one might assume) for FeS contains many more interactions between the valence atomic orbitals than in the simple H2 case. Three bonding (1 s, 2  1 p), three non-bonding (2  1 d, 2 s) and three antibonding molecular orbitals (2  2 p*, 3 s*) can be expected and their possible shapes are depicted in Fig. 3. The orbital character of 1 s and 3 s* is primarily dominated by S 3 pz and Fe 3 dz2 (with little Fe 4s character), whereas 1 p and 2 p* consist of an interaction between one of the other S 3p and other Fe 3d orbitals. The nonbonding 1 d level remains a Fe 3d level, since there is no symmetry-related orbital from S. The nonbonding 2 s shows symmetry-given interaction between Fe 4s and 3 dz2 orbitals. For such complex diatomic molecules, we express the LCAO ansatz quite generally: Xn Jj ðr Þ ¼ m¼1 cmj jm ðr Þ: (30) The AOs are solutions of the Schrödinger equation for hydrogen-like atoms. For an exact representation, an infinite number of basis function terms would be needed, but obviously this is not a practical idea in reality. If we intend to use Hartree-Fock theory for such complex molecules, the Roothaan-Hall formalism (see above) then leads to matrix eigenvalue equations FC ¼ SC3;

(31)

where F, the Fock matrix, is the matrix representation of the Fock operator, C is the square matrix of the expansion coefficients, S the overlap matrix of two orbitals and 3 is a diagonal matrix containing orbital energies.11-13,24,45 Not only can the LCAO approach be employed within the MO framework. Instead, in case of periodic (“infinite”) systems like crystals, the many AOs will find together to crystal orbitals (CO), which is also dubbed as tight binding approach in physics, as covered in the next sections.

3.07.1.5

Plane Waves and Bloch’s theorem

Translationally invariant, that is, periodic systems (¼ crystals), at least theoretically, contain an infinite number of atoms. At first sight, these extended systems seem to be nearly impossible to solve when using local orbitals, say using the HF approach. Plane waves (PW) are a way to get out of this misery simply because of their delocalized nature to describe periodic systems. Plane waves are solutions of the Schrödinger equation under application of a constant external potential to approximate the behavior of a wave function for the interatomic region or interstitial sites in a crystal by a single plane wave, however, this will not describe the atomic local characteristic in a satisfying way. Bloch’s theorem naturally introduces plane waves as

Jk ðr þ TÞ ¼ Jk ðr Þ eikT ;

(32)

so that by applying the translation T to a wave function (Bloch function) Jk(r) depending on a wave vector k, it is unchanged, except for a phase factor incorporating the periodicity of the system. The idea behind Bloch’s theorem can be better understood after having a look at Fig. 4: all knowledge about the system is contained in the unit cell (marked with blue), and the rest of the system can be replicated by exploiting the translational symmetry. Likewise, the total wave function can be expressed in terms of plane waves

Fig. 3

MO scheme for the iron sulfide molecule with its different orbitals.

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149

Fig. 4 Pictorial representation of Bloch’s theorem. A 2D, hexagonal sheet of atoms (dark circles). Only the (blue) unit cell needs to be known, the rest of the system can be constructed from the translational symmetry, described by the translation vector T. Reprinted from Comprehensive Inorganic Chemistry II, Vol. 9, V. L. Deringer, R. Dronskowski, Computational Methods for Solids, pp. 59–87, Copyright (2013), with permission from Elsevier.

eikr XN c k ðGÞeiGr U G¼0

Jk ðr Þ ¼ pffiffiffiffi

(33)

and including the reciprocal lattice vector G and the volume of the unit cell in real space U. ck are a set of PW basis set coefficients, not to be confused with the LCAO coefficients. This linear combination of plane waves improves the description of the wave function in vicinity of the atomic nuclei. The crystal orbital can also be expanded by a linear combination of atomic orbitals (LCAOCO), just like in the molecular case (see. Eq. 30), with the Bloch sum jmk(r) Xn 1 X Jk ðr Þ ¼ m¼1 cmk jmk ðr Þ; jmk ðr Þ ¼ pffiffiffiffi eikR jðr  RÞ (34) N R where R is the real-space lattice vector and N is the number of lattice sites. When comparing Eqs. (33) and (34), one realizes that any local function can be expressed as a linear combination of (ideally infinite) plane waves and vice versa.46–49

3.07.1.6

One-dimensional systems

Despite the generality of Bloch’s theorem and its associated plane waves of sine or cosine (i.e., essentially delocalized) character, it is straightforward to give it a very a local and chemical meaning. If we assume that atoms are building up the entire crystal, we can quickly write down a Bloch sum in which the extended wave function (the crystal orbital) is expanded in those atomic orbitals we already know from molecular quantum chemistry. Our friends from the physics community are doing essentially the same thing, and then this goes under the name “tight-binding approach”.50 Please note that this term does not imply that the atoms are tightly bound to each other; on the contrary, it means that the electrons are so tightly bound to its parent atoms that the original oneelectron atomic wave functions (the atomic orbitals) still serve us pretty well, at least well enough to handle the system. In other words, the chemical interaction between the atoms is the perturbation, hence this approach directly targets chemical bonding. For a one-dimensional system of hydrogen atoms, the linear-combination-of-atomic-orbitals-to-crystal-orbitals (LCAO-CO) Bloch construction then reads jk ¼

N X

eikna 4n ;

(35)

n¼1

in which we have expanded the atomic orbitals 4n (1s atomic orbitals in this case) into a crystal orbital jk which now depends on the quantum number k derived from reciprocal space. The interatomic distance is given as multiples of a (say, 1 Å), and the atomic orbitals are numbered from 1 to N (the total number of unit cells per crystal). It would be very time-consuming to calculate the course of jk over the entire reciprocal space but we might want to do that only at a few, “representative” k points or, simpler still, just look at the band’s energetic behavior at the center of reciprocal space called G (k ¼ 0  2p/a ¼ 0) and at its outer edge X (k ¼ 1/ 2  2p/a ¼ p/a), thereby defining the size of the so-called Brillouin zone (the unit cell of reciprocal space). At G, the exponential function equals unity, so the crystal orbital jk is nothing but the sum of the individual atomic orbitals, 41 þ 42 þ 43 þ /, clearly the bonding combination of atomic orbitals, and thus should be lowest in energy. At X, when k equals p/a, the exponential term alternates between þ1 and 1, so the crystal orbital will read  41 þ 42  43 þ /, the antibonding highest-energy combination. The qualitative plot including sketches of the crystal orbital is depicted in Fig. 5 (left), the one-dimensional analog of the molecular orbital diagram of the H2 molecule. Exactly in the middle of the Brillouin zone, between G and X, we have also indicated that the crystal orbital carries nodes at every third H atom. The dispersion of the band structure, that is, the energy difference between highest and lowest energy, increases if the interatomic distance shrinks because it scales with the orbital overlap or orbital interaction, to be precise.

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Chemical bonding with plane waves

Fig. 5 One-dimensional band structure of the infinite hydrogen chain spaced at 1 Å (left) and the infinite nitrogen chain spaced at 2 Å (right). The characters of the bands have been pictorially indicated. Reprinted from Dronskowski, R. Computational Chemistry of Solid State Materials: A Guide for Materials Scientists, Chemists, Physicists and Others. Wiley-VCH: Weinheim (2005) with permission from John Wiley and Sons.

The band structure of a one-dimensional chain of nitrogen atoms looks a little more complicated but follows the same procedure upon computational construction. Let us include the one 2s and the three 2p valence orbitals of nitrogen and let us also assume an NeN distance of 2 Å, then Fig. 5 (right) provides the respective band structure. Similar to the 1s crystal orbital of hydrogen, the 2s crystal orbital of nitrogen moves up in energy upon going from G to X but the dispersion is much smaller due to the wider NeN distance and the more contracted 2s atomic orbital. Likewise, the p-like crystal orbitals arising from the 2px and 2py atomic orbitals move up in energy but their dispersion is even weaker, simply due to the less efficient p-like orbital interaction. The crystal orbital that is generated by the 2pz orbitals of nitrogen, however, is quite different. Not only is its dispersion comparable to 2s, the trivial reason being the s-like orbital overlap, but the electronic band runs down from G to X, easily understandable by the orbital icons; this very crystal band is antibonding at G but bonding at X. In other words: while the orbital overlap controls the band’s dispersion, its topology (going up or down) is determined by the orbital symmetry.51 As regards symmetry, please also note that we have simplified things a little because a careful analysis shows that the 2s and the 2pz bands transform according to the same irreproducible representation (i.e., have identical symmetry properties), so they mix. As a consequence, the lower band dubbed 2s also carries a little 2pz contribution whereas the upper 2pz band also contains some 2s admixture. The closer they become in energy (that is, the more they approach the X point), the stronger the mixing. It is rather trivial to extend this to atoms with other orbital symmetries such as 3d, and it is also trivial to allow for more than one atom per unit cell because the LCAO-CO expansion then simply includes additional terms; nothing changes, at least not fundamentally.26 Likewise, we can go to higher dimensions such as two or three. For a quadratic system (say, a checkerboard of carbon atoms) the Brillouin zone of reciprocal space would also be quadratic, and we would analyze the various bands at more “special” points, not only at the very center which still reads G (kx ¼ ky ¼ 0) but also at X (kx ¼ p/a, ky ¼ 0) and Y (kx ¼ 0, ky ¼ p/a) and M (kx ¼ ky ¼ p/a), for example. For a three-dimensional (primitive) cubic crystal, the special points of reciprocal space would then be G (kx ¼ ky ¼ kz ¼ 0), X (kx ¼ p/a, ky ¼ kz ¼ 0), M (kx ¼ ky ¼ p/a, kz ¼ 0), and R (kx ¼ ky ¼ kz ¼ p/a), nothing to really worry about. And yet, a new phenomenon pops up for the simplest system, the one-dimensional chain of hydrogen atoms. Fig. 6C shows what we have seen before in Fig. 5 (left), namely the corresponding band structure of that hydrogen chain. Here, the lattice parameter a0 comprises just one H atom per unit cell. If we were to double this lattice parameter to a new a(¼ 2  a0 ) such that the unit cell contains two H atoms, there would also be two bands in the band structure, as shown in Fig. 6B. Mathematically, this is a trivial operation which may be graphically symbolized by “folding back” half of the crystal orbital seen on the right. Now, it is highly instructive to compare the doubly degenerate upper and lower crystal orbitals at the X point in (B), and they match the crystal orbital in the very center of the Brillouin zone of (C). The crystal-orbital icons at X indicate that both bands not only differ by a phase shift, they also have the same crystal symmetry because both are symmetric with respect to the hydrogen chain’s horizontal mirror plane. As a fundamental consequence, both bands must interact for this very electron count (1 electron per crystal orbital) of the H chain and will repel each other. We are witnessing what was first predicted by Peierls52 and goes under his name, the famous Peierls distortion, a spontaneous symmetry lowering arising for a half-filled band of a one-dimensional system. Because upper and lower bands “repel” each other (strongly so at X, but the weaker the more we immerse into the Brillouin zone), there results a distortion which leads to a shorter HeH distance (say, 0.8 Å) and a longer one (say, 1.2 Å). Alternatively expressed, the one-dimensional hydrogen chain collapses into a loose chain of H2 molecules, and this is the reason why the lower band has the character of the bonding sg molecular orbital whereas the upper band stands for the su* molecular orbital, both depicted in Fig. 5A.

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151

Fig. 6 As in the left part of Fig. 5 but drawn differently, with two H atoms per unit cell, subfigure (B), and again with one H atom per unit cell, subfigure (C). Because of energetic degeneracy for half-filling (Peierls case) the one-dimensional system structurally distorts to yield shorter (0.8 Å) and longer (1.2 Å) HeH distances, given in subfigure (A), and lowers its total energy. Reprinted from Dronskowski, R. Computational Chemistry of Solid State Materials: A Guide for Materials Scientists, Chemists, Physicists and Others. Wiley-VCH: Weinheim (2005) with permission from John Wiley and Sons.

Chemists will not at all be surprised that a one-dimensional hydrogen system will turn into isolated H2 molecules but this should not belittle the fundamentality of the Peierls scenario. It is a general phenomenon where some translational symmetry is lost and leads to a lower-energy, more stable system. It may also be regarded as the solid-state equivalent of the older and likewise famous Jahn-Teller instability53 sometimes found for molecular systems; here, the resulting energetic stabilization also results from coupling the nuclear coordinates with the electronic structure but the system is lowered in point-group symmetry. It has long been thought that the Peierls scenario is strictly valid only for one-dimensional systems but intense research has shown that Peierls distortions are found for two-dimensional as well and also for three-dimensional systems.54,55 Even some of the simple structures of the elemental solids may be regarded as having resulted from previous Peierls distortions without noticing the (high-symmetry) aristotypes from which they have originated. We simply “got used” to the low-symmetry structures and overlooked the high-symmetry ones. For example, the element tellurium with its helical chains of Te atoms is the result of a threedimensional Peierls distortion away from a simple cubic (a-Po type) structure, and here the bands have been filled by 2/3 and, hence, backfolded three times in reciprocal space.56

3.07.1.7

Concept of charges and bonding Indicators (COOP and COHP)

To more quantitatively characterize chemical bonds in terms of their covalent and ionic character, numerical bonding indicators come into play. For an ionic bond, it is pretty straightforward to use the ionic charges for quantification. The cohesive forces in an ionic compound stem from the electrostatic force (also called Coulomb force) caused by the interaction of like and unlike charged ions. As already alluded to in the introductory section, the formation of ions occurs when the difference in the electronegativity (DEN) of the particular atoms is so large that the valence electron(s) of one atom is transferred to the other atom, forming a cation and an anion, respectively. The electrostatic interaction between ions with unlike charges is attractive and the one between ions of like charge is repulsive. This charge concept has been around for a long time57 and two of the most famous wave function based methods to partition electrons and assign charges to atoms are the Mulliken and Löwdin population analyses.58–60 It has recently become possible to extract atomic charges and orbital populations from plane waves,61,62 too, so we will cover the population analyses in reciprocal space. Such technique for Mulliken and Löwdin population analysis is computationally cheap and yields reasonable charges.61,63,62 In more general terms, if the crystal wave function j(k) is constructed as a linear combination of atomic orbitals cm(k) in the LCAO-CO approximation,45,64 the so-called gross population (GP), a measure for the orbital occupation, can be calculated from the LCAO coefficients such that the Mulliken or Löwdin charge for an atom A is obtained from the difference of the number of the atom’s valence electrons N (when using pseudopotentials) and the gross (orbital) population GPm X qA ¼ N  GPm : (36) m˛A

Clearly, charge information is useful for understanding and designing new (functional) materials, like batteries or thermoelectrics, or to find new compounds for phase-change materials. In the quantum-chemical community, both Mulliken and Löwdin charges are often understood as being highly basis-set dependent due to the changing locality of the basis functions, but this problem no longer persists with plane waves as a consequence of their nonlocality.

152

Chemical bonding with plane waves

Simple textbook-like examples to illustrate the functionality of Mulliken (or Löwdin) charges are NaCl and CsCl, two iconic  and ionic cases. In both cases, depicted in Fig. 7, NaCl in its experimental, room-temperature structure with space group Fm3m  the cations Na and Cs show the expected charges of around þ1, whereas the Cl anion exhibits a charge of CsCl with Pm3m, around 1, depicting the typical ionic picture expected for alkaline metal halides. We reiterate that methods like the one of Mulliken or Löwdin are wave function-based. There are also (real-space) density-based charge analysis methods, such as the one by Bader within the Atoms in Molecules (AIM) framework.65,66 The density-based technique shall guarantee uniqueness, but comes with the cost of losing the wave function’s phase (sign) information. Instead, the topology of the electron density r(r) is used to partition the system into a set of volume elements V, corresponding to atoms. By calculating the so-called zero-flux surfaces, contour maps of the atomic electron density are obtained, and integration of the electron density within a certain (atomic) volume element for atom A with nuclear charge ZA, the charge arrives at Z rðr Þdr: qA;Bader ¼ ZA  (37) V˛A

Since the Bader charge analysis does not allow for atoms to overlap, caution is advised,61,63 especially in covalent compounds, where orbital overlap is significant. The impact of the electrostatic forces is captured by the Madelung energy,67–71 which may be used to estimate the stability of ionic compounds.61,72 In the case of molecular crystals or intermetallic compounds with small ionic contributions, the real charge is unclear, so Madelung energies cannot be accessed easily unless quantum-chemically derived charges are calculated. Coming back to the covalent bond, its idea is strongly linked to the Lewis model of the electron pair bond formed between two atoms with similar electronegativities.73 It gives a first quantitative measure of the bond strength in, say, organic molecules where CeC single, double, and triple bonds are possible. In the hands of a skilled chemist, this model enables the understanding and prediction of stability and reactivity of the molecule or certain functional groups. However, as electrons repel each other, a model that is based purely upon the electron density does not explain the quantum chemical situation to a satisfying extent. We also recall that the density lacks any phase information. Accordingly, distinguishing between bonding and anti-bonding interactions is not possible per se. As such, wave function-based methods are required. In molecular systems, the orbital overlap and Hamilton integrals Smn and Hmn are used to quantify the strength of a covalent bond. Z Z b n ds Smn ¼ fm fn ds; Hmn ¼ fm Hf (38) As solid-state materials are translationally invariant, this feature has to be respected when bonding indicators for these materials are formulated. Eq. (38) must somehow include the quantum number k that defines the position in the reciprocal space originating from Bloch’s theorem (section 3.07.1.5). The resulting band structures, introduced in section 3.07.1.6, will now serve to introduce the bonding indicators for the solid state. Before we dive into the concept of bonding, another difficulty of the band structure plots must be solved. For one-dimensional crystals such as the linear hydrogen chain, the choice of the path through the Brillouin zone seems obvious, from the G point to the X point. For two- and three-dimensional crystals, any path is equally valid because here, k is a vector with two or three dimensions, respectively. The solution to this issue is the integration of the band structure over the entire reciprocal space which results in the density of states (DOS). The DOS yields information on the number of states (actually, one-electron levels) in a given energy interval and its value is inversely proportional to the slope of the band structure.  1 vE (39) DOSðEÞf vk

Fig. 7 Mulliken charges of NaCl and CsCl. Reproduced with permission from Ertural, C.; Steinberg, S.; Dronskowski, R., Development of a Robust Tool to Extract Mulliken and Löwdin Charges From Plane Waves and Its Application to Solid-State Materials. RSC Adv. 2019, 9 (51), 29821– 29830dPublished by The Royal Society of Chemistry.

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153

For chemical bonding analysis, however, the DOS is not an appropriate tool as it only shows the position of levels, and not the way they interact. The information on bonding and anti-bonding can be introduced by including an additional quantity. The first approach is the crystal-orbital overlap population (COOP) by Hughbanks and Hoffmann which premiered in 1983.74 X   COOP ¼ Smn wk cm;jk cn;jk d 3 j ðkÞ  E (40) j;k

COOP is an overlap-weighted density of states and corresponds to the two-center case of the Mulliken population analysis in the solid. Positive values of the COOP are interpreted as bonding levels, while negative values are attributed to antibonding ones. When the COOP is integrated over the energy up to the Fermi-level, the ICOOP is achieved, to be understood as the bond population or number of electrons that occupy the covalent bond. Fig. 8 shows the band structure of the linear hydrogen chain from Fig. 5, but with color coded bonding information of the 1sorbitals. It consists of strictly bonding interactions at the G point that becomes fully antibonding at the X point. When the band structure is then integrated over reciprocal space, the density of states is achieved which, when multiplied with the overlap integral, results in the COOP plot on the right-hand side of Fig. 8. This plot can easily be explained by the orbital interactions shown in the band structure. At 8 eV, the HeH interaction has its maximum in the bonding region as all orbitals are in phase. With increasing energy, the COOP decreases until it reaches zero at the Fermi level (0 eV) despite the existence of states in the DOS at this point, a consequence of the non-bonding type of interaction. With further increasing energy, the COOP indicates an antibonding interaction that reaches its maximum at 30 eV; here, the phases of the 1s-orbitals are strictly alternating so the interaction between two neighboring orbitals is always antibonding. In sum, the interaction in the linear hydrogen chain is bonding as all bonding levels below the Fermi-level are occupied and the antibonding ones are unoccupied. Based on the COOP approach, the crystal-orbital Hamilton population (COHP)75 was developed within density-functional theory which utilizes the Hamilton integral (hence its name). X   COHP ¼ Hmn wk cm;jk cn;jk d 3 j ðkÞ  E (41) j;k

The COHP is a partitioning scheme for the band structure energy that consists of the Kohn-Sham eigenvalues. Its appearance is very similar to the COOP with one important exception. Since the COHP is an energetic measure for the covalent bond strength, negative values indicate a bonding behavior while positive values are considered anti-bonding. The energy-integrated COHP (the ICOHP), is an energetic measure for the bond strength, but it does not equal the experimental bond energy because it only covers effective one-particle energies. In order to result in a formalism analogous to the COOP, negative COHP are plotted as a function of the energy. The general procedure is completely analogous to one-dimensional systems as it can be seen in Fig. 9 for diamond which is a textbook example of covalency in the solid state. Fig. 9 shows the schematic path from the band structure to the final bonding indicators. The band structure is analogous to the one-dimensional case of the linear hydrogen chain with one important exception: the quantum number k is replaced by a threedimensional vector k. This leads to an arbitrariness in the set-up of the path through the Brillouin zone (see discussion before) which usually follows certain high-symmetry points. However, this is done solely by convention and any other path is just as correct. A more unambiguous representation goes by the density of states which is derived from the band structure by integration over the whole Brillouin zone. That way, all k-points are equally represented and the final result does not depend on their sequence. In the DOS, we encounter yet another convention: Chemists prefer to plot the energy as y-axis in order to illustrate the familiarity between DOS and band structure. Physicists, on the other side, plot the DOS with exchanged axes because mathematically, it is a function of the energy. As said before, the DOS only tells us where the states are and not what they do, so an additional quantity must be incorporated, either the overlap matrix element Smn or the Hamilton matrix element Hmn, resulting in the COOP and COHP techniques. At low energies, the orbital interaction of 2s and 2px/y/z is strictly bonding. In the framework of valence bond theory, this interaction is

Fig. 8 Graphical derivation of the COOP. Left: Band structure of the linear hydrogen chain. Middle: Density of states (DOS) of the linear hydrogen chain. Right: Crystal orbital overlap population (COOP). The energy is relative to the Fermi-level.

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Chemical bonding with plane waves

Fig. 9 Band structure, density of states, crystal orbital overlap population, and crystal orbital Hamilton population for diamond calculated using the LOBSTER program. The energy is shifted relative to the Fermi level. COOP and COHP are per bond.

formed by four sp3 orbitals. The molecular orbital theory deals with less localized s molecular orbitals that are formed by linear combination of one 2s and three 2px/y/z that are located on the carbon atoms. The bonding behavior persists up to the Fermi level, at least for the COHP. The COOP in Fig. 9 shows a very weak anti-bonding fraction, at least plausible by some over-estimation of anti-bonding seen above the Fermi level. In the COHP, this anti-bonding region is in the same order of magnitude, in fact it has the exact same integral as the bonding region, so both bonding and antibonding interaction cancel each other. This relation is quite reassuring as this is an analogy to molecular orbital theory where there is an antibonding (atomic orbital) interaction for each formed bonding AO-interaction. In its original definition, the COHP suffered from a problem connected to its origin in the band energy. Within DFT, the total energy of a neutral system is constant for any arbitrary shift of the external electrostatic potential (i.e., the energetic zero-point). This does not apply to the band energy since it covers only the electronic energy, which refers to a charged system. The potential of a real crystal depends on its surface as well as its volume, but both have no meaning in the theoretical description of a solid-state material as it extends into infinity without any surface. Thus, the electrostatic potential has no physical meaning in a DFT framework and is usually defined arbitrarily. As a consequence, the comparison of (I)COHP of different crystal structures is highly questionable as they differ significantly in their Madelung fields. To solve this issue, a covalency energy Ecov was developed that subtracts the potential shift from the total COHP and arrives at a quantity that is supposed to be invariant towards said shifts.76  X   Hmm þ Hnn Ecov m;n ¼ Hmn  cmj cnj d E  3 j Smn (42) 2 j However, this issue can alternatively be solved when an orthonormal basis set is employed for the calculation of the COHP. Let us consider a Hamilton matrix element for an arbitrary electrostatic potential

   b fn (43) Hmn ¼ fm  H between two orbitals fm and fn. If the electrostatic potential is then shifted by a constant value of V0, this can be represented in the Hamilton integral by means of perturbation theory:     

  V0 b þ V0 fn ¼ fm  H b fn þ fm jV0 jfn Hmn ¼ fm  H (44) ¼ Hmn þ Smn V0 ¼ Hmn The shift of the electrostatic potential does not affect the value of the Hamilton matrix element in case of an orthonormal basis set, that is Smn ¼ 0. This relation does not hold for on-site elements, as they have non-zero overlap:     

  V0 b þ V0 fm ¼ fm  H b fm þ fm jV0 jfm Hmm ¼ fm  H (45) ¼ Hmm þ Smm V0 ¼ Hmm þ V0 This way, it is possible to compare interatomic COHP and ICOHP of different crystal structures without methodological errors that could render the final results useless. Although the mathematical formalism discussed above is quite straightforward, we will consider a number of systems in order to support it. Fig. 10 shows the ICOHP for several molecular compounds in gaseous as well as crystalline phases. The ICOHP values of gaseous and crystalline molecules closely resemble a straight line, corroborating negligible impact of certainly differing electrostatic potentials. Another surprising feature of Fig. 10 is the separation of bonds that are qualitatively

Chemical bonding with plane waves

Fig. 10

155

ICOHP per bond for molecular compounds in crystalline and gaseous phase. The color code refers to the classically attributed bond order.

described as single, double and triple bonds. In the lower left corner, singly bonded molecules such as halogens, hydrogen halides, and SeO and PeO single bonds in sulfuric and phosphoric acid can be found with ICOHP values between 4.5 eV and 10 eV. The ICOHP value of diamond (9.7 eV) fits very well into this context, as it is qualitatively interpreted as a single bond. At higher ICOHP, up to 20 eV, double bonds can be found as they exist in sulfuric and phosphoric acid as well as molecular oxygen and carbon dioxide. At even larger ICOHP values, the CeN and NeN triple bonds of hydrogen cyanide and molecular nitrogen form the upper end of the present example. Qualitatively speaking, the ICOHP values of classic double bonds are roughly twice as large as the single bonds and the ICOHP of triple bonds are approximately three times the size of single bonds. To a chemist, this correspondence comes to no surprise because a double bond includes twice the number of electrons as a single bond. However, this correlation between ICOHP as a quantitative bonding indicator and the long-known yet qualitative concept of bond multiplicity support the usefulness of the COHP as a measure for covalent bond strength.

3.07.2

Part II: Methods

3.07.2.1

Basis functions

A DFT calculation always starts with trial KS wave functions built with a set of basis functions, but how do we choose the most suitable basis set for our calculations? Since different kinds of bases have different mathematical properties, the most suitable basis for a given system is the one which fits the system. For instance, as described previously, PWs are periodic and naturally follow Bloch’s theorem; therefore, they are suitable for crystalline systems with periodicity. On the other hand, local orbitals, such as atomic orbitals (AOs), are more appropriate for molecular systems because both are spatially localized. Nevertheless, it is not impossible to do the other way around, namely PWs for molecular systems and AOs for crystal systems, although this choice will be less efficient. PWs are basically a combination of cosine and sine functions, which, thanks to Euler’s identity, can be written compactly using a complex exponential: eiG $ r ¼ cos (G $ r) þ i sin (G $ r), here G and r, respectively, represent wavevectors and real-space positions. PWs also form a complete basis set, meaning that any real-space function in principle can be constructed by linearly combining PWs of different wavevectors. Another property of PWs is that they are orthonormal: Z Z 0 U U iG$r iG0 $r dre dreiðGG Þ$r ¼ dG;G0 ; e ¼ (46) 3 3 ð2pÞ ð2pÞ where dG,G0 is the Kronecker delta and U is a large volume. In terms of PWs, the KS wave functions can be written as 1 X PW jjk ðr Þ ¼ pffiffiffiffi C eiðkþGÞ$r ; U G jðkþGÞ

(47)

PW where j and k, respectively, denote the band and crystal-wavevector indices, whereas Cj(k þ G) are the PW coefficients. One can show that these wave functions also obey Bloch’s theorem where the translationally invariant term ujk(r) is given by

156

Chemical bonding with plane waves 1 X PW ujk ðr Þ ¼ pffiffiffiffiffiffiffiffiffi C eiG$r ; Ucell G jðkþGÞ

and Ucell is now the unit-cell volume. If we insert Eq. (47) into the original KS equations, they become:  X
Z2 PW jk þ Gj2 dGG0 þ VðG  G0 Þ CPW jðkþG0 Þ ¼ 3 j CjðkþGÞ : 2m 0

(48)

(49)

G

Here, the first term of the left side represents the kinetic energy, whereas V(GG0 ) is the Fourier transform (FT) of the potential given by Eq. (26). This simplicity in calculating the matrix elements of the kinetic energy and potential is one of the advantages of the PW basis. Furthermore, since the PW basis is complete, in principle, the obtained wave functions will be exact if infinitely many PWs are used. In practice, however, a limited number of PWs must suffice due to the limitation of the computer memory. This will cause some inaccuracy of wave functions, especially the parts close to nuclei, one of the drawbacks of PWs. Fortunately, various methods, so-called pseudopotential methods, have been developed to soften the potential by dividing the electrons into the core and valence parts, so the pseudopotentials in the end reduce the number of PWs. The resulting wave functions are called pseudo-wave functions, which only accurately describe the valence part. These methods will be described in detail in a following section. Moreover, the number of PWs is set up through a parameter called the cutoff energy (Ecut) that determines the maximum kinetic energy of PWs allowed in a calculation: Z2 jk þ Gj2  Ecut : 2m

(50)

This single parameter, which is another advantage of the PW basis, systematically controls the convergence of DFT calculations. Unlike PWs, AOs, however, do not have a general form. Nevertheless, AOs can always be separated into two parts, namely, the radial and angular parts: fnlm ðr Þhfnlm ðr; q; 4Þ ¼ Rnl ðr ÞYlm ðq; 4Þ:

(51)

where, r, q, f are, respectively, the radius, polar, and azimuthal angles, whereas n, l, m are the principle, azimuthal, and magnetic quantum numbers. Here, Ylm are spherical harmonics, which are orthonormal functions: Z (52) d4dqsinqYlm ðq4Þ Yl0 m0 ðq4Þ ¼ dll0 dmm0 : Cubic harmonics, however, are also often used for the angular parts because they yield real, instead of complex, values. The radial function Rnl(r) determines how localized or “contracted” an AO is, whereas the spherical harmonics determine the symmetry or character of the AO. Eigenfunctions of a hydrogen atom are examples of AOs whose derivation and forms can be found in any quantum chemistry textbooks. One important property of hydrogen-like AOs is that they have nodes, i.e., crossing points through the radial axis, which are proportional to the principle quantum number n, the number of nodes being nl 1 and l the angular momentum. Various sets of AOs have been mathematically developed since the birth of quantum mechanics. However, the only part that differentiates them is the radial functions. Some of the earliest basis sets were calculated by Slater in 1930,77 and unlike hydrogen-like AOs, Slater-type orbitals (STOs) do not have nodes since they are defined by taking the asymptotic form of the former at large distances: ð2zÞnþ1=2 Rnz ðr Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi r n1 ezr : ð2nÞ!

(53)

Here, n is now called an effective principal quantum number, whereas the (contraction) exponent z is a quantity that is proportional to the nuclear charge and inversely proportional to n. In his paper Slater also tabulated n and z for many elements, and since then calculations of more complex systems have become feasible. The parameters n and z, however, need to be readjusted whenever the potential in a given system deviates significantly from the corresponding atomic potential. Since AOs or local orbitals can also form a complete basis set, the KS wave functions can be constructed through a linear combination of atomic orbitals (LCAO): X ik$T jjk ðr Þ ¼ CLCAO fm ðr  TÞ: mjk e (54) mT

Here, m represents all the orbital labels, i.e., {n, l, m}, whereas T denotes the lattice translation vectors. For molecular systems, the LCAO sum over T is omitted and the systems have no crystal-wavevectors. Cmjk are the LCAO coefficients. An AO basis may or may not be orthogonal. Two non-orthogonal AOs separated by a distance R can have an overlap given by Z SRmn ¼ dr fm ðr Þ fn ðr  RÞ; (55)

Chemical bonding with plane waves

157

which is one of the ingredients of the COOP technique described in the previous section. The formulation of COOP is naturally facilitated by AOs because of the unambiguous sense of orbital assignment to the respective atoms; there are no atom labels assigned in any PWs. The corresponding Hamiltonian element in an AO basis is given by Z R Hmn ¼ dr fm ðr Þ HKS fn ðr  RÞ; (56) which is also an ingredient of another bonding indicator, i.e., COHP. Unlike using PWs, there is no easy way to calculate the elements of the KS Hamiltonian using AOs. Because the Hamiltonian also contains the potential terms, if expanded, Eq. (56) consists not only of one- and two-center integrals, but also of three-center integrals. The latter especially has become a challenge of AO-based calculations. Fortunately, there is a method first proposed by Boys78 to ease the calculations of multi-integrals, namely to approximate STOs by linearly combining so-called Gaussian-type orbitals (GTOs). The difference between STOs and primitive GTOs is in the radial function, for which GTOs use Gaussians: Rnz ðr Þ  ezr : 2

(57)

One of the advantages of GTOs lies in one of their properties, namely the product of two GTOs is also a GTO which can be integrated analytically. Hence, two- and three-center integrals of GTOs can be reduced into one-center integrals to be calculated analytically. Fig. 11 shows the radial functions of a numerically calculated 3s AO of an Na atom, its STO approximation, and a GTO. Unlike this hydrogen-like AO and STO, the GTO does not have any cusp (discontinuous derivative) at the nucleus because GTOs are natural solutions of a quantum harmonic oscillator which is quite different with Coulomb potentials. This downside makes GTOs poorly describe an electron’s behavior at the nucleus. Nevertheless, using the right number of GTOs, the behavior of valence electrons can still be captured quite accurately as depicted by Fig. 11 in which all the orbitals converge at large distance. Furthermore, due to the symmetry behavior of local orbitals, DFT calculations on molecules can usually be solved more efficiently using a right set of local orbitals than using PWs. Even a minimal basis set, not fully complete, often appears as “complete enough” to accommodate all the charges and to minimize the total energy. However, unlike the PW basis that uses one parameter to control completeness, a local basis does not have that single parameter, another difficulty of local orbitals. Whenever problems of incompleteness or over-completeness of the basis functions occur, some caution is advisable. Nevertheless, there are two ways to improve the accuracy of local-orbital-based calculations. The first one is by adding orbitals with the same symmetry, i.e., different exponents. This basis set is called a double- or multiple-z basis if it gets more than doubled. Another improvement can be done by “polarizing” the basis set, i.e., adding orbitals with higher angular momenta; these orbitals will allow further orbital mixing that in the end may lower the total energy. Up to this point, we have discussed how to do DFT calculations with a basis set that is chosen to give the most benefit to the study of our systems. Another idea is to harvest the advantages of both kinds of basis sets through a projection method that can transform both representations of the wave functions between each other. The method, which will be described in detail in a following section, allows us to take advantages of the simplicity of the PW basis for DFT calculations and, at the same time, enables us to accurately extract chemical information, such as chemical bonding, charge, etc., from any PW-based electronic structure.

3.07.2.2

Pseudopotentials in general

The sheer existence of the periodic table assures that valence electrons alone are an excellent starting point discussing chemical bonding. Since core electrons are chemically inert and seldomly contribute to interatomic properties, quantum chemists eventually came to realize that removing these core electrons and replacing them by effective, screened potentials will save a lot of computational resources since the computational effort increases fast with the atomic number. Often these so-called pseudopotentials (PP) allow to make calculations of heavy atoms possible to begin with. The whole idea is the separation of the exact wave function into a core and a valence part and to replace the core electrons by some weaker atom-centered potential, a pseudopotential, sometimes

Fig. 11 Schematic plots of a numerically calculated 3s radial orbital (full line) of a Na atom, its Slater-type approximation (STO, dashed line), and a combination of three Gaussian-type functions (GTO, dotted line). Reprinted from Dronskowski, R. Computational Chemistry of Solid State Materials: A Guide for Materials Scientists, Chemists, Physicists and Others. Wiley-VCH: Weinheim (2005) with permission from John Wiley and Sons.

158

Chemical bonding with plane waves

called effective-core potential (ECP), to reproduce the effects of the nucleus and the core electrons on the valence electrons. In 1935, Hellmann was the first to postulate this idea,79,80 thereby transferring the essence of the periodic table into quantum mechanics. In solid-state quantum chemistry and DFT, omitting the core electrons and replacing them by pseudopotentials has this time and memory space-saving effect since the number of KS orbitals that have to be evaluated can be reduced, which also reduces the time and storage necessary for the orbital information and further operations on them like orthogonalization. An additional gain for the usage of pseudopotentials concerns the short-range and long-range behavior of a crystal wave function. The behavior in the interstitial sites could not be more different than in the region near the atom, as we have already seen when discussing plane waves. The smooth interatomic region can easily be approximated with one or only few plane waves, whereas the oscillation of the wave function near the nucleus, caused by the orthogonality to the core states, ideally requires a very large number of plane waves to describe the locality of atoms correctly. To reduce this high demand on plane waves, and therefore computational resources, the core electrons are approximated with a smoother, nodeless wave function, as explained above. With fewer oscillations in the vicinity of the atom, the cutoff energy Ecut for generating PWs can be reduced which requires less computational effort later.48,81–84 There are a bunch of pseudopotential approaches for molecular and solid-state quantum chemistry.81,85–89 The two most important methods are norm-conserving85 and (sometimes nonsingular) soft89 PPs. In case of a norm-conserving PP, the electron density, generated by the pseudopotential wave functions and by the all-electron (AE) wave function must coincide. For a shape-consistent PP, the PP wave function and the AE must be identical to each other.85 In case of the (ultra-) soft PP the norm constraint is relaxed to reduce the number of PW needed even more (the smaller the number of PWs the softer the PP) but with the cost of introducing a generalized KS eigenvalue formulation that has to be solved during a solid-state calculation using soft PPs.89 A more generalized approach to PPs was introduced by Blöchl90 within the projector augmented wave (PAW) method, which b that maps the auxiliary PW unifies the ideas behind AE and PP approaches, namely by defining a linear transformation operator T 

pseudo-wave function J onto the true AE wave functions JAE 

JAE ¼ Tb J:

(58)

Within the well-known frozen core approximation, the PAW method exhibits a pseudopotential character. Details on this method will be provided in the next sections. To ensure the accuracy or suitability of a pseudopotential approach, the results from a calculation using PPs are compared to experimental values or an AE calculation, depending on the intention. Basic features that are analyzed for solids are the lattice parameters or the bulk modulus and, in case of molecules, the vibrational frequencies, bond distances and other geometrical details of the molecule are often inspected.48,84

3.07.2.3

Projector augmented wave method

As described above, PW-based DFT calculations can be very expensive due to the large number of PWs required to capture accurately the rapidly oscillating parts of the wave functions close to the nuclei. In order to reduce the computational costs one can replace a true potential with softer pseudopotentials as explained before. However, the pseudo wave functions obtained from a pseudopotential are softer than the true wave functions, meaning their radial parts do not have the correct nodal structure, and thus close to nuclei they do not behave correctly as the true ones do. In 1994 Blöchl proposed another alternative method called the projector augmented-wave (PAW) method to obtain the true wave functions while keeping the cost of using PWs low.90 The method is basically the generalization (or the marriage) of the pseudopotential method and the linear augmented-plane-wave (LAPW) method.91 Before we continue, at this point it is thought necessary to briefly introduce the Dirac notation to simplify the writing of equations so that the methods can be more conveniently described. The Dirac notation makes use of the angle brackets to represent a general vector/state and its Hermitian conjugate. An arbitrary state and its Hermitian conjugate represented in a real-space are defined using the Dirac notation as the following: f ðr Þhhrjf i f  ðr Þhh f jr i:

(59)

Furthermore, a projection or an inner product of two vectors can be represented as: Z dr f  ðr Þ g ðr Þhh f jg i: The last one also implies that

(60)

Z dr jr ihr j ¼ 1:

(61)

Now, back to our topic. The idea behind the PAW method is to obtain the all-electron/true wave functions ji, i.e., those with the 

correct nodal structure, through a linear transformation T of the pseudo wave functions ji , i.e.

Chemical bonding with plane waves  E ji ; jji i ¼ T e

159

(62)



where ji are purely represented in terms of PWs and given by Eq. (47). Now, how do we define the transformation? Since the parts of the pseudo wave functions that need corrections are the ones close to nuclei, we can then define spheres, each with a radius rc, surrounding the nuclei in which the transformation is only effective; outside the spheres the pseudo wave functions remain unchanged. Now inside any given sphere located at the position R, we can alternatively represent the pseudo wave functions in terms of smooth local orbitals which are called pseudo partial waves: E  E X  e ji ¼ fmR ; cmR e (63) m

for |r  R | < rc. The corresponding true wave functions in the same sphere represented by the all-electron (AE) partial waves, f, are given by X   cmR fmR ; jji i ¼ (64) m

The full wave functions for the whole space can then be written as  E  E X   cmR fmR  e fmR : ji þ jji i ¼ e m;R

(65)

Furthermore, because the PAW transformation is linear, the coefficients cmR can obtained by projecting the pseudo-wave functions inside the corresponding spheres onto some local functions, e p, which are dubbed the projector functions: E D ei : (66) pmR jj cmR ¼ e Since the pseudo-wave functions have to be recovered if Eq. (66) is inserted into Eq. (63), the projector functions should be biorthogonal to the pseudo partial waves, i.e. D E enR ¼ dmn : e pmR0 jf (67) After substituting the coefficients in Eq. (66) with Eq. (65) and comparing with Eq. (62), in the end the PAW transformation operator can be expressed as E D  X     fmR  e e pmR : T ¼1þ fmR (68) m;R

Furthermore, since the true wave functions have to be orthonormal, we have the following condition: D E D E e i jT y T j e j ¼ dij ; ji jjj ¼ j

(69)

where is T y the Hermitian conjugate of T . The AE partial waves for each chemical element are usually calculated by solving the Schrödinger equation for the isolated atoms; this method helps a quick convergence of KS wave functions. The pseudo partial waves and the projector functions, however, are not uniquely determined. Nevertheless, the pseudo partial waves are required to be identical to their AE counterparts outside the augmented sphere. They can be generated using polynomials92 or Bessel functions.93 Lastly, the projectors are determined by implementing the orthogonalization given by Eq. (67) through a procedure described by Blöchl90 or Vanderbilt.89

3.07.2.4

Plane-wave-based quantum chemistry programs

Due to its simplicity and several other advantages, the PW basis has gained enormous popularity among various DFT implementations. Most of these DFT computer programs are free for academic purposes, although some are also available commercially. Instead of describing all the DFT programs available out there, we would like to mention briefly three PW-based DFT programs in connection to the implementation of the projection method that will be described in the next section; its implementation as a computer program has so far enabled processing of DFT data from these three DFT programs. The first computer program is called the Vienna ab initio simulation package (VASP),93–96 which has been developed and maintained by the Kresse group of Vienna University, Austria. The program is written in Fortran and licensed as proprietary software and has very good support of pseudopotentials. It also offers various exchange-correlation functionals. Furthermore, the computational efficiency is achieved mainly through a parallel computing method called message passing interface (MPI), although the hybrid with another method called open multi-processing (openMP) has also been implemented.97 Several attempts have also been made to speed up VASP further using graphical processing units (GPUs).98–100 VASP uses a universal naming system for its input and output files. The next program is ABINIT which is developed by Gonze’s group at the Catholic University of Louvain-la-Neuve, Belgium, and distributed under the GNU general public license (GPL) or open-source.101–104 ABINIT is also written in Fortran and parallelized

160

Chemical bonding with plane waves

using MPI and openMP.105,106 The GPU implementation has also been experimented.101 ABINIT supports various kinds of exchange-correlations functionals and pseudopotentials/atomic datasets. The generation of PAW atomic datasets, furthermore, is collected in a library dubbed JTH atomic dataset107 and is formatted using the extensible markup language (XML), which is a very efficient format to allow an easy extraction of the PAW atomic data. The third DFT program is called Quantum ESPRESSO, which is also open-source.108–110 Its development is an open initiative through internationally collaborative work coordinated by the Quantum ESPRESSO foundation. The program used to be a standalone program called PWscf before the integration with other quantum programs by the same initiative. It is written mainly in Fortran, although C is also used to write some other parts, and like other two programs, it takes advantages of parallel computing through MPI and openMP. The GPU implementation using NVDIA machines is currently available only in the ground state subprogram, i.e., pw.x.110 Various exchange correlation functionals and pseudopotentials are also supported by the program. Ready-to-use pseudopotentials are available on its website, but many other groups also share their own built-in pseudopotentials publicly.111–115 Furthermore, the pseudopotentials for Quantum ESPRESSO follow a format called unified pseudopotential format (UPF) which is very similar to XML, and currently the developer is transforming UPF to the true XML format to improve its I/O efficiency.

3.07.2.5

Projection method

Although the PW basis is preferred in many DFT calculations due to its simplicity, many other essential analyses on electronic structure, such as population58 and chemical bonding26,51,74,75,116 analyses, require local-orbital descriptions of the electronic structure. Local orbitals can also be used as a basis to more compactly represent a tight-binding Hamiltonian for further advanced analyses or calculations.117,118 In order to facilitate this purpose, a projection method to transform the basis of the Kohn-Sham (KS) wave functions from the PW basis into a local orbital basis is required. Furthermore, due to their nature, only a small number of local orbitals, i.e., a minimal basis set, are usually utilized to represent KS wave functions. An early projection method was proposed by Sánchez-Portal et al. to reconstruct the PW-based DFT wave functions using various local-orbital bases.119,120 The main idea of this method is to project the PW-based KS wave functions onto each orbital in a given set of local orbitals to the get so-called transfer matrix E D e jk Tjmk ¼ cmk jj 1 X PW C ¼ pffiffiffiffi U G jðkþGÞ

Z

dr cjmk ðr Þ eiðkþGÞ$r

(70)

that basically describes the contribution of a local orbital cmk in the given PW-based wave function. Here, cmk is the reciprocal representation of a local orbital obtained through the Fourier transform: X     cmk ¼ p1 ffiffiffiffiffiffiffi eik$R cmR : (71) NR R Furthermore, we can also expand the KS wave functions in terms of local orbitals using Eq. (54) to obtain another transfer-matrix representation X

X  ¼ Smn;k CLCAO ; (72) cmk jcnk CLCAO Tjmk ¼ njk njk n

n

where Smn;k describes the overlap between two orbitals in the reciprocal space. The last equation forms a linear algebraic expression which can be solved, for instance with a matrix inversion method, to yield the LCAO coefficients. Furthermore, the orthonormality of the reconstructed KS wave functions can be quantified as E X D b jk ¼ j e jk jj e jk ¼ CLCAO Smn;k CLCAO : (73) O mjk njk mn

b jk should be one, but the reconstructed wave functions may deviate from the original ones. Therefore, in order to Ideally, O measure the average deviation, Sánchez-Portal et al. proposed a quantity called the spilling parameter,119,120  1 X  b jk ; SU h f 1O (74) Nj Nk jk jk where Nj and Nk, respectively, are the number of bands and k-points, and fjk is the occupation function. The spilling parameter basically checks the quality of a projection calculation with a given local basis set. Furthermore, since fjk ¼ 0 for all conduction bands, the parameter only considers the valence bands, that is, the occupied levels. The projection method described above suits well any PW-based calculations. However, it needs a modification if one wants to use it for the more accurate PAW method. Clearly, the PAW method operates by a PW basis, but the pseudo wave functions in this method are allowed to be very soft, and thus less precise, such that they only capture correctly the interstitial parts, while other auxiliary local functions are added up through augmentation to obtain the corresponding true wave functions:

Chemical bonding with plane waves  E  E X  E D E   e  fmR  e e jk : e pmR jj fmR jjk ¼ j jk þ

161

(75)

m;R

Consequently, the projection of any PAW wave function onto a local orbital, i.e., the transfer matrix, will yield two terms, namely the pseudo and augmented transfer matrices: E D aug Tjmk ¼ cmk jjjk ¼ T PS (76) jmk þ T jmk : Here, the pseudo and augmented transfer matrices are, respectively, given by E D e jk T PS ¼ cmk jj jmk

and aug

Tjmk ¼

X

n;R

 nR cmk jf

E D e jk ; e pnR jj

(77)

(78)

 E  nR i ¼ jfnR i  e where jf fnR . The pseudo transfer matrix is exactly the same as the one from the PW-based projection, i.e., Eq. (70), whereas the augmented one involves two scalar products. The first scalar product comes from the projection of the difference between the AE and pseudo partial waves onto a local basis function, whereas the second one is the projection of the pseudo wave functions onto projector functions; the latter is also often called the wave function characters. The first scalar product can be simplified by using the fact that the AE and pseudo partial waves are identical outside the augmented spheres. Consequently, the product will only involve integrations carried out inside individual augmented spheres, independent of unit cell positions, and constrained by their corresponding radius rc: X X

 



 0

0  nR ¼ p1  nR ¼ p1  n ¼ p1 n ; ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi ffiffiffiffiffiffiffieik$R cm jf eik$R cmR0 jf eik$R dRR0 cm jf cmk jf (79) NR R0 N R R0 NR where

 n ¼ cm jf

Zr c

 n ðr Þ: dr cm ðr Þf

(80)

0

By inserting (79) into (78), we then get aug

Tjmk ¼

X

n

D E X   n p1 e jk ; ffiffiffiffiffiffiffi pnR jj eik$R e cm jf NR R

(81)

which can be further simplified by realizing that the sum of projector functions over the translation vectors is just their Fourier transform: E X

D aug n e e jk ; (82) Tjmk ¼ pnk jj cm jf n

Furthermore, since the second scalar product has the same form as the pseudo transfer matrix, its expression in terms of PW coefficients has the same form as Eq. (70), in which we substitute c with e p. Once these two transfer matrices are determined, we can then sum them to get the total transfer matrix and use Eq. (72) to calculate the LCAO coefficient matrix. The projection quality can also be checked again by calculating the spilling parameter defined by Eq. (74). We can also reconstruct the elements of the KS Hamiltonian using our local basis functions, 2 3 X

 X E D  Hmn ¼ cm 4 jjk 3 jk jjk 5jcn i ¼ T jmk 3 jk T jmk ; (83) j

j

for COHP calculations. The aforementioned PAW-based projection method has been realized in a computer program called Local Orbital Basis Suite Towards Electronic-Structure Reconstruction, or LOBSTER for short.121,122 The program post-processes PAW-based DFT data by projecting the wave functions onto local orbitals/basis functions to reconstruct the same wave functions in terms of LCAO, and thus to enable chemical bonding and population analyses. Fig. 12 shows the diagrammatic description of the data flow in LOBSTER. The process starts with the data from a PAW-based DFT calculation; there are three DFT programs so far whose data LOBSTER can handle, namely, VASP,93–96 ABINIT,101–104 and Quantum ESPRESSO (QE).108–110 A respective interface routine of LOBSTER then loads the data into the computer memory. After cleaning them up, the data are then transferred to the projection routine. Additionally, basis functions from one of the three built-in basis sets are prepared and then also transferred to the projection routine; their setup can be controlled through an input file called lobsterin using several keywords, which can be found in the manual. Further description of LOBSTER’s basis sets is given in the next section.

162

Chemical bonding with plane waves

Fig. 12 Sketch of LOBSTER’s flowchart. The process starts from a system of interest with its Kohn-Sham wave functions ji and the corresponding eigenvalues 3 i calculated self-consistently by a DFT program using the projector-augmented wave (PAW) method. The wave functions and all related data are then acquired and loaded into the computer memory through I/O interfaces (red block) specifically designed to three DFT programs (VASP, ABINIT, and QE). A separate routine prepares a set of local basis functions cm that are selected semi-automatically from a built-in database. Wave functions and basis functions are then brought into the projection routine to determine the overlap matrix Smn and the transfer matrix Tmi between the delocalized and localized representations. From those, the projected coefficient and Hamiltonian matrices Cmi and Hmn, respectively, are accessible and enable various bond-analytic tools. The LOBSTER logo is copyrighted by the Chair of Solid-State and Quantum Chemistry at RWTH Aachen University. Reprinted from Nelson, R.; Ertural, C.; George, J.; Deringer, V. L.; Hautier, G.; Dronskowski, R. LOBSTER: Local Orbital Projections, Atomic Charges, and Chemical-Bonding Analysis From Projector-Augmented-Wave-Based Density-Functional Theory. J. Comput. Chem. 2020, 41 (21), 1931-1940 with permission from John Wiley and Sons.

Once the data of wave functions and basis functions are available in the memory, the calculations of the transfer, overlap, and LCAO coefficient matrices are performed, respectively. Finally, the resulted LCAO coefficients are used to calculate the reconstructed electronic structure, chemical bonding indicators, and population quantities. These results will be described in detail in the next sections. LOBSTER is written in object-oriented Cþþ and uses two external libraries (in addition to the standard Cþþ library), namely Boost123 to handle various algorithms and memory management and Eigen124 to efficiently deal with linear algebra, i.e., matrices, vectors, and related operations. To improve further the efficiency through parallelization, openMP has also been implemented in LOBSTER. The program can be downloaded free of charge for academic purposes at www.cohp.de.

3.07.2.6

Atomic basis sets in LOBSTER

There are many ways to build local basis functions for the projection methods. For instance, Sánchez-Portal et al.119,120 used two groups of local bases; the first one was obtained from numerical atomic calculations, whereas the second one was a group of Slatertype orbitals (STOs) built by linearly combining Gaussian-type orbitals (GTOs). LOBSTER, however, provides three atomic basis sets. The first one goes back to Bunge et al., published in 1993,125 hence for brevity it will be called the Bunge basis set. It supports atomic orbitals for atoms from He (Z ¼ 2) up to Xe (Z ¼ 54), and all the atomic orbitals were obtained by solving the Roothaan-Hartree-Fock (RHF) equations for atomic cases with energy error less than 0.6 meV.125 In more detail, a finite number of STOs were employed as a more primitive basis and linearly combined to build the radial part of the atomic eigenfunctions/orbitals X fnl ¼ cnlj Rj : (84) j

Here, Cnlj are the linear combination coefficients obtained from solving the RHF equations, whereas the radial part of STOs, Rj is given by Eq. (53). Note that the index j here covers both n and z in Eq. (53). Furthermore, although STOs themselves do not have nodal structures, the resulted atomic orbitals do have a correct nodal structure through the linear combination. The second basis set available in LOBSTER was produced by Koga et al. using the same HF method,126,127 and for shortness, we call it the Koga basis set. It provides atomic orbitals for elements up to Lr (Z ¼ 103). Furthermore, compared to the Bunge basis set, the Koga basis set uses more STOs to build the atomic orbitals, and thus gives more accurate atomic energy with the average error less than 0.05 meV for Z  54,126 and not more than 5 meV for 55  Z  103.127 Moreover, the calculated atomic orbitals in this basis were constrained to optimally meet the nuclear-electron cusp condition128–130 and the asymptotic long-range behavior.131–134 Although the previous two basis sets are already sufficient for most projection cases, some systems require unoccupied atomic orbitals for certain elements in order to correctly reconstruct the PAW wave functions. An example of this situation will be the hightemperature, body-centered cubic beryllium (b-Be).135 A PAW-based DFT calculation was done for this system with ABINIT101–104

Chemical bonding with plane waves

163

using the GGA-PBE exchange-correlation functional136 from the JTH table. Afterward, a projection calculation with LOBSTER121,122 proceeded with the reconstruction of the PAW electronic structure using local orbitals from the Bunge basis set.125 The atomic configuration of Be, i.e., 1s22s2, implies the sufficient use of the atomic orbitals 1s and 2s for the reconstruction. However, this set of orbitals turns out to be insufficient for the projection; 19% of the original charge was spilled through the projection. This high charge spilling can be significantly reduced by polarizing the basis, i.e., adding another (unoccupied) atomic orbital with a higher angular momentum, in this case 2p. This very 2p orbital was generated as part of the third and new basis set122 in LOBSTER. This new basis set takes the occupied orbitals as those of the Koga basis set. Additional (unoccupied) orbitals were then generated by fitting their radial part given by Eq. (84) to the radial part of the corresponding PAW wave function obtained by VASP using the PBE functional. The fitting then yielded the exponents and the linear combination coefficients for all STOs. The new basis set is named pbeVaspFit2015 after the generating method. For b-Be, adding the new 2p orbital into the projection calculation lowers the charge spilling significantly to 1.73%, and thus the reconstructed wave function are considerably improved. Since the square of a wave functions results in the corresponding charge density, simple plotting allows the visualization of the difference of the electron densities obtained from the PAW and the projection calculations. Fig. 13 shows the isosurface plots of the density difference before (left) and after (right) adding the 2p orbital for the fourth band at the G point. Moreover, the plots are fixed at 65% of the maximum density difference. As can be seen, the spilled charge is significantly reduced after the 2p orbital was included in the projection basis. This also confirms that in the solid state formerly unoccupied atomic orbitals may get involved in the bonding process, the well-known band broadening which makes Be metallic, and hence the formation of the valence bands. Another interesting thing worth noting here is that, although the atomic orbitals of the pbeVaspFit2015 basis set are based on the PAW wave functions of VASP, the orbitals are still very reliable for the construction of electronic structure calculated with the other DFT programs accommodated by LOBSTER, i.e., ABINIT and Quantum ESPRESSO.

3.07.2.7

The projected DOS, COOP, COHP, fatband, and k-dependent COHP and time-reversal symmetry

Using the LCAO coefficients cmjk obtained from the projection method, various quantities are directly accessible. The partial density of states (PDOS) is defined as X   PDOSm ðEÞ ¼ cmjk Skmn cnjk d 3 jk  E ; (85) njk

where Smnk is the overlap between orbitals m and n in reciprocal space such that the sum of PDOS over all orbitals m yields the total density of states (TDOS). To compare this DOS formula with the one based on the original PAW method, the TDOS and PDOS of diamond calculated with both methods are shown in Fig. 14A as taken from LOBSTER’s publication.121 The PAW-based PDOS were calculated by projecting the wave functions onto PAW spheres, and hence ignoring the interstitial parts. On the other hand, the LOBSTER projection (new method) considers all spaces. The PAW-based TDOS and PDOS were calculated using VASP,93–96 whereas the projection-based ones using LOBSTER.121,122 As we can see the TDOS from both methods are virtually the same and agree with each other that the system is an insulator. The PDOS, however, differ significantly. The PDOS from the PAW method (Fig. 14A (middle, colored shades)) are smaller than the ones from the projection method (Fig. 14A (right, colored shades)). By integrating each PDOS up to the Fermi level to yield the numbers of electrons occupying each orbital and then adding them, we can check the total electrons considered by each method; here the original number of electrons is 8 (four electrons per atom, two C atoms). For the PAW-based PDOS, we only get 5.11 electrons, whereas 7.99993 electrons are obtained using the projection-based formula. Therefore, compared to the PAW method, the projection method does not lose almost three electrons and allows for a more accurate distribution of electronic states. Furthermore, the crystal orbital overlap population (COOP) can be calculated by taking the overlap of a pair of atoms (orbitals) in real space and weighted with the sum of the products of the corresponding LCAO coefficients.

Fig. 13 Isosurface plots (Å3) at 65% of the maximum difference between the electron densities from the PAW and projection calculations for the fourth band of b-Be at G. The left side only uses 1s and 2s of the Bunge basis set, whereas the right side has included the new 2p orbital from the fitting procedure described in the text. Reprinted from Maintz, S.; Deringer, V. L.; Tchougréeff, A. L.; Dronskowski, R., LOBSTER: A Tool to Extract Chemical Bonding From Plane-Wave Based DFT. J. Comput. Chem. 2016, 37 (11), 1030-1035 with permission from John Wiley and Sons.

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Fig. 14 (A) Left: Total DOS for diamond computed using the PAW method in VASP. Middle: partial DOS using the PAW-based method of VASP. Carbon s and p contributions are indicated by red and cyan shading, respectively. The sum of both is given by a solid line. Right: As before but, this time, obtained from the analytical projection method we describe here. (B) Chemical-bonding analysis for the CeC bond in diamond. Left: Traditional COHP analysis performed by the TB-LMTO-ASA approach. Right: pCOOP and Hamilton populations (pCOHP), obtained with the projection method. As is convention, all plots show bonding (stabilizing) contributions to the right of the vertical line, and antibonding (destabilizing) contributions to the left. In all plots, the energy zero is chosen to coincide with the Fermi level 3 F. Reprinted from Maintz, S.; Deringer, V. L.; Tchougréeff, A. L.; Dronskowski, R. Analytic Projection From Plane-Wave and PAW Wavefunctions and Application to Chemical-Bonding Analysis in Solids. J. Comput. Chem. 2013, 34 (29), 2557–2567 with permission from John Wiley and Sons.

COOPmT;nT 0 ðEÞ ¼ SmT;nT 0

X   fjk cmT;jk cnT 0 ;jk d 3 jk  E : jk

(86)

where fjk is the occupation number of the state {j, k} or often just called the weight and cmT;jk ¼ cmjk eik$T

(87)

to simplify the writing. The analogous crystal orbital Hamilton population (COHP) is calculated in a similar manner by using the Hamiltonian in real space: X   COHPmT;nT 0 ðEÞ ¼ HmT;nT 0 fjk cmT;jk cnT 0 ;jk d 3 jk  E : (88) jk

k

Here, SmT;nT0 (HmT;nT0 ) is basically the Fourier transform of Smn (Hmnk ). Keep in mind that positive values of COOP (COHP) describe bonding (antibonding) interactions and vice versa. Since all the ingredients in the formulas above are obtained through the projection method, they are also called “projected COOP” (pCOOP) and “projected COHP” (pCOHP). Using diamond again as an example, we show in Fig. 14B the negative COHP of the CeC bond (left) calculated with the traditional method of the TightBinding Linear Muffin-Tin Orbital Atomic-Sphere Approximation (TB-LMTO-ASA for which COHP was originally developed),91,137 as well as the pCOOP (middle) and negative pCOHP (right) calculated using the equations above. All three qualitatively agree with each other in that the whole valence bands are dominated by bonding interactions. Furthermore, since the conduction bands are antibonding throughout, any additional electron density would shift the Fermi level to the conduction bands, and thus destabilize the system. We will demonstrate further how the projection-based chemical bonding analysis has been used to benefit material studies. The COOP/COHP formulas given above require a summation over all k-points in the first Brillouin zone (BZ). These calculations of COOP/COHP will become computationally more and more costly as the number of k-points increases. The computational cost, fortunately, can be reduced by a parallel technique such as openMP, which has been implemented since LOBSTER 1.1.122 In addition, one can also exploit symmetry to connect the wave functions of different k-points as to explicitly consider fewer number of k-points in the summation. One kind of symmetry that always exists in systems isolated from external fields, such as a magnetic field, is the time reversal (TR) symmetry. It describes the invariance of the aforementioned systems when the time is hypothetically reversed by the time-reversal operator t. The invariance of the systems can be shown mathematically using a so-called commutation relation between t and the Hamiltonian H: ½t; H ¼ tH  Ht ¼ 0:

(89)

For proof, please refer to any advanced textbooks of symmetry, such as Ref. 138. This commutation relation implies that Bloch states of the momenta k and  k share the same eigenenergy, i.e., 3 jk ¼ 3 j( k). Furthermore, an operation of t on a given Bloch wave function will transform the wave function into its complex conjugate. However, by inserting the complex conjugate into the KS equation, it can also be shown that the wave function’s complex conjugate is actually just another wave function with the opposite momentum. Therefore, we get the following relationship:

Chemical bonding with plane waves tjjk ðr Þ ¼ jjk ¼ jjðkÞ ðr Þ:

165 (90)

Since the COOP and COHP formulas only differ in the use of overlap and Hamiltonian, we will only derive the simplified formula of COOP; the same exact steps can be redone for COHP. The first step of the simplification is to split the summation of Eq. (86) into two groups, in which one group is the inversion of the other. Afterward, we realize that each k-point and its inversion are energetically equal: 2 3 COOPmT;nT 0 ðEÞ ¼ SmT;nT 0

6 7 7  X6 X  6 X 7 fjk cmT;jk cnT 0 ;jk þ fjk0 cmT;jk0 cnT 0 ;jk0 7d 3 jk  E : 6  6 7 0 j 4 k ¼k 5 BZ k˛ 2

(91)

Next, we recall from Eq. (54) how the wave functions are represented in terms of AOs. Using cubic harmonics, instead of spherical harmonics, AOs can be made real functions, and thus unaffected by the operator t. Then if we insert Eq. (90) into Eq. (54), it can be shown that the time-reversal of the wave functions consequently yields cmT;jk ¼ cmT;jðkÞ :

(92)

After putting Eq. (92) into Eq. (91), for every k-point we will find a term that adds with its complex conjugate; this addition just equals twice the real part of the same term. In the end, we arrive at o  n X  ef R c COOPmT;nT 0 ðEÞ ¼ SmT;nT 0 mT;jk cnT 0 ;jk d 3 jk  E ;  jk (93) BZ jk˛

2

where ef jk ¼ 2fjk , and Rfzg means the real part of the complex value z. The summation over k-points now only considers half of the BZ. Correspondingly, by substituting S with H, the COHP due to the TR symmetry becomes o  n X  ef R c COHPmT;nT 0 ðEÞ ¼ HmT;nT 0 cnT 0 ;jk d 3 jk  E : jk mT;jk  (94) BZ jk˛

2

These last two formulas have been implemented in LOBSTER 4.063 to make the calculations of the projection and postprojection twice more efficient. The efficiency applies not only to the calculation time but also to data storage. Moreover, since LOBSTER can now take only half of the BZ, the DFT data for the projection can now also be obtained twice more efficient. Now, back to chemistry! While COHP is useful in describing chemical bonding in energy space, the chemical bonding in real and reciprocal space is often direction-dependent, meaning bonding at some k-points may be of special interest or at least more significant than bonding at any other k-points; knowing this information can be crucial in studying a given material. Therefore, we define, based on the original definition of COHP, a new quantity called the k-dependent COHP that resolves COHP into individual contributions from each k-point and band: COHPmn;jk ¼ cm;jk Hmn;k cn;jk :

(95)

Additionally, we can also define a band character, or often called a fatband, to describe the character amount of an orbital on a given band: X FATBANDm;jk ¼ Cm;jk Smn;k Cn;jk : (96) n

To see the last two features in action let us now consider thallium fluoride (TlF). The compound has an orthorhombic (Pbcm) crystal structure with 8 atoms per unit cell (Fig. 15A). However, the orthorhombic structure of TlF is somewhat unusual because electrostatic repulsions among ions would make higher symmetric structures, such as rock-salt or CsCl structures, more favorable for TlF, just like in the TlCl case. More interestingly, a study by Häussermann et al. showed that a high pressure up to 40 GPa applied on TlF also did not trigger any structural change to a higher symmetric structure.139 To shed light on this puzzle, in addition to the electrostatic interactions, we need to consider orbital-orbital interactions that can be properly analyzed by means of chemical bonding analysis of individual bands, i.e., k-dependent COHP. In order to do that, the required DFT data were obtained using VASP within the PAW framework. Furthermore, to ensure the convergence a mesh of 763 kpoints and a cutoff energy of 520 eV were set up. The resulting PAW wavefunctions were then projected by LOBSTER to obtain the LCAO coefficients for calculating FATBAND and k-dependent COHP. Let us first have a look at the FATBAND of the orbitals Tl 6s and 6p on individual bands plotted, respectively, on the left and right sides of Fig. 15B. From the left plot we can see clearly that the s electrons of Tl reside in two groups of relatively dispersed valence bands, i.e., around 2 and 4 eV, in between is another group of valence bands basically formed by F 2p (not shown). From here we can infer that the electrons of Tl 6s are not as inert as it is commonly assumed. Furthermore, the right plot clearly indicates that Tl

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Fig. 15 (A) The crystal structure of Thallium fluoride (TlF). Contributions to TlF electronic structure from (B) fatbands of Tl 6s (left) and Tl 6p (right) as well as (C) k-dependent TleF COHP of 6s–2p (left) and 6p–2p (right). Reprinted from Nelson, R.; Ertural, C.; George, J.; Deringer, V. L.; Hautier, G.; Dronskowski, R., LOBSTER: Local Orbital Projections, Atomic Charges, and Chemical-Bonding Analysis From Projector-Augmented-WaveBased Density-Functional Theory. J. Comput. Chem. 2020, 41 (21), 1931-1940 with permission from John Wiley and Sons.

6p mostly contributes to the conduction bands. However, its little contribution can also be seen on some of the valence bands especially around 3 eV, although the interactions involving Tl 6p on those bands are not explicitly resolved. To explicitly resolve orbital-orbital interactions on individual bands we turn to k-dependent COHP. Fig. 15C shows the k-dependent COHP of Tl 6s–F 2p (left) and Tl 6p–F 2p (right) interactions. Negative (positive) values as usual describe bonding (antibonding) interactions. A broad view of these plots shows us that some narrow energy windows comprise a wide range of interactions such that nonbonding interactions will be superimposed by bonding/antibonding interactions and will not be noticeable in regular COHP. From the left plot we can see clearly that the valence bands around 2 and 4 eV, respectively, result from the bonding and antibonding interactions between Tl 6s and F 2p orbitals. This emphasizes the previous observation from the FATBAND that the inert pair of Tl 6s is not so inert after all. It is also obvious from the plot that the bonding and antibonding interactions are comparable in strength, and therefore cannot take a role in stabilizing the system. If these were the only interactions to consider, then the electrostatic interactions would dictate TlF to adopt a more symmetric structure. Surprisingly, as shown by the right plot, there exist bonding interactions between Tl 6p and F 2p orbitals spreading through the valence bands. These interactions counter the electrostatic interactions and are only possible because the less symmetric orthorhombic structure adopted by TlF allows for overlaps between Tl 6p and F 2p orbitals. However, this is not the only reason for the existence of Tl 6pF 2p interactions; they are possible also because Tl 6p and F 2p are relatively closer in energy. This is not the case in TlCl where the energy difference between Tl 6p and Cl 3p is larger and does not allow for cation p–anion p mixing. Therefore, only electrostatic interactions govern and TlCl adopts the high-symmetry CsCl structure. As a last remark we can also see clearly that if conductions bands are occupied, Tl 6pF 2p antibonding interactions will be activated and their strength will depend strongly on the direction in reciprocal space, with most destabilizing at G but rather weak in other directions. The k-dependent COHP feature is now available in LOBSTER 4.063 and has been employed in several studies. The feature was firstly employed as part of the machinery in a study to identify a novel strategy to enhance the thermoelectric material n-type Mg3Sb2.140 More specifically, the k-dependent COHP clearly shows that the conduction-band minimum (CBM) is dominated by covalent interactions. This finding, furthermore, along with the band structure and Fermi surface analyses, culminated in a new proposal to use partial substitution of Mg with more ionic cations to significantly improve the thermoelectric property; a more detailed description will be given in a following section. A high-throughput computational screening also utilized k-dependent COHP to help recognizing more efficiently a high Curie-temperature ferromagnetic semiconductor In2Mn2O7.141 The k-dependent COHP also played a similar role in another high-throughput screening project to help forecasting new diamond-like ABX2 materials with excellent and promising thermoelectric properties.142 The most recent application of the feature is to rationalize the trend in experimental band gaps of the Bi2MO4Cl (M ] Y, La, Bi) photocatalysts.143 In this study, the dominant interactions on the CBM revealed by the k-dependent COHP provide support for a molecular orbital model explaining the cause of the CBM (i.e., band gap) shift in Bi2MO4Cl. The model also hints at controllable conduction bands of Bi2MO4Cl to find derived exceptional photocatalysts.

3.07.2.8

Mulliken and Löwdin population analysis

When the square-integrable crystal wave function jj(k) is constructed as a linear combination of atomic orbitals cm(k) with the normalized, non-orthogonal basis set c in the LCAO-CO approximation45,64 like  E X    cmj ðkÞcm ðkÞ ; jj ðkÞ ¼ (97) m

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167

the LCAO-CO coefficient cmj(k) in reciprocal space can be used to compute the density matrix elements Pmn(k), which are essential for the population analyses. As already mentioned above, both Mulliken and Löwdin charges qA are calculated from the so-called gross population (GP), (Eq. 36), numerical quantities to determine the number of electrons, e.g., in an orbital or atom. In particular, GPm is the gross orbital population including all orbitals m belonging to atom A. Its calculation goes as follows: the electron density distribution in a molecule or solid is divided into a net population (NP) and an overlap population (OP) by using the normalization condition of a square-integrable wave function Z Z Z X X jj ðkÞjj ðkÞ ds ¼ 1 ¼ c2mj ðkÞ c2m ðkÞds þ 2 cmj ðkÞcnj ðkÞ cm ðkÞcn ðkÞ ds m

¼ ¼

X m

c2mj ðkÞ þ

X m 95% during the stochastic quenches. Such states are then also assigned to the minimum basin, and not to a transition region between several minima. As a consequence, such minimum regions often reach far above the standard energy barriers on the energy landscape. Those states, where the quenches lead to two or more local minima with non-negligible probabilities, e.g., 30% þ 70% or 50% þ 30% þ 20%, are assigned to various transition regions, depending on which minima are involved. Depending on the type of system such transition regions can be large or very small, and the distribution of various characteristic regions on the energy landscape yields valuable information about the stability of the isomers associated with individual local minima. We further note that, in principle, for every microstate belonging to a given minimum basin region, whose energy is above the energy of the lowest saddle point, there is always some highly unlikely quench path that will lead into another local minimum. Nevertheless, this probability will be vanishingly small due to the difficulty to find this pathdthe random walker would encounter an entropic barrier!d, and thus we do not treat this microstate as being part of a transition region and consider it as part of the minimum basin characteristic region, as mentioned above. Of course, one can combine the individual minima of the landscape into (structurally) related groups, such that large multiminima basins are formed containing, e.g., an ideal crystal structure together with all the equilibrium defect structures, each of which corresponds to a single local minimum. Then we can use these groups when analyzing the probability distributions ! ! Probð X Þ of the outcomes of the stochastic quenches for many configurations X ; and thus, characteristic regions can be alternatively defined on the basis of groups of minima instead of only based on individual minima. Such characteristic regions are related to the so-called inherent structures frequently studied in glassy systems,64,260,287 which correspond to the local minima one observes when performing gradient minimizations from many microstates along a Monte Carlo or molecular dynamics trajectory when simulating the time evolution of the chemical system. Usually, these minima are all slightly different, but in many instances, they can be grouped together.288 Each such grouping corresponds to one basin, which lies “below” part of thedinfinitely long, in principle, dtrajectory, and thus the set of points whose gradient minimizations end up in these minima is sometimes called an inherent region. Of course, such an overview over the local minima observed during a global optimization run will also be valuable for a general cost function problem, in order to judge the efficiency of the algorithm and analyze the complexity of the cost function landscape. One should note that, in the inherent structure analysis, each point along the trajectory is only associated with a single unique minimum, by construction. However, studies have shown that it is not uncommon that points very close to each other can be led by the gradient based minimizer into different local minima. Unless these can be nicely grouped into (structurally related) minima basins, this suggests that a more stochastic approach of associating general microstates with local minima would be more useful and realistic, naturally leading to the definition of the characteristic regions above: when ensembles of stochastic quenches at each microstate are used instead of gradient minimizations, there is usually no longer a unique minimum associated with a given stopping point along the trajectory, and one obtains the characteristic regions instead of inherent structure regions.

34 Of course, for practical purposes, one can only employ a finite number of quenches, for a finite number of starting points for the quenches. Furthermore, one will employ a discrete binning for the probability distributions that define the characteristic regions; else, the number of regions becomes essentially infinitely large. In practice, the choice of the bins and their widths will depend on the way the characteristic regions are subsequently employed. Nevertheless, the ! ! concept of the idealized characteristic regions where we associate every state X with a complete stochastic quench-probability distribution Probð X Þ is ! ! meaningful. In particular, Probð X Þ constitutes a high-dimensional (actually infinite-dimensional) function of X which should not only be continuous but also differentiable, at least outside special points like the saddle points of the landscape. For a metagraph, we still can define the stochastic-quench-probability ! distributions and the characteristic regions, but since the landscape is discrete, concepts like continuity or differentiability of Probð X Þ are not applicable in the standard usage.

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3.11.2.1.2.5 Locally ergodic regions Another class of very important subregions on the energy landscape of a chemical system are the so-called locally ergodic regions (LER), often also called stable regions or stable (macro)states. They correspond to subregions of the energy landscape that are stable in the sense that they are equilibrated, usually with regard to temperature, and we can treat the chemical system as being in a metastable thermodynamic state while it is residing inside such a region during its time evolution. Such stable states are usually only of limited interest in generic cost function landscapes,35 but are central elements of energy landscapes of chemical systems. However, in contrast to the pockets or the characteristic regions, locally ergodic regions are not “static” entities because the definition of a LER depends on the observational timescale, the dynamics on the landscape, anddfor chemical systemsdon the temperature (and other thermodynamic boundary conditions), and thus we will discuss them in more detail below in Sections 3.11.4.1.2 and 3.11.4.1.3.36 3.11.2.1.2.6 Densities of states and minima Another static property that can be very useful both in generic cost function problems and for chemical systems are the global and local densities of states of the system, where it is often useful to also generate densities of minima or saddle points, etc., because of their great relevance for understanding the time evolution of the complex system. Here, the local densities of states can be computed for various subregions such as lid-based pockets, (multi)-minima basinsdeven up to relatively high energies, if the characteristic regions associated with the basin reach high in energy above the first saddle pointsd, locally ergodic regions, or transition regions. Similarly, for very complex systems that are studied at relatively low temperatures, the global and local distribution of local minima for the physically relevant energy landscape are of interest. In contrast, the global and local distribution of saddle points would be particularly relevant at intermediary temperatures, since then we often expect the system to reside close to saddle points due to the flatness of the landscape in their neighborhood, together with the fact that the number of saddle points in complex high-dimensional systems usually greatly exceeds the number of local minima or maxima. Analogously to the local densities of states, a local distribution of local minima or saddle points as function of energy can be computed for various subregions. While such densities can be determined exactly for discrete landscapes, there are technical problems for continuous landscapes ranging from the normalizationdoverall or for the single minimum basinsdto the matching of densities of states belonging to different energy bands, or derived from combining simulations at different temperatures. 3.11.2.1.2.7 Barrier structure The barrier structure of the landscape controls the short- and long-time dynamics of the system. As far as static elements of the barrier structure are concerned, there are the cost or energy barriers, i.e., cost/energy differences, between local minima and all saddle points connected to them, plus the barriers associated with two connected saddle points. From this, one can try to construct a saddle point based visualization of the barrier structure,290–292 which is often suitable for short observational time scales tobs and low temperatures, where cost or energy barriers between individual minima dominate the dynamics of the system. However, for long tobs at finite temperatures, where locally ergodic regions correspond to whole minima basins and transition regions involve multiple paths over many different kinds of saddle points,37 it makes more sense to investigate the probability flows between various regions of configuration space, such as locally ergodic regions, i.e., the probabilities §AB(tstep) to reach a certain region A 3 S from region B 3 S during the time step tstep.38 For a given set of regions, the probability flows can be visualized in a compact notation as a transition probability matrix.258,294 Here, tstep corresponds to both the time between two measurements of the occupation probabilities in the various regions 35 Stable regions would be of interest when one analyzes in detail the way a global exploration or optimization algorithm works, since such regions can constitute traps for the walker(s) on the cost function landscape. 36 When we employ the full phase space, i.e., positions and velocities of all N atoms constitute a microstate in 6N dimensions, then quantities like the so-called normal hyperbolic manifolds50,289 contain valuable information about stable regions; in particular, they demonstrate that many microstates with atom positions near one minimum actually still belong to a neighbor minimum due to the velocities associated with the atoms. 37 In fact, we can classify the transition regions according to the set of locally ergodic regions they connect on the time scale tobs. 38 The probability flows are statistical quantities reflecting a stochastic dynamics on the energy landscape. Now, in chemical and physical systems, we usually think of a dynamics as a deterministic process, e.g., following Newton’s equations and realized via a molecular dynamics simulation. In principle, such a deterministic dynamics can be visualized as a special case of a stochastic dynamics, where there is only one neighbor state that is accessible from a given ! ! x N ; p 1 ;.; p N Þ ˛ microstate, for all microstates. However, to realize, e.g., Newton’s equations on an energy landscape, the microstates must be points ð! x 1 ; .;! ! x i and the p i are the position and momentum of particle i, respectively, i.e., the 3N-dimensional R6N in the 6N-dimensional phase space, where the ! configuration space of atom arrangements is not sufficient for this purpose. Similarly, the energy function would now be the total energy including both the kinetic and the potential energy associated with a point in phase space (note: this phase space is not related to the phase space or phase diagram in thermodynamics; instead, it refers to the use of phases in the formalism of classical and quantum mechanics). Conversely, we can visualize the potential energy landscape as a projection of the energy landscape over the phase space down to the space of atom configurations, where the kinetic energy is set to zero since ! we set the p i ¼ 0 during the projection. Then the probability flows (parametrized by potential energy or temperature) are the realization of a statistical ensemble of phase space trajectories for a given temperature (corresponding to the average kinetic energy) or (potential) energy. Since chemical systems are typically rather large with many degrees of freedom, and are usually visualized as being in contact with some kind of environment, even if only weakly, and even small systems such as molecules or clusters acquire some probabilistic aspects on typical observational time scales due to their quantum nature, focusing on only the positional degrees of freedom is usually a reasonable approach. As a consequence, a stochastic dynamics yields often a quite appropriate description of the time evolution of the macroscopic system. One should also note that studies of the behavior of phase space trajectories can result in valuable insights,293 e.g., illuminating the importance of entropic and kinetic aspects of the generalized barriers on the landscape, in addition to the more standard energetic ones.

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A, B, etc., of the real system, and to the length of the exploratory simulations used to measure the transition probabilities between the two regions A and B. Of course, these flows are completely determined once the landscape and its dynamics incorporated in the (deterministic or stochastic) exploration/evolution algorithm we employ to study the landscape, have been given.39 The logarithms of the inverse transition probabilities then serve as a measure of the generalized barriers separating locally ergodic regions.154 We note that these generalized barriers are, in general, complex combinations of energetic, entropic and kinetic barriers, but one can often extract some overall energetic and overall joint entropic þ kinetic barrier contribution, using, e.g., the threshold algorithm.273,295,296 The first static feature associated with entropic barriers is the ratio between the number of states within an energy slice that are associated with the minimum basin and the number of states that are associated with the exit region from the basin, i.e., which belong to the transition region(s) leading from the minimum basin to other regions of the landscape. Next, there is the number of connections or connecting paths between two minima basins or locally ergodic regions for a given energy slice, and finally the degree of connectivity of the states within a minima basin.40 These static entities can be translated into dynamic barriers via kinetic factors in the Markov transition matrix that can be constructed to describe the dynamics of the system on a long time scale using a master equation approach (c.f. Section 3.11.4.2.1).198 In this context, one often speaks of free-energy barriers,274,297,298 in particular if one studies transitions between specific pairs of locally ergodic regions or so-called free energy minima, since then often the standard Eyring-Polanyi theory expression for the free energy of activation299 can be re-cast in a sum of energy and entropy terms. However, it is important to be aware of the fact that entropy does not make sense at an instant in time, and thus defining a free-energy barrier always involves a choice of observation time,300 analogous to the requirement of a specified observation time for the existence of locally ergodic regions and free energy minima.41 Using a variety of algorithms to determine such static propertiesdusually together with various dynamic ones derived from, e.g., MD or MC simulationsd, one obtains a “complete” description of the landscape in terms of static properties such as local minima, maxima, and saddle pointsdincluding their curvaturesd, and the characteristic regions, together with the probability flows between various region of configuration space,258,273,294 As mentioned above, such regions are often basins around local minima or locally ergodic regions (for a given observational time scale) that are separated by generalized barriers on the landscape, which are extracted from the probability flows and incorporate entropic and kinetic features in addition to purely energetic aspects.154 Based on this information about the cost function landscape, one can actually optimize the global optimization search procedure. Furthermore, we can control the time evolution of the system, such that one will reach the global minimum or any other prescribed region of the landscape (perhaps corresponding to a desired metastable compound), in the most efficient fashion with the highest probability for a given (or randomly chosen) starting point on the landscape143,198,301.42

3.11.2.2 3.11.2.2.1

Representation of landscapes Projections on subsets of configuration space

As with many other intricate scientific problems, the availability of visual representations that highlight the most important features would be of great help studying complex cost function landscapes. However, these landscapes are high-dimensional and possess very complicated structures, and thus a reduction in the number of relevant coordinates that can be used to distinguish among algorithmically or physically relevant regions of the landscape or extract specific properties of interest is both highly desirable and poses a formidable challenge.268,286,302–307 For very small molecules, or molecules consisting of very few building blocks that can be treated as fixed, one can employ simple low-dimensional projections of the landscape, e.g., using two dihedral angles (c.f. Fig. 15). Similarly, low-dimensional cuts through the landscape can be used to elucidate the transition path during a conformational change or a chemical reaction309.43 For larger molecules, the so-called principal coordinate or principal component analysis has been applied to the set of configurations that represent local minima,268,302,303,305 with the goal to identify the most important coordinatesdor their

39 One should keep in mind that probability flows derived from constant temperature MC or MD simulations yield temperature dependent barriers. Extracting the static barriers from such data is non-trivial, even when many runs at many temperatures are available, and thus additional exploration tools such as the threshold algorithm273,295,296 are employed. 40 These connections between the microstates are sometimes associated with kinetic barriers as distinct from entropic barriers. However, cleanly separating entropic and kinetic barriers is usually only possible for very simple or artificial (discrete) landscapes. 41 As a consequence, the shape and structure of the “free-energy landscape” that can be constructed from locally ergodic regions and free-energy barriers depend on the given observational time scale.300 The observation time dependence is also reflected in the fact that the entropic barriers deduced from, e.g., the threshold runs also must exhibit some dependence on the length of the runs. Commonly, one chooses the run times tprob such that they are similar to the lifetimes of the locally ergodic regions of interestdfor the thermodynamic conditions envisioned for future applications of the generalized barriers in the context of, e.g., a kinetic network of the landscape143dbut at the same time long enough for the transition probabilities to have converged to stable values (as function of run time). 42 Essentially, this optimization of the time evolution of the system constitutes an optimal control problem, as can be seen in applications of finite time thermodynamics to, e.g., chemical systems.144,202,256 Of course, one might wonder why one would want to “optimize” the optimization if one has already found the global minimum and all the other quantities and points of interest on the landscape. The reason is that there are a multitude of similar chemical systems, which will exhibit structurally related energy landscapes. Using short exploration runs on such systems combined with the general features of the optimized search procedure will allow us to efficiently solve the global optimization problem for the new system. Similarly, information about such an optimized route to a particular local minimum/metastable modification will serve as guidance for the actual experiment and the synthesis of such metastable phases. 43 In principle, such cuts can also be used for high-dimensional systems, but, usually, not enough information about the transition path of interest is available.

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combinationsdin the system. This procedure involves a statistical analysis of, e.g., the atom-atom distances within the molecular conformations, which allows us to identify those linear combinations of distance vectors that contain the largest variance and thus are assumed to be most useful for classifying the many local minima. Such an analysis, or related clustering algorithms,310 have also been employed to organize the multitude of minima generated by some structure prediction algorithm into groups, with the goal to select “representative” configurations for a more detailed analysis and refinement. Of course, in practice, one would supplement such a grouping by a symmetry analysis of such molecules or clusters311 or use cluster comparison algorithms such as the CCL algorithm312 implemented in the structure analysis program KPLOT.313 Furthermore, these principal coordinates hint at paths for transitions in large molecules.268,303,305 A very similar approach291 is based on the so-called molecule optimal dynamic coordinates (MODC): these are the degrees of freedom, for which the participation in the fluctuations of the molecule is maximal, quite similar to the idea behind the principal coordinates. The MODC vary with time along the simulation trajectory, however; thus their application is usually restricted to short trajectories or pieces thereof that stay within one basin or at most a few dynamically connected basins or locally ergodic regions. For solids, this class of approaches appears to be most suited for amorphous or glassy compounds where no distinct stable structures are present. For crystalline systems, it is usually more straightforward to first determine the symmetries and space groups of the minimum configurations using algorithms such as SFND314 and RGS,315 and to subsequently compare the periodic structures in a pair-wise fashion using the structure comparison algorithm CMPZ,316 in order to group the minima into structure similarity clusters.44 Besides grouping of similar configurations, such symmetry information allow us to study possible transformation paths,52 since symmetry groups and their subgroups can serve as order parameters52 in thermodynamic systems. But one should be aware that with the exception of second order phase transitions one cannot expect that the simple “straightforward” path one would construct based on symmetry considerations, corresponds to the actual route on the landscape taken during the phase transformation: although such transformation routes appear very reasonable, in most cases they constitute only a zero’th order approximation, especially if the transition corresponds to a first order phase transition. Nevertheless, one often selects some “intuitively obvious” transition path as a first attempt, and defines a reaction coordinate along this path. While this may yield some insight into a possible reaction or transformation mechanism, a full unbiased landscape study is needed, in order to identify a wide range of possible transition paths together with their energetic, entropic and kinetic barriers.

3.11.2.2.2

Graph representations

While the projection methods depict low-dimensional cuts through the landscape, graph representations focus on landscape features that are expected to be dynamically relevant, and which can be analyzed or visualized in terms of separation of time scales approaches for the time evolution of the system. Of particular interest are the minima, pockets, characteristic regions and/or locally ergodic regions of the energy landscape and their connectivity, where the special points or regions are the nodes of a graph, whose edges reflect the path connectivity (usually below some energy lid) or the probability flows between the regions.317,318 Such graph representations of energy landscapes are extremely popular: they provide a simplified description of an energy landscape because they are independent of the dimensionality of the underlying landscape.45 Unsurprisingly, various types of tree graphs62,274,292,296,319–322 and other network-representations of landscapes291,323–331 have been invented and re-invented many times. And even without a visual image, the barrier structure and the probability flows can be represented in a straightforward fashion via transition probability matrices274,320,332,333 or transition maps,294 as a function of energy slice and/or as a function of temperature. Quite generally, the resulting graph/matrix representation of the landscape can serve as a starting point for a master-equation approach to modeling the dynamics on the energy landscape274,320,332,333 discussed below (c.f. Section 3.11.4.2.1). To illustrate some of the features of basic (tree) graphs, we consider a schematic (one-dimensional) energy landscape (see Fig. 4a). The most direct “brute force” approach to generate a graph representation of the landscape is to identify all local minima and saddle points (of all orders). In a second step, one identifies all the “direct” connections between the minima and saddle points, and uses this information to construct a network on top of the landscape. Removing the landscape and straightening the paths connecting the nodes yields the saddle þ minima graph of the landscape in two dimensions, where one would usually arrange the nodes in the y-direction according to their energy. However, two problems appear for non-trivial landscapes: for essentially all chemical systems larger than a small molecule or a few-atom cluster, the number of connections (and also often the number of nodes) is so large that the 2D plot becomes very confusing. More critical from a scientific point of view, the dynamics of the system is no longer well-described by a purely minima þ saddle point representation that takes only energy barriers into account and ignores the contribution of the entropic and kinetic barriers. Regarding the issue of visualization, we can simplify the graph by removing irrelevant saddle points and their connections.323 Another common approach uses dynamical arguments to justify merging various local minima and saddle points as long as they can be expected to be equilibrated at a given temperature; we will return to this issue later when discussing observation times. As a consequence, the graphs connect subregions of the landscape that are “far away” from each other, in the sense of having to traverse long paths on the landscape for each connection, and can contain many individual minima and saddle points. Such a lumping 44

These three algorithms are implemented in the structure drawing and analysis program KPLOT.313 By reducing the landscape to a set of non-overlapping regions and points, one obtains a discrete (coarsened) model of the landscape. This set of nodes can be embedded in two dimensions, together with the edges representing the connections between these graph vertices, as long as one accepts that the visual depictions of the edges are going to intersect, which is no problem, in principle. 45

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(B)

Energy

Energy

(A)

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States

States (D)

Energy

Energy

(C)

States

States

Fig. 4 Schematic (one-dimensional) energy landscape, together with three graph representations corresponding to different levels of coarsening of the landscape. (a) Complete landscape. (b) Stratified graph of the energy landscape. The size of the nodes in this type of multi-lump graph indicates the number of states inside the fraction of the energy slice that belongs to the node of the graph. Such a graph is the most generally applicable one for modeling the dynamics of the system on, e.g., the master equation level. However, for this representation to be appropriate, the horizontal connectivity between the nodes at the same energy should be much smaller than the connectivity within each node; else, the splitting of the slice into subnodes would be arbitrary and does not reflect the physics of the system. (c) Single lump (disconnectivity) tree-graph of the schematic energy landscape. No density of states is associated with the nodes. This representation is most useful for modeling the dynamics at very low temperatures where energetic barriers dominate the transition rates between different valleys of the land-scape and the equilibrium probability of being in a valley depends only on the energy of the minimum. (d) Multi-lump tree-graph representing each connected energy slice by a node. The size of a node in the multi-lump tree graph indicates the number of states inside the energy slice represented by the node. This representation is suitable for studying the dynamics and equilibration of the system at all temperatures where local free energies and entropic barriers between vertically connected nodes play a role. If large entropic and kinetic barriers are present inside a node, i.e., within a connected energy slice, a stratified graph description should be employed.

approach319 has the advantage that we can include in the lumped node all the microstates that are dynamically associated with the minima and saddle points.46 One popular, and easily automated approach employs the pockets of the landscape for sequences of energy lids: all the states that belong to a pocket for a given energy lid are merged into one (non-degenerate) node of the graph representation of the landscape, with the energy of the lid. As starting point, one usually picks a very high lid, such that all minima of the landscape are connected below this lid, or at least the whole region of interest of the landscape is inside the starting pocket. Next, a lid with a slightly lower cost or energy is used to find sub-pockets of the first pocket, and we merge the states in each of these sub-pockets into a node with the energy of the current lid, which is connected by an edge to the first node. This procedure is repeated for each of the sub-pockets until only the local minima remain as the leaves of the resulting tree graph. Finally, all those nodes in the graph, which are only connected to one node with higher cost and one with lower cost, are eliminated, i.e., we only consider those nodes in the graph, where a splitting of the landscape takes place. This minimal tree graph is often called a (single lump) lumped graph,295,296,319 disconnectivity graph320,321 or 1d-projection.62 An alternative way to generate such a graph (yielding the same tree graph) starts from the local minima. Instead of splitting pockets into sub-pockets, we consider each minimum as a leaf node of the tree. Next, we determine the lowest energy or cost, at which a transition between two minima has been observed, i.e., a path connecting the two minima exists below this energy. We now create a node at this energy and connect this new node to the two (or more) minimum nodes that are connected at this cost value. This process is repeated, until all minima are connected (see Fig. 4c). While in many tree graphs the nodes are treated as corresponding to only one “state”, regardless of the number of microstates contained in the pocket or energy slice, in the so-called multi-lump tree graphs, we divide the landscape and its pockets into ! a discrete set of energy slices SðL1 ; L2 ; R Þ between (lid) energies L1 and L2.47 All the N ! microstates inside each such conSðL ;L ; R Þ 1

nected slice are lumped into a node with weight or degeneracy N

2

! ; thus allowing a representation of the pocket (local)

SðL1 ;L2 ; R Þ

density of states in the graph representation (see Fig. 4d).

46 One should note, however, that such a lumping procedure is often executed without verifying that the underlying assumption of dealing with equilibrated states/minima/saddles really holds. While it is perfectly legal to construct such a graph even without this condition being true, one needs to check this condition before using such a simplified graph model to analyze the dynamics of the system. 47 Sometimes, we relax the constraint that these states must be connected only via paths wholly in the interval [L1,L2] and also allow connectivities via lower-energy paths, i.e., the constraint is that the paths must run below the higher lid L2.

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We can refine this multi-lump tree graph landscape representation further by taking the characteristic regions into account. For each energy slice, we can divide the set of microstates according to the characteristic regions they belong to, resulting in a division of each energy slice into a set of subnodes that correspond to various minimum and transition regions. As a consequence, we have (weak) horizontal connections between the sub-nodes inside the energy slices, and the graph is no longer a tree graph.48 Since one usually uses energy slices of equal width in this construction, one speaks of a stratified (lumped) graph representation of the landscape (see Fig. 4b).143,198 This construction is most useful when the transition regions of the landscape are very small, since then one can often drop the transition regions completely from the graph and just focus on the nodes associated with the minimum basin characteristic regions as function of energy slice.49 On the other hand, if there is a substantial fraction of the number of states inside the energy slices that belong to transition regions, there will be many vertical connections (towards lower energies) emanating from the transition regions and leading to different minima basins, thus complicating the analysis and modeling of the dynamics of the chemical system. These dynamical aspects are captured in the probability flows mentioned above, which directly reflect the statistical aspects of the dynamics of the relaxations and transformations on the landscape. Of course, the flows observed in the experiment or a MD/MC simulation depend on the temperature of the system, but we can extract temperature-independent information by studying the connectivities within each energy slice using random walkers restricted to this slice. The resulting transition probability matrices can be visualized as transition maps as function of cost value,258,259,294 which show the minima basins and the amount of probability flow as function of energy range. Furthermore, in the case of chemical systems, we can combine stable regions with information about the (generalized) barriers separating them by constructing a free energy landscape in a graph-like fashion consisting of discrete locally ergodic regions that are separated by generalized barriers (on a given time scale tprob, and at a given temperature). These barriers incorporate both energetic and entropic aspects and are computed from the probability flows between the regions,258,294 which are determined using global exploration algorithms such as, e.g., the threshold algorithm.273,295,296 This time scale dependence of the free energy barriers can be seen in the changes in the free energy landscape as function of observation time as mentioned earlier (c.f. Section 3.11.2.1.2).300 In addition to using weighted nodes that reflect the number of states represented by the node, one can include so-called “weighted edges” into the graph description, trying to capture some of the dynamical features of the landscape. Once we are on the level of lumped graphs of any kind, the edges connecting the various nodes are representative of many paths connecting the microstates belonging to the nodes. Thus, one can assign kinetic factors to such edges representing the ease or speed with which probability can flow between the nodes.155 Another way is to use weighted edges to indicate the energy range (i.e., the saddle energies) that need to be crossed between individual local minima.334 A completely different type of landscape representations via trees are those generated on the basis of the RRT (rapidly exploring random trees) algorithm.335–338 These trees can be constructed for continuous configuration spaces, where straight lines and distances can be defined. To generate such a tree, one starts from a set of initial microstates and picks a random direction along which one moves in a straight line until one hits an (energy) barrier. Here, one moves a minimum distance from the current microstate (usually, one also defines a maximum distance, too); if one hits a barrier before the minimum distance is reached, one stops explorations in this direction and tries again from the current state. If one fails finding a new point for a given number of attempts, one goes back on the tree to the previous microstate. In this fashion, one generates a tree-like structure emanating from every point in the initial set of microstates. In addition, one checks, whether the point one has found is within some cutoff distance to an already generated point in any of the graph pieces that are being generated in paralleldif so, one also rejects the newly found point, or merges it with the point already present on one of the other graph pieces. Assuming that the move has been successful, the point that has been reached defines the next point on the graph. From this point, one repeats the procedure until whatever pocket one is exploring is finished, and one returns to the original point and attempts to continue from there in more directions. This graph building process takes place in parallel from many initial microstates, until no new points can be generated. Usually this means that all the (tree) graph pieces have merged into a single graph (which is often no longer a tree graph). Of course, there is some freedom in defining what is a barrier in this context. The simplest version consists of dividing the set of configurations into a binary grouping of allowed states (C ¼ 0) and non-allowed states (C ¼ N), and thus one rejects all the disallowed points. For chemical systems, it makes more sense to either generate trees using a set of finite energy thresholds (using the so-called threshold-RRT algorithm339) to define allowed and forbidden points thus generating a sequence of lid-dependent graphs, or to use a Boltzmann factor type of acceptance criterion for a permitted point (using the so-called T-RRT algorithm340), thus generating a stochastic version of graphs as function of temperature. We note that these points do not map out the structure of the local

48 Note that this stratified graph does not correspond to the “all minima þ saddles” graph mentioned earlier, since its division into horizontal sub-nodes is only very indirectly connected to the saddle points of the landscape. 49 Frequently, the probability flows, i.e., the horizontal connectedness of the nodes of the resulting stratified graph are very small, and one can even assume that the system equilibrates within each minimum region up to high energies before a noticeable amount of probability has flown into the neighbor minimum basin. In this context, we note the so-called return energies or return barriers,294 which can be included in the (tree) graph pictures as an indication of the entropic barriers surrounding a minimum. They show the energy, up to which, say, 90% of the random walks (of a given length) starting in a given minimum never leave the basin. These energies are often much higher than the energy barriers at which, in principle, a transition between two minima would be possible; this observation also correlates with the fact that the characteristic regions of minima basins often extend far above the energy of the first saddle point that would allow the system to leave the basin. Keep in mind that the return barriers are associated with one minimum; even if only two minima A and B are connected, the energy of the return barrier for minimum A could be very different from the one for minimum B. In contrast, the saddle point energyddefining the size of the energy barrierdis the same, of course.

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minima þ saddle basins as such, but the points generated from an original starting state mostly represent the “boundaries” of pockets in the landscape regardless of how many local minima are inside such a graph or sub-graph. Clearly, local minima can be obtained via local minimizations from the nodes of the graph. This leads to configuration space filling graphs, but they can only be visually depicted if one focusses on a few important degrees of freedom, such as the well-know f-j torsion angle plots of proteins.341

3.11.2.2.3

Order parameters

The analysis and visualization methods discussed above use the cost or energy landscape on the level of microstates as the starting point, and proceed by identifying important dynamically relevant microstates and regions of the landscape, and explore the limits of coarsened representations of the landscape. In physics and chemistry, one alternatively approaches this issue from the opposite limit, i.e., beginning with the question of what kind of macroscopic phases can exist on the landscape, and how these might be classified. A popular way starts with a thermodynamic and symmetry inspired classification scheme in terms of the so-called order parameters52 mi, where i ¼ 1, ., M lists the M different order parameters that are used for the classification. In principle, we are free to define an order parameter m in any way we find useful, but usually one chooses some (in principle) measurable quantity (or quantities) of the system that can be used to distinguish among different (thermodynamic) phases or modifications, such as a space group or a cell parameter of a crystal,52 the magnetic moment of a material,52 or the number of native contacts observed in a protein.342 Given our choice of order parameter, we can now collect all microstates that have the same value of the order parameter(s) and label this set of microstates by the value of the order parameter(s).50 In a statistical mechanical analysis, we would now compute the free energy F(m) by summing over all the microstates with the same value of m, weighted appropriately according to the statistical ensemble that is associated with the thermodynamic boundary condition of the system. For each value of the thermodynamic variables, T, p, etc., we can plot the “free energy” as function of the order parameter and analyze F(m) regarding its minima, maxima and saddle points as function of m. One is particularly interested in how the minima of F(m) vary as function of temperature, pressure, magnetic field, etc., since this allows us to study phase transitions of various orders that can take place in the system. While this appears to be a very useful approach in studying an energy landscape, there are several conceptual issues and problems inherent in an order parameter based simplification of the landscape. For one, we note that, e.g., the use of symmetry such as incorporated in a space group, which is very helpful in analyzing structural phase transitions, would not work, if we try to define the order parameter as a property of a microstate. The reason is that the full symmetry only appears for a very few microstates, which carry practically no statistical weight for non-zero temperatures. In fact, in many chemical systems, the high-symmetry phase only appears as the average over many microstates. As a consequence, many intuitively useful order parameters are actually defined on the continuum level of the system, i.e., they average over fluctuations on the atomic level over whole material points (containing usually thousands of atoms) and finite observation times. Thus, all sets of states, for which, e.g., the cell parameters averaged over many units cells (of slightly fluctuating size) have the same value would be associated with the same order parameter “state”. This can lead to complications, since the set of all microstates that yield the same order parameter value might not constitute a stable (locally ergodic) region or correspond to a characteristic region; this is especially true for order parameter values that do not correspond to thermodynamic phases that are associated with equilibrated minima basins. After all, the many microstates that do not belong to a (meta)stable phase are just lumped together according to the order parameter values but usually might not be equilibrated on any time scale shorter than a global equilibration time scale. Thus, this predominantly thermodynamic picture of the simplified energy landscape classified according to an order parameter is only really suitable, if we are dealing with observation times for which global equilibrium holds, i.e., fluctuations can be ignored. Quite generally, we always need to clarify, in principle, whether we can treat all configurations with the same value of the order parameter as if they were a locally ergodic region for which a local free energy is properly defineddat least on some observational time scales343.51 Similarly, the “stability” that comes to mind when hearing the term “free energy minimum” might be spurious.52 It might well be that these minimum regions are not locally ergodic at all, and that the local maximum of the free energy as function of order parameter, which looks like a free energy barrier, has nothing to do with the actual dynamics.53 Only if the order parameter based free energy landscape description reflects the true barrier structure of the underlying energy landscape, are we able to interpret the dynamics of the chemical system in terms of F(m). 50

Note that some order parameters cannot be measured as suchdor only very indirectly, such as a quantum mechanical phase factor -, or might not be defined for an individual microstate, i.e., it might be a collective property only realizable on the level of a material point ! r that encompasses a multitude of microstates, such as a conductivity or permeability (c.f. Section 3.11.5.1). 51 Conversely, for many choices of an order parameter, the microstates that constitute a locally ergodic region on the landscape might actually exhibit a wide range of order parameter values. 52 We are faced with analogous problems, if we happen to choose an inappropriate set of reduced coordinates when trying to simplify the energy landscape itself.309 P 53 A simple example is the Ising ferromagnet si ¼ 1 with the energy function, E ¼  J < i, j >n.n.sisj, but where a reversal of all spins is a likely possible move (in addition to the usual single spin flips). Selecting the magnetization per spin, Mz ˛ [ 1, þ 1], as an intuitively sensible order parameter, m ¼ Mz, there are two (energetically degenerate) local minima of the free energy F(Mz) close to Mz ¼ 1 at low temperatures, which appear to be separated by a local maximum in the free energy at Mz ¼ 0. However, according to the dynamics on the energy landscape, the system can easily jump between the two minimum regions at no cost (since the Hamiltonian is symmetric with respect to a flip of all spins). Thus, the states with Mz ¼ 1 belong to the same locally ergodic region, and the set of microstates associated with the local maximum in F(Mz) at Mz ¼ 0 does not represent a relevant barrier. We note the danger in choosing an order parameter without taking the underlying kinetics of the system into account; if we had picked the absolute value of the magnetization as the order parameter, m ¼ r Mz r ˛ [0,þ 1], then the dynamics implied in F(m) would have been consistent with the actual dynamics of the magnet system.

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3.11.3

Energy landscapes in inorganic chemistry

Energy landscapes of isolated chemical systems

Turning to features of energy landscapes of chemical systems, we recall that our main interest is in the time evolution and the dynamics of a chemical system. Here, we keep in mind that a major aspect of the time evolution of a chemical system are its thermodynamically stable and metastable states as subregions of the configuration space that remain unchanged in time. These locally ergodic regions are characterized by the fact that they are in global or local thermodynamic and statistical equilibrium on appropriate observational time scales, and thus do not exhibit any changes in their statistical properties, such as energy, X-ray diffraction spectrum, etc., as far as their time evolution is concerned. We mainly consider the configurations of the system on the atomic level; extensions to include electronic degrees of freedom are possible, in principle, but require a different approach to characterizing the chemical system, and will not be discussed here in greater detail. We begin the discussion of chemical systems with the isolated system, which does not have any contact with any external environment. From the point of view of thermodynamics and statistical mechanics, this suggests that we are treating the system in the microcanonical ensemble, i.e., there is no exchange of energy, volume, particles, etc., with the environment. Now, from a classical mechanical point of view, isolation means that during the time evolution of the system, i.e., during its walk through the state space on the energy landscape, certain properties of the system do not change, such as the total energy, the total angular momentum, the total translational momentum, and furthermore, no change in atom content is allowed. The reason is that these are constants of motion: the energy because of invariance of the Hamiltonian under time translation, the total momentum due to the global translational symmetry of the Hamiltonian (i.e., no external force acts on the finite system and the internal forces balance by Newton’s third law) and the total angular momentum accounts for the global rotational symmetry of the Hamiltonian. While their constancy is in agreement with the thermodynamic interpretation of non-interaction, we do have a problem with the volume since its constancy is not associated with a symmetry of the Hamiltonian. Fixing the volume is a thermodynamic boundary condition, but intuitively this appears to be due to an interaction with the environment that forces the atoms to stay within a prescribed given volume, which corresponds to an applied external pressure.54 On the other hand, placing the (finite number of) atoms into an infinitely large volume, corresponding to zero pressure, would imply that the thermodynamic ground state consists of a distribution of isolated atoms within the infinitely large volume, at least for finite energies/atom above the ground state. Usually one addresses this issue by demanding that the atoms must not leave a reasonably large but fixed volume,55 and similarly by setting a maximum volume of the simulation cell when describing a solid/liquid by a periodic approximant; in this fashion, the pressure is non-zero but remains very small and can be set to zero for most practical purposes during, e.g., molecular dynamics or Monte Carlo simulations.56 Concerning the interaction with constant external electromagnetic fields, we include them into “contact with the environment”, and thus for the purpose of this section, we ! ! are dealing with zero pressure P ¼ 0, no magnetic or electric fields B ¼ 0; E ¼ 0; etc. when we discuss an isolated chemical system. ! !! ! !! Regarding the state space of a system of N atoms, in the classical limit, the state X ¼ ð R ; P Þ or X ¼ ð R ; V Þ of the system at each ! ! ! ! ! ! ! moment in time is given by two 3N-dimensional vectors, R ¼ ð x 1 ; .; x N Þ and P ¼ ð p 1 ; .; p N Þ ¼ ðm1 ! v 1 ; .;mN ! v N Þ or V ¼ ! ! ð v 1 ; .; v N Þ; giving the positions and momenta (or velocities) of all atoms, which can evolve as function of time. This yields the full classical description of a state as a point in the so-called phase space,344 which is the 6N-dimensional space of both the N posi! tion vectors ! x i and the N velocity or momentum vectors ! v i or p i ; respectively.57

Recall that in the microcanonical ensemble, the pressure in the system is defined via the relation p ¼  (v E/vV)N,S. Depending on what kind of structures we expect or want to investigate for the system, the diameter of the sphere might be on the order of the sum of the 1/3 diameters of all the atoms, D
Natomdatom, or proportional to the average atom diameter datom multiplied the N1/3 atom, D
Natom datom. However, one must be P aware that we might prejudice the system as far as its thermodynamic ground state is concerned, since, e.g., if V z iVatom(i), then the system can only be in a liquid or solid state, while allowing atoms inside a very large sphere, we might be too close to an infinite volume, resulting in a gaseous distribution of isolated atoms as the thermodynamic state. 56 Still, this issue must be confronted when simulating, e.g., the stability of molecules in outer space. conj 57 In the general mathematical formulation of classical mechanics,344 generalized coordinated qi, velocities q_i and (conjugate) momenta pi ¼ vLðqi ; q_i Þ=vq_i P conj are used in the Lagrangian ðLðqi ; q_i ÞÞ and Hamiltonian ðHðqi ; pi Þ ¼ i q_i vL=vq_i  LÞ formalism, respectively. (The conjugate momentum is also called the !conj !mech generalized momentum or the canonical momentum.) Note that usually the conjugate momentum equals the mechanical momentum p ¼ p ¼ m! v 54 55

of the particle; this is not the case, if the (non-conservative) potential depends on the velocity, as in the case of friction potentials or in magnetic fields. For ! a free particle i with charge Qi in a magnetic field B ; the potential in the Lagrangian Lð! r i; ! v i Þ ¼ Tð! v i Þ–Uð! r i; ! v i Þ takes the form ! ! ! !conj ! ! Qi ! ! ! viþ Umagn ð r i ; v i Þ ¼  c A ð r i Þ v i ; where A is the vector potential of the magnetic field, B ¼ V  A : The conjugate momentum is then given by p i ¼ m! mech ! 2 2 ! Qi ! Qi ! 2 1 1 1 !conj A : If we employ the Hamiltonian formalism, then we incorporate the magnetic field in the kinetic energy term m ð v Þ ¼ ð p Þ ¼ ð p  i i i i c c AÞ ; 2 2mi 2mi since the Hamiltonian is a function of position and (conjugate) momentum and not of the velocity of the particle.

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P P !mech The energy of this system is given by the sum of the kinetic and potential energies, Ekin ¼ ð1=2Þ i mi ð! v i Þ2 ¼ ð1=2Þ i ð p i Þ2 = ! ! mi and Epot ¼ Epot ð x 1 ; .; x N Þ; respectively, and the dynamics follows from Newton’s equations, with the forces computed by taking the gradient of the potential energy. As long as we can invoke the Born-Oppenheimer approximation,58 implying that the atoms/ions can be treated as essentially classical particles, and neglect zero point vibrationsdthe usual cased, this picture also applies in a quantum mechanical treatment after integrating out the electronic degrees of freedom.58 Since the dependence of the kinetic energy on the (mechanical) momenta is just a quadratic function and the potential energy (in non-magnetic systems) only depends on the positions of the atoms, the dynamics of the system is given once the hypersurface of the potential energy over the 3N-dimensional space of all atom arrangements, the so-called configuration space of the system, is known. This potential energy hypersurface over the space of atom configurations is the cost function landscape of a chemical system that is usually meant when one talks of the energy landscape of a chemical system.21,59,89,101–103,118,123,147,260,345 Nevertheless, one should keep in mind that in the case of a non-zero magnetic field, e.g., due to the magnetic interactions between the atoms, the field would enter the Hamiltonian not as a function of atom positions but via the kinetic energy term in the Hamiltonian, both for classical and quantum mechanical systems.344,346 As a consequence, it is no longer possible to restrict ! ! the state space to only the position vectors R ; and the (conjugate) momenta P have to be included.59 Similarly, it might be necessary to go beyond the Born-Oppenheimer approximation, and include not only the vibrational degrees of freedom in the energy calculation but also the corrections (or “exact” contribution if feasible) from the electron-phonon coupling. As mentioned earlier, this can lead to new electronic ground states that might induce also a different ground state atom arrangement compared to the oneelectron approximation. Finally, we might want to study processes, which include contributions from excited electronic states, and thus we need to add the energy landscapes of electronically excited states, preferably again in the Born-Oppenheimer approximation. Here, a major problem is that, e.g., the lowest electronic excited state will often switch its nature as function of atom arrangementdkeep in mind that all possible atom arrangements are included in the state space, and thus, e.g., both metallic and insulating states for solidsd, and similar switches are possible and expected when re-arranging the atoms inside a molecule. However, what is frequently ignored or treated as “obvious”, is the moveclass of the chemical system. As mentioned in the preceding Section 3.11.2.1.1 about general features of cost function landscapes, if we want to describe the time evolution of the system or study the stability of metastable phases, it is important to keep in mind that a physical or chemical system has a “natural” or “inherent” time evolution based on the laws of physics, which depends on, and thus is prescribed by, the initial state, the (effective) interactions between the constituents of the systems (atoms, spins, etc.), and the applied external forces and thermodynamic boundary conditions, where the external parameters can vary with time, of course. In such a case, we have a naturally defined set of states and neighborhoods, as function of evolution time, which cannot be modified, in principle.60 In order to ensure that the moveclass corresponds to the physical time evolution, we only allow very small changes in the atom positions or the cell parameters.61 This requirement of physical realism also presents us with a natural set of properties and features of the system’s landscape which we want to determine via various exploration algorithms. To identify these, we note that our physical measurements in the experiment always depend on the observational time scales tobs over which the measurements are performed, and on the initial state of the system (including the values of the external parameters).62 We again remark upon the importance of keeping apart the features of the true landscape of the real physical system, and those aspects accessible in model systems when using simulation algorithms, as far as their time evolution is concerned.

3.11.3.1

Finite system

While the above comments regarding energy landscapes of isolated chemical systems apply to every type of system, some aspects depend on whether we are dealing with a system consisting of a finite (small or large) number of atoms, or whether the number is essentially infinite. In general, when we are dealing with a finite number of atoms, we would place them into a closed volume in space, such as a sphere with fixed radius, and thus, the microstates of the system consist of all atom arrangements with position 58

In a full quantum mechanical description, we also need to take additional electronic and spin degrees of freedom into account. ! ! !! If the interaction with the magnetic field can be restricted to localized spin states ! s at the atom positions, one can employ X ¼ ð R ; S Þ instead, where S ¼ ! ! ð s ; .; s Þ:

59

1

60

N

Of course, we often make simplifications and approximations when describing or modeling the system before applying the laws of physics, but that is a conceptually different aspect. 61 Keep in mind that in the case of the reduced set of coordinates of a periodic approximant of a solid, we can only model a limited type of evolution because of the enforced periodicity of the solid. Thus, for many processes such as nucleation and growth, we must employ very large simulation cells containing thousands of atoms. However, in such systems, the number of local minima on the energy landscape usually exceeds our ability to register all the minima and the barriers among them, and some coarsened picture is needed to allow an understanding of the processes on an energy landscape level. Furthermore, in many instances, the transformation will occur at the surface of the finite piece of material or only at an interface inside the (granular) material; describing such a process using energy landscape analysis would again require the use of large simulation cells to capture the configuration space of the surface or the interface. 62 If the system is in (local) thermodynamic equilibrium on the time scale of observation, then the influence of the initial state is suppressed, and we can use the laws of statistical mechanics to compute the properties of the system in its current (metastable) state.

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vectors inside this sphere. On the other hand, no such restrictions would be made on the velocities of these atoms.63 By choosing a sufficiently large enclosing sphere, we would ensure that all states of matterdsolid, liquid and gaseousdwould be part of the energy landscape; which of the states is realized depends on the total energy selected. We also note that the chemical system should be invariant under an overall rotation of all the atoms, and thus all microstates that differ by an overall rotation in R3 around the origin are treated as identical states. Similarly, we would eliminate an overall translational motion; however, as long as we employ an enclosing sphere, the origin of the coordinate system is fixed, and thus such translational motion of all the atoms would be taken care of by demanding that all the displacements of the center of mass of the total cluster, molecule, etc., under investigation, which occur in the moveclass employed would be subtracted before executing the move.64 Now, in addition to rotation and translation of the molecule as a rigid body,65 we can have discrete rotations or reflections etc., which map the molecule onto itself. The reason is the identity of atoms of the same type. The same is true when exchanging (permuting) atoms that are of the same type. In all cases, we regain the same molecule. This means that a smooth path that exchanges two atoms by infinitesimal moves can lead to physically identical states! Thus, the topology of the energy landscape becomes rather complex, because the usual way we deal with the (classical) state space assumes that all atoms are labeled individually, even if they have the same properties. This issue is usually ignored in the discussion of the energy landscape of chemical systems, because most studies focus on identifying local minima as candidates for feasible isomers or polymorphs or study local energy barriers between nearby local minima. But this identity under permutations, possibly combined with various symmetry operations, must be considered when discussing the global topology of the energy landscape.

3.11.3.1.1

Continuous state space

In the general case, where we employ the phase space or atom arrangement space for the atoms as the state space of the energy landscape description, we are dealing with a continuous energy function. But depending on how we implement the constraint to force the atoms to stay within a closed (spherical) region, there can be discontinuities or issues of differentiability of the energy function. For example, if we enforce the constraint by restricting the state space to only microstates with all atoms inside the sphere, we can only compute a one-sided derivative, in principle, once one of the atoms is located on the boundary of the sphere. Furthermore, in certain situations such as a net charge in the system, e.g., if we are interested in the landscape of a charged cluster or when modeling a Coulomb explosion,232 it can well be that it is energetically favorable for (some of) the atoms to stay away from each other as far as possible, and thus, we would be dealing with a local minimum at the boundary of state space. In such a situation a local (or even the global) minimum of the landscape cannot be identified via the zero-gradient plus positive curvature condition. In order to ensure differentiability everywhere, one often uses the full configuration space R3N allowing all atom arrangements, in principle, and incorporates the spherical constraint via a so-called penalty function; in this case, one would choose a functional form that is zero until close to the spherical boundary at the radius RS, and then rapidly increases to extremely high values of energy, e.g., P x i jÞ where Vi ðj! x i jÞ ¼ ðj! x i jRS Þn Þ for r! x i r > RS and Vi ¼ 0 for r! x i r  RS otherwise, with A being a large positive Vpen ¼ A i Vi ðj! number, and n a large even positive integer. Furthermore, we chose Vi ðj! x i jÞ such that Vpen is not only continuous but also differentiable, at least up to a sufficiently high degree. Any chemical system consisting of N atoms, including molecules, cluster, crystalline and amorphous solids, liquids and gases, belongs into the category of systems with a continuous state space. Frequently, one explicitly or implicitly applies restrictions to the system when studying its energy landscape based on the specific issues at hand: for example, in the case of molecules one often enforces a bond-network via molecular modeling potentials which behave in many ways like the penalty function mentioned earlier insofar as they guarantee continuity and differentiability of the energy function and ensure the existence of the networkdat least in a visual sense. However, when employing a more realistic energy function such as an ab initio energy function, one might quickly notice that the seemingly stable bond-network is only a small energy barrier away from some rearrangements, such as a redistribution of the hydrogen atoms in a radical molecule. Thus, enforcing such a network via penalty functions on top of the ab initio energy function can lead to serious errors in the analysis of the dynamics of the molecule, and therefore requires careful considerations. On the other hand, by, e.g., treating certain sub-groups of atoms in the molecule as rigid building blocks, one can greatly reduce the complexity of the state space and allow global explorations on systems, which would be close to impossible when allowing all atom positions to vary freely. Here, as everywhere, the model employed must fit the properties studied for the system, and contain all relevant degrees of freedom.

63 Here, we are concerned with the microstates; of course, in a configuration where an atom is located on the surface and its velocity has an outward-pointing component, this atom will be reflected, i.e., the microstate will evolve into a microstate with the atom in the same position on the surface but with an opposite value for the velocity component involved. 64 We can similarly remove any overall rotational component of the moves; however, this is usually more difficult to compute, and thus one frequently ignores such an overall rotational movement. 65 Note: we can also have an energy contribution from the rotation energy of the molecule due to its rotation with respect to the “masses of the universe”,52 but we would usually ignore this for our chemical system unless we are interested in the rotational eigenstates of the molecule. In contrast, the translation does not give a contribution to the (kinetic) energy due to special relativity, i.e., we always can go into a coordinate system, in which the isolated chemical system is at rest.

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In contrast, molecules that correspond to clusters are usually not expected to exhibit a singular stable bond network, and one studies the many possible isomers and the transitions between different atom arrangements without adding any restrictions except the ones implied by the choice of energy function when modeling the interactions among the atoms. If one chooses, e.g., an empirical or semi-empirical energy function, this can prejudice the system and result in different minima and barriers compared to the ab initio energy landscape.259 In principle, the energy landscape of a macroscopic chemical system, which can be in a solid, liquid or gaseous state, is just a special case of a very large but still finite system with N
NAvogadro
1023 atoms, where the material is treated as a gigantic atom cluster. We first note that the energy landscape of such a system will include locally ergodic regions corresponding to real metastable macroscopic crystals in the solid state on a given observational time scale, and also even larger regions that correspond to the liquid statedbut only for elevated temperatures or large total energies in the case of the microcanonical descriptiond, which contain all (!) the feasible (at low temperature) solid (crystalline and amorphous) atom configurations as subsets of the liquid region. The many local minima of these “solid” configurations and their local minima-basin neighborhoods correspond to the inherent structures one observes when performing local minimizations along a molecular dynamics or Monte Carlo trajectory in the liquid state, while the many higher-energy “transition regions” make up the majority of the microstates inside the “liquid” locally ergodic region (c.f. Sections 3.11.2.1.2 and 3.11.4.1.2). Finally, at even higher temperature, essentially all states on the energy landscape are part of the locally ergodic region even for short observation times, and we are in the gaseous state. Of course, on time scales larger than the global equilibration time of the chemical system, there is only one ergodic region, which encompasses the whole of state space. But on shorter time scales and at low temperatures, we find many locally ergodic regions of the macroscopic system that contain mesoscopic (non-equilibrium) defects like grain boundaries, dislocations, etc.; each of these defects are stable on observational time scales for which a given grain is in a thermodynamically (meta)stable state and would be identified as an individual crystallite (including equilibrium point defects). Such mesoscopic defects lead to another level of complexity of the energy landscape as far as locally ergodic regions are concerned, and it is only our point of view of instinctively focusing on the crystalline grains instead of the mesoscopic defect structure, which lets us ignore this aspect of the energy landscape.66 The great advantage of studying macroscopic (solid) materials as a large cluster is the realism once the cluster size exceeds several thousand atoms, such that the global minimum begins to look like a cut-out from a macroscopic crystalline solid. In particular, the influence of the surface on the existence and stability of locally ergodic regions and their corresponding metastable compounds can be studied, including transformations and nucleation and growth phenomena. The disadvantage is the other side of the same medal: the number of atoms are so large that one will have great difficulties to obtain a complete set of, e.g., local minima and saddle points for such a mesoscopic cluster containing thousands or tens of thousands of atoms, and surely not for a macroscopic cluster that is large enough to contain, e.g., mesoscopic defects and interfaces between different phases. In particular, the surface of the cluster may continue to have a great influence on the transformations of the system such as the formation of crystalline nuclei from the melt, or the stability of mesoscopic defects. Thus, one must carefully weigh these advantages and disadvantages when choosing the model description for macroscopic chemical systems as a very large cluster or as a crystalline approximant.

3.11.3.1.2

Discrete state space

Quite frequently, it makes sense to reduce the complexity of the energy landscape by reducing the number of degrees of freedom in the system. One way mentioned above is by merging many individual atoms into a rigid (or only slightly flexible) building group, such that only the relative distances and orientations of these groups enter the description of a microstate; in certain situations when using empirical potentials, one can also reduce the complexity of the energy function by eliminating the interactions between the atoms within each rigid building group. Another way to make a landscape more tractable is to reduce the number of degrees of freedom of individual atoms by, e.g., keeping their positions fixed and considering only, e.g., their spin degree of freedom when studying magnetic systems such as spin glasses. In particular, if the spins are not treated as vectors but as quantized with respect to a certain axis, the state space of these N spins becomes a discrete set of isolated points in a high-dimensional space, constituting, e.g., for the Ising-model,53 the corners of a hypercube where the edges correspond to the moveclass of only switching one spin at a time.67 An important class of discrete approximations of the full state space are encountered when modeling solids such as alloys or multinary salts, which can be described as the distribution of different types of atoms on a given (periodic) lattice, at least in an approximate fashion. For example, two or more types of metal atoms of similar atomic radius can occupy all the positions on a lattice such as a fcc, hcp, or bcc lattice. Other examples are the distribution of several types of small cations over the octahedral 66 For a materials engineer, these defects are of greatest importance347dboth for reasons of principle and for technological applicationsdof course, but for most solid state chemists such defects are usually more of an irritant, at least up to now. 67 Note that this hypercube only reflects the state space and the neighborhood relation. The energies of the microstates will depend on the interaction parameters Jij between the spins si and sj of atoms i and j, and thus the “heights” of the corners of the hypercube could lead to a massive distortion of the cube. P If we were to choose the magnetization M ¼ isi a as the height axis instead of the energy, then the shape of the hypercube would be recognizable.

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and/or tetrahedral holes of various close packings of (spherical) anions.68 In both cases, each microstate of the discrete configuration space consists of a distinct distribution of the atoms over the lattice or hole sites,69 and the moveclass will consist of atom exchanges, i.e., jumps of pairs, triplets, etc., of atoms.70 An important issue is that for discrete energy landscapes there exists no natural dynamics or moveclass for the system. Viewed from the full energy landscape where every atom could occupy any position in R3, each of these atom arrangements on the lattice constitutes a local minimum atom arrangement, in particular, if we perform a local minimization for each arrangementdthus, they would correspond, e.g., to the defect structures associated with a crystalline compound. However, when using such models and restricted landscapes, we are interested in much longer time scales, on which these minima no longer constitute individual locally ergodic regions. Thus allowing such exchange moves defines a subset of such lattice-based atom arrangements, which constitute local minima as far as this moveclass is concerned. By comparison with (local) molecular dynamics or Monte Carlo simulations, one can establish the time scales texch on which various atom exchange moves are expected to take place, and thus one can model the dynamics of the system on time scales large compared to texch by performing a Monte Carlo simulation using this moveclass.71 Such a dynamics is often called a kinetic Monte Carlo simulation, where one selects the moves according to the time scale on which they are expected to occur; for more details, we refer to the literature.348 Now the choice of the possible exchange moves will depend on whether we are interested in the dynamics within a certain observation time window, or whether the focus is on establishing equilibrium properties such as the global density of states or the average density of various atom types as function of spatial region with each region encompassing, e.g., hundreds of possible lattice sites.

3.11.3.2

Infinite system

While every chemical system consists of a finite number of atoms, one often analyzes macroscopic systems in the thermodynamic limit (N / N and V / N while their ratio r ¼ N V is constant, or converges fast to a constant with N / N), or tries to approximate the large but finite system by a system of infinite size when computing its properties. One reason for such an approach is that it ensures that surface effects are truly negligible as one usually can expect in the thermodynamic limit.72 In practice, one is again forced to employ approximations, which fall into two classes, periodic approximants, and nonperiodic ones.

3.11.3.2.1

Periodic approximants

In the case of solids, where the number of particles is on the order of Avogadro’s number NAv, one tries to ignore surface effects and reduce the number of degrees of freedom by assuming periodic boundary conditions (with variable size and shape of the periodic ! cell), such that the configuration is defined by X ¼ ð! x 1 ; .; ! x Ncell ; a; b; c; a; b; gÞ; where Ncell is the number of atoms in the peri! odic cell, and x ¼ ðx ; y ; z Þ with x , y , z ˛ [0, 1) are the fractional coordinates of the atom i inside the cell, and (a, b, c, a, b, g) i

i

i

i

i

i

i

are the variable cell parameters. Since xi ¼ 1 corresponds to xi ¼ 0 (and analogously for yi, zi) because of the periodic boundary conditions, and, furthermore, (infinitely) many choices of the cell parameters correspond to the same infinite crystal on the same level of periodic approximation, the topology of this landscape is the one of a complex high-dimensional torus. This complicated topology exists on top of the permutation induced topology mentioned earlier that is due to the fact that exchanging two identical atoms via a path in configuration space results in the same quantum mechanical state of the system. The great advantage of this approach is that, for energy landscapes dominated by crystalline phases, few degrees of freedom are sufficient to represent the ideal crystalline modifications that are at the heart of the most important locally ergodic regions associated with the real crystal including the equilibrium defects. Furthermore, the energies (and some other properties) of these crystalline structures can be computed relatively quickly even on ab initio level as long as the number of atoms inside the periodically repeated cell is small.

68 In principle, any multinary atom arrangement can serve as a template for such a distribution of different types of atoms over this set of given discrete atom positions; both crystalline and amorphous structures can be used, in principle. 69 One distinguishes here two types of situations: on-lattice and off-lattice atom positions. In the case of “on-lattice” atom positions, the atoms must be located exactly on the site of the perfect lattice, while in the “off-lattice” positions one performs a local minimization for every such atom distribution over the lattice, in order to eliminate spurious energy contributions due to the stresses in the lattice when using atoms of different size and charge, etc. 70 An important special case are situations with occupational deficiencies, where some of the lattice sites are not occupied at all, leading to vacancies as far as lattice sites are concerned. Such vacancies can lead to fast atom movement by hopping of atoms to open sites compared to the atom exchange that must take place via interstitial defect configurations and this will most likely be relatively slow; this high atom mobility can, e.g., facilitate ionic conduction. Note that one often introduces such vacancies as a “pseudo”-atom into the state space description. Usually, the vacancy is only a convenient placeholder and does not interact with the atoms, the atoms of the background matrix if present, or other vacancies. But sometimes, a model with such effective vacancy-vacancy interactions can be a useful efficient description of, e.g., the elastic energy contributions in the system. 71 However, as one will observe when performing the MD simulations to find this moveclass, there will be many additional local minimum configurations associated with structural defects that violate the assumption that the atoms are only located on sites of the prescribed lattice. This needs to be kept in mind when interpreting the results of the very long kinetic Monte Carlo simulations; e.g., the lattice might actually be unstable, and a different lattice or the melt could be the thermodynamically stable state. Furthermore, the time scales of atom-atom and atom-vacancy exchanges will be different, and also depend on the distances between the atoms that are to be exchanged. This would be taken into account by, e.g., a probabilistic moveclass where long-distance jumps are attempted more rarely. 72 Of course, there are exceptions, e.g., systems consisting of a fixed finite number of layers where only the area goes to infinity in the thermodynamic limit, such that the volume is always proportional to the area, but such special cases need to be analyzed separately.

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The disadvantage is that the barriers one computes are heavily weighted towards energy barriers, and estimates of entropic barriers can be difficult to obtain. On the other hand, if one repeats the analysis for larger numbers of atoms inside the periodic cell, the energetic barriers will be reduced and the entropic barriers will become more prominent.73 Clearly, the larger the periodic approximant, i.e., the more atoms are in the (variable) periodic unit cell, the better the approximation will be as far as including various point defects, variations of composition, etc. are concerned. But we can never catch all the fluctuations of the infinite system. Furthermore, we note that for a fixed periodic cell that is often used to describe systems with amorphous structures without any periodicity, even very large approximants can be insufficient to catch the full behavior of the system. Actually, even the global minimum might still be inaccessible, if the fixed periodic cell is incompatible with the symmetries of the crystalline structure that constitutes the global minimum on the energy landscape of the infinite system. On the other hand, once the cell contains several hundred or thousand atoms, it might be large enough to represent a nucleation and growth process that takes place in a first order transition. For smaller approximants, the transformations we can model are either second-order transitions following zero vibration modes or only rough local approximations of the true transition in case this transformation can be realized by a sequence of synchronized local barrier crossings and atom rearrangements. Finally, we recall that, in reality, even a macroscopic essentially infinite system is just a very large but finite system, and thus it has a surface. But the use of periodic cells excludes a true surface, unless we artificially create a semi-infinite approximant, where a large empty space is included in the periodic cell, making the system look like a giant infinite slab of finite thickness. Such an approach can be useful when studying the behavior of the surface of a solid, but it requires quite substantial constraints on the system.74

3.11.3.2.2

Non-periodic approximants

The alternative to the periodic approximants to the macroscopic (infinite) system is to use a sequence of the finite cluster-type system discussed above (c.f. Section 3.11.3.1), for an increasing number of atoms. In this approach, one tries to extract those qualitative and quantitative properties of the system that scale with the number of particles and then extrapolate these properties to the infinite system. This method of using very big clusters is more realistic, but, as pointed out earlier, we need very large clusters before the surface no longer plays a role if we want to represent the “infinite periodic crystal”; it often takes hundreds of atoms before a cluster corresponding to a cut-out from the global minimum bulk structure has the lowest energy among all the clusters of a given size,349,350 and for other properties thousands of atoms might be needed.75

3.11.3.3

Dimensionality of chemical systems

Since chemical systems exist in the real three-dimensional world, both the positions and velocities of the atoms correspond to threedimensional vectors. However, one often deals with situations, which can be described, at least in an approximate fashion, as lowerdimensional systems. We distinguish between exact or pure 1D and 2D systemsdlocations and movement of the atoms are only possible in R1 and R2, respectively, dand quasi-1D, quasi-2D and, as a limiting case, quasi-0D systems, where the lower dimensionality only is apparent in the “infinite system limit”; in the quasi-1D, -2D and -0D cases, location and movement in the “additional” directions is extremely restricted. Such low-dimensional systems can also correspond to restrictions of atoms to some general closed or open (locally two-dimensional) surface such as the surface of a sphere or a cylinder.

73

Perhaps the most impressive effect of an entropic barrier is the stabilization of a macroscopic metastable modification of a chemical compound: In the limit of an infinite number of atoms inside the simulation cell, the energetic barriers per atom that separate two crystalline phases becomes vanishingly small compared to any finite temperature. Nevertheless, the metastable compound will not transform quickly into the thermodynamically more stable phase, because it is protected by a huge entropic barrier, which represents the difficulty for the system to find the lowest energy path between the two phases by random trial moves, i.e., random fluctuations in the spatial temperature distribution of the system. In contrast, one could steer the macroscopic system systematically from one phase to the other by “concentrating” all the kinetic energy provided by the temperature into a local set of few atomsdthus raising its local temperature to an arbitrarily high value sufficient to overcome all energy barriers between the metastable and the thermodynamically stable phase in this local region -, which then easily can be transformed from one structure to the other (this is similar to the way one uses zone-melting to achieve localized controlled (re-)crystallization of a material with many defects). But this means that it is not an energy barrier that protects the metastable phase but an entropic barrier: the difficulty to access the one (or the few) low energy transformation paths on the landscape during the usual stochastic time evolution of the macroscopic system. 74 For example, one might have to forbid periodic interactions in the direction vertical to the surface, thus reducing the 3D solid to a 2D-periodic slab of finite thickness. 75 In this context, we note that so far we have only considered “microscopic” energy landscapes, in the sense that the microstates have been defined via variables referring to atomic, or, at most, building group degrees of freedom. However, one can envision energy landscapes in chemical systems which refer to mesoscopic units, such as material points in the context of the continuum approximation of solids and liquids,351 as the basic unit elements and their characteristics as the defining degrees of freedom of a configuration of the system. Such a landscape could be the appropriate one for dealing with mesoscopic defects such as dislocations or grain boundaries, and the energies would be a combination of the (free) energies of the individual material pointdassumed to possess a certain crystal structure or averaged amorphous structuredplus the elastic interaction energies between the adjacent material points. Here, the “structure” Strð! r Þ of the material point ! r would be one of the discrete (or continuous, in principle,) valued variables characterizing the material point, on the same level as its other properties such as density rð! r Þ; displacement ! u ð! r Þ or conductivity sð! r Þ: This would constitute an intermediary landscape between the atomic-level energy landscape and the thermodynamic space where all atomic level information has been eliminated, and even spatially varying order parameter information has been incorporated into the free energies of macroscopic pure and mixed phases in thermodynamics.

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In practice, such situations frequently occur when the atoms or molecules are deposited on some substrate, leading to the formation of monolayers, or islands, or atom chains on the substrate, or when atoms are intercalated inside a layered compound, or fill nano-tubes or enter metal-organic framework compounds, just to name a few. Here, we will just point out some of the most important issues associated with such systems and their energy landscapes; for more information and discussion, we refer to the literature.163

3.11.3.3.1

Existence and embedding of 2D and 1D systems

Clearly, the atoms remain three-dimensional entities even in pure one- or two-dimensional real chemical systems, and therefore the quantum mechanical energy calculations will be the same as for three-dimensional systems. Of course, we can employ empirical potentials that are restricted to two or one dimensions or versions of quantum mechanical codes that have been rewritten for two- or one-dimensional systems, but this is not a primary concern here. We are interested in systems consisting of real atoms that are arranged in, e.g., planar monolayers or linear chains, and treating them as (approximately) 2D and 1D systems, respectively, is just an idealized representation. However, embedding these low-dimensional atom configurations into the real three-dimensional world can pose serious problems. The reason is that an atom arrangement that corresponds to a minimum when the atoms are restricted to one or two dimensions, will usually become unstable once the atoms are allowed to move in all three spatial dimensions.76 The description of a low-dimensional system is often quite straightforward, e.g., we can just keep one of the coordinates fixed to some value for all N atoms i, e.g., zi ¼ 0 for a monolayer, or ri ¼ R for a sphere with radius R.77 As a consequence, the effective state space is now R2N or (S2(R))N, respectively, where S2(R) is the surface of a sphere with radius R in three dimensions. We note that the low-dimensional chemical system usually would inherit the local topology of the 3D system, R3N, restricted to the lower dimension of the space, at least locally.78 Furthermore, the global topology and geometry of the low-dimensional system does not necessarily have to be similar to RN or R2N.79

3.11.3.3.2

Quasi-1D and quasi-2D systems

In contrast, quasi-1D and quasi-2D systems are explicitly three-dimensional, although they appear to be “pure” 2D or 1D systems in the limit of infinitely large quasi-2D- or quasi-1D-systems when the ratio of thickness to area or diameter to length goes to zero, respectively, corresponding to a variant of the thermodynamic limit. Such quasi-low-dimensional systems are either freestandingdincluding the pure 1D- and 2D systems as limiting cases for monolayers and monatomic wiresdor systems in direct contact with some kind of substrate, such as the surface of a slab, or the interior of a metal-organic framework (MOF) or a zeolite, which usually exerts some kind of constraint forces establishing the low-dimensional character of the system. An example of freestanding quasi-1D systems would be long wires with a diameter of only a few atoms.80 A special case of hollow wires are (single wall) tubes, with the atoms being located on the surface of a cylinder.81 A very large class of systems that have a quasi-1D character are single long chainlike molecules, like peptides, proteins, DNA, etc., which have been intensively studied because of their importance in biology and medicine.358 Although many of such molecules have a strong tendency to deviate from a straight-line type arrangement, the study of optimal arrangements of the residues and monomers along the structural back-bone of these chain molecules can often be considered a quasi-1D problem in a first approximation.82 One group of quasi-2D systems can be free-standing stacks of two or more (not necessarily identical) atomic monolayers, which are close enough to interact, e.g., double layer graphene.363 Another class of quasi-2D systems are the exfoliated building blocks of layered compounds such as the transition metal dichalcogenides, e.g., a single layer of edge-connected MoS2-octahedra. Along the stacking directions, these slabs only weakly interact (predominantly via van der Waals forces).364,365 Although such quasi-2D slabs are usually more rigid than the pure 2D monolayers, they are still unstable against shape deformations in the thermodynamic limit, when their area grows to infinite size, and the same is true for quasi-1D systems, of course.83 76 Note that the release of a constraint in a constrained optimization problem frequently destabilizes the minimum obtained while the constraints had been active. For example, it can be shown that, at non-zero temperatures, two-dimensional solids are unstable against (infinitely) long-wavelength perturbations,52 leading at least to a long-wavelength deformation into the third dimension, and sometimes even to a loss of their periodic order. Thus, even systems like graphene exhibit long-wavelength oscillations about an average “two-dimensional” structure.352,353 77 Note that this would be the constraint on the state space associated with the so-called Thomson problem: Distribute N (identical) charges on the surface of a fixed sphere such that their electrostatic energy is minimized.354,355 78 However, as discussed in the famous “Flatland” novel,356 the atom movements on the given surface (or along the given line) are now quite limited compared to those in 3D. For example, in one dimension, the sequence of the atoms is fixed unless “tunneling” through an atom is allowed as a move. 79 For example, if we grow a monolayer of N atoms on the outside of a cylindrical nanotube, then the topology of the landscape would be (C1 5 R)N where C1 is the unit circle and C1 5 R is the topology seen by a single atom. 80 Note that there is no requirement for the wires to be cylindrical, or even to have a constant cross section. 81 Note that the diameter of the tube can vary over some range, the tubes may be multi-walled tubes,357 or several layers of atoms might be allowed on the surface of the tube, etc. 82 Quite generally, very long chain-moleculesdeven without biological relevancedtend to fill “space” to various degrees; they are often modelled as self-avoiding random walks.359 Depending on the environment (temperature, pH-value, etc.) one observes transitions between more open and more compact structures, as, e.g., in the well-known protein folding transition.360–362 83 Regarding this instability of local minima of the quasi-1D/2D systems, one should distinguish between those, which are only unstable in the large system (thermodynamic) limit, and those instances where already small molecules or small periodic unit cells are distorted when the constraint is removed. An example are carbon or silicon monolayers, where nearly all carbon-based structures were stable against short wavelength distortions in the third dimension, while many of the silicon based layers were unstable.119

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In practice, one often encounters quasi-1D and quasi-2D systems when studying deposits on a surface or inclusions, e.g., inside a MOF or an intercalation compound. These atoms or molecules are constrained to remain in contact with the substrate, such that the deposit constitutes a “one-dimensional” or “two-dimensional” system after the substrate material has been subtracted. A special case are the mobile atoms that make up the surface layer of a material, which are studied in the context of surface reconstruction dynamics, surface relaxation, and surface transition366.84

3.11.3.3.3

Clusters and molecules on surfaces

3.11.3.3.4

Choice of energy function and state space

Placing few atomic clusters or flexible molecules of medium to large size on top of the surface of a rigid or flexible substrate at very low concentrations constitutes another class of low-dimensional systems. In these systems where not even one monolayer can form, both the behavior of individual clusters or molecules on the surface, and the structure and dynamics of multi-molecule/multi-atom islands (ranging from dimers to several hundreds of molecules or atoms) is of interest367,368.85 Clearly, many possible combinations of zero-, one- and two-dimensional objects can be used to realize low-dimensional systems and their corresponding energy landscapes: chains of atoms on a planar surface,369,370 stackings of nanotubes,371 shapes and self-assembly of chain-like molecules, and also the dynamics of molecules adsorbed on tubes or clusters,372,373 atoms or ions on various curved surfaces,374 and even on objects with unusual global topologies such as a Möbius strip, or along the edges of terraces on surfaces, etc. Many of these systems have already been studied, both from a theoretical point of view and in the experiment, but some are purely theoretical up to now.

Although the reduction in spatial dimensions seems to suggest that these low-dimensional systems are less complex, this is frequently not true, in particular concerning the technical issues associated with the realization in a model or simulation. For example, a large number of atoms is usually involved in modeling the substrate in a somewhat realistic fashion, and the interactions among the deposited atoms and molecules, among atoms inside the substrate, and between the deposit and the surface exhibit many different types of bonding mechanisms which require a treatment on ab initio level, in principle. This again necessitates a multi-stage approach in many cases, where we perform the global exploration with a combination of simplified energy functions and then refine the results (minima, barriers, transition paths, etc.) using ab initio methods.86 Concerning the constraints that establish the low dimensionality of the system, we usually use one of two options as mentioned earlier: hard constraints or soft constraints (c.f. Section 3.11.3.3.1). To enforce the low dimensionality via a hard constraint, one usually explicitly excludes any microstate that violates the constraint, e.g., one uses only atom position vectors with the x- and y-coordinates in a pure (planar) 2D-system, or only the z-coordinates and the polar angles Q for the quasi-1D system of atoms on a cylindrical surface.87 In contrast, as mentioned earlier, soft constraints are usually implemented as a penalty function, which disfavors any deviation by assigning a large energy penalty.88 Of course, mixtures of soft and hard constraints might also be employed, depending on the type of problem under investigation.

3.11.3.3.5

Choice of moveclass

As we have discussed earlier, if our goal is the reproduction of the physical time evolution of the chemical system, then the moveclass should be as “physical” as possible. In the case of the low-dimensional systems, this can lead to conflicts between the lowdimensionality of the model system and the paths the real system follows due to its embedding in three dimensions. On the one hand, the moveclass should respect the constraints defining the low-dimensional system, to be consistent. But in real chemical systems, the processes involved in changes of the molecule’s shape, in the re-arrangement of atoms that create structures on a surface, or in the growth of a low-dimensional phase, often exploit the freedom to move in three dimensions. But these kind of moves are not permitted

While the 3D stability of the quasi-2D/1D system is not a great concern, because the constraining force field is always present and not an artificial constraint, one often encounters undesired minimum structures that consist of multiple layers of atoms or molecules. This is especially critical if the global minimum of the quasi-2D atom þ substrate system consists of a three-dimensional thick film or a cluster and not of a monolayer of the atoms and molecules. 85 While most surfaces studied are planar, this is not a necessary condition, and many interesting properties of a chemical system can be studied as function of surface shape. 86 If reliable (structural) information is available from the experiment, which obviates the global exploration stage, then one can study a small region of the landscape, involving only one or two related local minima, directly on the ab initio level. While this is very popular in combined experimental and theoretical studies of, e.g., biomolecules on surfaces,164 starting from rather detailed experimental information greatly reduces the predictive power of landscape studies. 87 A well-known reduction scheme is the description of a chain molecule via its set of dihedral angles between the rigid monomers or residues of the chain, keeping all the atom-atom distances fixed along the chain.375 If we only allow changes in the dihedral angles while keeping the bond lengths fixed, then the resulting landscape would have the global topology of a high-dimensional torus because of the periodicity of the angle-coordinates. 88 Note that constraining a pure low-dimensional system with a penalty function has the disadvantage that we always slightly violate the constraints, i.e., our system is never perfectly two- or one-dimensional. Thus, we might not be able to identify the true minima, etc., of the pure low-dimensional system. On the other hand, in this fashion the embedding into three dimensions can be achieved in a nice gradual fashion by slowly reducing the size of the penalty. 84

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in the low-dimensional landscape. For example, the transport of atoms or molecules on a surface can take place via the gas phase or via a solution that covers the substrate, on which the monolayer of atoms or molecules is supposed to grow and evolve.89 Since the constraints can be quite complex, this can lead to a variety of desired or undesirable outcomes, depending on their implementation and their physical realism. For example, when modeling intercalation layers of a chemical system, one might want to force the atoms to stay within their original layer, to preserve composition inside each layer. But if we want to study, e.g., the staging transition,376–378 then jump moves between different layers must be allowed, and the same holds true if we want to investigate, e.g., a possible segregation of different atom types into one or more layers. In practice, such jumps between layers usually occur via a gas or solution phase enclosing the material, and less via solid state diffusion,90 but within our lowdimensional model the atom transfer would be realized by jump moves between two layers. Such intercalation or intergrowth compounds can be analyzed as quasi-two-dimensional systems, with highly complex energy landscapes, where the energy function involves interactions both within each layer and between layers.91 A related problem appears for a molecule or an atom on a realistic atomic surface in contrast to an idealized plane. Since the real surface is not perfectly flat on the atomic level,92 we must allow the atoms or molecules some freedom of movement orthogonal to the surface, both regarding their optimal locations and when simulating, e.g., the process of diffusion on the surface. But once the atoms are no longer restricted to move only in the x-y plane, we must be aware of the possibility that the atoms will move further away from the surface than just (a fraction of) an Angstrom into the gas phase, especially if we use elevated temperatures in our exploration and simulation algorithms. As a consequence, the atoms might climb on top of (mono-layer) islands already formed by the other atoms on the surface. But that would “ruin” the purpose of studying the formation of a single monolayer via modeling the atoms as a quasi-2D system.93 Of course, we can introduce some height limit the atoms or molecules are not allowed to exceed, but that is rather unsatisfactory, since tuning this limit introduces unpleasant fudge parameters into the model. Another problem connected to the realistic substrate surface arises when we want to identify the true (periodic) 2D-structure of an (infinitely large) assembly of atoms or molecules on a crystalline substrate. Intuitively, one would like to use a periodic 2Dapproximant for the atoms on the surface, in analogy to the 3D-crystal. However, since the substrate has its own (unchangeable) surface periodicity, the periodic 2D-unit cell that contains the atoms or molecules of interest and which we want to vary during our exploration of the energy landscape, must be compatible with the periodicity of the surface. Thus, the 2D cell can only change in discrete steps, and we must hope that the true cell of the periodic 2D system can at least approximately agree with a supercell of the substrate surface lattice, without having to use gigantic unit cells.94 As an alternative, one could take a very large surface slab and place many molecules or atoms on top of this realistic surface. As long as the unit cell of the optimal (infinite) atom layer is small, it might be possible to identify it by analyzing the local minimum structures we obtain for this finite system.95

3.11.3.4

Energy functions

As mentioned above, in the study of chemical systems, we mainly deal with two different classes of cost functions: (Potential) energy functions described by various kinds of empirical potentials or ab initio energy functions, and those consisting of or including objective functions not based on approximations of the energy but on comparisons with experimental data (or more refined, e.g., ab initio energies) of some kind, or combinations of those two types of cost functions.

89 Moving an atom between locations on the surface via, e.g., the gas phase, as seen from the point of view of a person restricted to the low-dimensional system, would correspond to the atom vanishing at one location and re-appearing at another place far away some finite time later. As far as the moveclass is concerned, this constitutes a straightforward jump-move. But we lose insight into the physical processes involved, both regarding details of the emission of the atom into the gas phase and the re-absorption on the surface, and concerning the question what kind of re-arrangements of the atoms remaining on the surface take place during the time that elapses between the emission and re-adsorption while the atom is in the gas phase; to take the second concern into account, one could place the atom into a holding pen, let the system evolve for a certain time, and then release the atom back onto the surface. In practice, we often assign such a move a very low probability to occur in the moveclass, as in the kinetic Monte Carlo approach mentioned earlier, where the probability distribution of such a jump movedas function of, e.g., distance -, is derived from, e.g., three-dimensional MD or MC simulations. 90 The exchange can also take place via grain boundaries between different crystallites, but then the relatively high mobility of the atoms inside these interfaces makes the grain boundary similar to a gas phase environment. 91 In addition to the direct interaction terms of atoms belonging to the same layer, the effective energy function of such multi-layer systems will usually exhibit both direct (two-body) interaction terms between atoms located in different layers, e.g., via Coulomb forces or van der Waals forces, and indirect (elastic) interaction terms mediated by the matrix of the host compound.377 92 We are not considering here the presence of mesoscopic features of the surface, such as terraces or islands of substrate atoms. Such features would be explicitly included as part of the model system, and thus do not constitute an unwelcome complication. 93 One person’s ruin is another person’s gain: By observing that the system prefers multilayer arrangements, we realize the limitations of the models we are employing, leading to improved approaches in studying such systems. 94 If we can assume that the interaction of the substrate with the atoms or molecules is rather weak or averages out over the size of the molecule, we can replace the true substrate surface with an averaged out attractive slab surface without any atomic resolution. While possibly being reasonable for large molecules on metallic surfaces, it can be problematic for small molecules on ionic surfaces, and for atoms on any surface.119 95 Note the similarity of this approach with the approximation of a macroscopic bulk material by a large cluster. Again, one would need a large computational effort to ensure that the shape of the circumference of the 2D-atom arrangement does not affect the ranking of the candidates for the 2D-periodic monolayer on the surface.

Energy landscapes in inorganic chemistry 3.11.3.4.1

Empirical potentials

3.11.3.4.2

Beyond empirical potentials

291

While one would clearly prefer to evaluate the energy on ab initio level,96 when studying a chemical system, the global exploration of the ab initio energy landscape of a system containing hundreds or thousands of atoms pushes the computational limits for most systems. Thus, one frequently employs simplified energy functions where the actual choice (simplified ab initio, DFTB, semiempirical, empirical many-particle potentials, few-body potentials, etc.) depends on the computational resources available, and on the required level of global/local accuracy, transferability, global/local applicability, and robustness. Here, global applicability means that no unphysical regions are allowed with a finite energy, e.g., no two atoms are allowed to be at the same position, or there must not be any microstates with infinite negative energy.97 Furthermore, the resulting landscape should be accurate in the sense that the physically and chemically interesting modifications should be present as local minima, and the barriers between them should be of reasonable size. Similarly, the landscape should be robust, i.e., it should not change its overall barrier structure upon small changes in the parameters of the potential.379 The potential should be able to represent the different relevant modifications of the system, while the functional form should be transferable between related chemical systems.98 The speed requirement suggests that preferably the energy should contain only two-body, at most three-body terms. The types of empirical potentials available to model the landscape of solid materials range from two-body potentials like the Born-Mayer-potential,380 the Buckingham-potential,381 simple robust Coulomb-plus-Lennard-Jones potentials379,99 without or with environment dependent radii,386,387 over dipole388 and quadrupole389 shell models, to various kinds of breathing potentials,390–397 and potentials that include three-, and four-body interaction terms. Finally, there are many-body potentials such as the Gupta potentials,398 the Sutton-Chen potentials,399 the Finnis-Sinclair potentials400 or the closely related embedded atom methods.401 Determining satisfactory parameters for a given chemical system and type of potential is usually quite challenging, especially if no experimental data are available. Hence, one employs fits to ab initio energies,402–404 or extrapolates parameters deduced from related systems, such as typical bond lengths379,386,405–407 or average ionic radii145.100 In this fashion, rather simple empirical potentials have been developed in recent years,255,408 which combine accuracy, global applicability (within the unavoidable limitations) and robustness. An alternative approach has been developed in the past 15 years,409,410 where one employs a neural network to optimize the implicit parameters in a generic empirical neural-network based energy function. Here, the network is trained on a set of ab initio energies of (randomly chosen) atom configurations. The transferability of the parameters between different chemical systems is still an issue, but with the new developments in the field of machine learning, it is expected that such neural network based energy functions will be increasingly used for energy landscape problems.411

For the final energy ranking of, e.g., minimum and saddle point configurations obtained using empirical potentials, one needs to employ ab-initio methods, many of which are nowadays available for both theoretically and experimentally oriented researchers.412–417 Choosing an appropriate functional, pseudo-potential and/or basis set is non-trivial, but the availability of such functionals, together with basis sets and pseudo-potentials for most atoms of interest, allows us to relatively easily compare the results for different ab initio methods and functionals, pseudo-potentials and basis sets.111 As long as the system is not too large, the global exploration of the ab initio energy landscape is feasible,418–420 including the determination of the energy barriers. The computation of entropic barriers on full ab initio level is usually too expensive computationally; however, this can already be done using less refined methods such as density functional tight binding (DFTB).421 The greatest advantage is that ab initio energies are usually more accurate than empirical potential based energies due to the general applicability of the functionals employed, and they can be used for essentially all types of bonding present in chemical systems.101 The major disadvantage of using ab initio energy functions remains the computational cost, which can easily slow the calculations by a factor of 103–105 compared to calculations based on empirical potentials,418–420 in particular since general atom configurations exhibit none of the symmetries that are usually used to accelerate the calculations.102

96 One should be aware that even the most refined ab initio energy function still contains many approximations and parameters, which sometimes are based on experimental information in an indirect fashion. 97 Quite a number of empirical potentials, which have been fine-tuned to reproduce the experimental properties for configurations close to the experimentally observed structure, become strongly unphysical when one moves further away from this minimum. Besides unphysical configurations, one sometimes finds that there exist lower energy minima on the landscape than the one that had been chosen as the reference for fitting the parameters of the potential. 98 We note that in order to overcome the limitations of the simple potentials as much as possible while still retaining their advantages, one should repeat the global explorations for slightly varied values of the parameters characterizing the potential,21,379 since experience has shown, that the most important structure candidates usually are quite robust against minor changes of the parameters of the potential. 99 The infinite-range 1/r Coulomb-interactions (and similarly other infinite range power-law based potentials) are usually evaluated using some convergence procedure.382–385 100 Furthermore, one can make the radius parameter depend on the atom’s current environment.386,407 101 In contrast, if we are not assured of what kind of bonds will be formed by the atoms, choosing a certain model potential to describe the interactions would introduce a dangerous bias in the calculations. 102 Here, one often encounters a convergence problem, when dealing with random atom arrangements which are far away from realistic solid state structures. In particular for gas-like atom arrangements, one should probably employ Hartree-Fock calculations, since the relatively large band gaps typically produced by this method make convergence easier compared to DFT calculations.118,418–420

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An alternative that combines the generality of the quantum mechanical calculations with the speed of the empirical potential calculations is the density functional tight binding method (DFTB).422–424 This method is robust and globally applicable for a given chemical system, and it is about three orders of magnitude faster than standard DFT methods. However, the accuracy is limited due to the approximations involved and due to the quality of the atom-atom tight-binding parameters employed in the energy function. One problem, up to now, is the lack of a complete set of parameters for all pairs of atoms, since these need to be fitted to the appropriate ab initio calculations. Nevertheless, this method is independent of interaction type103 and very fast, and thus would be the method of choice when searching not only for local minima but studying the energetic and entropic barriers of an energy landscape.104 In recent years, approaches like the Quantum Monte Carlo method have experienced a renaissance,425–428 and single loop (summation) level perturbation methods have made the transition from analytical calculations to numerical codes.429 But notwithstanding the great advances that have been made in the development of faster and more accurate codes, one should repeat the calculations with several (ab initio) methods to compare results of the global explorations if one investigates unknown chemical systems for which no comparisons to experiments are available.105 In this context, we note that there is a practical issue with the availability of the desired combination of energy function and exploration algorithm, since not all energy evaluation codes include all types of exploration algorithms, and, conversely, exploration algorithm packages usually contain only a limited number of energy functions. Since many empirical potentials, and, in particular, the ab initio energy functions are very complex and cannot be quickly implemented inside an exploration code, one often employs interfaces to some external code, to which the energy calculation, and also often the computation of a gradient or a local minimization, are delegated.106 A disadvantage is that starting the external code, writing input for this code and parsing and analyzing the output for every step of the exploration algorithm often adds an order of magnitude to the computation time. Some ab initio codes actually offer the possibility to implement an exploration or simulation method, although this requires some knowledge of the internal set-up of the ab initio program, together with access to the source code.107

3.11.3.4.3

Energy functions with building units and molecules

In general, each atom of a N-atom chemical system has complete freedom of movement, i.e., the state space is R3N. But this implies that we need to calculate the energy for a N-atom system, and explore a 3N-dimensional landscape. However, in many situations, we do not want to study the full landscape but only investigate some aspects of the system that can be described with a greatly reduced set of (generalized) coordinates. For example, a molecule can frequently be regarded as an indestructible unit within a molecular crystal, and if, in addition, parts (or the whole) of the molecule can be treated as rigid, the size of the configuration space is greatly reduced.108 In addition, the atomatom level interactions between the atoms belonging to different molecules can sometimes be replaced by an effective interaction between the molecules. As mentioned above, a fast globally applicable potential that contains covalent, ionic and van der Waals forces at the same time is difficult to construct and parametrize. Thus, subsuming, e.g., the energies of the covalent bonds in P a constant energy contribution of a rigid building unit a, Ea ¼ < i, j > ˛ aVij ¼ constant, is of great practical use.109 Similarly, we often eliminate the large numbers of water or solvent molecules that surround, e.g., a protein, by introducing an effective medium or pseudo-particles that interact with the protein in a simplified fashion. Another way to accelerate the energy calculation for many multi-atom systems consists of using several energy functions of different degrees of accuracy,110 depending on how close the atoms are to the place in the material where, e.g., a chemical reaction of interest takes place. For example, an ab initio energy function is used in the close neighborhood of the reaction site, while further away, a simple but fast energy function is employed, and even farther away, the influence of the rest of the chemical system is represented by some continuous medium exerting an average force on the atoms at the interaction site.

At least on the level of DFT and tight-binding; van der Waals terms might have to be “added by hand”, but this is often also necessary for other ab initio methods. In this context, we remark that one can also select weaker convergence criteria, and minimalistic basis sets, etc., to speed up the ab initio calculations during the global search, where one subsequently refines the minima structures and energies using high-quality ab initio calculations.418 105 Keep in mind that the true ground state of the system might require calculations that go beyond the one-electron approximation and include both electron-electron correlation and electron-phonon interactions; furthermore, this state would need to be identified without prior information about its existence or structural and electronic properties. 106 For example, one code which employs many such interfaces is the G42þ global landscape exploration package,119 which contains interfaces to a number external codes such as CRYSTAL,415 QuantumEspresso,416 Gaussian,417 DemonNano,424 GULP,430 AMBER,431 and several others. 107 For example, the threshold algorithm has been implemented directly in the DFTB DemonNano code,424 accelerating the performance by at least an order of magnitude compared to calling DemonNano from the threshold algorithm implemented in G42þ .421 108 Of course, ideally, these constraints arise naturally from high energy barriers on the N-atom energy landscape that prevent the atoms in the molecule from changing their relative positions. 109 For very complicated compounds, e.g., proteins, a whole hierarchy of building units (atoms, side chains, residues, a-helices/b-sheets/coils, domains) can be identified,246 leading to a sequence of ever coarser approximations of the full energy landscapes. This is reflected in the large number of empirical force fields modeling the interactions among atoms/building units/solvent molecules, e.g., UNITAT,432 OPLS,433 AMBER,434 CHARMM,435 GROMOS87,436 TIP3P,437 DISCOVER,438 ECEPP,439 ECEPP/2,440 ECEPP/3,441 MM3,442 where sometimes effective molecule-environment interactions443,444 are also incorporated. 110 This procedure is often called the QM/MM approach.445 103

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Energy landscapes in inorganic chemistry 3.11.3.4.4

293

Special cost function landscapes, involving non-energy terms in the cost function

As an alternative to rigid building units, we can use again penalty terms to enforce the general shape and bond topology of the building unit. In contrast to the rigid constraints, this allows the atoms in the building unit to move somewhat, making the building unit flexible.111 In the past, energy-like terms based on chemical intuition, such as bond-valence terms,447 have been added to the cost function of the system, since they are easy and fast to evaluate and reflect phenomenological insights into classes of chemical systems.112 On the other hand, adding easy-to-calculate penalty-like terms based on such “chemical rules” can be quite useful when constructing cost functions for inverse problems, e.g., when performing structure solution from experimental data. Now, if the major contribution to the cost functiondthe difference between, e.g., a computed and measured diffractogramdis not sufficient to identify a chemically reasonable structure, one usually would add a regular (possibly computationally expensive) potential energy function term to provide additional guidance to the exploration algorithm (c.f. Section 3.11.7.2.3). But often already adding a “cheap” chemical intuition driven term guarantees that the low-cost structure solutions do not violate common chemical and physical sense.

3.11.4

Time and energy landscapes

3.11.4.1

Introduction of observation time

3.11.4.1.1

Measurements in experiment and simulations; the concept of ergodicity

In an abstract sense, the cost function landscape is an interesting mathematical entity and can be investigated as such, where the focus is not only on, e.g., the critical points such as minima, maxima and (for continuous landscapes) saddle points, or, for metagraphs, on the average number and distribution of edges between discrete microstates or the distribution of energies over the microstates, but also on issues like the global topology, the number or density of paths of given length between two given states, or the number of disconnected landscape pockets for a given value of energy or lid. Outside of pure mathematics, however, the focus is on the landscape as a tool in solving some “real world” problem like an optimization or optimal control problem, or studying the behavior of a chemical, physical, biological, sociological or economic system. Trying to understand a real system involves measuring its properties, computing and predicting these propertiesdincluding those not accessible to the experimentd, and, finally, to develop the ability to control the system in some fashion. By its very nature, any measurement involves observing the time evolution of the various observables of the system for a certain time. But since the configuration space of the system under investigation encompasses all feasible (relevant) states of the system, any measurement corresponds to following the trajectory of the system on its energy landscape, while registering various quantities of interest such as the energy, atom positions, or the magnetization, just to name a few.113 The trajectory is governed by (more or less) fundamental laws such as Newton’s equations for a classical chemical system, or empirical rules of the market in economics, or the algorithm of a global optimization procedure. In every case, the dynamics is incorporated in or should at least be consistent with the moveclass of the landscape, regardless of whether we talk about the real physical energy landscape or about the model approximation on paper or in the computer. In principle, the outcome of every such set of measurements would depend on the specific trajectory the system traversed during the experiment, and thus, in general, computing the result of the measurement would require a reproduction of the complete trajectory. However, in particular for many (finite and closed)114 chemical and physical systems, the laws of equilibrium statistical mechanics apply. Thus, for a known probability distribution over the microstates of the system, we can use statistical methods to compute an equilibrium property of the system as an ensemble average of the corresponding observable O, hOiens and compare the result with the time average of the measurements along the (infinitely long) trajectory hOitobs / N.115 If the statistical and time 111 In this context, we mention the so-called statistical potentials,53,446 which describe the temperature dependent interaction between atoms or building units by introducing “effective” or “average” energies that are only defined on larger observational time scales. These potentials are designed to reproduce various statistical quantities, e.g., distances between non-adjacent building units observed in experiments or MD simulations. 112 This chemistry-driven approach is nowadays rarely employed, since the quality of the empirical potentials has improved and ab initio energy calculations are pretty standard by now, and thus the limitations of the phenomenological potentials become a problem. 113 We keep in mind that many quantities that are measured in the experiment can only be obtained as time and spatial averages over material points; examples are electric and magnetic permeabilities, susceptibilities, heat and electric conductivities, etc., since they do not make sense or are not defined on the level of an instantaneous atom configuration or quantum mechanical state. 114 For infinite systems, in the thermodynamic limit, we consider the appropriate extrapolations of the measured quantities. 115 In practice, one often employs averages over many such long trajectories starting from different initial microstates of the systemdequivalent to repeating the experiment. In this fashion one can obtain a better estimate of the behavior of the system at infinitely long times when the (finite) system is definitely in global equilibrium but which are inaccessible for simulations. In this context, we note the similarity to the concept of self-averaging. This concept is employed to justify the assumption that we can treat even extremely large yet homogeneous chemical systems such as a macroscopic crystal as being ergodic although a single trajectory on the landscape cannot explore the whole configuration space within the observation time or even during the time that has passed since the preparation of the material. In the self-averaging procedure we mentally divide the homogeneous piece of material C in small copies Ci whose configuration space is small enough to allow a trajectory to sample the equilibrium distribution of microstates sufficiently densely, i.e., tobs [seq ðCi Þ, while still being large enough for the copy Ci to be able to serve as a substitute for the full material C as far as the properties of interest are concerned. Of course, each Ci will deviate from the full material C according to the statistical fluctuations in, e.g., the composition. But by averaging over the properties of the copies, we can recover the properties of the full system including such fluctuations, in an approximate fashion. Of course, we cannot deal with situations, where the system would prefer to exhibit, e.g., a decomposition into two or more phases with macroscopically different compositions. But in this case, the system is no longer homogeneous and the pre-condition of homogeneity needed for self-averaging is violated.

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averages are equal, rhOitobs / N  hOiens r ¼ 0, we call the system (globally) ergodic, with respect to the observable O.54 Of course, the exact agreement will only occur in the limit of infinitely long trajectories, but the world around us is full of examples, where equality holds in an approximate sense even for finite measurement times. Furthermore, we notice that there are a multitude of metastable chemical systems such as metastable compounds or isomers or conformations that can exist for a very long time and can be characterized by seemingly unchanging values of various observables, although they are not the thermodynamically stable state of the system. Clearly, this demonstrates the need to consider the issue of time scales involved in the time evolution of a system, the performance of measurements, and the simulation of this evolution.89,149,154,300,343,448–451 The most important time scale is the observation time tobs, over which we follow the trajectory of the system. Of course, the starting point of the trajectory where we begin the observation is also important, but leads to complications because we usually do not have enough information in the experiment to identify the initial microstate configuration. However, for systems in equilibrium, the precise initial configuration should be irrelevant. Thus, one often introduces an initialization time tinit when simulating a trajectory, after which the effects of the choice of starting point are assumed to have become negligible. The size of tinit depends on the quantity we want to observe, or on the kind of properties a generic starting point should have.116 We note that such an initialization time would appear both in the real physical system and in the simulations. global , after which the system is in global equiCharacteristic time scales for the system itself are the global equilibration time seq 117 librium, the local equilibration times seq(R) for a subregion R after which this subregion is in equilibrium, and the escape time sesc(R) after which the system has moved out of the subregion R; in the latter two cases, we assume that the system had started its time evolution inside the region R. In experiments, usually some time passes between the original preparation of the (chemical or physical) system and the start of the actual measurement. This elapsed time during which the system can relax according to the more-or-less well-defined thermodynamic (and possibly other) boundary conditions is called the aging time sage or waiting time tw (in simulations). For thermodynamically stable phases, this aging process is not of great importance. But for systems prepared in a metastable phase or a nonequilibrium state, the length of the aging phase can have a strong influence on the subsequent measurement(s) on the time scale tobs.270,271 In glassy systems, for example, we find that if tobs  sage, then our measurements suggest that the system is in thermal equilibrium. In contrast, for tobs [ sage, we observe non-equilibrium behavior such as drifts in the observables instead of fluctuations about an equilibrium value, or peaks in the specific heat similar to those one would usually associate with a phase transition. Often, one exposes the system to interactions with the environment that vary with time, and thus lead to a time dependence of the energy landscape. The characteristic timescale on which noticeable changes in the energy landscape take place is called tvar, and the relative size of all the other time scales with respect to tvar is of great importance regarding our ability to model the system in terms of equilibrium properties or for the real system to even develop such properties. A special case in this regard are macroscopic chemical systems, where we deal with an additional important time scale, the time s(B) eq on which a thermodynamic boundary condition B, e.g., an externally prescribed temperature or pressure, can establish itself throughout the system. After all, in a real physical system, the change in B usually occurs through the surface of the material via, e.g., heating or externally applied mechanical forces. As a consequence, there exist gradients in the quantity B from the surface into the interior of the material, which thus induce appropriate fluxes that aim to eliminate the imbalances, e.g., heat or deformation currents, and it will take the time s(B) eq before a constant value of the quantity B is established throughout the system.118 Concerning computational explorations of an energy landscape, we also encounter various relevant (computational or algo(A) rithmic) time scales, such as the time scales t(A) aver or tsample on which we average or at which frequency we sample a quantity A along the trajectory, respectively. The issue here is one of time resolution, since the simulation might only agree with the real physical time evolution of the system on time scales larger than tmodel, where tmodel denotes the time scale beyond which we can assume that our model dynamics reflects the behavior of the system with respect to various observables of interest.119 Furthermore, the mathematical implementation of the dynamics, i.e., the simulation algorithm, might introduce artificial short-time correlations in the quantity A that are unphysical on time scales smaller than t(A) corr. Thus, we would expect an unbiased distribution of the values of A from the (A) simulation only for sampling times t(A) sample > tmodel, tcorr.

116 The shortest initialization time would correspond to the correlation time t(A) corr along simulated trajectories introduced below, after which the simulation of beyond which we can assume that the the system can be considered as properly started, while the other extreme is the global thermal equilibration time sglobal eq system is in a microstate typical for a globally equilibrated system. In-between initialization times would be the local equilibration times of sub-regions of the landscape, or the timescale on which a certain property is equilibrated even while the system or the subregion is not yet fully equilibrated with respect to all observables. 117 Usually, we refer to statistical and thermodynamic equilibrium, since this is most relevant for chemical systems, but we can also consider equilibrium with regard to specific properties or based on other suitable criteria. 118 Usually, one assumes that such equalization processes proceed as exponential relaxations, B(t)
Beq þ (Binit  Beq) exp (t/s(B) eq ). But in non-equilibrium situations, such processes can also appear via a power-law behavior or via a multitude of exponential relaxations exhibiting a wide spectrum of relaxation times; in that case, s(B) eq becomes poorly defined as a single entity. 119 An example would be the use of MC vs. MD simulations: While for the MD simulation we would expect a good agreement with the behavior of the real system on time scales of the individual MD steps tMD and thus tmodel ¼ O(tMD)
10 14  10 15sec, this is not the case for MC simulations. Here, we only expect agreement with experiment after many MC-steps, each corresponding to, e.g., the small shift of an atom, i.e., on time scales for which at least local equilibrium for the atomic degrees of freedom has been established, e.g., tmodel ¼ O(1/uyib)
10 10  10 12sec, where uyib are characteristic (high) vibrational frequencies of the material.

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295

A special case of an important exploration time scale is tprob, which indicates the length of an exploration run before one checks, whether one still is in the starting region, e.g., a minimum basin, or has reached another region, e.g., another minimum basin or a transition region. Such exploration runs measure the probability flows between important regions of interest on the landscape corresponding to, e.g., metastable compounds. Thus, for the observations obtained to be useful, it is important to choose tprob in such a way that it is comparable to the escape time from the various stable regions on the landscape,120 but tprob should still be much shorter than the global equilibration time sglobal of the system.121 In the context of a master equation dynamics (c.f. Section eq 3.11.4.2.1), where we follow the time evolution of the system on larger time scales described by transition probabilities, each step corresponds to an elapsed time tstep which usually equals or is comparable to the exploration time scale tprob. def In this context, we note that we need to keep two observational time scales separate: The time scale tobs , which we have in mind def exp , on which the when we define the existence of a locally ergodic region (LER) via the relation sesc [ tobs [ seq, and the time scale tobs exp def actual experiment or simulation takes place. Usually, tobs ¼ tobs (and we simply speak of tobs) when we study the properties of the metastable compound. But if we want to investigate transitions among locally ergodic regions (assured to exist on the time scale def exp def def exp ), then we use observation times tobs
sesc [ tobs . Since it is usually obvious whether we refer to tobs or tobs , we just use tobs in tobs 122 most of the text.

3.11.4.1.2

Locally ergodic regions for an isolated chemical system

In the preceding subsection, we mentioned two characteristic local time scales of an energy landscape, the equilibration time seq(R) and the escape time sesc(R) for subregions R. We note that for an arbitrarily (!) chosen set of microstates that together are defined to be a subregion R, the escape time is most likely to be much shorter than the equilibration time. This implies that we cannot even define an equilibration time for this region, unless we force the system to reside inside the region until equilibrium has been established. Conversely, this implies that regions R for which sesc(R) [ seq(R) are very special. In particular, for observation times tobs in-between these two time scales, sesc(R) [ tobs [ seq(R), the system will be in local equilibrium during the measurement along a trajectory that starts inside region R because of tobs [ seq(R), but the trajectory will not contain states outside the region R because of sesc(R) [ tobs. As a consequence, the time average over the trajectory will equal the statistical average over the region R, analogous to the definition of global ergodicity for the whole system. We therefore call such regions locally ergodic regions def with the region R, such that for measurements on the time scale tobs, and we can associate a “defining” observational time scale tobs def 63,101 on the time scale tobs
tobs the region is locally ergodic. We note that this concept of a locally ergodic region can be extended to general cost function landscapes if we want to explore the behavior, i.e., the time evolution, of the system represented by the landscape according to some (empirical) dynamics.123 Here, too, we can encounter “stable” regions where the system can reach a statistical equilibrium such that the statistical average of some observables equals the time average as long as we observe the system on time scales longer than the local equilibration time but shorter than the time scale on which the system becomes destabilized. To analyze this central energy landscape concept in more detail, we first consider an isolated chemical system. We recall that an isolated system is characterized by a given constant total energy, total angular momentum, total translational momentum, no change in atom content, and constant volume. Now, the concept of a stable region124 requires the definition of a dynamics of the system, such as, e.g., a MC walk or MD simulation under the condition of constant total energy.125 Here, the condition of

120 Here, we need to distinguish between measurements of probability flows as function of temperature and as function of the energy slice in which the random walk takes place. In the former case, we need to cross both energetic and entropic barriers, and tprob should exceed sesc(T), in order to yield non-trivial results. In the second case, we only deal with entropic barriers, and thus tprob can be relatively short compared to the first case. 121 In practice, one would perform many checks during the exploration at sample times tsample, e.g., tsample ¼ tprob/10 or tsample ¼ tprob/100, since the escape times from the various starting regions will vary considerably. Furthermore, one would repeat such explorations many times for each starting point, in order to get a good statistically valid sample. Finally, we note that we can introduce a time dependence into the definition of probability flows by varying tprob, which allows us to study barriers between neighboring basins and pockets on the energy landscape as function of exploration time. 122 def As an aside, we note that changes in tobs are also relevant, in principle, for the definition and measurement of the probability flows, insofar as such changes implicitly modify the set of LERs of interest and thus the structure of the relevant probability flows. But, recall that the probability flows first of all depend on def def > seq; else, if tprob < tobs  sesc, the system would remain always in the starting minimum the time scale tprob used to define the flow. Here, tprob
sesc  tobs def during the explorations. Thus, the flow depends only indirectly on tobs in the sense that we consider only flows that contribute small amounts of escape def . Note that if tprob [ sesc, the probability flow “empties” the locally ergodic regions, i.e., it reduces the probability to find the system probability, sesc
tprob  tobs global in this region, much too quickly, even if one samples at a rate of, e.g., tsample ¼ tprob/100. We note that for the extreme case tprob [ seq [ sesc(R), the probability flows correspond to the transition probabilities to “instantaneously” reach a LER R according to its equilibrium probability p(R), from any starting point on the landscape, within one (large) time step tstep ¼ tprob. 123 Examples could be human or animal migration patterns that are stable over some years (the cost function might be some “happiness” of “fitness” function for a population or individual members thereof), or the stable state of a business that generates a fluctuating but on average constant income for its owner. 124 Such regions are often also called “stable states”, “subcanonical ensembles” and ‘“wells”, respectively.448,452–455 Intuitively, it is clear what this means, but the precise definition of what constitutes a stable state varies to a certain extent; we are not going to discuss these subtleties here. 125 When using MC-walks as the dynamics, the time resolution of the MC steps and their physical realism should be in a realistic ratio to the given observation time, such that the resulting trajectory represents a proper time evolution of the system on the time scale of observation of interest (recall the discussion of tmodel in Section 3.11.4.1.1). This can also include major jumps on the landscape if the observation time scale of interest is much longer than the time the system would need to make an equivalent large change using, e.g., MD simulations. In this context, it is important to keep in mind that the different observables can have different equilibration times.

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constant energy (and volume and number of particles) means that, from a statistical mechanical point of view, we are in the microcanonical ensemble52,53.126 The time evolution of a classical system, i.e., its trajectory in phase spacedthe 6N-dimensional space of all atom positions and velocities (or momenta) defined earlierd, is determined by the positions and velocities of all the atoms at some initial time t1, together with Newton’s equations and the energy function of the system, from which the forces on the atoms can be calculated.127 A physical measurement or simulation thereof consists of the time average of some observable O, hOit1 ;t2 ¼ R ! 0 ! 0 1 t2 0 tobs t1 Oð R ðt Þ; V ðt ÞÞdt , over a time interval [t1, t2] of length tobs ¼ t2  t1 along this trajectory, under the condition that the total energy of the system remains constant. As mentioned above, although the initial state is never known in macroscopic chemical systems, many properties can be reproducibly measured and calculated nevertheless in many situations, since they are essentially independent of the starting point t1 of the trajectory in phase/configuration space, hOit1, t2 ¼ hOitobs ¼ (t2  t1), within the accuracy am of our measurement. (O) (O) Furthermore, if tobs > seq where seq is the equilibration time with respect to the observable O,128 and the average over the R !! ! ! Oð V ; R Þd V d R trajectory is essentially indistinguishable from the statistical average, the so-called ensemble average, hOiens ðEÞ ¼ R ! ! dV d R 101,343,459 within the accuracy am of the measurement process, rhOitobs  hOiens r < am, then we call the system ergodic (with respect to the observable O, up to the accuracy am, on the time scale tobs). Note that in the case of the microcanonical ensemble discussed R ! ! here, the normalization integral d V d R ¼ g ðE; V; N; .Þ equals the number of microstates of the system g(E, V, N, .), which have the prescribed total energy E, volume V, etc.129 As we have mentioned, in many complex systems, such an approximate equality of time and ensemble average can also occur for local subregions R, where the trajectory is assumed to remain in region R for the whole observation time interval, and the ensemble  R ! ! ! !  R Oð V ; R Þd V d R   1 t2 ! 0 ! 0 0 average is only taken over the region R, rhOitobs ðRÞ  hOiens ðRÞr ¼ tobs t1 Oð R ðt Þ; V ðt Þ; RÞdt  R R ! !  < a, as long as   dV d R R R ! ! 130 we do not demand that a ¼ 0. Here, the normalization of the ensemble distribution R d V d R equals the number of states inside the region R, g(E, V, .; R). In terms of the time-evolution of the system, the trajectory explores R densely enough that all statistically significant states of the system within R are visited with a frequency that agrees with the statistical (ensemble) distribution.

126

The construction of stable regions for general cost functions would be analogous to the case of the microcanonical ensemble described here. In reality, the world is not classical, and thus the simple picture of well-defined neighborhoods of states, and a time evolution that can be visualized as classical single walker trajectories or as probability flows represented by ensembles of walkers becomes problematic. After all, in quantum systems essentially all (micro)states of the system can be connected by non-zero probability amplitudes, rendering the definition of a local minimum questionable. Furthermore, interference phenomena between different paths connecting two states will appear. As a consequence, many classical features, such as what we mean by a “local minimum” or locally ergodic region, become fuzzy. But, clearly, locally ergodic regions exist in physical and chemical systems. Again, we can resort to ! time scale based arguments to suggest a possible approach to address this issue: we treat a given microstate X as a local minimum of the quantum system with ! ! an escape time sesc ð X ; cÞ, if for all tobs  sesc ð X ; cÞ the likelihood to reach a state with lower energy is below a given number c  1. The same argument can be used to consider a set of microstates R with regard to the possibility of it being a locally ergodic region, i.e., we define equilibration and escape times for the set def R, and define again a local ergodicity via sesc(R) [ tobs [ seq(R), which now might be denoted a “quantum ergodicity”. This way, quantum and statistical averages are combined in the definitions of the relevant time scales and local ergodicity. In practice, one can define “core” and “edge” states of a region R,456 that correspond to the interior and the boundary of an ergodic region, respectively. For more discussions on quantum ergodicity and related issues, we refer to the literature.456,457 global 128 However, we note that global equilibrium often cannot be obtained within the time span of the experiment seq [ tobs. Now, if the system exhibits exponentially fast relaxation dynamics to global equilibrium, this allows us to apply our statistical mechanical tools even for finite observation times, but a “practically defining” feature of many complex systems is that such a fast relaxation dynamics does not exist.450 In fact, there are different types of complex systems, some of which will be ‘complex’ on all time scales of practical interest, but will exhibit such an exponential relaxation at extremely long observation times (e.g. glasses like SiO2 where a crystalline ground state exists and is accessible, in principle), while other systems will never show a global exponential relaxation behavior (such as ideal spin glasses in the computer). Yet in some macroscopic systems, while equilibrium among spatially separated regions is not reached within tobs, one can often treat these regions as only weakly interacting parts of the whole system, and consider them locally (spatial) or partially (with respect to some degrees of freedomdsomewhat analogous to the equilibrium with respect to some particular observable O) equilibrated. Then a measurement consists of an average over these pieces of the system. If these pieces can be assumed to behave like “small” representatives of the whole system, such an average would correspond to an “ensemble” average, and we would speak of a “self-averaging system”, as mentioned earlier (c.f. Section 3.11.4.1.1). A related type of self-average concerns the distribution of some system parameters, e.g., local environments an atom might see.458 Again, we would be able to use formulas from, e.g., equilibrium statistical mechanics, if in a macroscopic system the theoretically expected/prescribed distribution is realized. One often assumes that amorphous and glassy systems exhibit one of these two types of self-averaging, but the same might well be true for, e.g., the mesoscopic (line and interface) defect distribution in crystalline solids, or the dynamics of proteins. 129 In principle, it can happen that the condition of constant energy, etc., can result in a split of the configuration space into disconnected regions, i.e., the trajectories generated according to Newton’s equations cannot reach all states in the configuration space (recall: the configuration space equals the phase space) that lie on the (E ¼ constant) cut through the landscape. In that case, we have to restrict the ensemble average to the connected subregion which is explored by a given trajectory. 130 We use a instead of am, since now the uncertainty involves not only measurement limits or limits on the information about the starting point, but also limits due to the possibility that the trajectory leaves the region R within the given finite observation time. 127

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def This defines the concept of local ergodicity343 of a sub-region as function of observation time scale tobs(¼ tobs ). We start the search for locally ergodic regions on the landscape at very short observational time scales and very small subregions of phase space, and then repeat the search for increasing tobs.131 For each observational time scale, we search for those subregions R of the phase space, for which time averages along trajectories restricted to the region R of length tobs yield the same result as the ensemble average for the region R to an accuracy a. But since we consider subregions of the full landscape, we know that the system can leave R, in principle. Thus, as a second criterion for local ergodicity, we also check that the probability of the trajectory to leave the region R during tobs is very small, i.e., smaller than some small number b. The shortest observation time for which this inequality holds, defines the equilibration time of the region, seq(R; a). Similarly, the escape time sesc(R; b) is defined as the “largest” observation time, for which the likelihood that a typical trajectory of length tobs that starts inside R will leave R is smaller than b.132 Combining these two criteria yields the earlier characterization of a locally ergodic region R on the time scale tobs : seq(R; a)  tobs  sesc(R; b).101,343 We note that this definition of a locally ergodic region is a constructive one, i.e., we cannot identify the microstates whose union forms the locally ergodic region by just “looking” at def which together allow us them, but we need to perform many test simulations from many starting points of defined lengths tobs def 133 to identify the regions that are locally ergodic on given time scales tobs . A limiting case of locally ergodic regions are the socalled marginally ergodic regions,89 where the system leaves the ergodic region on essentially the same time scale on which it achieves equilibrium, sesc(R; b) z seq(R; a). Although at first glance they appear to be rather useless and unlikely to exist, marginally ergodic regions are actually of great relevance for the analysis of the aging and relaxation dynamics of glassy materials134 whose landscape consists of nested sequences of marginally ergodic regions, . seq(Rn)  sesc(Rn) z seq(Rn þ 1)  sesc(Rn þ 1)..89 As a consequence, a system that ages will appear to be in local equilibrium for an observation time on the order of its relaxation (or waiting) time tw that has elapsed since it had been forced into the non-equilibrium state (e.g., by quenching from the melt). But for longer measurements, the data will show increasing deviations from the equilibrium behavior. def ) can be divided into various kinds of The remainder of the landscape that does not belong to locally ergodic regions (for given tobs transition regions connecting locally ergodic regions. Frequently, these transition regions can be split into subregions that exhibit some def , but extracting dynamical information from the probability flows inside transition kind of local stability on time scales tobs < tobs regions needs to be handled with care. Note that just as a locally ergodic region can contain many local minima (which are equilibrated def ) and saddle points, the transition regions also encompass many saddle points and local among each other on the time scale tobs minima, but they are usually not able to equilibrate on time scales of interest before escaping from the region. It is important to def keep in mind that, for a given value of tobs , there exist many locally ergodic regions, and, conversely, a given region can only be treated as locally ergodic for a finite range of observation times. As we increase the observational time scale, new locally ergodic regions will appear and old ones will vanish, usually via a merger with another locally ergodic region, which can be a joint enterprise, where the system will spend similar amounts of time within each of the originally separate regions, or can be like an unfriendly take-over, where the eliminated region becomes a small side region of a large ergodic region. The latter case usually happens for macroscopically large chemical systems, where the likelihood to be in the larger region is exponentially larger than to be in the smaller region.135

131 The concept of local ergodicity is actually complementary to the concept of broken ergodicity introduced by Palmer460: upon shortening of the observational time scale, ergodic regions of the landscape can lose their ergodicity. But it may also be the case that instead of a complete loss of ergodicity, the phase space of the system can be split into pieces, such that these subregions are still ergodic by themselves. On shorter time scales, the ergodicity of these pieces is again broken, etc. Once such a split is no longer possible, the sequence of ergodicity breaking stops. This process of ergodicity breaking starts at the infinite observational time scale, at which the system is globally ergodic, and then descends to shorter observational time scales. This results in a descending ergodic hierarchy upon the decrease in time scale. Note that it may well be that already after breaking the global ergodicity the system cannot be split in such a way that ergodicity on finite (or shorter) time scales is present. Furthermore, broken ergodicity only argues via equilibration times, i.e., the break of ergodicity occurs when tobs drops below seq(R). In contrast, the concept of local ergodicity begins with very short observation times, identifies the locally ergodic regions via both equilibration and escape times, and then follows these regions as the observation time is increased through their creation, disappearance and mergers, i.e., we are concerned with an ascending hierarchy of locally ergodic regions that appear when increasing the time scale. 132 Note that, in practice, we would perform an average over the time averages of many trajectories of length tobs, with different starting points inside R when computing the equilibration and escape times for the region. Thus, due to the probabilistic nature of their definition, the numerical values of the characteristic times should only be treated as approximate, and in extreme cases only as order of magnitude estimates. 133 Note that seq(R; a) increases if we decrease the value of a, since for a smaller value of a, we require longer observation times to achieve higher agreement between ensemble and time averages. Similarly, sesc(R; b) decreases, if we decrease the value of b, because if we accept only a smaller percentage of trajectories that leave the locally ergodic region, the length of the trajectories must be reduced. Thus, if we were to demand either a or b to become infinitesimally smalldrequiring perfect agreement between the two averages or zero probability of a trajectory leaving the region -, no locally ergodic regions would be able to exist. 134 One should note that glassiness implies the glassy type of dynamics with a continuous alteration of the structure and a monotonous but logarithmically slow decrease in energy including aging phenomena. But although structural glasses are usually amorphous as far as their atom arrangement is concerned, an amorphous compound as such is not a glass, and could, in principle, correspond to a thermodynamically (meta)stable state. 135 def Note that, by construction, the locally ergodic regions that exist at a given time scale tobs are non-overlapping. Furthermore, the microstates belonging to a locally equilibrated region remain in equilibrium on all time scales larger than seq(R; a), even if the observation time scale exceeds the escape time sesc(R; b) from this region. While this means that the relative likelihood of occurrence along the trajectory does not change for these states within R, the absolute value of the likelihood of occurrence will diminish since the system will leave the region with a certain probability for tobs  sesc(R). Furthermore, depending on type of system and observational time scale, the transition regions between the locally ergodic regions do not noticeably contribute to the total free energy of the system (e.g., at total energies where we mostly deal with a crystalline material), or they can actually encompass large thermodynamically relevant regions of configuration space (e.g., at high total energies where a liquid is more likely to be found). In this context, we note that one practical issue is the definition of ! the boundary of a locally ergodic region. Formally, we define it as the set of points X surf ðRÞ, for which the probability to enter the region R equals 1/2. It is clear that the boundary will be a fuzzy entity, but as long as the phase space/configuration space volume of the boundary is small compared to the volume of the region itself, this definition is applicable in practice as the likelihood to be at or near the boundary for tobs  sesc will be rather small.

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As indicated earlier, these locally ergodic regions correspond to the (meta)stable modifications of the system that can be present on the given time scale of observation for an isolated chemical system. In particular, we can do statistical mechanics restricted to the region R, which implies that it is meaningful to compute a local entropy S(R) ¼ kB ln (g(E, V, .; R)). Similarly, for very large systems with very many atoms, we can assign a temperature to the system; actually, there will be a “local” temperature for each locally ergodic region on the time scale tobs, and for tobs / N, a global temperature can be defined via the standard thermodynamic ! ! relation T ¼ dE/dS53.136 If for a very large system we can write the total energy as a sum of independent terms E ¼ Epot ð R Þ þ Ekin ð V Þ, we can usually approximate the temperature of the region from the average kinetic energy, T ðRÞ ¼ ð32NkB Þ1 hEkin iensðRÞ for tobs > teq(R).137 It is important to keep in mind that once the system is prepared inside a locally ergodic region R, it is not “aware” that other locally ergodic regions might exist with lower local free energy for observation times tobs  sesc(R). Nevertheless, one often discusses the concept of “minimization of (local) free energy” in thermodynamics when discussing which phase of a system is expected to be observed; for the microcanonical ensemble this would correspond to the maximization of the local entropy, i.e., the system is expected to be observed in the locally ergodic region which contains the largest number of microstates. While this appears eminently reasonable, it contradicts our observation that once a system is prepared inside a locally region R it will not leave this region on time scales shorter than sesc(R). In fact, the principle of maximization of local entropy or minimization of local free energy applies under the condition that we had allowed the system to relax for an initialization time that is much larger than the global equilibration global time, tinit [ seq . In that case, for observations that subsequently take place on the time scale tobs, the probability p(R) to be found in region R is proportional to pðRÞ ¼

expðSðRÞ=kB Þ g ðE;V;.Þ ,

i.e., pðRÞ ¼

expðSðRÞ=kB Þ gðE;V;.Þ ,

where g(E, V, .) is the total number of microstates with P Rp(R) z 1, i.e., the likelihood to be in a tran-

total energy E. It is clear that for this concept to be useful, we usually assume that sition region on the time scale tobs is negligible.

3.11.4.1.3

Locally ergodic regions in general statistical ensembles

The definition of local ergodicity given above is very general and works for all statistical ensembles and dynamics in systems that can be described using the formalism of statistical mechanics. Only minimal changes are necessary when we include an interaction with the environment, e.g., heat exchange or volume exchange with reservoirs at given temperature or pressure such that instead of constant energy or volume the system has now a constant temperature or pressure: In the ensemble average, each microstate i with energy Ei is weighted by an appropriate factor, e.g., exp.( Ei/(kBT)) for the canonical (N, V, T) ensemble at a given temperature T, and, similarly, the time evolution occurs under the appropriate (thermodynamic) boundary conditions, e.g., at constant temperature T and pressure p for a Gibbs ensemble (N, p, T) molecular dynamics simulation. ! ! But temperature and other thermodynamic parameters like pressure p, external electric and magnetic fields E and B , respectively, etc., enter in the energy landscape description in very different ways. While the temperature influences the dynamics on the landscape in a stochastic fashion via the average kinetic energy hEkini ¼ 3/2NkBT, the other quantities modify the energy function itself. As we will see in the next section, enforcing, e.g., a constant pressure only changes the cost function from the energy to the enthalpy, and thus the above analysis of the locally ergodic regions for the microcanonical ensemble directly carries over. The reweighting of the microstates in the ensemble average is trivial, although the dynamics becomes complicated since we need to introduce, e.g., the constant pressure requirement via a coupling to a volume reservoir.463 In contrast, if we introduce a heat exchange with the environment such that the system is at a constant temperature Tdwhich is nearly always the case in the study of chemical or physical systems138d, then we need to introduce the appropriate Boltzmann weighting factors into the ensemble average formula, and similarly the dynamics would couple to a heat reservoir.464 For a given fixed temperature, the general procedure of identifying locally ergodic regions is completely analogous to the case of the microcanonical ensemble, i.e., we find those subregions of the energy landscape, where the time average approximately equals the ensemble average on the time scale tobs. If this holds true, then the ergodic theorem tells us that the time averages of observables

In this context, we mention the concept of “slow time” that has been introduced to address fundamental questions about the validity of thermodynamics in the limit of infinite time.461 There exist systems, where the dynamics of the system becomes overwhelmed by fluctuations, which is not the usual equilibrium limit of statistical mechanics. One considers the systems on observational time scales with very low time resolution, i.e., the time steps between individual measurements, tsample, are very large. In particular, the thermodynamic limit is no longer properly defined, and thus quantities like temperature can become meaningless. On the other hand, new (mathematical) structures and features can appear, or become visible, in the slow-time regime that were hidden when performing measurements with very high time resolution (i.e., for very small values of tsample), reminiscent of the emergence of new phenomena in complex systems. Here, we will not discuss this issue, and the relationship between slow-time and chaos and the properties an energy landscape might require to exhibit slow-time behavior, and refer the reader to the literature461,462 on this emergent new field of research. !! 137 Recall the earlier discussion of the reduction of the energy landscape for chemical systems from the total energy landscape Eð R ; V Þ to the potential energy ! landscape Epot ð R Þ. 138 Planets that are too large to reach a constant temperature on our observational time scales, or isolated molecules in outer space would be notable exceptions. 136

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R ! ! ! 1 t2 Oð! Oð R ðt Þ; V ðt ÞÞ along a trajectory of length tobs ¼ t2  t1, hOitobs ðR; T Þ ¼ tobs R ðt 0 Þ; V ðt 0 ÞÞdt 0 inside the locally ergodic region R t1 R !! !! ! ! Oð V ; R ÞexpðEð V ; R Þ=kB TÞd V d R equal the (Boltzmann) ensemble average of this observable hOiens ðR; T Þ ¼ R R restricted to the !! ! ! expðEð V ; R Þ=kB TÞd V d R R region R, rhOitobs(R; T)  hOiens(R; T) r < a.139 Again, we can compute for every locally ergodic region R the local free energydanalogous to the local entropy for the microP P canonical ensembledF(R, T) ¼  kBT ln Z(R, T) ¼  kBT ln j ˛ R exp ( E(j)/kBT), with Z(R, T) ¼ j ˛ R exp ( E(j)/kBT), and thus apply the usual laws of thermodynamics to the system as long as it remains within the region R. Furthermore, both the equilibration time and the escape time will depend on temperature. In particular, Arrhenius law, sesc f exp (EB/kBT) for energy barrier def controlled processes, often applies and thus the set of locally ergodic regions for a given value of tobs strongly depends on temperature. On time scales where global equilibrium holds, we can again determine the modification i most likely to be encountered at P a given moment during an infinitely long observation by calculating the probabilities p(Ri) ¼ j ˛ Ri p(j), where pð jÞ ¼ expðEð jÞ=kB TÞ , Z

for all regions Ri and finding the maximum. This is equivalent to finding the minimum of the local free energies F(Ri) with respect to the locally ergodic regions Ri.140 But in the case of bulk systems, the phase with the lowest free energy usually overwhelms the other metastable phases, and once this phase has been reached, the likelihood for the system to leave this phase due to a fluctuation becomes essentially zero. On the other hand, for finite chemical systems, such as clusters or molecules, the likelihood to find the system in another isomer besides the global minimum is not infinitesimal but finite (though small), along a sufficiently long trajectory. Fig. 5 shows a schematic view of such a long-time trajectory on a landscape with many competing locally ergodic regions (one of them even stabilized def on which individual measurements take place during only by entropic barriers). The regions are locally ergodic on the time scale tobs the long-time experiment, and some (but not all) of the regions are visited during the long exploration/simulation time def 141 texp [ tobs . To further illustrate the concept of local ergodicity, let us choose as a typical chemical observable the X-ray diffractogram of a compound. If tobs can be chosen such that we get a reproducible diffractogram of a modification Ri, tobs [ seq(Ri), before the substance is transformed to another modification or disintegrates, tobs  sesc(Ri), then we would claim that we are dealing with a metastable compound on the time scale tobs.142 We recall that once metastability has been established, we can restrict ourselves to such a locally ergodic region Ri and analyze the properties of the modification it represents, using the full apparatus of equilibrium statistical mechanics and solid state theory. In particular, the knowledge of all the regions that are locally ergodic for the observation time we have used or are planning to use in our powder diffraction experiment, provides us all the feasible structure candidates of the system under investigation for our measurement conditions. In principle, we can calculate for each candidate the powder diffractogram, and select the one, which agrees best with the experimental one.143

Recall that equality, i.e., a ¼ 0 only holds for global ergodicity in the limit tobs / N. As discussed for the microcanonical ensemble, the concept of “minimizing the (local) free energy”, in order to decide, which phase is the thermodyglobal namically stable one and should be observed, only applies, if we have allowed the system to equilibrate globally over a relaxation time trelax [ seq before     RÞ ¼ exp  FkðBRTÞ =exp  Fktotal to be found in region R during a subsequent measurement for the measurement takes place, such that the probability pðRÞ ¼ ZZðtotal BT 139 140

an observation time tobs is maximal for the minimal local free energy F(R), where seq(Ri)  tobs  sesc(Ri) for all regions Ri of interest. We again assume here P P that iZ(Ri) z Ztotal such that ip(Ri) z 1, i.e., we assume that we are not dealing with a situation where marginally ergodic regions and transition regions dominate the free energy. 141 Picture a long-time experiment on an isolated small cluster in vacuum over many months, i.e., texp ¼ 10 years, where every month, we perform ten 1-hour def ¼ 1 hour. As long as sesc from the current region of the landscape is larger than, say, a week, the 10 measurements would long measurements over 1 day, tobs yield the same resultsdwithin statistical fluctuationsdwhich would agree with the predicted ensemble average for this locally ergodic region, and we would register this region as a particular metastable isomer. But over the time of a year, we might visit many such regions, and find that various locally ergodic regions     Ftotal iÞ associated with the different isomers Ri of the cluster, are encountered according to their Boltzmann probability pðRi Þ ¼ exp  FðR kB T =exp  kB T . 142 We have to be careful not to run into a seeming paradox here if we are dealing with phases stabilized by configurational entropy. An example might be the measurement of the powder diffractogram of a solid solution, after the system has spent several months relaxing from an originally homogeneous metastable quenched solid solution phase into the thermodynamically stable state consisting of two phases, one being A-rich and the other B-rich. The experiment might only require a couple of hours measurement time. But the short measurement does not yield an average distribution over all the relevant configurations of these two phases, but only measures a small sample of the structural realizations of the A- and B-rich solid solution phases. Similarly, if we let a crystalline system relax for long enough (at low temperatures) such that the ideal crystal configuration has equilibrated with the (point) detect configurations, defect idealcryst trelax [ srealcryst [ sesc , sesc all of which correspond to individual local minima, then a super-precision measurement on a very short time scale eq defect realcryst  seq , e.g., a few pico seconds, where the system cannot move from one minimum to a neighbor one, will not observe the structure of the tobs  sesc real crystal, but just one of the many defect configurations. Only when taking many such instantaneous measurements spread out over a long enough time interval tint [ trelax can we expect that the average over these measurements will yield the diffractogram of the real crystal centered on the structure of the ideal crystal, instead of only an average over a couple of “defect structures”. Alternatively, self-averaging can come to the rescue (c.f. Section 3.11.4.1.1), when the spatial width of our probe beam is so large that it includes many “copies” of the full system, whose average approximates the true ensemble average of the solid solution phase or the real crystal. 143 Of course, the system should not transition between two modifications while we perform our measurements.

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Fig. 5 Schematic complex energy landscape. Red height lines indicate mountains and green lines valleys, respectively. Here, we imagine that we exp are performing a long-time experiment over a time tobs , during which the system can evolve among various metastable phases, indicated by the random path the system follows (marked in blue). During this time, we periodically perform individual experiments, e.g., measure the X-ray def . For those modifications of the system that are locally ergodic on this time scale, we diffractogram, each of which takes a time on the order of tobs would obtain the structure of a locally equilibrated polymorph; the other measurements would yield data that cannot be interpreted as a single (meta) exp exp def stable phase. Note that the time tobs it takes for the system to evolve is much longer than the time of the individual measurements, tobs [ tobx . def of a given individual measurement, for the temperature at which the random walk takes place, Regions that are locally ergodic, on the time scale tobs are enclosed by blue squares; those which are visited are depicted with a solid line, and those that are not visited with a dashed line, respectively. def def Regions R enclosed by purple squares are locally equilibrated, but are no longer locally ergodic on the time scale tobs , since tobs > sesc(R). The def def region R enclosed in a black square is not yet equilibrated on the time scale tobs , since tobs < seq(R), although the two sub-regions R1 and R2 def def (enclosed by the dashed blue squares), R1, 2 3 R, are locally ergodic on the time scale tobs , sesc(R1, 2) [ tobs [ seq(R1, 2). Note that one of the locally ergodic regions is entropically stabilized; all the other regions are mainly stabilized by energy barriers.

We remark that the determination of all the locally ergodic regions for a given chemical system for arbitrary values of temperdef ature T and many observational time scales tobs is usually still beyond our capabilities. Nevertheless, experience has shown that the locally ergodic regions of many modifications in solids or isomers of large molecules at low temperatures below the melting point contain many structurally related non-degenerate local minima of the potential energy, once we include the equilibrium defects. Thus, in many cases, a reasonable approach towards identifying candidates for locally ergodic regions consists in finding the local minima of the potential energy of the chemical system, determining their structural similarities, analyzing the barriers separating them, and studying the probability flows among them.

3.11.4.2

Time evolution and master equation dynamics

A characteristic, nearly defining feature of complex energy landscapes is the existence of a multitude of relevant time scales spanning many orders of magnitude. Thus, the time evolution of a system with such a landscape, on very long time scales, is usually impossible to reproduce on the microstate level.144 However, one can frequently construct a dynamically coarse-grained picture of the landscape: one replaces the configuration space consisting of the individual microstates by a set of discrete nodes a, Scg ¼ {a}, that are connected via edges, and then employs a stochastic dynamics on this node graph which approximates the true dynamics in a probabilistic sense on large time scales. In practice, one follows the probability Pa(t) of the system to be found in a node a, i.e., to be in a microstate which contributes to the coarse grained node a, as function of time t, for all nodes in the node-graph. These nodes possess degeneracies ga that equal the number of microstates contributing to a node a. Furthermore, the edges of the node graph between nodes a and b possess weights fab that are proportional to the number of connections between the microstates belonging to node a and those belonging to node b. Such kinetic factors will enter the description of the coarse grained dynamics of the system.145 144 For very low temperatures, the tree graphs based on local minima and their lowest-energy saddle points can be used to model the (local) dynamics on the landscape in an approximate fashion, since all escape times are much larger than the equilibration times, and the escape times are energy barrier dominated. Thus, an Arrhenius style hopping dynamics can be used to visualize the time evolution of the system. But already for somewhat higher temperatures, we need to take entropic barriers into account. This information is only partly contained within a saddle-point network, even if the widths of the saddles are known. Furthermore, we quickly note that kinetic and entropic barriers become relevant even at very low temperatures once saddle points with nearly identical energies compete and thus the kinetic and entropic barriers describing the likelihood to find one of the two saddles starting from the minimum via a random walk become important. 145 Since there are many connections among microstates within a node a, there is also a non-zero “in-node” kinetic factor faa s 0, in general.

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! Often, one uses the short-hand vector-like notation P ðt Þ for the set of probabilities of all the nodes. Here, one should keep in P mind that for each allowed probability vector, we require aPa(t) ¼ 1 and 0  Pa(t)  1 for all times t, since the Pa(t) constitute a probability distribution. Furthermore, one frequently considers only discrete time steps when modeling the dynamics on the node graph, i.e., t is measured in integer units of some basic time step tcg. In order for this time scale to be consistent with the coarsegraining, one usually assumes that the distribution of probability over the microstates within the node a corresponds to some kind of statistical (thermodynamic) equilibrium distribution, and thus tcg  seq(a).146

3.11.4.2.1

Master equation dynamics

The above coarse-graining allows us to transform the complicated dynamics of an ensemble of walkers on the landscape (represent! ing the evolution of the probability distribution P ðt Þ) on long time scales into a “simple” matrix multiplication problem ! !! ! ! ! ! ! ! v P ¼ A P , or ! P t¼ðnþ1Þtcg ¼ G P t¼nt cg , with A ¼ G  E ( E is the unit matrix in node space). This so-called master equation vt approach465 employing a stochastic Markov dynamics can describe the dynamics on very long time scales compared with those accessible in, e.g., a molecular dynamics simulation296.147 P Component-wise, the time development of Pa(t) is then computed as Pa(t þ tcg) ¼ bGab(T) Pb(t).148 For physical or chemical systems, the transition probabilities Gab(T) depend on the temperature T149; for general cost function landscapes, some other control parameters might also exist.150 Besides incorporating conservation of probability as mentioned above, the Gab(T) have to P ensure that the stationary distribution is the Boltzmann distribution Pa(eq)(T) ¼ ga exp ( Ea/T)/Z, where Z ¼ aga exp ( Ea/T) is 151 the partition function at temperature T, and ga is the degeneracy of node a. The stationary distribution Pa(st)(T) is defined via the relation Pa(st)(t þ tcg; T) ¼ Pa(st)(t; T), i.e., it is constant in (discrete) time. ! P Since Pa(t þ tcg) ¼ bGab(T)Pb(t), it follows that Pa(st)(T) must be an eigenvector of the transition probability matrix G ðT Þ with P P eigenvalue 1: Pa(st)(t þ tcg) ¼ bGab(T)Pb(st)(t) ¼ Pa(st)(t) 5 1 , Pa(st)(t) ¼ bGab(T)Pb(st). Now, we demand that Pa(st)(T) ¼ Pa(eq)(T) P and thus it follows that Pa(eq)(T) ¼ bGab(T)Pb(eq)(T). Usually, we can assume that for very long times, every probability distribution Pa(t) converges to the stationary distribution, limt / N Pa(t) ¼ Pa(st)(T) ¼ Pa(eq)(T).152 A commonly used choice for the transition

146

There are many subtle aspects to such a coarse-graining; we refer to the literature for more details.198,348 Keep in mind that the Master equation dynamics does not refer to the time-evolution of a simple system, but always describes a statistical ensemble thatdas a wholedfollows a stochastic time evolution. Even if we start with Pa(0) ¼ 1 for a ¼ a0 and Pa(0) ¼ 0 for a s a0, the subsequent dynamics reflects the evolution of an (in principle infinitely large) ensemble of independent random walkers which just happen to start in the same node. ! P 148 To ensure that Pa(t þ tcg) is still a probability distribution, the matrix G must fulfill the conditions aGab(T) ¼ 1 for all b and Gab(T)  0 for all a, b. 149 If the temperature, or other parameters that enter the transition probabilities, depend on time, then the elements of the Markov matrix Gab are time-dependent. As a consequence, by modifying the transition probabilities, one can control the dynamics of the system. 150 In this context we note that this “Boltzmannization” procedure of introducing temperature into the dynamics needs to be modified, if we want to describe a microcanonical ensemble based (constant energy) dynamics instead of a constant temperature dynamics. In this case, the equilibrium distribution for P a given total energy Et equals Pa(eq)(Et) ¼ ga(Et)/Nt, where Nt ¼ ggg(Et) is the total number of microstates in the system with energy Et. The detailed balance relation is given by Gab(Et)gb(Et) ¼ Gba(Et)ga(Et). We note that formally Gab(Et) ¼ fab/gb(Et) fulfills the detailed balance condition, where fab ¼ fba are the kinetic factors that incorporate the connectivity between the microstates of the nodes. Alternatively, Gab(Et) ¼ fabga(Et) also agrees with the detailed balance condition; the factor ga(Et)gb(Et) by which these two expressions differ can be absorbed into the kinetic factors fab. We note that these kinetic factors do not affect the stationary distribution of the stochastic process, as long as fab ¼ fba. Furthermore, we note that the degeneracies of the nodes are functions of Et. A particularly interesting special case are systems, where the total energy can be written as a sum of the potential energy that only depends on the position ! P ð p Þ2 coordinates, and the kinetic energy Ekin ¼ i¼1;N 2mii . In this case, the degeneracy of the node a with potential energy Eapot < Et becomes gajoint(Et, Eapot) ¼ pot pot kin kin pot kin kin ga (Ea )ga (Ea ¼ (Et  Ea )), where ga (Ea ¼ (Et  Eapot))
(Et  Eapot))3N/21dE and dE is the width of the energy slice for the node a.53 In this case, one can measure the density of states associated with the potential energy landscape, and add-on the contribution of the kinetic energy such that Eakin þ Eapot ¼ Et. Since the coarsening process implies that the system is essentially equilibrated inside the node a on the time scale tcg, the contribution of the momentum degrees of freedom to the transition probabilities can be taken into account via the joint density of states and does not require a special analysis of the kinetic factors fab in momentum space. Of course, if this assumption does not hold, one would need to follow the trajectories also in momentum space, but then it is not clear that a master equation description would be appropriate. 151 The Boltzmann distribution over nodes can be considered a coarse-grained version of the Boltzmann distribution over microstates as long as we can assume that the microstates within a node are equilibrated on the time scale tag. ! 152 Note that not every matrix G ðT Þ that fulfills the conditions as a transition probability matrix also leads to a convergence of every Pa(t) to the stationary     ! 0 1 1=2 distribution. A counter example would be G ðT Þ ¼ which has the stationary distribution , only positive entries and each column sums to 1 0 1=2     ! ! ! ! 0 1 1 0 and ð G ðT ÞÞn¼even ¼ for odd and even one. But the limit limn/N ð G ðT ÞÞn is not defined, as ð G ðT ÞÞn oscillates between ð G ðT ÞÞn¼odd ¼ 1 0 0 1 ! ! st values of n, respectively. In this context, we also note that if every Pa(t) converges to Pa(st)(T), then limn/N ð G ðT ÞÞn ¼ G ðT Þ where st ! ðst Þ ðeqÞ ð G ðT ÞÞab ¼ Pa ðT Þ ¼ Pa ðT Þ for every b. In this context, we note that for stochastic simulations and explorations aimed at establishing equilibrium properties, it is important to ensure that the dynamics fulfills the so-called detailed balance condition,198,348 i.e., Gab(T ¼ N)gb ¼ Gba(T ¼ N)ga for the transition matrix elements Gab(T) between nodes b and a at infinite temperature, and the analogous condition for finite temperatures, Gab(T)Pb(eq)(T) ¼ Gba(T)Pa(eq)(T), where ga is the degeneracy for state a and Pa(eq) is the equilibrium probability distribution at temperature T. If the goal is to only identify, e.g., local minima, this condition is not required for the exploration algorithm. 147

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probabilities in physical or chemical systems is the Metropolis dynamics,153 where for the coarse-grained dynamics the transition probabilities can be constructed from the analysis of the probability flows on the energy landscape.154 For more details on the construction of a coarse-grained model of an energy landscape and the corresponding transition probability matrices, we refer to the literature.198,274,296

3.11.4.2.2

Equilibration trees

The analysis of the time evolution of a system with a complex energy landscape using a master equation approach can lead to many deep insights into the behavior of the system, including the ability to design ways to steer the system towards certain (locally ergodic) regions in an optimal fashion. If one does not apply external controls, the time evolution will often correspond to a relaxation of the system towards thermal equilibrium for constant thermodynamic boundary conditions, such as fixed temperature and pressure. Due to the presence of many relaxation time scales differing by many orders of magnitude, the classical approach of searching for the “last” (exponential) time scale on which the system approaches the thermodynamically stable state is only of limited use.155 For one, the system might never reach the statistical global equilibrium state for feasible experimental and simulation time scales, e.g., for a glass, and secondly, the time scales of interest as far as applications are concerned, are much shorter than the largest time scale of relaxation. But once we try to determine all the eigenvalues and their corresponding eigenvectors, the situation quickly becomes quite inscrutable. To allow an analysis of the process, how the different nodes of the coarse-grained model of the landscape (or, in principle, also the microstates themselves, of course) begin to form larger equilibrated subregions on the level of the coarse-grained description, the concept of the so-called equilibration trees156 has been developed.82,274 Starting point is the observation that for a timeindependent energy landscape, two states or nodes that are in statistical equilibrium (on a certain level of accuracy aeq–tree) will remain so throughout the remainder of the time evolution of the system. This is surely true for states corresponding to or belonging to locally ergodic regions, but also applies to states belonging to transition regions.157 Since the stationary probability distribution is unique and can be assumed to be knowndafter all, we know all the nodes with their energy158 by constructiond, we can construct for every pair of nodes a and b the ratio of their equilibrium probabilities, Pa(eq)(T)/Pb(eq)(T) and compare this with the ratio of their current probabilities Pa(t)/Pb(t) at the time t, r[(Pa(t)/Pb(t))  (Pa(eq)(T)/Pb(eq)(T))]/(Paeq(T)/Pb(eq)(T)) r < aeq–tree . The time seq–tree(a, b) for which this ratio remains smaller than aeq–tree for all times t > seq–tree(a, b) is defined as the equilibration 153 To start the Metropolis algorithm, an initial state a0 is chosen, usually, but not necessarily, at random. Then at each step of the algorithm, a neighbor a0 of the current state ak is selected at random (according to the moveclass) to become the candidate for the next state. This candidate state is accepted as the next state according to an acceptance probability criterion: Paccept ¼ 1 if DE  0 and Paccept ¼ exp.( DE/T) if DE  0, where DE ¼ E(a0 )  E(ak). If this candidate is accepted, then ak þ 1 ¼ a0 ; else, the next state is the same as the old state, akþ1 ¼ ak. To implement this probabilistic decision rule we generate a random number r uniformly in the interval [0,1], and compare it with the Boltzmann factor exp( DE/T). If r < exp ( DE/T), then ak þ 1 ¼ a0 , else ak þ 1 ¼ ak. With this Markov process description of the Metropolis algorithm, we are now able to model the stochastic dynamics of an ensemble of walkers on energy landscapes as function of temperature in terms of the transition probabilities Gab(T). This procedure constructs the transition matrix from the edge weights fab, node weights ga and the acceptance probabilities, and is often called a “Boltzmannization” procedure.274 A general expression for Gab(T), which automatically fulfills the detailed balance condition would be: Gab(T) ¼ fab/gb exp ((Ea  Eb)/kBT) if Ea > Eb and Gab(T) ¼ fab/gb if Ea  Eb, as long as a s b. For a ¼ b, we P P have Gaa ¼ 1  gsaGga, which follows from the probability conservation of the stochastic process, gGga ¼ 1 for all a. As for the microcanonical ensemble, we use again the fact that the kinetic factors fulfill the relation fab ¼ fba when writing down this formal expression. Unless we have detailed microstate level information about the kinetic factors, the actual magnitudes of the transition probabilities are often taken from simulations or landscape explorations at various temperatures, e.g., T ¼ N during a threshold run, together with information about the degeneracies of the nodes and the connectivity of the node graph; these are then used to adjust the kinetic factors, from which the Boltzmannized version of the stochastic matrix Gab (T) is constructed for general T. Of course, other acceptance criteria can be used, but these usually result in stationary equilibrium distributions that are different from the Boltzmann distribution. 154 Of course, such a stochastic dynamics can also be performed on the level of the microstates; a typical example is the classical Monte Carlo dynamics using the Metropolis algorithm.466 155 When one considers the eigenvalue spectrum of the Markov matrix A for a typical chemical or physical system in the continuous time master equation picture, there will be one eigenvalue that equals zero, l0 ¼ 0dits eigenvector corresponds to the stationary equilibrium probability distribution -, and many !ðevÞ negative eigenvalues li ¼  1/tirel < 0. The closer li is to zero, the longer is its corresponding relaxation time scale tirel. Note that all the eigenvectors P i for P (ey) li < 0 are “null-vectors” in the sense that a(Pi )a ¼ 0, i.e., they only describe a re-distribution of probability but are not probability vectors themselves. 156 Sometimes, these equilibration trees have been called “free energy trees”, since they describe the monotonic merger of regions encompassing ever larger statistically equilibrated subregions of the (coarse-grained) graph model of the landscape, and thus the merger of regions separated by increasingly large “free energy barriers”. However, since this adds more to the confusion about what should be called a free energy tree in the first place, the more standard name “equilibration tree” is more well-defined and suitable. 157 One can focus the equilibration tree analysis on the physically and chemically interesting nodes of the graph, eliminating uninteresting states belonging to transition regions, if desired. Mathematically, one can, of course, construct landscapes, where states might be “in equilibrium” only accidentally on a certain time scale before dropping out of equilibrium, and only later reaching true statistical equilibrium after more (simulation) time has elapsed. 158 For a node, whose microstates exhibit a spread in energy, we can use the average energy of the microstates belonging to the node.

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time between the two nodes a and b, and this is indicated by a merger of these two nodes.159 Plotting the set of nodes vs. the (logarithm) of these equilibration times,160 then yields the so-called equilibration tree of the system.274,296 We note that the structure of this tree does not necessarily reflect the structure of the tree-representation of the landscape derived from the analysis of the lidbased pockets.161

3.11.4.3

Time dependence of cost function/energy landscapes

Both in general cost function problems and in the case of chemical materials in (strong) contact with the environment, we often ! encounter energy landscapes that vary with time, Eð X ; tÞ. All three elementsdcost/energy function, move class and configuration spacedcan depend on time, either separately or together. As a consequence, major features of the landscape such as generalized barriers or locally ergodic regions, will also vary with time, greatly complicating the analysis of the systems. However, if we can control the way the cost function landscape changes with time, we can use optimal control theory, analogously to finite time thermodynamics,256 to guide the time evolution of the system, e.g., to let it settle in a specific region in configuration space that corresponds to a desired compound or modification. Of course, we control the energy landscape every day in the laboratorydat least in a rough fashiond: varying the pressure changes the enthalpy function of the landscape, or raising or lowering the temperature alters the probability flows on the landscape. Usually, we endeavor to have the system stay close to local or at least partial equilibrium,162 since we expect that this will allow us a better control of the system and the processes involved.163 Else, complex feedback loops between the response of the system and the control might occur, which depend not only on the instanta! neous values of ð X ; tÞ but on the whole (or parts of the) trajectory of the system since the beginning of the process.164 In the following, we will discuss some of the most important aspects of time-dependent energy landscapes; for some further discussions, we refer to the literature.241

3.11.4.3.1

Variation of state space and moveclass

Since changes in state space automatically imply changes in the local (and possibly global) topology of the landscape, we discuss these two types of modifications together. Changes of only the moveclass are quite frequently employed to fine-tune global exploration and optimization algorithms; for example, the size of attempted atom moves, or the frequency with which certain moves are attempted, might vary as function of simulation time. In contrast, the neighborhood relation in a chemical system reflects the underlying physics of the system that controls the evolutiondon a certain time scaledthat cannot change without a concomitant change of the energy function. Exceptions are systems where the dynamics is externally driven by some (time-varying) effective force that we do not account for in the energy function, and which is therefore only reflected in the moveclass of the landscape. Now, the actual interaction of the external force would be on physical time scales (e.g., femto seconds), while the model dynamics would be on a much larger time scale. Thus, we are justified to ignore the fast changes in the external force-system interaction when setting up the landscape for our model system. As a consequence, variations of only the moveclass in a physical time evolution can make sense if they are supposed to reflect the behavior of the chemical or physical system on (moderately) long time scales 159 As an alternative, we can use every point in configuration space (or some representative sample of them) as starting point of the time evolution of an ensemble of walkers that obey the given time evolution laws of the system, and then observe how each initial state begins to equilibrate with its neighbor states, forming locally ergodic regions in the process, which subsequently merge into larger equilibrated regions, etc. This ensemble of walkers then approximates the stochastic master equation time evolution. 160 Like the relaxation time scales of the system, the related equilibration time scales are spread over many orders of magnitude, and thus it is more instructive to plot the logarithm of the times where nodes merge on the y-axis instead of the absolute times. Furthermore, there is a built-in dependence of these equilibration times on the accuracy aeq-tree and, though usually only weakly, on the initial probability distribution of the system, and thus it makes more sense to use a logarithmic scale. 161 In the Traveling Salesman example discussed in reference,274 the equilibration tree exhibited multiple sub-branchings even suggesting a self-similar structure, while the lid-based disconnectivity tree consisted of a simple main trunk to which all side-pockets were directly connected without any notable sub-branches. 162 We speak of partial equilibrium if only a (large) fraction of the degrees of freedom in the system are in equilibrium, e.g., all the degrees of freedom that are orthogonal to the reaction coordinate(s) of a chemical process. 163 If we are far from equilibrium, we usually encounter control problems, but sometimes a successful outcome is achieved nevertheless, if the non-equilibrium probability flows drive the system in the “right” direction, even when no detailed control via manipulation of close-to-equilibrium states is possible. ! ! 164 An example would be a TSP problem with NC cities, where the travel time Tð X ; tÞ needed to cover the route X changes as function of time, and the speed v is so small that we cannot finish the whole route before the effective lengths of the individual distances Li(ti) between the cities have noticeably changed. In this case, we have a “quasi-external” parameter (the interior time), which controls the route scheduling, and the cost function is “path-dependent”. But, due to ! the straightforward relationship between the effective path lengths Li (ti) and the “internal” time points ti, each state X (¼ selected city route) of the system has a well-defined length, and we can thus globally optimize the corresponding TSP-problem. In this way, we have “incorporated” the time aspect into the state itself, because the state corresponds to the whole route. Ideally, we would to the same for any time dependent energy function in generaldalso for physical or chemical systems -, but this is usually much too complex and difficult to do, as it would correspond to solving the time-evolution on the time dependent energy landscape for each trajectory of the system.

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exceeding the time scale of the elementary processes which would be involved in variations of the energy. Similarly, changes in state space, e.g., in the number or type of atoms in the system, or the removal of constraints, are also usually accompanied by changes in the energy or cost function.165 Since the moveclass implicitly defines what we call a local minimum and influences the generalized barriers and probability flows, changes of the move class also change the locally ergodic regions on the landscape, and alter the corresponding free energy landscape. As mentioned above, when we modify the configuration space S as function of time, the moveclass must change implicitly, too. The most dramatic change in the state space of chemical systems is usually the addition or removal of atoms. We can treat this change in the number of atoms as a switch between two configuration spaces, but we could also introduce a giant state space (analogous to the extended Hilbert-Fock space in elementary particle or many-body physics467,468), where the creation and annihilation of atoms is part of the moveclass, and the microstates can exhibit different numbers and types of atoms or molecules. A somewhat disconcerting consequence is that the locally ergodic regions also change, in principle, when adding/removing an atom. However, if we work in a state space where changes in the number of atoms are allowed, locally ergodic regions encompassing microstates with different numbers of atoms can exist.166 Other important changes of the configuration space of chemical systems would be adding or removing restrictions ! regarding the volume Vð X Þ that the (periodic or non-periodic) simulation cell that contains the atoms is allowed to 167 or relaxing the requirements that, e.g., restrict atoms to a two-dimensional plane, by creating new microstates have, with, e.g., a z-component for every atom position vector. Similarly, we could allow the orientation of magnetic dipoles to vary instead of forcing them to stay aligned to, e.g., the z-axis. Another change of states would be to release the fixed charge value of ions in a materialdtypically one of the parameters of the empirical potentialdto become a variable degree of freedom, where the contribution to the energy associated with the creation of ions is given by the ionization energies and electron affinities of the ions.21,379

3.11.4.3.2

Variation of the cost function

Finally, the probably most common variation of the energy landscape with time concerns the energy function. One can distinguish two basic types of changes of the cost function for chemical systems: modifications of the atomic interactionsdin the extreme case transforming one chemical system into another one (sometimes called “computational alchemy”469–472), and changes in the external forces applied to the system including variations of the thermodynamic parameters, such as electric and magnetic fields, or mechanical stresses. This variation of the energy function is usually not accompanied by changes in the configuration space or the moveclass. ! ! The microstate X given by, e.g., the atom coordinates, and the neighborhood of the states N ð X Þ are only rarely affected by ! ! the modification of the energy or cost functions Eð X Þ or Cð X Þ, respectively.168 Both minima and the barriers of the landscape will change their shapes and sizes, and even vanish or appear as the cost function is varied. Clearly, this affects the probability flows in the system, and thus the equilibration and escape times, and the existence, of feasible LERs. As a consequence, both the set of the thermodynamically (meta)stable states of the system and the direction and time scales of chemical processes possible ! ! in the system change as function of Eð X Þ. Varying Eð X Þ thus gives us some control over the chemical system, aiding in the synthesis of desired modifications, or allows us to efficiently compute the difference in free energies between two chemical systems.473

3.11.4.3.3

Fast vs. slow variation

The practical consequences of the change of the landscape with time depend on two aspects of the variation: the magnitude of the exp for our change,169 and the time scale tvar on which a noticeable variation takes place compared to the time scale of observation tobs exp experiment or other processes taking place in the system. If tvar [ tobs , then we can deal with the time-dependent landscape by studying a sequence of only slightly modified versions. For each such time slice we can pretend that the landscape is fixed on 165 Keep in mind that the energy function yields the value of the energy for a given microstate. This function will change if there are more atoms in the system, even if we use the same empirical potential or functionaldsometimes also denoted as the energy “function” of the system!dto describe the interactions between the atoms. 166 If we have an order parameter, which classifies all (relevant) LERs,343 then we might be able to group the LERs for different numbers of atomsdand even for some ranges of compositiondtogether according to the order parameter value. This might work for LERs corresponding to real crystalline phasesdi.e., the LER consists of a minimum representing the ideal crystal plus all minima associated with equilibrium defect configurations -, on relatively long observation times, which can be classified based on, e.g., (averaged) symmetries, cell parameters and atom positions inside the unit cell, which is possible for a slowly varying energy landscape. But this order parameter approach becomes essentially impossible for fast variations of the landscape that allow only stable regions on very short observation times, where single local minima corresponding to individual defect arrangements constitute the only LERs of the system. 167 Besides the maximum limits on the volume discussed earlier, we can also have minimum bounds on the volume; e.g., for a zeolite we might prescribe a maximally allowed density. 168 However, if the changes in the energy function consist of adding a new type of interaction, e.g., a magnetic field, we would need to add a (new) spin degree of freedom in the definition of the microstate. 169 Note that any change in the cost function (or state space or neighborhood relation), no matter how tiny, results in a new landscape, in principle, but the system’s behavior might not be affected for all practical purposes.

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the time scale of observation, and compute those properties of the system that are of interest for small enough time scales of obserexp vation, such as the locally ergodic regions or probability flows. Here, tobs is the observational time scale we are interested in for our 170 chemical system from an experimental point of view. exp Thus, the properties and landscape features of the system such as locally ergodic regions that can exist on the time scale tobs will undergo an adiabatic slow evolution, such that at each moment in time we can treat the landscape as being constant as long as we exp 171 are only interested in features present on the time scale tobs . In contrast, for fast variations of the landscape with time, we will only be able to define and discuss LERs and “controlled” probdef ability flows on very short observational times scales tobs that are much smaller than our actual time scales of interest def exp exp tobs  tvar  tobs . In such a case, the system is highly non-equilibrated on the relevant time scale of observation tobs , and, in general, we never reach local equilibrium and, usually, also do not reach a stationary state. The results of our experiments will strongly depend on the starting state, and we never establish any real control over the system. As discussed below, if the fast variation exp (c.f. with time is periodic (!) in time, some kind of “recurring” state of the system might appear for large observation times tobs Section 3.11.4.3.4) and, similarly, landscapes with fast but very small variations might yield reproducible features on the time scale exp tobs (c.f. Section 3.11.4.3.5). In the following, we consider some special cases of time variations of the energy landscape: def  seq(R), the escape time from a LER shrinks to the order of tvar, sesc(R) z seq(R) z tvar: In this case, (a) In the case of tvar z tobs the locally ergodic regions are reduced to being at best marginally ergodic, but without the nesting property associated with the aging behavior seen in glasses. As far as metastable modifications are concerned, we most likely will observe a more or less def . However, since the well-defined but possibly quite random sequence of LERs that are capable of existence on the time scale tobs size of the transition regions is also very likely to grow, possibly quite substantially, the time the system spends outside the LERs is expected to increase considerably. (b) If the energy landscape varies so fast and by such large amounts that not even local minimizations (via, e.g., a stochastic quenchdboth in experiments and in simulations with realistic move classesd) that require a time tmin on the average can finish in a controlled fashion, tvar  tmin, it is not clear, whether the instantaneous energy landscape will yield much insight beyond its availability to compute, e.g., the atomic (Newtonian) forces at any given moment. In this case, extracting landscape features such as instantaneous local minima or barriers, is of limited value, as far as the analysis of the dynamics of the system is concerned. (c) A less extreme case are non-periodic but not extremely rapid variations of the landscape with a moderately large amplitude, i.e., seq(R) [ tvar [ tmin. As long as the amplitude of the variation is relatively small, the “averaged” occupation probability distribution over regions of the configuration space as a whole or over the set of local minima might be meaningful. However, in contrast to the case of periodic variation (c.f. Section 3.11.4.3.4), there is most likely no “reference” set of LERs that reappear periodically. Thus, we must extract the “constant” distribution over “quasi-stationary” (but not-equilibrated) regions of the state space from computer experiments, i.e., many simulations starting from a large number of starting points on the landscape at the same initial time t0 (¼ 0).172 (d) In general, the time scale of variation of the energy landscape need not be uniform but can exhibit many (alternating) time periods with rapid and slow change of the landscape. During those time intervals where the landscape changes only slowly, exp tvar [ tobs , seq(R), sesc(R), we are dealing with an adiabatic evolution of the landscape with time discussed above. In contrast, when the switch between landscapes takes place in an arbitrary yet rapid fashion, several possibilities need to be considered: large single sudden jumps, or sets of many smaller jumps, where in-between these smaller jumps again a smooth and rather small variation of the landscape appliesdanalogous to an adiabatic transitiondor the landscape remains fixed until the next jump occurs.173 This suggests to study the adiabatic and the jump-like time-regimes separately.

Here, we note that as long as we deal with large but solitary jumps in the shape of the energy landscape that are separated by relatively long time intervals during which no changes in the landscape occur, the dynamics resembles the relaxation processes of a quenched system, regardless of the magnitude of the jump-like change of the landscape: For any jump, the probability distribution over the microstates of the system more or less equals the equilibrium probability distribution inherited from the preceding stage, where the system had been allowed to relax on a constant landscape. This distribution, of course, usually greatly differs from the equilibration distribution for the new landscape after the jump, but usually (by assumption), the next jump will only occur after the system had been able to relax enough to allow us to extract some relevant features of the relaxation behavior, ranging from partial

170 When we discuss a time varying energy landscape, we use the notation tobs to indicate an abstract “generic” observational time scale in contrast to the exp experimental time scale of interest tobs . 171 Since the size of the LER will decrease/increase only slowly and smoothly as function of time, the appearance and disappearance of LERs can usually be handled without major discontinuities. 172 Unsurprisingly, the dynamics of such a chaotically varying energy landscape exhibits many analogies to the dynamics found in many classical chaotic systems, at least from a statistical point of view.474,475 173 By assumption, the time between two jumps should be sufficient to allow for a substantial relaxation of the system before the next jump occurs. In particular, if many jumps occur rapidly after one another, the size of the change in the landscape should be small enough to allow for relaxation processes, even if only a short time elapses before the next jump occurs. In the limit of a continuous variation instead of the step-wise one discussed here, we can try to approximate the continuous change by an appropriate sequence of jumps and relaxations, such that the ratio of change per time remains approximately constant.

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relaxation to full (local) equilibration. We note that this adjustment/relaxation can actually be quite fast, because the temperature of the system might be quite high; in contrast to standard quenches in the experiment, the jump to the new landscape occurs via a change in the energy function (or state space or moveclass) and not through a change in temperature.174

3.11.4.3.4

Periodic variation

Periodic variations of the energy landscape with time, due to, e.g., periodic oscillations in the applied pressure or the electromagnetic field, where after one period tp ¼ tvar the landscape is completely the same as before, are an important special case.175 If we now check the landscape at periodic time points, t, t þ tp, t þ 2tp, ., we might expect to find various kinds of attractors characterizing the dynamics of this driven system. Similar to the case of a classical system with a periodic driving force, we hope and expect that the system will reach some “steady-state” like behavior with a stable distribution of probability over all the (instantaneously feasible) LERs in the system when observed in time steps of the period of the variation. Of course, we can only consider LERs that can exist on def the time scale tobs  tvar.176 This kind of distribution might correspond to a limit cycle in a classical dynamical system, but it might also show properties of a strange attractor,474 depending on the type of energy function and its time variation. Similar to the case of non-periodic variations of the landscape, we need to simulate the trajectories of many walkers to obtain the probability flows over many periods of the system, starting in various LERs, and investigate whether the probability distribution represented by these walkers becomes periodic in agreement with the driving force (¼ landscape variation), after some initialization phase. However, it might happen that the probability distribution over the configuration space might not represent an occupation of locally ergodic regions at all. Possible reasons might be that the variation of the energy landscape is very large such that no consistent set of LERs can be identified as stable (or at least as approximately stable) between successive times of observation in the first place, or that the relevant state space volume associated with LERs is much smaller than the one belonging to transition regions. In this case, we might still have a stable occupation probability distribution over the microstates of the system, but no correlation with LERs, for which we could perform statistical mechanical analyses of their properties.

3.11.4.3.5

Noisy cost function landscapes

Fast non-periodic or chaotic variations of the energy function, but with only a small amplitude in the fluctuations of the landscape parameters around some average values, are another important special case. For such modulations, the barrier structure of the landscape, and thus also the LERs and the corresponding metastable compounds and isomers of a chemical system remain essentially unchanged.177 Especially if the variation of the landscape is caused by fluctuations in the energy function, a “noisy” landscape is sometimes assigned a noise-temperature Tnoise proportional to the (square root) of the size of the fluctuations in the energy of the system. As a consequence, simulations on the average landscape performed at temperatures T > Tnoise represent the true dynamics on the noisy landscape reasonably well for observation times tobs [ tvar.178 Noisy landscapes are also encountered in the optimization of the parameters in a neural network (NN) for machine learning applications.257,476,477 The quality of the outcome of a training set of examples to be solved by the network can be interpreted as a cost function of the parameters of the network. Since each member of the training set presents a slightly different task to the neural network, and we use different training sets or sequences of examples over a long training time, the cost function for a given set of NN parameters changes as the training set employed varies with time by small essentially random amounts, such that the cost function landscape acquires a “noise”-like feature in the process.

3.11.4.4

Robustness of cost function landscapes

This variation of the energy landscape with time raises an important issue: the robustness of the energy or cost function landscape with respect to modifications. Now, in the case of the time variation of the landscape, we are dealing with a “real” change in the cost function or energy landscape, and it is of great interest to study the properties of the landscape as function of the parameters (state 174 In contrast, when the system is quenched from the melt into a glassy state, the energy landscape of the system does not change. Only the accessible part of the complete landscape seen by the system while relaxing is different: at high temperatures, the walker moves on the landscape high above the deep minima basins and is only weakly affected by the underlying energy barriers and deep minima. On the other hand, at low temperatures, the system basically must hop among the deep minima, slowly climbing over the large barriers separating the minima, and, as a consequence, can get stuck in, e.g., amorphous atom arrangements. Such a rapid change in the accessible landscape when the temperature changes by only a small amount has been suggested as the mechanism of the glass transition: In the low-energy range of the landscape of glass formers, such as in lattice networks,63,75 where they often exhibit exponential growth in the local density of states, g(E) f exp (a(E  E0)), and in the density of minima in the low-energy range of the landscape, a “phase” transition between the low-energy and high-energy regions of the landscape can occur via the so-called exponential trapping at a temperature Tc ¼ 1/a.63 175 Here, we are not concerned with the net increase of the energy in the system due to the absorption of sound waves or electromagnetic radiation. 176 exp In the limit of very long observation times, tobs [ tyar, we would expect to have, at each moment t0 in a time interval of length tp between two measuring times t0 ˛ [t, t þ tp], a specific distribution, which evolves into the next “stable” distribution a short time afterwards as function of t0 , and always repeats after another time interval tp has elapsed. An example would be a variation of the periodic pressure, such that the system oscillates between high-pressure and standard-pressure phases. 177 Note that for the special case of constant state space and moveclass, this picture of a landscape whose energy slightly fluctuates with time can be adapted to describe the exchange of energy with a reservoir at a given temperature. 178 These noise temperatures are also remotely related to the effective temperatures which one sometimes assigns to rapidly quenched glassy systems, and which are meant to reflect the amount of disorder in the amorphous structure in terms of a single compact parameter.261

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space, cost function, moveclass) that can vary with time for a chemical system or optimization problem, for a given physical or algorithmic process causing the changes in the landscape. But the same questions arise when we consider the landscape we have defined in the computer: How closely does our model landscape represent the real physical landscape, and how robust is our description when modifying the parameters of the model? Quite frequently, for example, in the case of the potential energy, the landscape is actually explicitly parameterized in terms of, e.g., two-body or many-body interaction strengths, ionic charges and radii, etc., in the case of an empirical potential, or of, e.g., pseudo-potential parameters and/or basis set parameters used in ab initio energy functions. Yet neither the empirical potential nor the ab initio energy function encompasses the complete physics of the chemical system on a quantitative and often not even qualitative level. When dealing with these limitations of our model description of a chemical system, there are several points one needs to keep in mind: On a qualitative level, does the approximation of using, e.g., an empirical potential change the set of local minima and locally ergodic regions and their connectivity compared to, e.g., the ab initio energy landscape or the experimentally accessible physical landscape, i.e., would we obtain a similar tree graph description of the landscape on the microstate or on the LER level? Connected to this issue but being more quantitative is the question, whether the energy ranking of the minima, or the free energy ranking of the LERs, is unchanged when switching to a more accurate energy function. Finally, supposing that the minima are the same on both landscapes, are the energetic and entropic barriers separating them also similar? Experience has shown that within their range of intuitive validity empirical potentials can provide a good overview over the set of promising minima structures corresponding to metastable compounds, but one should not expect an, e.g., ionic potential to properly describe structures that are not based on dense packings of cations and anions, such as, e.g., molecules based on covalent bonds between the atoms. On the other hand, even many intermetallic compounds exhibit structures analogous to ionic ones. This is simply due to the small but noticeable charge transfer between the different metal atoms and reflects the constraints on dense packings because of the different atomic radii of the metal atomsdjust like in ionic systems where we find different preferred coordination polyhedra due to the different combinations of ionic radii. Although the energy ranking for large sets of minima is often not really reliable, one frequently observes that if all the (meta) stable modifications found in the experiment correspond to local minima on the empirical potential landscape, then even the ranking by energy is quite acceptable for the low-energy structures.179 However, comparison studies of empirical potential and ab initio energy landscapes for (small) cluster systems259 have yielded instances, where the empirical potential landscape completely misses several low-energy minima of the ab initio landscape, and, similarly, the tree graphs of the two landscapes exhibit both qualitative and quantitative differences. If we stay within a family of empirical potential landscapes, we note that the change in the landscape as function of the parameters of the potential is quite smooth. Many major minimum structures remain as minima on the landscape, while the energy ranking among them changes, until the modification of the potential, e.g., an increase or decrease in the ionic radii in the case of an ionic potential, becomes so large that some of the minima become unstable and vanish, perhaps after being replaced by minima that had not existed on the original landscape.113,379 To test this robustness, one would repeat the landscape explorations for slightly different parameters.180 Such explorations for many slightly different choices of parameter values are of particular importance when studying the landscape of non-yet-synthesized chemical compounds, since we have no experimental structure data to which one could fit the potential.181

3.11.5

Energy landscapes in interaction with the environment

The environment and the interaction of the system with the environment are very important for chemical systems.182 Typically, such an interaction results in static or dynamic changes of the energy landscape compared to the isolated chemical system. In general, these interactions fall into three categories. The most straightforward case is realized, if the applied interactions, fields or forces are constant in time (and space), the system is in thermodynamic equilibrium with the environment, and no (macroscopic) fluxes of any quantities such as charge, heat, atoms, etc. are present. Of course, this implies that the system is studied on observational time scales much larger than the equilibration times of the system, in particular the time it takes for the externally applied forces to make themselves felt throughout the material. However, even constant applied fields can induce a non-vanishing flux of some quantities as long as sources and sinks at the surface of the material are provided. In this case, the material will not be in equilibrium but in 179 Note that inorganic and organic (crystalline) compounds differ greatly in this regard, since the energy ranking in molecular crystals depends on subtle differences in the van der Waals and hydrogen-bridge bonding contributions to the energy among the many polymorphs, which otherwise have nearly the same energy. In contrast, the interaction terms in inorganic compounds are quite large and long-range, leading to rather distinct structures separated by relatively large energy differences. Of course, for large periodic approximants containing many atoms in the periodically repeated cell, there will be many atom configurations incorporating (equilibrium) point defects, which differ only slightly in energy from the perfect crystal, but since they are easily recognizable as defects, they do not pose a problem. 180 For general cost functions, one would similarly check, whether the landscape and its local minima are robust against variations in the parameters entering the model that describes the original problem of interest. 181 For the same reason, the ab initio calculations for an unknown chemical system should be repeated using different methods, e.g., Hartree-Fock, DFT, and hybrid methods, in order to gain an overview of the spread in the energy differences and the possible rankings among the different structures candidates.111 In this context, one should never underestimate the unconscious bias induced by the existence of a known isomer, which influences, e.g., the choice of density functional when modeling the system, and which can make the researcher rely on the empirical potential or functional way beyond their range of applicability. 182 This is different in many investigations in physics, where one usually attempts to keep the number of relevant variables to a minimum and avoids interaction with the environment as much as possibledapart from probes for controlled interactions with the system.

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a stationary steady-state situation. Usually, this implies that some gradients of, e.g., particle concentrations or temperature, etc., are present in the system. Quite generally, for those systems that are at least in stationary equilibrium, we can take our cue from thermodynamic considerations when proposing minimally complex extensions of the energy landscape of the isolated system that can capture many of the features of a chemical system whose interaction with the environment does not depend on time. Finally, the environment can be variable in time, e.g., due to fields or forces that are themselves time-dependent. This commonly results in non-vanishing fluxes that also vary with time. As a consequence, the system is in a state of non-equilibrium, especially on short observational time scales, where the effects of the time variation cannot be averaged out.183 Since we have already analyzed general time-dependent cost functions in Section 3.11.4.3, in this section only some aspects of time-dependent interactions specific to chemical systems will be discussed.

3.11.5.1

General aspects

In practice, interactions with the environment take place via the presence of external electromagnetic fields, or through external atoms belonging to the environment that impact the originally isolated chemical system in form of direct contact on the atomic or mesoscopic level. The direct contact can take the form of, e.g., a device that applies an external mechanical force on the surface of the chemical system (on the mesoscopic level),184 thus generating a pressure or shear forces throughout the material. Alternatively, the surface of the system might be exposed to a liquid or gas (on the atomic level), whose atoms or molecules can enter the system via diffusion or chemical reaction on the surface. Finally, the direct contact can lead to the creation of, e.g., an electric current through the material via generation of a potential difference between two parts of the surface of the chemical system; whether we treat this on the atomic or mesoscopic level would depend on the phenomena we are interested in. Thus, in general, we must extend the state space of the isolated chemical system by adding “all” the relevant atoms of the environment, in order to be able to account for, e.g., deformation and compression, corrosion processes, growth of layers from a solvent on the surface, or incorporation of atoms belonging to the enclosing container into the system. We note that this picture of a formerly isolated chemical system interacting with the environment presupposes the existence of a surface that separates the chemical system from the environment.185 Since we previously had assumed that the state space of the isolated system was of infinite extent (R3N) where all atoms could be anywhere in R3, in principle, this suggests that the most elementary interaction with the environment is just the spatial confinement of the atoms constituting the chemical system to a finite region in (three-dimensional) space. Such spatial confinement can reflect a general enclosure of the chemical system by an (inert) container, or more specific background structures such as a metal-organic framework or a layered compound where the chemical system of interest can be inserted into the background matrix. Another consequence of splitting the “world” into the “system” and the “environment” is the introduction of a time scale of equilibration, tsplit, which defines the range of observation times, for which we can still keep the system distinct from the environment as far as the exchange of atoms between the system and the environment is concerned. Furthermore, in many instances, we no longer consider the initial microstate of the system as a random arrangement of atoms (inside the prescribed volume of the system), but use a particular atom configuration, e.g., the ideal crystal modification, or an ideal thermodynamic state, e.g., a real crystal where the initial microstate belongs to the locally ergodic region associated with this (metatable) equilibrium phase of the system.186 The physical (collisional) impact of the atoms belonging to the environment on the surface of the system, the absorption of electromagnetic radiation, and the heat and radiation187 generated by chemical reactions, can produce an exchange of energy between the material and the environment, resulting in the establishment of a temperature and possibly heat flows in the system. In an analogous fashion, there can be an exchange of atoms and molecules with the environment, leading to (diffusion) currents into, out of, inside and through the chemical system, and the establishment of chemical potential(s) mi for the system for different atom and molecular species i.188 183 While we often can assume that the applied field penetrates completely into the material and is constant both in time and space, e.g., the magnetic field or the pressure are the same everywhere in the system, this is not true in general. One example is the screening of the electric field inside a metal, with a frequency dependent penetration depth.478 Similarly, if a field or a thermodynamic boundary condition applied at the surface of the material varies with time, there will be a delay until this change has been realized throughout the material. The same issue arises, e.g., if we deposit atoms from the gas phase on the surface of a cold substrate with finite thermal conductivity. Since the heat of absorption cannot be eliminated quickly enough, the (top layers of the) deposit might have a (considerably) higher temperature than the nominal substrate temperature, and thus some annealing might take place already during the supposedly quenched deposition on the substrate.479 184 Of course, the mesoscopic level contact is based on atom-atom interactions, but such typical solid-solid contacts are often described on the level of continuum models, and the formulas employed are justified only on that level. 185 A special case is an environment that penetrates the chemical system of interest, such as the solvent surrounding and infiltrating a protein. But even in that case, one would be able to distinguish between the system of interest and the environment, although the energy landscape of the combined system would appear to be dominated by the much larger number of solvent atoms. 186 The simplest environment we can imagine is the empty space, i.e., a vacuum, into which our finite chemical system, e.g., a solid crystal, is placed; perhaps with some temperature that can be visualized as an average energy density of electromagnetic radiation. In this case, tsplit would be the time scale on which the solid evaporates into the (thermodynamically preferred) gaseous state. 187 Keep in mind that any system will emit black-body radiation, and thus energy can be transferred from the system to the environment even without atoms being involved as energy carriers. 188 Defining a temperature, pressure or chemical potential presupposes that we are considering the system on time scales, on which the system has established thermodynamic and statistical equilibrium with the environment. This does not mean that the system is globally equilibrated in itself, e.g., it could be in a glassy state or in a metatstable phase.

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In general, the interaction will be time-dependent, and, furthermore, depend on position ! r ˛ R3 , such that its inclusion in the energy landscape description of the chemical system-plus-environment can become highly complex. However, for the special case of time-independent interactions with the environment, one can often establish a thermodynamic equilibrium between the system and the environment, after some equilibration time if necessary. Then we can describe the effect of the environment in a minimal! ! ! istic fashion via a set of thermodynamic parameters that are constant in time, such as T, p, mi, E , B , s , etc. We note that these ! ! parameters can still depend on position r , e.g., Tð r Þ, etc., even in equilibrium, due to the thermodynamic boundary conditions at the surface of the system. The existence of gradients of thermodynamic parameters results in corresponding currents, which either eliminate the gradients after some time, or lead to a steady-state situation, where a constant flow of some quantity such as heat, atoms/ions, or electric charges (carried by electrons, holes, ions), through the system is established. Such minimalistic descriptions of the interaction with the environment that are valid for observational time scales larger than ð AÞ r Þ of a thermodynamic parameter A at a material point ! r (encompassing thousands of both the local equilibration time s ð! eq

(A) atoms, typically) and globally for the whole material seq , respectively, are extremely convenient and allow us to include many interactions in a relatively straightforward fashion into the energy landscape of the chemical system.189 The spatial variability of the various forces and thermodynamic parameters within the chemical system, which represent the environment in an average fashion, cannot always be taken into account on the atomic level.190 The reason is that quantities like the electric polarization or the magnetization, or the elastic deformation field are only defined on the mesoscopic level in the continuum approximation351,478.191 As a consequence, introducing such a spatial variation into the energy landscape necessitates the computation of properties like the susceptibilities, permeabilities, electric and thermal conductivities, etc. on the so-called material point level of resolution.192 !!! For example, for spatially varying electric and magnetic fields, we need to replace the term M ð X Þ B in the extended energy function by R ! ! ! ! ! ! 1 ! M ð X ; r Þ B ð r ÞdV and an analogous expression for the electric field term, where Vð X Þ is the volume associated with the ! Vð X Þ Vð X Þ ! configuration X of the N atoms of the chemical system. Here, the integral is over all the material points of the material in the continuum ! approximation.351,478 Of course, this might require additional computations for each configuration X , because we need to evaluate the !!! local magnetization M ð X ; r Þ for each material point ! r in the continuum approximation. In general, we also need to compute the electric ! ! ! and magnetic permeabilities 3 ð X ; ! r Þ and mð X ; ! r Þ as function of position ! r . Furthermore, there can be non-linear relations between H !! !! ! ! ! and B or D and E , such that 3 ð X ; r Þ or mð X ; r Þ would explicitly depend on the magnetic and electric fields.

In this context, we note that the continuum approximation implies a (minimal) spatial and time average over all the atoms and their movements inside a material point (with a side length of ca.  1 nm for solids or liquids).193 In order to associate this with

ð AÞ (A) Note that this equilibration time seq ð! r Þ or seq is still of relevance after the chemical system has reached equilibrium with the environment, because it also A corresponds to the time scale sfluct on which equilibrium fluctuations of the thermodynamic parameters occur. Of course, these equilibration (or relaxation) ð AÞ r Þ that refers to the time scale on which timescales can be different for different thermodynamic quantities A. Furthermore, the local equilibration time s ð! 189

eq

(A) a local thermodynamic gradient between, e.g., two material points is eliminated, is different from the macroscopic equilibration time seq that governs how, e. g., heat diffuses inside a macroscopic system after applying heat sources and sinks on the surface of the material, until a constant temperature (profile) is established, since this will depend on the size of the system. An extreme example would be the Earth itself, where billions of years are needed to bring the planet to a constant temperature; the cooling rate is estimated as ca. 5  10 K/108 years.480 190 For “infinite” periodic systems where we employ periodic approximants, one would attempt to compute the interaction on the level of the atoms contained within one such periodic cell, and also try to determine the corresponding shape and size of the cell. 191 Of course, we can compute, e.g., the induced dipole moment of an individual atom or molecule, but this does not capture the effect the newly polarized atoms have on their neighbors, i.e., this does not yield the effective electric field inside the material, which is taken into account in the macroscopic electric polarization. 192 In practice, it is highly non-trivial to compute, from first principles, quantities like electric or thermal conductivity, or the electric or magnetic permeability, possibly as function of material point position ! r in the continuum approximation. It might require performing MD-simulation experiments of “macroscopic” (or at least mesoscopic) versions of the (solid) system, in order to deduce these quantities from the simulations. Usually, one would not invest the time and ! ! effort, but, in principle, such an analysis for a given configuration X is possible, and should be performed as part of the evaluation of the cost function Cð X Þ. 193 A material point is the elementary unit of the continuum approximation in physics. It contains just enough atoms to assume that the (relatively short) time averages of various properties of the material point are well-defined, and the fluctuations about these averages can be ignored in the continuum limit. The typical size of such material points in solids and liquids is ca. 1–10 nm, and in gases about an order of magnitude larger. Note that the number of atoms inside the material point usually varies about some average value; furthermore, the “coordinate” ! r of the material point serves the dual purpose of labeling the material point and representing its position (at least at the beginning of the experiment) thus indicating, which material points are “neighbors”, and also allowing the definition of spatial derivatives. On longer time and length scales, the properties of the material points can change, of course, but the change, both in space and time, is so weak that we can define derivatives with respect to time and space on the level of the material points, yielding the field of hydrodynamics. For more details, c.f.351

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! ! a given microstate atom configuration X , one would need to assign for all X that contribute to the same material point ! r the same !! !! ! ! moments M ð X ; ! r Þ and P ð X ; ! r Þ and also the same average permeabilities 3 ð X ; ! r Þ and mð X ; ! r Þ. Of course, ideally, the external

field acts directly on the atoms/ions, allowing us to cut out the middle man of the continuum approximation, but this is not always the case and can involve additional complications as mentioned above.194

3.11.5.2 3.11.5.2.1

Temperature General aspects

Among the interactions of the chemical system, temperature greatly differs conceptually from pressure, mechanical stresses, and electromagnetic fields (to be discussed later in this section in more detail). Including the latter ones in an energy landscape descrip! ! tion is quite straightforward and can be achieved by extending the potential energy Epot ð X Þ by adding terms that depend on X , e.g., ! pVð X Þ, resulting in an extended (potential) energy or cost function. In particular, we can still characterize this landscape in terms of its local minima, saddle points, characteristic regions and generalized barriers.195 Of course, temperature is also defined for the microcanonical ensemble that describes the isolated system with constant energy, but as a derivative quantity. Using it as a thermodynamic boundary condition of the system changes the “rules-of-the-game”, since now all microstates of the system are accessible in the time evolution of the chemical system. The easiest case is T ¼ 0, which implies that there is no kinetic energy in equilibrium, and where the only “dynamics” is a descent into the nearest local minimum.196 Here, only the potential energy and not the total energy is the most useful choice as a cost function for chemical systems. In contrast, for ! ! non-zero temperatures, we can use either the potential energy as function of X ¼ R ¼ ð! x 1 ; .; ! x N Þ or the total energy as function ! !! ! ! ! of X ¼ ð R ; P Þ where P ¼ ð p 1 ; .; p N Þ. But again, one usually drops the kinetic energy term, since studying the time evolution of the system via constant temperature simulations only makes sense on time scales for which hEkinitobs ¼ hEkiniens ¼ 3/2NkBT. In general, the time evolution of the chemical system corresponds to a complicated trajectory in state space. In particular, none of the microstates observed during an experiment or a MC/MD simulation at a given temperature perfectly agrees with the ideal crystal structure of a solid compound, irrespective of the choice of energy function. On time scales tobs [ seq(Rrealcryst), the locally ergodic region that corresponds to a real crystal does not consist of a single atom configuration but encompasses a large number of microstates representing, e.g., lattice vibrations and other excitations such as point defects, whose ensemble average corresponds to the crystal structure actually observed in the experiment. As mentioned earlier when introducing the concept of local ergodicity (c.f. Sections 3.11.4.1.2 and 3.11.4.1.3), for any given observational time scale tobs, the state space of the chemical system is split into a large number of disjoint locally ergodic regions, with the remainder of the configuration space consisting of transition regions connecting the locally ergodic regions that correspond to kinetically stable compounds. On very short time scales, the locally ergodic regions will consist of individual local minima, but at elevated temperatures and on longer time scales, locally ergodic regions will typically encompass many local minima, e.g., the perfect crystalline minimum plus the minima corresponding to equilibrium defects of this structure, or compounds with controlled local or global disorder.197 Other very important examples are locally ergodic regions that contain all the local minima associated

194 On the other hand, this way to deal with the system on time scales larger than the material point equilibration time can lead to the definition of energy landscapes over the space of material points. However, such descriptions are not yet generally possibledand would require many order-parameter-like properties associated with the material point beyond the total charge, density, etc. known from standard continuum physics, such as “average structures” or complex contact interactions. Still, they might allow us to discretize the configuration space, leading to something analogous to a classical density functional type of description of the chemical system on the energy landscape level. ! ! 195 One might think that one could try to add a “potential entropy” term like TSð X Þ to the energy Eð X Þ for non-zero temperature, in analogy to the way one ! ! adds a pVð X Þ term to take the pressure into account. But the entropy of a single configuration X is zero, since the “degeneracy” of a single microstate equals ! ! ! gð X Þ ¼ 1, and thus Sð X Þ ¼ kB lngð X Þ ¼ 0. After all, the entropy is an ensemble quantity, like the free energy, and therefore only makes sense when computed for a locally ergodic region on time scales larger than the local equilibration time. 196 Any microstate that is not a local minimum of the energy corresponds to a non-equilibrium situation, and will move downhill, i.e., relax, to a local minimum. 197 def defects We recall that often the time scales tobs [ seq [ sesc(Ridealcryst), sesc(Rdef) on which the system can establish local equilibrium between the minimum corresponding to a perfect crystalline atom arrangement Rcryst and the defect minima Rdef even at very low temperatures (and on which the free energy includes a contribution due to the various equilibrium defects), are much too long to be used for a measurement. In particular, if the actual defects measurement were to be performed on much shorter time scales tobs  sesc(Rcryst), sesc(Rdef)  seq , the defect contribution to the free energy would not be observed.

Energy landscapes in inorganic chemistry

311

Fig. 6 Schematic overview of metastable phases as function of observation time for two different temperatures. Phases at the higher and lower temperature are marked in red and black, respectively. Since the life times of the phases can vary over a very large range, we plot the logarithm of the observational time scale. C refers to individual local minima, which for most chemical systems are locally ergodic only on very short time scales for finite temperatures. A1 and A2 are glassy phases, where the dashed horizontal line indicates their marginal ergodicity until they finally transform into a more stable ordered phase, and M is a supercooled melt, which usually would not show any aging, but remain metastable until a sudden crystallization sets in. B1 and B2 are solid solution phases, where the dashed line indicates that they might appear to be marginally ergodic in the experiment because of very slow equilibration, until they finally reach their locally ergodic state as a metastable phase that exhibits controlled disorder. Finally, P1–P4 are ordered crystalline phases, where P1 is the thermodynamic ground state at the lower temperature. As we can see, the stability range is not correlated with the local free energy G(p, T). Note how the stability range and the free energy ranking of the various modifications change as function of temperature (e.g., the modifications P1 and P2 switch their ranking), since both the local free energies, and the equilibration and escape times, of the locally ergodic regions depend on temperature (usually, the free energy of a phase becomes more negative as the temperature is increased). At the elevated temperature, some phases, such as the solid solution phase B1, no longer reach local ergodicity before transforming into some other phase, while the second solid solution phase B2 nearly becomes the thermodynamically stable phase, and, similarly, the supercooled melt can remain stable for a longer time, on average.

with, for example, the rotation of complex anions in a solid or the oscillation of individual atoms in double-well potentials, or the many local minima representing possible atom arrangements belonging to solid solutions or alloy phases.198 At even higher temperatures, essentially all locally ergodic regions and many of the transition regions merge on the observational time scale. This usually would indicate that melting occurs, and the system is no longer found in the solid but in the liquid state (or even the gaseous state). This situation can be visualized as in Fig. 6, which presents a qualitative sketch of Gibbs free energies of some typical locally ergodic regions found in a solid as function of observation time, for one given pressure and two different temperatures. In this figure, C corresponds to individual defect configurationsdnote that the ideal crystal is just one minimum “in the crowd” and only distinguished from the associated (point) defect configurations by having a slightly lower energyd, M represents a supercooled melt, A1-A2 correspond to glassy phases, B1-B2 represent solid solution phases, and P1-P4 are real crystalline modifications, where P1 is the thermodynamically stable phase and P2-P4 are metastable phases, for the lower temperature (shown in black). The range of observation times for which a phase R is locally ergodic is indicated by the horizontal line, where seq(R) and sesc(R) mark the left and the right endpoints of this time interval. We indicate the marginal ergodicity of the glassy phases by using a dashed line, which ends when some ordered crystalline modification begins to nucleate inside the amorphous phase. Note that the supercooled melt does not show aging: it remains in the locally ergodic region that exhibits liquid-like propertiesdin particular, a very low viscosity

198 Non-equilibrium defects cannot be part of a locally ergodic region, by definition. In practice, we can actually classify the regions that are locally ergodic on the time scale tobs by the type and number of kinetically stable non-equilibrium defects (e.g., the set of pinned dislocations, or of grain boundaries in a crystal). Similarly, amorphous and glassy phases are not locally ergodic on time scales larger than those needed to leave a local minimum configuration, but are often marginally ergodic.

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Energy landscapes in inorganic chemistry

compared to the glassy stated, and the transition to a lower free energy state takes place very abruptly.199 A special case of such a long-lived supercooled melt in inorganic chemistry is elemental gallium where the density of the melt is larger than the one of the thermodynamically stable a-phase but lower than the ones of the metastable crystalline modifications, leading to very interesting transformations among the metastable phases, the melt and the stable crystalline phase, as function of temperature treatment (and waiting time)481.200 Taking the lower temperature (phases marked in black), we see that on the observation time scale t1 (indicated by a vertical dashed line), we would only find individual local minima as locally ergodic regions. In contrast, for observation times t2, the individual local minima are no longer stable for the temperature we have assumed, and we are already in the time range where glassy systems exhibit aging phenomena (indicated by the dashed horizontal lines).201 We also see the supercooled melt. However, the time has not been sufficient, to treat the solid solution phases as locally ergodic, although the crystalline phases are already thermodynamically metastable. For observation times t3, we would be able to observe the glassy phases A1-A2, the solid solution phases B1-B2, but no longer the supercooled melt. Furthermore, the crystalline modifications P1, P2 and P3 are phases that are in local thermal equilibrium, assuming that we have been given the system in this phase in the first place. For t4, only the glassy state A1, the second lowest free energy phase P2 and the crystalline ground state P1 are present as marginally ergodic or locally ergodic regions. Finally, for t5, we are essentially at infinite times, and only the thermodynamically stable phase P1 remains. Here, we always keep in mind that we would not know of the existence of competing phases with lower local free energy, as long as our observation time is smaller than the escape times from these regions of the energy landscape. At higher temperature (marked in red), both the free energy ranking of the phases, and the time intervals over which they are locally ergodic change. Typically, the local free energies are lower, i.e., more negative, at higher temperatures (due to thedTS(R) ! term in the local free energy Gð R Þ), the equilibration times are somewhat shorter, especially for the solid solution phases, and the escape times are much shorter, especially for those phases, which are mainly protected by energetic barriers. In particular, we note that in this schematic example system, the high-temperature ground state is now modification P2, closely followed by the solid solution phase B2. Clearly, at even higher temperatures, the melt will appear as a locally (and globally) ergodic region in the diagram, essentially replacing the supercooled melt.

3.11.5.2.2

High vs. low temperature

At low temperatures, we can expect most systems to exhibit long-lived locally ergodic regions corresponding to metastable solid modifications on our observational time scale, which together dominate the sum over states at the given temperature, P aZ(Ra) z Ztotal. In contrast, at very high temperatures, the full configuration space of the landscape will be sampled on typical observation times, including the gaseous and the liquid phases. For infinite systems, this means, that periodic approximants or large clusters containing very many atoms will be needed to describe the phases and time evolution of the macroscopic chemical system. In particular, the time evolution can no longer be visualized as a hopping between locally ergodic regions or as aging in a nested global , the probability flows will be dominated by kinetic and sequence of marginally ergodic regions. In particular, for tobs < seq entropic barriers and less so by energy barriers. For temperatures T ˛ [Tmelt, Tgas], where the liquid phase is the thermodynamically stable one, intermittently stable regions dominate the landscape on rather short observational time scales.64 On the level of atom configurations, we deal with local clusterings of atoms, reminiscent of cage models of liquids.482 As far as the microstates of the system are concerned, we recall that they correspond to the arrangements of all (!) atoms in the system; therefore, each stable atom cluster corresponds to a partial equilibrium of this sub-group of degrees of freedom in the microstates on very short time scales. But not all atoms are part of such a cluster at any given moment in time, and thus we never observe all degrees of freedom equilibrated in such a fashion at the same time. As a consequence, we only see marginally ergodic regions at best, and individual local minima are so unstable that they become irrelevant. Similarly, whole minima basins will not be locally ergodic on typical observational time scales. Many models for liquids have been developed over the past decades, and their energy landscapes have been extensively studied,66,91,483 but we will not discuss them in detail. For temperatures below but relatively close to the melting point, Tmelt/2 < T < Tmelt, the time evolution not only of crystalline high-temperature phases but also of glass formers is controlled by their complex energy landscapes. In this temperature range, aging phenomena are particularly prominent,202 and nested marginally ergodic regions are the paradigm for understanding the dynamics of such chemical systems on the energy landscape.89,484 Since the beginning of landscape studies of chemical systems,59 energy

199 As usual in chemistry, the situation is not always clear-cut: for high-viscosity polymer melts aging-like phenomena are expected, since the line between the melt and the glassy polymer is often not very well-defined. 200 An appropriate energy landscape description for this system could be sketched on the (macroscopic) level, where the density and the complexity of the various phases and modifications would serve as an organizing principle. We also note that the high density of the melt assists in stabilizing the (super-cooled) melt, since the kind of large volume fluctuations needed to create a viable nucleus of the a-modification are statistically very unlikely for the isolated system.481 201 Note that as the system ages, its free energy keeps decreasing at a (very) slow rate. Thus, these dashed lines should be slightly tilted. The same holds true for the dashed lines in the approach to equilibrium for the solid solution phases B1-B2. 202 Recall that for many systems, the glass transition temperature lies in this temperature range, Tglass z 2/3 Tmelt.261

Energy landscapes in inorganic chemistry

313

landscapes of glasses have been investigated for both discrete and continuous model systems,62,75 trying to elucidate the similarities and differences between liquids and glassy systems.66 If the chemical system does not (easily) form glassy phases, we often observe high-temperature crystalline modifications that are either associated with a single dominant minimum like the low-temperature real crystals,203 or multi-minima basins that include multiple (symmetry-related) stable regions, instead of individual crystal structure minima, i.e., systems exhibiting controlled disorder.204

3.11.5.2.3

System with a thermal gradient

The temperature we establish for the chemical system need not be the same throughout the material, even in a stationary state, i.e., it can depend on position Tð! r Þ. One particularly important case is the existence of a macroscopic temperature gradient, which might be due to enforcing different temperatures at opposite surfaces of the material. While we do not find theoretical landscape investigations for systems with such a gradient in the literature, we note that MD simulations of such systems have been performed. Starting with a fixed temperature difference at opposite surfaces, a temperature gradient was established, together with a constant heat flow through the material.486,487 Furthermore, a master-equation approach488 has been used to study phase separation in thin films by describing a binary mixture via an Ising spin model.489 In addition, on a macroscopic level, phase field simulations have been employed to analyze the interface structure in diffusion couples in the presence of a temperature gradient.490 On the other hand, experimental studies of the influence of temperature gradients on phase transformations and phase separations exist,491–495 which suggest a rich field for global energy landscape studies of systems that exhibit such gradients.205 Besides establishing a heat flow through the system, the thermal gradient will also generate different temperatures in different (spatial) slices of the chemical system. As a consequence, if the system can possess many high-temperature phases, different crystalline modifications might appear along the direction of the gradient, depending on the temperature of each slice. Recalling that a microstate represents the positions of all Ntotal atoms in the system, a locally ergodic region for the total system would exhibit spatial separation, in the sense that a microstate belonging to this region would be a composite or union of the subsets of Na atoms P that belong to different spatial regions a ( aNa ¼ Ntotal), such that the microstates restricted to the Na atoms inside each of these regions a (at different temperatures) participate in a distinct locally ergodic region Ra restricted to this slice.206

3.11.5.3 3.11.5.3.1

Mechanical forces Pressure

The simplest example of an external mechanical force applied to a chemical system is a constant (hydrostatic) isotropic pressure throughout the whole material. If the system is in equilibrium with the environment, then the most natural extended energy func! ! ! ! ! tion is the so-called “potential enthalpy”, Hpot ð X Þ ¼ Epot ð X Þ þ pVð X Þ, with X ¼ R .21,101 This approach is prototypical for extending the (potential) energy for a system in (equilibrium) contact with the environment, in a minimalistic fashion. In classical thermodynamics, we use a Legendre transformation of the energy E to the enthalpy H ¼ E þ pV when switching from volume to vE .53 To avoid possible confusion, we note that in statistical mechanics and therpressure as the thermodynamic variable p ¼  vV modynamics, E and V are functions of pressure after the Legendre transformation, and to switch from statistical mechanics to ther! modynamics, one computes the ensemble average over all microstates X . But the (potential) enthalpy function of the extended ! energy landscape is a function of X , just as in the case of the (potential) energy landscape, and p is just an external parameter.207 !! ! Of course, we can equally well use the total energy Eð R ; P Þ instead of the potential enthalpy Epot ð R Þ. Exploring this enthalpy land! !! scape over the phase space X ¼ ð R ; P Þ ˛ R6N for given values of p yields those modifications of the chemical system as locally ergodic regions Rdcharacterized by the local entropy S(R) ¼ kB ln g(R)d, which are favored at a prescribed pressure p.208 As in the case of the microcanonical ensemble, we can define for each locally ergodic region on the enthalpy landscape its own temperature. Clearly, once we add temperature as an external thermodynamic variable, we would usually switch to the potential enthalpy function that only depends on the positions of the atoms, since now the expectation value of the energy associated with the kinetic

203

In this case, we can identify (metastable) high-temperature modifications via the usual global search for minima on the energy landscape.21,101,102,116,120 Clearly, predicting such phases without prior information is considerably more involved than finding structure candidates that correspond to single local minima.88,485 205 Note that not only temperature gradients play a role in the experiment, but also applied mechanical forces, e.g., due to the substrates onto which the film is deposited. 206 Note that we encounter the same kind of “splitting” of the microstate when we calculate properties Að! r Þ on the material point level, where for a given ! microstate X only the atoms with coordinates inside the material point labeled by ! r contribute. 204

207

As a consequence, the pV term can be treated as a penalty-type term, favoring configurations with small volume. One often analyzes the enthalpy landscape as function of pressure and studies, how the LERs change as function of pressure concerning their existence, size and stability. 208

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Energy landscapes in inorganic chemistry

degrees of freedom becomes proportional to the temperature, and thus it makes sense to consider only the sets of atom positions as the microstates in the state space of the system. We can compute local free energies (also called local Gibbs free energies or free enthalpies) as function of pressure for those regions in configuration space that are stable for the given observation time. For low temperatures, these locally ergodic regions will usually be neighborhoods of local minima or minima basins, and thus identifying local minima of the potential enthalpy landscape generates possible candidates for metastable modifications of the chemical system at non-zero pressures. Furthermore, after letting the system relax into global equilibrium we expect to most likely encounter the phase with minimal local Gibbs free energy when performing a measurement on time scale tobs. We finally comment on the difference of a pressure being applied to finite or infinite systems. In the case of a finite system, we can model the pressure as a constant inward force/area term on the surface of the chemical system, with an associated energy contribution for the atoms throughout the system,209 while in the case of the periodic approximants to the infinite system, one uses a term ! ! ! pV total ð X total Þ1pV cell ð X cell ÞNC , where NC equals the “number” of periodically repeated cells in the system; X total refers to the ! description of the system by all atoms in the system while X cell refers to the Ncell atoms inside the periodically repeated cell plus the parameters of this cell.210

3.11.5.3.2

General stress tensor

! ! The isotropic pressure discussed above is a special case of the general external stress field s .241,351 Since in general s involves an anisotropic set of forces, e.g., a shear force or an uni-axial pressure gradient, the use of a simple periodic approximant misses important effects. The chemical (macroscopic) system will respond to the applied stress by adjusting its (macroscopic) shape, in addition to changes in its volume. In particular for large stresses, the elastic approximation will no longer be appropriate and plastic deformations will occur that can only be represented on the mesoscopic level since they involve, e.g., dislocations, grain boundaries or crack formation. This effect can only be treated when we employ energy landscapes with at least mesoscopic numbers of atoms in the cluster or periodic approximant for the whole solid. As long as the periodic approximant contains only a small number of atoms,211 we can only model a deformation of the periodic cell. ! ! ! ! To create an extended energy function, we proceed in analogy to the case of the pVð X Þ term, by adding a termd s u ð X Þ to the ! ! ! potential energy function, and we assume that the stress s and the strain u ð X Þ are the same throughout the material. Note that from the point of view of thermodynamics, the components of the stress and the deformation (strain) tensors are conjugate intensive and extensive variables496.212 The minus sign is due to the fact that the stress tensor has the opposite sign compared to the ! ! ! pressure; the definition of the mechanical pressure is pmech ¼  (1/3)(sxx þ syy þ szz).213 In general, s u ð X Þ is shorthand for P ! ! the component-wise product of the stress tensor and the strain tensor and the volume of the system, ij sij uij ð X ÞVð X Þ.214 Furthermore, we note that the definition of the elastic energy of a material implies a zero of this energy to occur for an initial equilibrium configuration for a given applied stress (e.g., zero stress), and the strain tensor describes the resulting deformation due to the newly applied stress.215

209 This is implemented in, e.g., the landscape exploration package G42þ and ensures that the atoms of the system stay in a compact (spherical) region.119 A both technical and conceptual issue arises because the pressure is again a macroscopic quantity when visualized in terms of applied mechanical forces, i.e., we really need to define a surface for these forces to act upon. For a material point, this is possible but for individual atoms we must assume some kind of effective surface, which, for its definition, would require some time scale over which we can imagine many collisions to take place with the environment atoms that generate the external pressure. 210 Of course, once one switches to the energy per atom or per simulation cell by dividing the potential enthalpy by Ntotal ¼ Ncell NC or by NC, respectively, the factor NC drops out of the expression. 211 Of course, periodic approximants containing millions of atoms can capture plastic behavior. But then the number of, e.g., low-energy local minima becomes overwhelmingly large, making a systematic exploration of the whole landscape very difficult. ! ! 212 For infinite solids, u ð X Þ is the mapping of the macroscopic deformation tensor into a deformation of the periodic approximant, i.e., the periodic cell, for a given atom configuration ð! x 1 ; .; ! x Ncell Þ.497,498 213 The non-zero components of the strain tensor for an isotropically expanded system, where the material points (and thus also the atoms) are displaced by an P amount proportional to their distance from the center, ! u ¼ k! x , are uxx ¼ uyy ¼ uzz ¼ k, and thus !V ijsij uij dV ¼  pmech3kV. Note that the trace of the deformation tensor corresponds to the relative increase of the volume of the system. Furthermore, the formula for the definition of the deformation part of the u of the material strain tensor (excluding rotations of the body), uij ¼ (1/2)(viuj þ vjui), as the symmetrized derivative matrix of the displacement vector ! point, is only valid for small deformations. 214 The multiplication with the volume takes into account that the strain tensor is defined for material points and thus an integration over all material points is needed; for a homogeneous systemdwhere the strain in each material point is the same -, and with a constant macroscopic stress tensor, this is equivalent to multiplication with the volume. R P 215 The total elastic energy is found by first computing the work of deformation for an infinitesimal strain, dWdeform ¼ V ij sij ð! r Þduij ð! r ÞdV, and integrating this expression until the final (finite) strain has been reached. For the case of a linearly elastic material, Hooke’s law allows us to perform this computation directly, yielding an expression proportional to the square of the finite strain.

Energy landscapes in inorganic chemistry

315

! We note that it might be necessary for spatially varying stress fields, s ð! r Þ, to explicitly integrate the energy term ! ! ! !! 351 Of course, the reaction of the material to the applied stress might not s ð r Þ u ð X ; r Þ over the whole macroscopic solid. P be linear, i.e., the strain tensor might be a non-linear function of the stress instead of following Hooke’s law sij ¼ kllijklukl where lijkl is the elasticity tensor. This is analogous to the nonlinearities that appear due to induced electric and magnetic dipole moments when applying external electric and magnetic fields. Such external stress fields can be due to enforced grain boundaries or dislocations inserted into an ideal (or real, i.e., with equilibrium point defects) crystalline structure. In this case, one can perform MD or MC calculations, in order to analyze the behavior of such macroscopic defects, their movement through the system, and their life-time. However, since modeling such defects required very large cluster-type or periodic approximants of the chemical system, containing tens or hundreds of thousands of atoms, the usual global landscape exploration methods are only of limited use. Nevertheless, in principle, such situations are amenable to an energy landscape analysis, since on large enough observational time scales one can treat, e.g., one or more individual dislocations and their random thermal movements (or movements due to the application of some shear forces) as the exploration of a locally ergodic region, and the annihilation or creation of additional dislocations and/or grain boundaries as the escape from this region and as a switch to the equilibration into a larger ergodic region. Again, we see the outline of a mesoscopic energy landscape that can be used to study chemical systems that are spatially inhomogeneous on the mesoscopic or continuum level. Furthermore, we note that the application of an external shear force can result in the creation of a dislocation current through the system,499 which can be analyzed analogously to the introduction of an electric current, as far as its effect on the formation of various steady state phases of the system is concerned.

3.11.5.4

Electromagnetic forces

For the incorporation of electric and magnetic fields into an extended energy function of a chemical system, we are again guided by ! !!! ! !! thermodynamics in adding terms –M ð X Þ B and  P el ð X Þ E to the potential energy or enthalpy, resulting in a cost function Cð X Þ ¼ ! !!! ! !! Epot ð X Þ  M ð X Þ B  P el ð X Þ E .216 Thus, we need to compute the magnetization or polarization of the system associated with ! a microstate X ¼ ð! x ; .; ! x Þ. 1

N

In the most straightforward case, this can be done by computing or employing given (permanent) atomic electric and magnetic P! ! ! ! P! ! ! dipole moments, p ð! x Þ and ! m ð! x Þ, P ¼ p ð x Þ and M ¼ m ð x Þ; if the magnetic moment of the system comes from el

i

i

el

i

el

i

i

i

band electrons, (relativistic) spin polarized energy calculations where the electronic bands are split due to spin-orbit coupling terms can allow us to compute an effective magnetic moment associated with a microstate. However, if the moments cannot be computed for individual microstates, we might need to calculate the moments in the continuum approximation.478 If the applied field is large enough to induce a dipole moment for an atom (or a molecule) without a permanent moment, we will obtain non-linear terms (in the field) in the extended energy function. Besides these interactions of the electromagnetic field with the atoms, there is an additional contribution to the energy term from ! ! 1 mð! 1 3 ð! X Þð B Þ2 and 8p X Þð E Þ2 , respectively, of the fields themselves, which is always the magnetic and electric energy densities 8p ! present.217 This electromagnetic field energy density is multiplied by Vð X Þ, and thus appears in the cost function formally like i h ! !!! !!! ! ! ! ! 1 mð! 1 3 ð! X Þð B Þ2 þ 8p X Þð E Þ2 Vð X Þ  M ð X Þ B  P ð X Þ E , in addition to the mechana pressure term, Cð X Þ ¼ Epot ð X Þ þ pmech þ 8p ical pressure pmech. Thus, this contribution by the fields to the extended potential energy function also favors modifications that exhibit a high density.

! ! Clearly, if the electric or magnetic field becomes large enough, modifications with large polarization P el or magnetization M , respectively, are the preferred ones, just like a high pressure favors high-density modifications. ! ! 217 Note that mð X Þ ¼ 1ðs0Þ and 3 ð X Þ ¼ 1ðs0Þ are non-zero even if the atoms do not interact with the fields directly. However, keep in mind the screening ! effect inside a metal: in that case 3 ¼ N, but E inside ¼ 0 (on the continuum level, for a homogeneous piece of material). Thus, no electric field density is present inside the material, but the energy density outside the material is increased, and needs to be taken into account for very large fields. Analogously, the superconducting state expels the magnetic field,500 again increasing the magnetic energy density outside the material. As a consequence, energy minimization leads for very large fields, i.e., larger than the critical magnetic field of a superconductor, to a destruction of the superconducting state and thus a large magnetic field penetrates the material nevertheless. Similarly, for very large electric fields, even a metal will be penetrated by the electric field. 216

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3.11.5.4.1

Special aspects associated with the electric field

An important special case is the presence of mobile ions in the material while applying a constant electric field.218 In that case, we need to distinguish two different situations: (a) The chemical system consists of a finite set of atoms, such as a molecule or cluster of N atoms. In this case, we can write down ! a term q E ,! r for every ion i with charge q in the energy function, where ! r is the distance of the ion from the center of mass i

i

i

i

of the cluster.219 Clearly, this term favors the movement of every ion towards (þ/)-infinity, and thus the global minimum would consist of a separation of all positive from the negative ions, which are moved to infinity in opposite directions (and at infinity, they would move away from each other due to Coulomb repulsion).220 Since this would happen even for very weak electric fields, one would want to exclude this effect by restricting the size of the volume that can be occupied by the atoms belonging to the cluster as discussed earlier; e.g., the Natom atoms are only allowed inside a sphere with some maximal diameter pffiffiffiffiffiffiffiffiffiffiffi Dmax
3 Natom datom , where datom is the diameter of an atom in the cluster. As long as the energy gain by separating the positive P  ! and negative charges,
qi E Dmax , remains smaller than the cohesive energy of the cluster (without electric field applied), i

the compact low-energy isomers would still be among the relevant minima of the system, although the shape of the cluster in the electric field might be distorted due to the electric field, and the same holds true for molecules that exhibit regions with positive or negative charge accumulation.221 (b) The second case concerns the periodic approximants of an infinite solid where the unit cell is allowed to change size and shape. To avoid spurious infinities when we couple the electric field to the charges of the ions, we need to ensure that the size of the periodic unit cell does not grow infinitely large or its shape becomes extremely distorted. After all, the content of the periodic cell plays the same role as a finite cluster. Thus, we must ensure that the energy contribution of the dipole moment generated inside the cell by the displacements of the ions towards opposite sides of the cell, must remain small compared to the cohesive energy.222 Clearly, for large fields, we cannot avoid such a charge separation. Even if no ionic charges are present in the system and we only deal with atom centered dipole moments in an insulator, very large fields might remove the valence electrons from the atom, effectively destroying the localized dipole moments and creating an ion-electron plasma in the system, or, at ! least, switching the system from an insulating to a conducting state for the same configuration X .223 Finally, we note that if the system is finite and pieces of the surface are at different electric potentials and connected to other materials that can serve as sources and sinks of the ions inside the chemical system of interest, then the electric field might generate a stationary state of the material with a constant ionic current. For such a steady-state current in thermodynamic equilibrium, the occupation of the channels inside the material through which the ions travel can become averaged out on the equilibrium time scale. From an energy landscape perspective, all the microstates corresponding to atom configurations where the mobile atoms reside at different positions inside the channel, are in local equilibrium. On this long time scale, a new kind of locally ergodic region corresponding to a phase with a “smeared-out” occupation of the channel region might come into existence. Of course, in metals, an electric field will generate an electric current, if source and drain contacts with the environment exist; otherwise there will be buildup and depletion of electrons on opposite surfaces of the system, to counterbalance the field inside the metal. Finally, for very large electric fields in an insulator, we can observe a metal-insulator transition, i.e., electrons are moved from the valence band to the conduction band in an insulator, making the system electronically conducting in the process.224

3.11.5.4.2

Special aspects of the magnetic field

As mentioned above, the magnetic field can straightforwardly interact with permanent or induced magnetic dipoles. In particular, this applies to the magnetic moments associated with the spins of free electrons and ions with nonzero total spin components, and to nuclear magnetic moments, too. Of course, the energy function should contain terms describing the interactions between the dipoles themselves, either on the ab initio or the model Hamiltonian level.58

218 Such individual charges that directly interact with the electric field would appear in the model description, e.g., when one uses empirical energy functions that include ions instead of atoms, or allow charge transfer between the atoms. 219 In contrast to the (induced) dipoles, the standard energy function already takes the Coulomb interaction between the ions into account. 220 Although this is unlikely to happen if we start short simulations at low temperatures for a compact aggregate of atoms with total charge zero, more often than not one observes that some of the atoms or ions evaporate from the surface and diffuse towards infinity for longer simulations. P  ! 221 qi E dclust is larger than the energy of a cluster with diameter dclust, then the global minimum will correspond to a broken-up On the other hand, if

i

molecule, reminiscent of a so-called Coulomb explosion that can occur when too much excess charge is placed on an originally neutral small molecule or cluster.232 222 Similarly, one would prescribe limits on the shape of the periodic cell to stop it from becoming highly anisotropic and flat (which is another way the system can try to minimize its energy in the presence of an electric field). 223 We note that constant electric fields have been implemented in some ab initio codes, but there are subtle aspects in treating the Stark-effect in the band structure eigenstates of the Hamiltonian.58 In principle, this can also be done for external magnetic fields, but these are more difficult to handle due to the fact that the simple translational symmetry invariance in the band structure calculation is lost58; for suggestions how to address this issue, we refer to the literature.501 Here, we are more concerned with the general concepts for an extended energy landscape, and not with the technical aspects of implementing such interactions in an ab initio code. 224 Here, we are not discussing the details of the electric current, i.e., whether it takes place via hopping (frequently appropriate for amorphous solids) or via the usual band conduction (most likely to occur for crystalline materials).

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A major difference to the electric field is the non-existence of magnetic monopoles, and thus there is no potential energy term ! analogous to the electrostatic potential energy qi fð! r i Þ experienced by a charge qi in an electric field E ¼ Vf generated by an electrostatic potential fð! r Þ.225 Instead, the magnetic field enters the Hamiltonian of a charged particle via the kinetic energy term which mech conj ! ! ! ! 1 ð! 1 ð! gets modified by the magnetic vector potential A , where B ¼ V  A : Ekin ¼ 12 mð! y i Þ2 ¼ 2m p i Þ2 /2m p i  qci A Þ2 , as dis500 cussed earlier (c.f. Section 3.11.3). The Bohr-van Leeuwen theorem shows thatdfor classical free particles in a magnetic fielddthere is no contribution of the magnetic field to the free energy of the particle, and thus we have no magnetic moment associated with the particle. But on the quantum mechanical level we obtain a contribution to the energy of an electron gas in a (constant) magnetic field through the so-called Landau levels and the energies associated with them.346 Furthermore, the magnetic field will change the band structure in the ab initio calculations,58 complicating the band structure calculations considerably.226

3.11.5.4.3

Electromagnetic radiation

Finally, we comment on the presence of time-varying electromagnetic fields. First, there are the periodically varying fields, whichdwhen observed on time scales that are many multiples of the periods of the fieldsdcan be treated as a source (or drain) of energy flow into (or out of) the system. Usually, the electromagnetic field will add energy to the systemdeither homogeneously via, e.g., excitation of phonons or excitation of electrons between bands, or locally at special places (at the surface via surface phonons etc., or at certain atoms/ions/molecules which can locally absorb radiation of certain frequencies very efficiently).227 This energy is then either re-radiated (via fluorescence, phosphorescence, etc.) or converted into heat inside the material, until a temperature has been reached at which black-body radiation can carry away as much energy as is imparted by the incoming radiation, or at which a (conductive or convective) heat flow through the interface to some other material belonging to the environment in contact with the chemical system of interest, becomes possible, such that a stationary state with respect to energy exchange with the environment is established. Here, we note that it might be necessary to include the electronic excited states energy landscape of the system into the description as is frequently done in the study of molecular reactions,502,503 in order to be able to properly model the absorption of the electromagnetic radiation within the context of the energy landscape picture. We also recall that the presence of photons in a solid can lead to the creation of hybrid phonon-photon, electron-photon and exciton-photon quasi-particlesdfrequently called polaritonsdvia the interaction of the photon with a (dipole) polarization wave propagating through the crystal,504,505 which can change the energy of the system, and thus modify the energy landscape. In the steady-state, on time scales larger than the period of the electromagnetic fields, we can explore the landscape and identify the locally ergodic regionsdcorresponding to possibly interesting metastable phases of the chemical system, at the temperature that is established due to the influx of radiation energy, analogous to the case of metastable phases in the presence of thermal gradients.

3.11.5.5

Changes in composition

The addition (or removal) of atoms to (or from) a chemical system, respectively, is one of the most important types of interactions with the environment.228 Since here the environment serves as a source or drain of atoms, the distinction between the chemical system and the environment is no longer clear-cut.229 Furthermore, on long time scales, it can easily happen that the chemical system whose interaction with the environment is being studied, has become unrecognizable after, e.g., a chemical reaction of an oxide A2O (the original chemical system) placed into contact with another oxide B2O (the original environment), results in a new material ABO. In such a case, one might prefer to include the environment into the definition of the chemical system, and just start the exploration of the landscape from microstates where all A-atoms and B-atoms are confined to different spatial regions of the combined system. On the other hand, on shorter time scales, the atom exchange could be considered a perturbation of the original chemical system, and thus keeping system and environment separate would be a sensible proposition. Similarly,

We do not consider electromotive forces, i.e., electric fields that are induced by time-varying magnetic fields according to Faraday’s law.478 A constant electric field also affects the band structure calculations,58 but not quite to the extent as a magnetic field does. In both cases, one important technical issue it to what extent the spatial variation of the electric and magnetic fields is compatible with the unit cell of the periodic approximants one employs to model the (infinite) solid. 227 Incorporating an atom level description of local absorption of radiation energy into a classical energy landscape approach to chemical systems is not straightforward. Only on larger time scalesdas mentioned alreadydwould this absorption translate into (re-)arrangements of atoms in the system. Nearly instantaneous breaking of bonds caused by radiation with frequencies specially tuned to the energy level structure of the bond would be a possible exception. Here, we can implement the presence of the radiation field into the moveclass of the standard energy landscape of the isolated system, by including bond-breaking moves (or increase the probability of attempting such a move). In practice, we would allow the system to “tunnel” through a high energy and entropic barrier, because the very specific path to this bond-breaking transition usually would not be easily reached via thermal excitations of the system. 228 A special case would be radioactive decay, removing atoms of one species and replacing them by the fission products plus a-particles (or helium atoms after addition of electrons to the a-particles). 229 In the case of steady-state currents (c.f. Sections 3.11.5.4 and 3.11.5.6), the net composition of the chemical system does not change, and thus we can employ the same state space for the description of the landscape. 225

226

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while the chemical system might be greatly changed due to the interaction with the environment, the latter might remain essentially unchanged, e.g., if a metal (the chemical system) is placed into an (infinitely) large oxygen gas bubble (the environment), such that only the concentration of oxygen in the material is varied, while the concentration of oxygen in the bubble stays the same. Changing the number and/or types of atoms in the chemical system, whose energy landscape we want to study, has major consequences, both technically and conceptually. We note that the configuration space of the system now contains microstates with different numbers of atoms and different atom species in the chemical system. We can address this issue by extending our state space to include microstates for different numbers of atoms, analogous to a Fock space in quantum mechanics506 but staying in the classical picture, and also allow atoms to change their chemical identity when appropriate. As a consequence, the neighborhood relation (moveclass) for these “extended” microstates of the landscape includes switches between states with different numbers of atoms, which in turn affects our definition of what constitutes a local minimum of the landscape, since now we deal with a combination of continuous and discrete degrees of freedom.230 In particular, we need to take the variation in the number of atoms into account when we identify the locally ergodic regions of the system. Typically, the sets of atom arrangements that constitute the LERs will exhibit a spread in the composition of the atom configurations, at least for bulk systems. From a thermodynamic point of view, we can consider the chemical system to be in equilibrium or out of equilibrium with respect to the environment. In principle, we can also have a steady-state situation, where the atom exchange involves macroscopic flows of atoms or electric currents through the chemical system.231 When one incorporates interactions that vary the composition of the system, into the energy landscape, one needs to be aware of the fact that this process will occur by moving atoms through the interface between system and environment. Short-circuiting this process, in order to simplify the energy landscape description, while often necessary, can lead to various side-effects that need to be kept in mind when interpreting the results of landscape based investigations. If we do not want to combine the chemical system and the environment into a single larger system, then three general approximate approaches can be distinguished, of different levels of complexity and temporal resolution. A first compromise is to describe the interface between the system and the environment in a complete fashion by including all the relevant atoms or degrees of freedom of the interface into the landscape, while keeping the rest of the chemical system and the environment separate. While being very expensive, this might often be the only solution when we are dealing with non-equilibrium situations yet still want to model the dynamics of the system in detail on short observational time scales. In the other extreme, we again take our cue from thermodynamics and construct a minimalistic extension of the landscape by ! ! representing the environment through a term mi in the cost function, mi Ni ð X Þ, for species i, and configuration X that is inspired by ! ! ! P ! the chemical potential.21,507 In this fashion we get an extended energy function Cð X Þ ¼ Eð X Þ þ pVð X Þ þ i mi Ni ð X Þ.232 The interpretation of mi is a “price” we have to “pay” for removing an atom of species i from the environment. From a thermodynamic perspective, we treat the environment as an (infinite) reservoir or reference state, mi ¼  miref where miref is the chemical potential ! of the atom inside the reservoir.233 Alternatively, we can extend the energy to a cost function by adding a term mðiÞ Ni ð X Þ to P ðiÞ ! ! ! ! the (microcanonical) energy, Cð X Þ ¼ Eð X Þ þ pVð X Þ  i m Ni ð X Þ, where m(i) ¼ v E/v Ni is the chemical potential of species i in the isolated chemical system, analogous to the introduction of the chemical potential in thermodynamics. Since in the minimalistic picture the isolated system is assumed to be in equilibrium with the environment,234 we have m(i) ¼ miref, and thus mi ¼  m(i) in the expression for the extended energy function for a system in equilibrium with the environment regarding particle exchange.235

230 We note that this is again a matter of time scales: On very short time scales, we can treat the system as being restricted to a cross section of the landscape with constant number N and types of atoms. In this case, a minimum is defined in the usual fashion as for the (continuous) landscape of the isolated chemical system. On time scales, where complex moves take place that can change the number of atoms, many of these local minima might be unstable against the switch to the (Fock-state) landscape cross section with a larger (N þ 1) or smaller (N1) number of atoms. This is most noticeable if we are dealing with small clusters of atoms, where, e.g., the global minimum structure can massively change when adding or removing an atom. 231 We can also have a quasi-stationary situation, where the system is in equilibrium with the environment in a small spatial sub-region, e.g., at the surface of the material. In principle, this region can “move” through the material, e.g., as the progress of a corrosion process. Depending on the time scale of interest, such quasi-stationary set-ups can be useful in elucidating various aspects of the chemical process of interest, on different time and length scales. ! 232 Since adding/removing atoms usually involves changes in the volume of the material, we are adding the pVð X Þ term to the cost function. 233 This interpretation of mi is analogous to a penalty term that favors or disfavors the addition of atoms to the chemical system. Note that the chemical potential of the reservoir will vary as function of temperature and pressure. Furthermore, one needs to decide whether one wants to use an atom in the gas phase, an atom in the element in its solid ground state, or in some other state such as in a molecule or in an educt (e.g., in some binary compound), as the reference state. 234 While true for many situations, this assumption does not always hold, e.g., when the chemical system is bombarded by particle radiation, or when surface reactions take place such as in corrosion processes. Furthermore, it is often highly non-trivial to compute the chemical potential, since it is a thermodynamic equilibrium property and does not depend on individual atom configurations. Nevertheless, for a number of cases, (approximate) analytical expressions are available for the chemical potential, e.g., for a gas or a solid of the species i at given pressure and temperature. 235 Depending on whether one employs m(i) or mi as the “chemical potential” parameter in the cost function, the additional term enters with a minus or a plus sign, respectively. Usually, the “price-to-pay” would be a positive number mi ¼  miref > 0, since, in most situations, it requires energy to remove an atom from its reference state. Here, we mostly use mi, in order to keep in mind that in many applications we are not in an equilibrium situation, in contrast to the case of applied temperature or pressure.

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We note, that we can, in principle, treat mi as a thermodynamic parameter we are free to vary, like the applied pressure, and, thus, we can analyze the extended energy landscape as function of mi. Usually, mi would be fixed by the specific environment the system interacts with, but we often might have some freedom in adjusting the chemical potential by modifying the experimental set-up. Unless the chemical system and environment are fully in equilibrium and the system itself is in global equilibrium, we would expect that the chemical potential will usually vary as function of position ! r (and time) for real chemical systems, because the interaction with the environment takes place at the surface of the chemical system thus generating differences in concentration ! of the various atom species as function of position. Therefore, the term mi Ni ð X Þ in the cost function must be replaced, in general, R !! ! 1 by an integral over all material points, ! ! Ni ð X ; r Þmi ð r ÞdV. The situation is now similar to the system with a thermal Vð X Þ Vð X Þ gradient, since now different spatial regions of the material can exhibit local thermodynamic equilibrium for the local value of ! r Þ. As a consequence, the LERs for the whole material could then consist of atom arrangements X the chemical potential m ð! i

that form different modifications, because the local equilibrium composition of the various atom species will vary for different spatial regions of the material. In principle, a spatially varying chemical potential can also occur if the overall number of atoms R !! 236 1 Again, the LERs would correspond to “unions” of different modifications that of a species ! ! Ni ð X ; r ÞdV is constant. Vð X Þ Vð X Þ r Þ. are (meta)stable for different values of the chemical potential mi ð! r Þ of an atom species i will result in currents of this type of atoms inside the Clearly, differences in the chemical potential mi ð! material, since it “pays” for the atoms to move until mi ð! r Þ is constant throughout the system (in equilibrium). If we enforce r Þ between opposite surfaces of the material by adding and extracting atoms of species i, then a permaa permanent difference in mi ð! nent (constant) diffusion flux will be established inside the system, and the material will be in a steady-state. A third approach in-between the two ways discussed so far for dealing with the environment from the perspective of the energy landscape, is to add the reservoir not as a fictive entity but as a “real” external reference system, where a large number of atoms are in equilibrium for given thermodynamic conditions (T, p, .). Depending on the type of environment-chemical system interactions, many reference states can be chosen, e.g., an element in gaseous, liquid, or solid form, or some simple binary or ternary compound. Now we compute the chemical potential from the (free) energy loss of the reservoir upon removing the atom.237 Our extended ! energy function now corresponds to a grand canonical energy as function of (extended) microstate X . Using such an in-situ approach to evaluate the chemical potential can be employed if the chemical potential cannot be analytically computed as function of, e.g., pressure or temperature (at least not with the desired accuracy). Here, we remark again on the importance of time scales when studying the energy landscape of chemical systems. Due to the slow diffusion of atoms in a solid, it can take a very long time, before a prescribed potential has been established throughout the chemical system.238 In principle, this is analogous to the application of an external pressure or temperature as thermodynamic boundary conditions, but the time scales involved are usually quite different. A prescribed pressure will spread through the system with a speed on the order of the speed of sound, and thus, for typical sample sizes and measurement time scales in the experiment, exp (p) [ seq and while heat transfer is often slower than the propagation of the pressure, one nevertheless can usually assume that tobs exp (p) (p) (T) tobs [ seq Here, seq and seq are the time scales on which an applied pressure or temperature are established throughout the 239 system, respectively. But unless we study gases or liquids, the addition/removal of atoms usually takes place via diffusion through a real interface (surface, grain boundary, etc.) between the chemical system and the environment, followed by bulk diffusion. The time scale controlling the establishment of the chemical potential inside the material is the one for solid-state diffusion, which is often extremely slow. In the case of, e.g., a given temperature throughout or on the surface of a system, one faces the question of whether we want to analyze processes such as phase transitions or the nucleation and growth of solid modifications induced by such an external temperature, by following the probability flows, represented by the system’s trajectories, on the energy landscape, or whether we just want

236 This means that there is no net exchange of atoms between the material and the environment; in the extreme case, we would have no atom exchange at all, i.e., the system would be isolated as far as compositional changes are concerned. 237 Depending on the type of investigation, we would either compute the instantaneous change in energy or a time-average of the (free) energy change of the reference state. In both cases, the fluctuations in the reservoir, as function of temperature, pressure, etc., are taken into account when we consider the time evolution of the system over many forward-and-backward transfers of an atom between the system and the reference state. A number of simulation packages508 offer this type of in-situ simulations of the chemical potential. 238 Of course, we can prescribe this value by fiat in a simulation, but in reality, our chemical system of interest is first brought into contact with the environment, followed by the exchange of atoms until eventually equilibrium might be reached and a (constant) chemical potential or gradient thereof is established. This is the long time scale we refer to. 239 We note that there are many instances where we want to study the reaction of the system to the sudden change in the external applied pressure or temperature, in form of a shock or a heat pulse, at some small region on the surface of the material. In this case, we would investigate the relaxation behavior of the (macroscopic) system, which is strongly influenced by the barrier structure of the landscape.

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to find the minima of the landscape (as candidates for LERs). Similarly, in the case of atom exchange with the environment, we need to decide on whether we are interested in studying the processes involved in the real interaction with the environment, or whether our goal is to find the stable states of the system, i.e., the globally and locally ergodic regions of the chemical system for a given (family of) value(s) of the chemical potential. In the latter case, the crucial issue will be to employ a moveclass that efficiently incorporates variations in composition. Introducing short-cuts for a system with constant chemical potential, we can encounter problems regarding the elimination of intermediary phases that might appear during the actual process and which might be of interest in practice, and furthermore we usually forfeit insight into the mechanisms of phase formation and the response of the system to vari(m) ations in the environment.240 However, if such processes that take place on much shorter time scales, are of interest, i.e., tobs  seq (m) (seq is the time scale of establishing the chemical potential in the material), then we will face non-equilibrium situations, and, e.g., the choice of initial configuration of the combination system þ environment will be important. Yet even in this class of problems, one can often apply separation of time scale methods, since often local (spatial) equilibrium will be present.509

3.11.5.5.1

Variation of composition within an isolated chemical system

Compositional changes do not only appear via exchange of atoms with the environment, but can also occur within an isolated chemical system during relaxation processes that require long time scales, or in situations where a species based description of the system is more efficient than an atom based one. 3.11.5.5.1.1 Order-disorder phase transitions A classical example are the rearrangement processes that occur during an order-disorder phase transition in a binary or quasi-binary system. Here, the system is often prepared in a metastable state where the two different atom species constituting the solid solution are originally at essentially random locations on the sites of an overall lattice due to a fast quench to very low temperatures from the high-temperature phase above the miscibility gap in the phase diagram.241 To equilibrate the system, one can either explore the landscape via atom-exchange moves on the lattice (possibly plus short local energy minimizations) of a macroscopic model of the (infinite) system, using the energy landscape of the isolated system. Alternatively, we can accelerate the separation of the atom distribution into high- and low-concentration regions by introducing chemical potentials (as function of position) that favor low and high concentrations of a certain atom species in different spatial regions of the material. In this fashion, the entropic barrier separating the original random atom distribution from the equilibrated atom distribution can be reduced.242 3.11.5.5.1.2 Chemical reactions In addition to atoms, we can introduce whole molecules or complex ions as species, each with their own chemical potential. By modifying the moveclass such that it becomes possible to, e.g., exchange one carbon atom plus two oxygen atoms (that are spatially close together) by one CO2 molecule, we now are constructing an energy landscape for the efficient realization of chemical reactions in a spatially distributed system on a time scale, where such reactions can be visualized as taking only “one-step” between configurations in the state space.243 Of course, the underlying elementary time step of this move as part of the moveclass is much larger than the time scale on which the actual reaction of a carbon atom with two oxygen atoms takes place.244 On the other hand, we would allow moves that correspond to diffusion of the atoms and molecules between different spatial regions; here, too, one might want to replace individual atom moves by net diffusion moves between, e.g., different material points. As far as the energy landscape is concerned, we can still work with the original (ab initio) energy function, without introducing a chemical potential, but we will most likely now have to discretize the configuration space, e.g., on the level of material points, in order to capture the macroscopic spatial scales on which such reactions will take place throughout the gas, liquid or solid material.245 For the description of such a diffusion-reaction system, we can furthermore include the dependence on temperature and pressure, both regarding the reaction time scale and the possibility of an optimal control of the time evolution of the system. Such an energy landscape that incorporates chemical reactions is most likely to be useful for systems where the gas phase is important, but it might also be suitable when describing the catalysis on the surface or in a constrained region inside some background structure.

240 This would be analogous to the way we modify the landscape in subtle and not so subtle ways to perform an efficient global optimization, possibly eliminating some minima in the process and losing information about the stability of the regions. 241 We disregard here the fact that during a fast quench many structural defects will be generated in the assumed basic underlying lattice of the solid solution. 242 We need to keep the various time scales in mind, both regarding the processes we really want to observe, and with respect to the realism of the moves; after all, the exchange moves represent by themselves already quite an acceleration of the atomistic dynamics we would see during a molecular dynamics simulation. 243 Of course, we would also include a move that corresponds to the break-up of such a molecule, since in thermodynamic equilibrium both CO2 and C and O should be present; in addition, one would need to consider other molecules, such as CO and O2, and the reactions generating them. 244 If we want to analyze an individual chemical reaction, we would have to stay with the usual moveclass that allows only tiny atom displacements, possibly together with introducing the excited states of the atoms as part of the configuration space. Another approach would be to use the excited state landscape(s) of the chemical system. As is common in the analysis of chemical reactions, each atom arrangement would be associated with several possible energy levels of the system (on ab initio level), yielding a multi-valued landscape with several sheets. The moveclass would then contain moves on each of the sheets of the landscape, and moves corresponding to switches among the sheets, each with a certain probability. 245 For long-time simulations, this suggests that the dynamics could be described by a master equation approach, where the time evolution of the distribution of atoms and molecules over the space occupied by the chemical system is being analyzed. Such methods are well-known in physical chemistry.510,511

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3.11.5.5.1.3 Ionization Another application involving a change of “species” in the system are changes in the ionization state of atoms; such an approach would be suitable when the real energy function is approximated by an empirical potential with parameters, such as ionization state dependent ionic radii, that can be employed to compute the energy for different ionization states of the atoms. Here, the “price-topay” for transferring an electron from atom A to atom B consists in the ionization energies or electron affinities of atoms A and B, depending on whether we are dealing with positive or negative ions, respectively.21 As far as the configuration state of the system is concerned, we can either add another degree of freedom to the description of the atom idin addition to the position vector ! x i we have the charge qi of the atomd, or we define the various ions as additional species that can be exchanged from a reservoir with the corresponding atom (and other ions belonging to the same atom type)dthe parameter mi, a ¼ Eiion(a) now refers to atoms of type i and their ionization state, a ¼ .,  2,  1, 0, þ 1, þ 2, ., and Eiion(a / b) ¼ mi,b  mi,a is the ionization energy (or electron affinity) difference associated with a switch from state a to state b for atom i.246

3.11.5.5.2

Insertion/removal of atoms from the system in non-equilibrium situations

As mentioned earlier, the exchange of atoms between the chemical system of interest and the environment is usually not an instantaneous process. Furthermore, in many cases, thermodynamic equilibrium is not reached at all on time scales of interest, e.g., during slow solid-state reactions at the interface of two bulk compounds. In the extreme case, the interaction with the environment will lead to a complete elimination of the original chemical system, such as an alkali metal reacting with the oxygen in the atmosphere, the transformation of a layered compound into an intercalation compound with a different composition, or the dissolution of a salt in a solvent. Thus, it is important to identify, for each system under consideration, the relevant experimental or environmental time scales, on which the processes we want to study take place. These would be compared with the time scales on which locally ergodic regions can exist on the extended energy landscape; this might especially be true for instances where the addition of new atoms is very slow, and thus for certain time-slices the system can be considered as out of equilibrium but at a constant overall composition. Furthermore, one would analyze whether one can employ a minimalistic description via (time-dependent) chemical potentials to describe the (incomplete) relaxation or possible steady-state behavior of the system, or whether local (spatial) equilibrium can be assumed at, e.g., the surface where the atoms from the environment enter the chemical system, even though we are dealing with concentration gradients from the surface to the center of the material.247 In the following, several prototypical examples are discussed illuminating various non-equilibrium aspects of compositional changes due to contact with the environment, and their consequences for their description within an extended energy landscape picture.

3.11.5.5.3

Particle radiation and/or radioactive decay as source of particles

In general, there are two ways for new atoms to appear in the chemical system: creation (everywhere) inside the solid due to radioactive decay mentioned earlier, or due to a particle flow248 through the surface of the system.249 Depending on the intensity of the stream of particles impacting the material, there will be a certain penetration depth from the surface, within which we assume the particles to appear at a certain rate, determined by the parameters of the environment. In the case of particle radiation, we need to keep track of two aspects: For one, we deposit energy into the system via the impact of the atoms or ions on the surface and the heat of friction generated while stopping their momenta.250 The second effect is a change in the composition of the system upon permanent incorporation of the atoms into the material; if surface atoms are ejected during the absorption process, this also contributes to the compositional change. We could try to accommodate this change in composition in the energy landscape picture in a minimalistic fashion by the introduction of an appropriate chemical potential. But we note that we change the composition of the system “by force” into one direction onlydatoms are only added and none are released (except upon ejection during high-energy impacts, but the net result is usually an increase in the number of atoms unless we use sputtering geometries or particles with extremely high energies)d, and thus the parameter mi would be zero (or actually highly negative). Therefore, there would be no price to pay for adding an atom, and, from a thermodynamic point of view, we massively favor the “generation”

246 Usually, one only changes the oxidation state in integer units, but, in principle, one can also change the effective charge in a continuous fashion. In that case, one needs a polynomial approximation of the ionization energies/electron affinities as function of the effective charge of the atom. 247 In this context, we note that adding and removing atoms “inside” the system, i.e., inside a periodic simulation cell for a periodic approximant of a solid or liquid system, instead of through the surface requires a careful consideration of, e.g., the creation of empty space needed to accommodate the added atoms. For one, we generate additional pDV terms, and if we do not create such a space, we encounter extremely high interaction terms due to the fact that we insert an atom “on top” of the (often tightly packed) atoms already in the cell. Experience has shown that an efficient way to achieve such an insertion is by combining it with an enlargement of the periodic cell, possibly followed by a local optimization.21,507 248 While electrons, positrons, muons, neutrons, etc., all count as particles, here we are considering only atoms, ions, or molecules. 249 Here, we are not considering the transformation of several atoms into a molecule, or the split of a molecule into its constituent atoms during chemical reactions, discussed in the previous Section 3.11.5.5.1 250 When realizing this process via, e.g., MD simulations, we often observe reflections of the atoms from the surface (with slightly reduced kinetic energy), and the final deposition of the atoms on the substrate might occur after several collisions. On the other hand, for high-energy particles, the repulsive interaction with the surface atoms is not large enough to repel the incoming atoms, and the atoms penetrate the material and come to rest below the surface (while possibly ejecting some surface atoms in the process).

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of atoms inside the material regardless of the local stresses created by this process. As long as we do not release these inserted or generated atoms back into the environment, there is no steady-state, and we are dealing with a non-equilibrium situation.251 Various aspects of the chemical processes involved in particle radiation can be investigated using the underlying energy landscape. On time scales where the spatial distribution of foreign atoms in the material can be assumed to be approximately constant, we can study short-term local equilibrium properties and the relaxation behavior to equilibrium. This would be a situation where the system might split into slices or regions of different compositions, and the energy landscapes of each such slice could be considered an isolated system, at least on short time scales. At the other extreme of very long time scales, we would find the steady state where implantation and emission are in balance.252 At intermediary times, we would explore the time evolution of the system in a non-equilibrium, non-steady-state situation. Now, the landscape varies relatively fast compared to, e.g., the equilibration with respect to the chemical potential, and possibly temperature, tvar  seq(T), seq(m). In such a situation, concepts like globaldand even localdergodicity and equilibration are most likely not applicable (c.f. Section 3.11.4.3). However, we might deal with a special case of marginal ergodicity, with or without aging phenomena, i.e., the marginally ergodic regions are most likely not nested, because the composition keeps changing, but one should not exclude this possibility, of course. In this case, we would measure, through simulations of ensembles of walkers, the probability flows on a landscape that varies with time, in order to gain information about (the limits on) the stability of (formerly) locally ergodic regions, or about the structural changes that are the effect of the changing composition. 3.11.5.5.3.1 Corrosion Another example of an essentially “open-ended” interaction with the environment are corrosion processes. In this case, the chemical reactions take place at the surface separating the chemical system from the environment.253 The atom level mechanisms of various local reactions, such as oxidation of a few atoms belonging to a metal surface, can clearly be investigated using energy landscapes that are restricted to the few atoms involved in the reaction. Such energy landscape studies of chemical reactions are rather common,512–517 and take place at local equilibrium conditions as far as the corrosion process is concerned. On the other hand, if we want to go beyond such local events, we are dealing with a system out of equilibrium. The surface structure, i.e., the environment-chemical system interface, undergoes a spatially inhomogeneous evolution on large time scales. Again, standard energy landscape concepts do not apply, and we appropriate methods discussed in Section 3.11.4.3 for time dependent energy landscapes. If the corrosion process is rather slow, time-slices for which the landscape is approximately constant in time, but where the material varies in space as far as its composition and the (thermodynamic) boundary conditions are concerned, should be useful. In particular, we could analyze such quasi-stable slices/regions with respect to relaxation related phenomena, and regarding possible nucleations of new phases, and the spatial shape of the interface. Clearly, this would involve very large simulation cells, but this would be the price for realism in a non-equilibrium system. 3.11.5.5.3.2 Defects in equilibrium with the environment Another class of systems where there is a balance between equilibrium and non-equilibrium are processes, where atoms or molecules diffuse into and out of the system, while modifying thedoverall stable, howeverdstructure of the compound; intercalation compounds, filling of metal-organic frameworks, or diffusional substitution of one atom species by another would be examples. In certain situations belonging to this class of systems interacting with the environment, the use of a chemical potential can be very helpful, since in the long run a thermodynamic equilibrium is established. An example would be the study of defect compounds as function of composition, where we can assume that we are on time scales, for which the amount of, e.g., hydrogen or oxygen atoms/ions inside the material can be assumed to be in thermodynamic equilibrium with a reservoir of gaseous hydrogen or oxygen molecules. Instead of dealing with the many additional degrees of freedom required by an atom level description of the reservoir, its presence is incorporated in the chemical potential. Another example is, e.g., the use of a partial pressure based chemical potential of a solid in equilibrium with, e.g., an oxygen atmosphere.518 In this situation, the gaseous state of the oxygen atoms which are in equilibrium with the solid is properly accounted for by the (adjustable) reference chemical potential, since in this case the dependence of the chemical potential on, e.g., the temperature or the overall pressure, is well-established from experiments for many gas species.

251 After a very long time, the rare emission of atoms due to surface evaporation might balance the stream of incoming particles, resulting in a stationary state of the system, where the flow of atoms being implanted into the material is compensated by a release of the atoms driven by an exceedingly high concentration of the atoms inside the chemical system. In this steady-state situation, we would find a constant (time) averaged distribution of these particles over the material, with a probably quite high overall concentration. 252 Note that we can have quite interesting currents and steady-state distributions of the foreign atoms over the system: As a source of the currents, we might find some regions several atom layers below the surface, from which atoms are moving towards the center and then to the opposite (to the particle stream) side of the material, and a second major current direction back towards the surface area where the particle stream hits the material, not to mention atom currents parallel to the layer where the atoms are implanted. 253 Such processes will also occur at interfaces such as grain boundaries of a material, which are “infiltrated” by the foreign atoms from the environment that perpetrate the corrosion.

Energy landscapes in inorganic chemistry 3.11.5.6

323

Enforced currents

In the preceding subsections, we have discussed spatial variations of external (thermodynamic) parameters such as thermal gradients, constant electric fields or gradients in the chemical potential. These result in various currents, which try to eliminate these gradients, and if some source(s) and sink(s) for the carriers of the current exist (usually on the surface of the material), then we reach a (thermodynamic) stationary state of the system. We can turn this situation around by enforcing constant fluxes in the system by injecting and extracting a current on opposite surfaces of the system. Such constant currents, e.g., electric currents, thermal currents or (particle) diffusion currents, constitute another class of interactions between the environment and the chemical system of interest. They also are a type of thermodynamic boundary condition, but for a steady-state and not a classical equilibrium state; thus, our usual Legendre formalism cannot be applied directly when trying to integrate these boundary conditions in a minimalistic fashion into the energy landscape.254 Instead, we take our inspiration how to incorporate constant fluxes into the energy landscape picture from non-equilibrium thermodynamics,519–522 since the currents lead to a stationary equilibrium with the system being in a steady state, on sufficiently long time scales. In this analysis, the currents are the primary entity characterizing the interaction of the environment with the chemical system. In contrast, the electric field, thermal gradient, etc., which we measure in the presence of the current, are to be considered a side-effect of the given current, even though we would apply the corresponding fields or gradients, in order to establish the current in the first place.255 The presence of constant fluxes in the system has several consequences. There can be a direct coupling of the carriers to the (atomic) degrees of freedom via collisions with the atoms in the system.256 This is closely related to the second effect, the generation of heat in the system via dissipation of the electric or diffusion current. As a side-effect, the temperature of the system will change, even if it is not connected to an external heat reservoir (at some external temperature), until some equilibrium is reached with the heat released from the material (via radiation, convection or conduction), similar to the case of radioactive decay or particle radiation discussed above (Subsection 3.11.5.5.2). Conversely, the temperature gradient associated with a heat current can generate an electric field (the Seebeck-effect53,500). In this context, we recall the Onsager relations53,500 which couple the thermal, diffusive and electric currents. The final effect of an enforced electric current is the build-up of a large magnetic field inside the material. The energy density associated with this field (and the electric field driving the current) results in a pressure-like term as discussed in Subsection 3.11.5.4. The (supposedly non-thermal) effect of the electric current on phase transitions in orbital liquids,523257 might be an example for direct coupling of an electric current to the atom-level degrees of freedom; however, usually, the current itself does not strongly ! interact with a microstate X :258 In fact, the contribution to the potential energy of the system comes from the electric field driving the current, which exerts a force on the atoms of the material, as discussed above. ! In the case of large (constant) electric current densities j el ; the contribution to the electric and magnetic field energy from the ! ! ! ! fields associated with the current can become quite substantial. Given j el ; we can deduce the electric field E ¼ ðsel ð X ÞÞ1 j el

254 While the product of conjugate pairs of thermodynamic variables yield a quantity with units of energy or entropy (depending on whether we are using the energy or the entropy as the fundamental function in the microcanonical ensemble), the analogous product of a current with its conjugate thermodynamic force generates an energy or entropy flow, which does not represent a term in the standard thermodynamic potentials; however, it can be associated with dissipative phenomena. 255 Note that electric and diffusion currents require physical contact of the environment with the material such that the carriers can enter and leave the system via a source and a drain. Thus, if the material is isolated such that no steady-state current can flow but we still somehow enforce a current inside the material, there will be a build-up and depletion of electric charge or diffusing atoms of a certain species at opposite surfaces. In contrast, no exchange of material particles with the environment is needed for a thermal current as heat can also be carried by phonons or photons (radiation). In principle, we can use electron beams and field emission to “close” the electric current loop even without physical contact, in analogy to thermal radiation, but we do not consider such special cases. 256 In physical treatments, the collisions are often modelled in the quasi-particle picture of the solid, where the (quasi-particle) carriers, such as band-electrons, phonons, polarons, ion-type polarons, etc., interact with other quasi-particles in the material.,58,500 Since the quasi-particles are collective entities involving many atoms each (and thus a set of such quasiparticles would be linear combinations of our usual microstates), such collisions would involve many atomsdand implicitly many microstatesdat the same time. For simplicity, we will keep using the language of “classical” atoms or ions to describe the landscape of the chemical system in the presence of currents. 257 Presumably, the current would couple to the electronic degrees of freedom, which induces a phase transition on the electronic level; however, this is primarily an electronic effect, and thus is not directly connected with the classical energy landscape as function of atom-configuration based microstates we are interested in. As mentioned earlier, including the electronic degrees of freedom explicitly in the construction of an energy landscape is not trivial, since the quantum mechanical aspects make it difficult to set up a suitable state space and moveclass. Still, nucleation-and-growth behavior has been observed in studies on orbital ordering.524 Since this implies that first order phase transitions can occur in electronic phase transitions, it appears that the corresponding quantum system exhibits a complex landscape with a multi-basin structure, and, presumably, a hierarchy of time scales controlling the equilibration processes among the microstates of the system. 258 Phase transitions in the presence of electric currents have been investigated; in these studies,519 where a rapid change in the electric conductivity is observed to be associated with the transition, the focus is on the existence and growth of nuclei of the second phase, together with the question whether hysteresis might occur in the system. The general thermodynamic formalism of steady-state energy landscapes is applicable to these systems, and the same holds for the use of electric currents in alloy processing,521,522 which take place on the steady-state extension of the energy landscape of alloy-forming systems in the presence of rather large electric currents.

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! driving the current. We note that the electric conductivity depends on the microstate X ;259 and the size of the necessary electric field ! ! is inversely proportional to the conductivity for a given j el : Thus, a constant current j el implies a contribution by the electric field ! h i 2 ! ! X Þ ðs ð! 1 ! to the energy of magnitude Eel ð X Þ ¼ 3 ð8p j el Vð X Þ: Similarly, the magnetic field associated with the electric current el X ÞÞ ! !2 ! ! X Þ B Vð X Þ:260 Since the current is assumed to be fixed, the energy contribution for the contributes an energy of Emag ð X Þ ¼ mð8p ! ! ! atom configuration X depends on the magnetic permeability mð X Þ; electric permeability 3 ð X Þ and the electric conductivity ! ! sel ð X Þ; which will vary with X ; in general. ! ! ! Turning to a constant heat current j Q ; this implies the presence of a thermal gradient in the material, j Q ¼ stherm ð X ÞVT;500 again resulting in a sequence of constant temperature slices when describing the (macroscopic) solid.261 As discussed above, each slice can be in (local) equilibrium at the temperature of the slice, and exhibit the usual locally ergodic regions on the observational time scale of interest. From an energy landscape perspective, the locally ergodic regions associated with the whole (!) solid would approximately correspond to a “union” of atom arrangements in each slice, which belong to the locally ergodic regions of the individual slices. The combined microstates of the full chemical system also contain the atoms at the interfaces, of course, in case the modifications present in adjacent slices have different structures. Several special cases can appear, when combining the locally ergodic regions for all the slices into a locally ergodic region for the full solid. For example, the locally ergodic regions of all slices might belong to the same phase (just at different temperatures), i.e., the whole solid is one homogeneous phase. Alternatively, we might find a low-temperature phase at the low temperature end and a high-temperature phase at the high-temperature end. The globally ergodic region for the complete chemical system could be one of these cases, but there might even be one or more intermediary phases at temperature (zones) in-between the applied highest and lowest temperatures on the surface, Thigh and Tlow.262 We note that, in contrast to the electromagnetic field energy associated with the electric current, no term is added to the extended energy function that would represent the thermal gradient. Diffusion of atoms (or molecules) through the material is a third class of currents that can be prescribed for a chemical system in interaction with the environment.263 For a constant particle current to be present, the same number of atoms of a given species must enter and !ð AÞ leave the system. The driving force of diffusion currents j with atoms of type A as carriers, are imbalances in the concentration n ð! rÞ A

D

of the atom species A between different regions of the material, resulting in chemical potential differences.264 As discussed in Section 3.11.5.5 a steady-state diffusion current requires a source and a sink on the surface of the material, corresponding to a high and low value of the chemical potential, respectively.265 This chemical potential difference can be explicitly included in the extended cost function. !ð AÞ The relation between the strength of the (prescribed, i.e., fixed) diffusion current j D and the concentration differences neces!ð AÞ ! ! ! ¼ Dð X ; ! r ÞVn ð X ; ! r Þ:53,500 Just as in the case sary is characterized by the diffusion constant Dð X ; ! r Þ that enters Fick’s law, j D

A

In general, the conductivities and permeabilities will be tensors, and can depend on position ! r. ! In general, the magnetic field B generated by the current I flowing through the material must be computed from Maxwell’s equations or Biot-Savart’s law,478 and will depend on the shape of the material. ! 261 As can be seen in, e.g., MD simulations,486 the temperature is approximately constant within each such slice. Furthermore, we note that stherm ð X ; ! r Þ can ! ! vary as function of position r for a given microstate X . 262 Due to the macroscopic (or at least mesoscopic) size of the slices needed to make them stable enough to exhibit LERs restricted to the slice, we must study the complete landscape consisting of NAv atoms, while a small periodic approximant would not be sufficient. As mentioned already, this might yield satisfactory results if the interfaces between neighboring slices (at different temperatures) do not destabilize the phases in each slice. 263 Other types of currents in (solid) materials are, e.g., spin-polarized currents,525 for which an analogous analysis can be performed, of course; such currents will not be discussed here. In this context, we note that while we always imply a solid material in our discussion of enforced currents, we can also consider prescribed currents in liquid materials, and investigate their energy landscapedafter all, the energy landscape of the chemical system also contains the microstates and locally ergodic regions that represent the liquid and gaseous states of the material, and, on the macroscopic level, slices with, e.g., different temperatures or composition are to be expected. Thus, the presence of an electric field or a heat current might influence the local structure of the liquid, and similarly, the melting temperature might depend on the applied fields and enforced currents. However, such local structural changes when moving from slice to slice are expected to be rather smooth, in contrast to the exciting phenomenon of a sequence of structurally very distinct modifications we can observe in solid materials. Thus, we mainly focus on the solid state in our discussion of the energy landscape of chemical systems in interaction with the environment. 264 Recall that the (local) concentration nA ð! r Þ of an atom species A is a function of the chemical potential mA ð! r Þ; for gas-like systems, this relationship is often ! an exponential function, mA ð r Þ
kB Tln ðnA ð! r ÞÞ:53,500 265 Note the analogy to ion conduction, where the electric field, i.e., a difference in electric potentials drives the current instead of a difference in chemical potentials. In this context, we note that one frequently incorporates the electric potential into the chemical potential. This so-called “electrochemical potential” is useful for analyzing the diffusion of ions in both an electric field and concentration gradient500; this connection is also reflected in the Onsager relations53,500 mentioned in the context of the coupling of thermal and electric currents where again the same carriers (electrons, ions) are responsible for both currents, at least partly. 259 260

Energy landscapes in inorganic chemistry

325

! ! of the electric or heat conductivity, computing Dð X ; ! r Þ for a given microstate X will involve simulations (or models) on the material point level. Again, there can appear an induced heat current and heat production, due to the ability of the atoms to transport not only “atom identity” but also heat, and due to the existence of friction effects on the mesoscopic level. But there is no new energy density term associated with the chemical potential for a stationary state with a constant diffusion current, in contrast to the energy of the magnetic field associated with the electric current. In conclusion of this section, we remark that a full integration of the environment into the energy landscape approach is a complex task with many open questions and challenges, even if one only aims for a minimalistic implementation guided by equilibrium and non-equilibrium thermodynamics. Since the mesoscopic properties of the material such as permeabilities, conductiv! ities, etc. are usually defined on the continuum level, computing them for a given atom arrangement X is quite involved, even if we can use periodic approximants and linear response theory. Thus, including enforced constant currents into the extended energy landscape of a solid is not easy in practice. Nevertheless, the underlying concept is consistent and allows us to generalize the energy landscape description of a chemical system in interaction with the environment to include steady-state situations with constant currents flowing through the system, to incorporate the presence of static external elastic and electromagnetic fields, and to take into account the sources and sinks of atoms that change the composition of the chemical system.

3.11.6

General methods to explore and classify cost function and energy landscapes

In this section, we will give a short overview over some of the most popular and generic methods that have been developed to study cost function and energy landscapes and to identify the most important features of such landscapes, keeping an eye on the special aspects of those computational methods that are specific to chemical systems. The largest number of such global and local exploration methods have been developed to identify local minima of the landscape,266 followed by the search for saddle points, the identification and analysis of landscape pockets, and the measurement of the local and global densities of states. More specific for chemical systems are the search for and analysis of transition paths, the identification of locally ergodic regions, the computation of local and global free energies leading to the construction of phase diagrams, and optimal control methods for chemical processes. Furthermore, the natural moveclass might need to be preserved during the global exploration, if we want to find metastable compoundsdin addition to the global minimum which corresponds to the thermodynamically stable compound at T ¼ 0dor identify the true energy and entropy barriers or compute the rate constants for various transitions between locally ergodic regions. Thus we might need special algorithms tailor-made for chemical systems or special adaptations of generic exploration algorithms to the case of chemical systems.267 In general, such exploration and optimization algorithms can be divided into local and global approaches.535,536 The former explore only a limited region of the landscape, in most cases by following some downhill route from a starting point, locating one or more local minima in the process, or sampling the state space near such a minimum. On the other hand, the latter procedures can, in principle, locate the global minimum, elucidate the global and local barrier landscape, and measure the global and local densities of states. In this context, we note that many algorithms are multi-purpose ones, i.e., they can be employed for several tasks, such as finding local minima and/or transition paths, and/or compute densities of states, and/or calculate free energy differences. Since the number of local minima of energy landscapes of chemical systems usually grows exponentially, if not factorially, with the number of atoms in the system123 and these minima are located on a complex barrier landscape such that the number of saddle points and possible transition routes among the minima increases at the same or an even faster pace, the use of global methods is necessary for their identification and analysis. However, a typical set of global optimization runs for the determination of the local minima in a chemical system, or of global explorations for the identification of barriers and characteristic regions, and for the measurement of probability flows, involve millions or even billions of energy evaluations and/or gradient minimizations, each of which can be very costly in chemical systems. Even with modern computers, this tends to be very time-consuming, and therefore multi-stage procedures are very common21,89,101,102,537: First, a global search on an empirical, or simplified ab initio, energy or cost function landscape identifies local minima and/or saddle points, etc. In a second step, these are classified, and finally all, or only the most promising ones, are locally optimized on the full quantum mechanical level.111,113,386 In this process, many different levels of approximation for the description of the energy landscape are applied: these concern the choice of energy function (e.g., ab initio energies,418,419,538,539 empirical interaction potentials between atoms, or energy functions that describe the interactions between whole groups of atoms109,540), the way the microstates are defined (e.g., based on single atoms, groups of atoms,109 or nodes in bond networks541), and the moveclass according to which the walker moves in state space during the global exploration and optimization.

266 Note that for many tasks in chemistry, identifying low-lying states, in particular local energy minima, might be completely sufficient, without knowledge of the stabilizing barriers being required, since other information about the systemdoften based on chemical intuition or experimental datadassert the existence of the structure that is associated with the local minimum. This will surely be true when global optimization methods are used to assist structure determination, since here additional criteria like powder diffraction data exist, which are used to judge the quality of the structure candidate found by the global optimization. 267 For example, there exist a large number of suggestions how to modify the energy landscape in order to simplify the barrier structure for optimization purposes, or for identifying transition paths. Such modifications can involve introducing penalty terms or filling of basins,526 increasing the number of degrees of freedom or dimensions,527–529 or creating a smooth mean-field like energy landscape that serves as a guiding surface for the optimization algorithm.530–533 Other procedures that have been employed are the diffusion equation method (DEM) and the distance scaling method (DSM),534 just to name a few. Since we are usually interested in identifying all low-lying minima because they constitute candidates for metastable phases in the chemical system, one must be careful when using such acceleration schemes, to avoid eliminating chemically and physically relevant minima.

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Quite generally, we note that these global and local exploration methods fall into several categories, depending on the type of exploration moves employed. For one, there are stochastic explorations employing single walkers, interacting groups of random walkers, and ensembles of walkers.268 Next, we can use deterministic exploration methods, again applied to single or many walkers, and finally, there are multiple combinations of the two types of procedures, e.g., global stochastic and/or deterministic explorations are combined with deterministic and/or stochastic local minimizations or explorations. Turning to the stochastic landscape explorations,542 we note that the basic stochastic move of a walker from its current state i to a neighbor state j is a two step process: first the neighbor state is selected based on the moveclass of the energy landscape, and then one decides according to some criterion, whether the move attempt is accepted or whether the random walker should stay at its current position. Typically, the acceptance rule involves the energies or costs of the current state Ei and the neighbor state Ej, but many other criteria can be employed, of course.269 For chemical systems, the Metropolis criterion466 already mentioned in the context of the master equation dynamics (c.f. Section 3.11.4.2.1) is the most common acceptance criterion, where the move is accepted if the ratio of the Boltzmann factors of the two states is larger than a random number r chosen uniformly from the interval [0,1], exp.((Ej  Ei)/kBT) > r. Here, T is the temperature of the systems during the Monte Carlo random walk and controls the acceptance rate; for general systems or cost functions we use c ¼ kBT as a generic control parameter (without any physical meaning),270 and it can be shown that in the infinite time limit this random walk visits all states in the system according to the Boltzmann probability distribution at temperature T.271 Clearly, both the moveclass and the acceptance criterion play a very important role in the efficiency of the search or exploration algorithm, since they allow us to focus on those regions of the landscape, which are most relevant for the statistical evaluation of the system’s properties via the MC simulation.272 Thus, one often speaks of “importance sampling” when employing Monte Carlo simulations.458 In contrast, if the walker follows a unique trajectory that is completely determined by a strategy that does not involve any random elements, we speak of a deterministic exploration or simulation. In classical chemical or physical systems described by a continuous landscape, a deterministic walker moves on the energy landscape according to the equations of motion based on the laws of physics, such as Newton’s equations in a molecular dynamics simulation. In many hybrid approaches, stochastic and deterministic elements are combined, but one can also use combinations of, e.g., different stochastic procedures combining different moveclasses and acceptance criteria for the use in different stages of the exploration. The most common version of hybrid methods involve stochastic global explorations, where after a certain number of steps by the random walker, a deterministic local minimization is performed. In the extreme case, each move to a neighbor state according to the stochastic moveclass is followed by such a local minimization, such that the walker effectively performs a “walk on local minima”. Conversely, one finds examples of molecular dynamics simulations, where every once in a while adusually largedMonte Carlo jump move is inserted, in order to accelerate the movement of the walker, e.g., out of a local minimum basin. Of course, all the algorithms mentioned have been implemented in their own special fashion by the many research groups using them. Most of the corresponding computer programs or scripts are specialized to one method only, although a few contain several algorithms as sub-modules.273 Furthermore, several of the codes or scripts can interface to external programs that provide, e.g., energy calculations on ab initio level or with complex empirical potentials, or sophisticated local minimization techniques.274 Quite generally, the number of algorithms and codes dealing with the exploration of energy landscapes has steadily increased over the past decades. Many of the algorithms found in the literature are variations or combinations of the basic algorithms and

268 In an ensemble of walkers, moves take place on the ensemble level, i.e., not only the choice of feasible neighbor states involves two or more walkers simultaneously but also the acceptance criterion acts on the ensemble level. In contrast, for groups of walkers, the walkers interact, e.g., by repelling each other, but each walker’s trial moves are generated and accepted on an individual basis. However, in the literature, the term ensemble is commonly used to describe both types of multi-walker approaches. 269 For example, for an ensemble of random walkers, we might want to ensure that they stay close together or are widely spread out over the landscape, in order to provide us with a diverse sample of, e.g., the local minima. In that case, the instantaneous distance in configuration space to the other walkers can be one of the criteria for accepting of rejecting the attempted move. 270 If only downhill moves are accepted (c ¼ kBT ¼ 0), we speak of a stochastic quench, in analogy to the quenching of a material to a very low temperature. 271 For this to be true, the moveclass must be microscopically reversible, i.e., if state A is a neighbor of state B, then B is also a neighbor of state A, and it must fulfill the detailed balance condition (c.f. Section 3.11.4.2.1). 272 Other acceptance criteria are based on power law distributions instead of the Boltzmann distribution,543–545 or use a simple acceptance threshold L(c) that varies according to the control parameter c, i.e., the move is accepted if Ej Ei < L(c).546 However, for such acceptance rules, the time averages of observables along the trajectories are no longer equal to the statistical mechanical averages. Note that, mathematically speaking, the system might still be ergodic with respect to, e.g., the power law distribution, according to which the algorithm has performed the sampling. However, it is not ergodic with respect to the Boltzmann distribution that is exhibited by physical and chemical systems in thermodynamic (statistical mechanical) equilibrium. 273 Some examples of codes specializing in (global) energy landscape explorations are: the G42 þ code119 that contains several modules implementing a number of global and local exploration algorithms in various degrees of sophistication, such as MC simulations, simulated annealing,547 multi-walker annealing, evolutionary search,548 thermal cycling,549 basin hopping,550 threshold explorations,296 ergodicity search,187 parallel tempering,551 or prescribed path explorations,307 for infinite (using periodic approximants) and finite systems in one-, two- and three-dimensions, where the energy is computed either with via various built-in empirical potentials or through interfaces to a variety of ab initio codes (e.g., CRYSTAL,415 Quantum Espresso,416 Gaussian,417 or deMonNano424) and to several codes using empirical energy functions such as GULP430,552 or AMBER431; the GMIN-code553 that implements a variety of algorithms for, e.g., global and local searches for minima and saddle points and the computation of thermodynamic properties, which is especially suited for clusters and molecules; the GULP-program430,552 that incorporates both force-based methods and genetic algorithms554 and contains many types of empirical potentials, and can be applied for bulk materials and clusters; the KLMC (“Knowledge Led Master Code”) program141 that is applicable to clusters and periodic systems in different dimensions, and contains several global exploration methods, various analysis tools, and interfaces to, e.g., GULP, FHI-AIMS,555 VASP,412 and NWCHEM.556 274 The reader is referred to the individual publications for more details and contact information regarding the authors of the various programs.

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approaches shortly described in this section, and we refer to the general literature for further details and recent refinements. Similarly, providing information about specific features of the algorithms and their practical implementation that are associated with making the codes work efficiently and reliably, goes beyond the purview of this chapter.

3.11.6.1

Global optimization techniques

Since the availability of computers, a plethora of numerical methods for finding the extrema of a cost function have been developed, many of which have been adapted and applied to the solution of discrete and continuous optimization problems in chemistry. Hence, in this subsection, we can only give a short overview over some of the most common approaches employed in chemistry.275 A quick scan of the literature will show that these are just the tip of the proverbial iceberg; however, many of the algorithms found in the literature tend to be variants or combinations of the basic versions presented in this subsection. While producing results of similar quality in many casesdc.f. the “First Blind Test for Crystal Structure Prediction”557deach of the algorithms mentioned below has its own strengths and weaknesses one needs to take into account when exploring a landscape.276 A similar outcome was also observed in a large but unpublished systematic comparison study for cluster (KMgF3 and KZnF3) and bulk crystal (Zn2OS) structure prediction with up to 40 atoms per simulation cell, where simulated annealing, multi-quench, multiple-starting-point quenches, basin hopping, thermal cycling, multi-walker-interacting simulated annealing, and genetic evolutionary optimization methods yielded results of similar quality,559 but where, as in the “Blind Test” mentioned above, the multiple-starting-point quenches again noticeably fell behind once the number of atoms in the system increased. Fig. 7 shows the deviation of the average minimum energies found for a given computational effort and size of the system at the example of (KMgF3)n clusters for three of the exploration methods. Regarding the quality of such a global optimization algorithm, we are most concerned about the efficiency of the search.277 For stochastic optimizations, one usually tries to optimize the moveclass and to fine-tune those parameters that are used to control or modify the acceptance criterion as function of time, i.e., the progress of the algorithm, sometimes called the temperature program or schedule of the algorithm. Based on thermodynamic considerations, various such programs or self-adapting schedules have been proposed.560–564 However, designing an efficient moveclass without much detailed a-priori knowledge about the landscape is still more an art than a science.

3.11.6.1.1

Exhaustive methods

So-called exhaustive searches belong to the most basic global optimization methods. Here, one explores the energy landscape in a deterministic fashion that can be proven to lead to the global minimum. Since these methods are not stochastic, they usually require that the energy landscape is either discrete or can be easily discretized or divided into sub-regions, e.g., via fast local minimizations.278 The most simple approach is a straightforward enumeration of the whole state space, but more efficient approaches that systematically exclude sub-regions of the state space of feasible solutions can be quite effective.279 Examples are the branchand-bound methods323,565 and branch- and-cut algorithms.536 In this context, we also mention the lid algorithm82,100,274 that completely enumerates all states within pockets below energy lids on the landscape (c.f. Section 3.11.2.1.2), including all the local minima, of course.280 However, the size of the state space of a chemical system is usually very challenging for all exhaustive methods, because even for discrete approximations of the state space, the number of microstates grows factorially with the number of atoms/simulation cell.

3.11.6.1.2

Multiple local minimization methods

Perhaps the easiest methods to code and fine-tune are multiple-quench/minimization approaches. Here, one repeats a local minimization proceduredperhaps available in pre-packaged formdfor a very large number of starting points. The philosophy behind this approach is that by using a good local minimization algorithm based on the first and second derivatives of the energy535 and many different starting points should produce all important local minima, including the global one. This approach is most suitable for landscapes with manydbut still extremely few compared to the number of microstatesdrather widely separated minima, since 275 Keep in mind that, in chemistry, we are not only interested in the global minimumdall minima with low energies and sufficiently high barriers surrounding them are of importance since they represent potentially valuable metastable compounds. 276 The test involved three of the most widely and basic global optimization approaches: a simulated annealing algorithm, an evolutionary algorithm, and a multi-starting point local minimization algorithm. We note that, subsequently, structures with even lower energies were found, using the thermal cycling approach.558 Furthermore, in this blind test, simulated annealing and the genetic algorithm yielded comparable results, while the truly unbiased random starting point minimization methods proved to be the weakest ones in all test cases, as one would expect from general considerations for global optimizations of highly complex cost function landscapes. 277 Another aspect are the undesired side-effects of simplifications and modifications of the energy landscape, which are often introduced to accelerate the search procedure, or make it feasible in the first place. 278 In contrast, stochastic methods that achieve exhaustive enumerations of, e.g., the local minima on the landscape are either not as efficient or cannot guarantee a successful listing of all minima. To achieve a success probability that is extremely close to one, in order to guarantee the identification of the global minimum (or achieve some other task), stochastic exhaustive algorithms typically require a massive overkill in the number of steps, i.e., numerical operations, compared to even a simple enumeration procedure. 279 One should keep in mind, however, that these methods are usually fine-tuned to identify the global minimum; thus they need to be run in a recursive fashion, if one is interested in other local minima, too. 280 As happens a lot in this field, this algorithm has been re-invented several times.566,567

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then local optimizations from widely dispersed starting points will deliver with a high probability all minima, including the global one, for a reasonable computational effort. However, in most instances, the landscape possesses extremely many minima, and local optimizations tend to get stuck in uninteresting high-lying minima. In that case, multiple local optimizations are not very effective, unless one has an efficient way of choosing starting points for the type of system of interest, as discussed below.281 Ideally, the choice of starting points (approximately) exhaustively covers the state space; in practice, their selection is either systematic in some fashion568 or at random.569,570 However, if we have some prior information about the system, such as expected structural properties of the minima, we can be guided by chemical intuition,571 such as generating network models or dense packings of spheres,25,572–576 or we extract structures from databases140,577–582.282 A close alternative is to run long stochastic or molecular dynamics simulations, and then perform stochastic quenches or gradient minimizations from various stopping points along the trajectory.584,585 However, sufficiently long trajectories that can guarantee us to visit all important minima (regions) of the landscape are often similarly time-consuming as a systematic (quasi-exhaustive) generation of starting points. Thus, these trajectories have to be not only very long but also need to employ various kinds of accelerated dynamics procedures (c.f. Section 3.11.6.1.5) to avoid getting stuck in uninteresting (or already explored) regions of state space.

281 When judging the outcome of multiple-quench procedure based searches presented in the literature, one needs to keep in mind that a random generation of starting points is frequently asserted in the outline of the paper, implying a selection from the full configuration space of the chemical system. Yet in practice the set of configurations from which the starting points have been selected at random had already been massively reduced beforehand compared to the full configuration space of the system by some preselection based on “chemical intuition” or other criteria. As a consequence, most of the high-lying minima that usually greatly reduce the efficiency of the multiple-quench methods have effectively been eliminated before the select-and-minimize process started. 282 The main problem with relying on a database to provide structure candidates is the fact that usually many energetically very low-lying minimum structures exist on energy landscapes, which have no representatives in the databases. Regarding the topological network approach, it is clear that, in principle, any atom configuration can be “drawn” as a network of atoms with connections between neighbor atoms. Thus, in principle, one could construct all possible atom-networks, as long as one is willing to include a gigantic number of different types of bond-connectivities or weird coordination polyhedra when generating the topological network. But since we have no way to distinguish nonsensical configurations just based on the topological conditions (recall: the topology had nothing to do with atom-atom distance; only when we embed the network into R3 do we establish distances between atoms), the number of hypothetical candidates would grow exponentially fast with the number of atoms/cell.583 Since we need to check all candidates by local minimizations, this would also overwhelm the computational resources, even if we use very simple fast potentials for the first checking stage, to screen out the absolutely crazy modifications. We realize that taken to the extreme, this approach would be equivalent to scanning the configuration space more or less densely; we note that this quasi-exhaustive method has been employed for the prediction of possible structures of molecular crystals consisting of (rigid) unbreakable molecules, with very few molecules in the periodic approximant.568

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Stochastic (Monte Carlo) and molecular dynamics based methods

One might assume that performing very long Monte-Carlo or molecular dynamics simulations at constant temperature would be an obvious way to identify all minima of interest, because every possible minimum should be encountered according to its Boltzmann probability. But unless we know where to searchdsimilar to the case of the random starting point problem in the multi-quench procedure discussed above (c.f. Section 3.11.6.1.2)d, and thus start the runs in those regions, one spends most of the computing time in high-energy regions of the landscape. For typical landscapes belonging to complex chemical systems such as large molecules (clusters, proteins, etc.), glassy systems, or non-trivial crystalline solids, even performing frequent local minimizations is not effective enough. To concentrate the ensemble one samples during a MC-walk or (equilibrium) molecular dynamics simulation into the region of interest on the landscape at low energies,283 one could try to lower the temperature, although one then faces much higher effective barriers impeding the movement between minima basins. Nevertheless, the expectation that for an optimal decrease of the control temperature T (e.g., one could try to stay close to equilibrium), the random walk will reach the global minimum, has inspired the so-called (stochastic) simulated annealing algorithm151,193,547,587.284 Over the past decades, this method has proven to be quite good in finding low-lying minima for all kind of cost functions, where no prior information about the structure of the landscape had been available.285 As mentioned before, stochastic global optimizations can employ large changes (jump moves) in the atom configuration during each move, to quickly access distant regions of the landscape. One efficient variant where every large move is followed by a local minimization is often called basin hopping.550,590–592 There also exist extensions of basin hopping,159 where not the energy but the free energy around a local minimum (computed from the Hessian in the harmonic approximation) serves as the cost function, for a given external temperature. Here, the goal is to find the local minimum with the largest free energy “on the fly”, and not via an additional analysis step after all the local minima have been identified, thus hopefully avoiding the exploration of regions of the landscape with unfavorable free energies. Besides using an optimal moveclass, we can adjust other features of simulated annealing to raise its efficiency.193,593 These include adaptive modifications of the general type of acceptance criterion of the random walk mentioned above,466,543,545,546 and the temperature schedule T(n), as function of the number n of moves (for a MC walker) or simulation time windows (for a MD walker). For most schedules T(n) decreases monotonically with n, but some algorithms alternate between high and low temperatures, e.g., temperature cycling,549,558,594 and there exist adaptive schedules563,564 using feedback from properties of the already explored regions of the landscape.286 In order to efficiently exploit information about the landscape obtained while the algorithm is running, one often uses multiwalker implementations.593,597,598 A multitude of such approaches has been developed, e.g., the Demon-algorithm,599 or averaged landscape methods531–533,600 such as conformation-family Monte Carlo,601 superposition state molecular dynamics,602 SWARM molecular dynamics,603 or particle swarm optimization.604,605 Other approaches employ ensembles at several temperatures (and pressures) such as multi-overlap dynamics,606–608 parallel tempering,551 and J-walking609 where different walkers run at different temperatures and periodically switch microstates (or temperatures), in order to overcome barriers more efficiently. Another approach is the so-called multi-canonical (ensemble) algorithm,610,611 where the overlap among the probability distributions at many different temperatures is employed for a more homogeneous exploration of the whole landscape.287

283 We note that as long as we employ a physically relevant moveclass for the stochastic walkers, it is usually permitted to replace the MC-simulation of a chemical system by a MD- simulation of “equivalent length” at constant temperature, and conversely, for large enough observation times; here the equivalency actually defines the length of the MD-simulation needed. We know that MD is better suited to reproduce the actual time-evolution of a configuration, while MC can access larger regions in configuration space within a limited computation time due to the availability of non-physical moves. Depending on the problem at hand, we usually pick either one of the two methods, but there are also studies and algorithms, where MC- and MD-stages alternate during the simulation.586 284 Note that it can be proved that a logarithmically slow schedule T(t) f A/(1 þ ln t) guarantees that one will find the global minimum with probability approximating one.588 Of course, this is too slow in practice, but it provides a limiting temperature schedule. 285 In contrast to molecular dynamics or gradient based approaches, simple stochastic methods that rely only on energy evaluations and not on derivatives are very powerful as a first search method to use for an unknown system, since they can be applied to both discrete and continuous landscapes, and they are compatible with moveclasses that include major discontinuous rearrangements of the atom configurations resulting in large jumps in state space. However, since it is usually advantageous to exploit all easily accessible information available about the system, combining the stochastic methods with periodically performing local minimizations that employ first (and second) derivative information about a continuous landscape often proves to increase their efficiency.557,589 286 Based on general optimality considerations in finite time thermodynamics,595 the so-called constant thermodynamic speed schedule has been proposed as an optimal temperature programdat least for rather slow ensembles of walkers that remain relatively close to (local) thermodynamic equilibrium. Using some assumptions about the relaxation behavior of the system to equilibrium, one finds that the decrease in temperature should be inversely proportional to the ensemble specific heat of the system, i.e., to the energy fluctuations.563,564,596 287 Finally, another ensemble-based exploration approach is the extremal optimization method,612–614 which produces a very wide spread in the energy distribution of the local minima.

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3.11.6.1.4

Genetic and evolutionary algorithms

Another group of multi-walker global optimization methods that describe a deterministic or stochastic evolution of an interacting ensemble of copies of the chemical system, are the so-called genetic algorithms,554,615–621 which are inspired by biological evolution.288 Over the past two decades, genetic or evolutionary algorithms have become very popular in chemistry.135,137,538,548,552,622–630 These interacting walkers not only exchange information about the regions of the landscape exploreddas is the case in the group-of-walker algorithmsdbut also generate new configurations on the ensemble level; commonly, these sets of walkers are called generations consisting of parents and their children. The moveclass includes both the standard changes of individual walkers (often called mutations) and ensemble moves where a new walker (microstate) can be generated by choosing some of the atom coordinates from the two (or more) parent microstates; the latter moves are derived by imitating the mixing of parental genes in the context of biology and evolution (frequently called crossover moves). Whether a new walker is accepted into the ensemble, replacing one of the parent walkers in the process, is then decided according to some selection scheme. Usually, this decision consists of considering the whole enlarged ensemble (all children and all parents of the same generation), i.e., both the child and its parents can become part of the new generation.289 The availability of potentially very effective moves and selection criteria involving many walkers at the same time, is the basis of this method. Important for the success are the proper choice of the ensemble, the moveclass, the acceptance/survival criterion, and the encoding of the configurations, such that moves involving several configurations can be effective.290 In order to accelerate and fine-tune this reduction process, a preparatory step, where one assigns a fitness value to each walker according to some rule, is often inserted before the actual selection takes place. Examples of fitnesses are the ratio of the (potential) energy of the atom configuration to the average energy of the population,554 the rank of the walker by energy instead of its actual energy,631 or the outcome of a so-called stochastic universal sampling.632 Some variations use the rank assigned via some sampling method such as a tournament633 with pair-wise comparisons of (randomly) chosen members of the enlarged ensemble, or employ a “greedy” version where only those walkers with the lowest energies survive, use probabilistic survival criteria analogous to the Metropolis criterion, or allow (some) new configurations to survive for at least one generation.291 In particular for chemical systems, one often employs hybrid moves, i.e., one performs a local minimization for every child walker before an (energy based) fitness is assigned,619,635 because without such a minimization essentially all cross-over moves would be rejected as energetically unfavorable.548,636 An important technical aspect is the way the members of the ensemble are encoded for the purpose of cross-over moves, i.e., either as a genotype637 corresponding to a discrete encoding of an atom configuration, or as a phenotype548,638,639 where we employ the actual atom configuration.

3.11.6.1.5

Jump methods, taboo methods, accelerated dynamics

3.11.6.1.5.1 Jump methods and accelerated dynamics Another class of optimizations stress the importance of taking large steps, or, equivalently, taking many small steps with high acceptance rates, on the energy landscapes while avoiding the complete randomness of the multiple starting point quench approach. They can be considered extreme variants of annealing schedules, or basin hopping algorithms. Common ways to achieve “jumps” on the landscape are to design special moves (e.g., the so-called “snake-step” in polymer simulations640), to intersperse local minimizations between large Monte Carlo steps,611 or to massively raise the acceptance rate by, e.g., a sudden re-heating of the systemdexamples are the bounce algorithm,641 the thermal cycling mentioned above,549 or sequential multi-quench642dsuch that the walker can cover much larger distances within a given number of MC-steps or MD simulation time. Similar local heat pulses are also used in the context of evolutionary algorithms630 discussed above (c.f. Section 3.11.6.1.4). We note that many of the methods that use such activation-relaxation steps643 have been independently re-invented or seen a new life as slightly modified versions of the original algorithms. To keep track of the changes, they are usually given a new name, leading to much confusion for the non-specialist. Thus, here, we only list some of the most popular approaches. Many methods that are conceptually quite closely related to explicit jump move algorithms effectively modify the energy landscape,156,526,534,644–650 for example, via adjusting the energy function by locally elevating visited areas,526 lowering barriers relative to the local minima,646 or via changes in the moveclass such as in stochastic tunneling613,647,651,652 or dynamic-lattice searching.653 While MC based methods can work to a large extent by changing only the moveclass and the acceptance criterion, landscape modifications via changes of the energy function are more relevant for accelerated molecular dynamics runs. Finally, we note that such landscape variation based methods are always adaptive, in some fashion, to the progress of the global (or local) exploration.

288 In biological terminology, one would speak of the system evolving towards larger fitness values. In the context of general global optimization, the fitness would just be the negative of the cost; however, in biology, fitness corresponds to the likelihood to produce surviving offspring, and has no connection to an energy function. 289 On a more abstract level, one can actually consider the ensemble of walkers itself as a microstate of a multi-walker configuration space. Each generation of a genetic algorithm search would then correspond to one such multi-walker microstate, and we can analyze progress of the algorithm as a stochastic (or deterministic) walk in the multi-walker state space. 290 Since most of the ensemble moves have no physical equivalents, this procedure can have difficulties identifying other physically relevant minima besides the global minimum, or estimating the stability of local minima. 291 Frequently, some of the newly generated walkers or some of the parent configurations are kept in the ensemble, even if their energies are not favorable, in order to maintain a high diversity of the ensemble.634 Such effectively multi-generational selection strategies are common in some classes of evolutionary optimizations.616,617

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3.11.6.1.5.2 Taboo searches A different category of global optimization methods are the so-called taboo searches.654–656 Here, regions of the landscape that have already been visited during the MC/MD simulations are not allowed to be entered or visited again. In some way, this class of methods derives from exhaustive searches, where one keeps track of the states already visited. For example, one can use rigid exclusion constraints via penalty terms,526 to exclude regions already seen by the walker. In the original version, tabu-search654 proceeds by selecting always the state with the lowest energy among all neighbors of the current microstate as the one to move to.292 Here, all the states that have already been encountered are kept in (computer) memory and are (if possible) effectively removed from state space, similar to a branch-and-cut move; else, they need to be individually “memorized” and associated with a penalty or a constraint flag. As a consequence, all moves that would lead to the excluded microstates are removed from the set of feasible moves (tabu), allowing the algorithm to climb out of local minima since the low-energy states are no longer accessible. A crucial limitation of any taboo search is the size of the set of states already visited that can be kept in the available computer memory.293 This computational limitation also applied to the use of penalty terms, which accumulate over time along the trajectory.294 Combining a taboo search with quenches and large moves can alleviate the memory problem to a certain extent by using a basin hopping-like scheme of large moves, and only keeping track of the minima.657 Taboo search elements are frequently included in more complex search algorithms; an example would be the metadynamics approach.658,659

3.11.6.1.6

Lid methods

Another class of global optimization schemes are based on using a threshold or a lid, to restrict the region being explored by the algorithm. The walker in the Monte Carlo or molecular dynamics simulation is not permitted to cross into those parts of the landscape where the cost or (potential) energy exceeds a prescribed value (the lid or threshold). For a random walker that stays below the lid, every attempted move is accepted (as if we were running a MC simulation at infinite temperature), while for a MD simulation one uses a high temperature and if the trajectory crosses the potential energy threshold, one resets the walker to an earlier microstate along the trajectory and reassigns new velocities to the atoms.273 Probably the earliest such optimization method is the deluge algorithm,660 where the lid seen by the random walker that must not be crossed during the random walk, is slowly lowered from very high lid values, until the walker gets stuck in a deep minimum.295 Another example is the threshold algorithm,273,295,296 which had originally been developed to realize the pocket search in the lid algorithm274 on continuous landscapes. The MC or MD walker is allowed to move below the threshold for a sequence of energy lids. Besides sampling the local densities of states, one checks whether favorable new local minima have been accessed by periodically performing one or more quench runs from stopping points along the trajectories. While the original threshold algorithm employs only a physical moveclass without jump moves for the random walker, in order to be able to determine the physically relevant barrier structure of the landscape, for the purpose of global optimizations, other variants have been developed such as the threshold-minimization algorithm for the investigation of flexible molecules661 and the threshold genetic algorithm.662

3.11.6.1.7

Rapidly exploring random tree (RRT) based methods

Of special interest in recent years has been a new class of search methods originating from robotics, the so-called rapidly growing (exploring) random tree methods663 that have been developed to deal with the so-called (robot) motion planning problem.664 Relying on the analogy between problems in robotics and structural biology, methods originally developed to structure and optimize robot motions have been extended and applied to the simulation of molecular systems, and the generation of “space-filling” tree structures as mentioned earlier (c.f. Section 3.11.2.2.2). In particular, computationally efficient methods have been developed for sampling and exploring the energy landscape of macromolecules, in particular biomolecules (c.f.337 for a survey). Algorithms have been proposed to study the landscape of flexible segments in proteins, e.g., protein loops,665–668 and to simulate their motions.335,669,670 This involved efficiently crossing the barriers between different regions of the energy landscape. Due to their ability to quickly sample the basins and barrier structure of a chain molecule, robotics-inspired algorithms are also suitable, in combination with various local optimization tools, for the global optimization of highly flexible biomolecules.336,338,671 Here, we also mention the related so-called IGLOO algorithm, which has been successfully employed to find minimum structures on the energy landscape of molecules on surfaces.164,221

Both spellings, “taboo” and “tabu” are common in the literature. In practice, one starts removing the oldest configurations from the list, hoping that one has moved far enough on the landscape such that the microstates removed are no longer among the neighbors of the current/recent set of configurations. Due to this need to keep an enormous list of all the microstates already accessed, taboo-searches are more appropriate for discrete configuration spaces. Similarly, the need to keep all already visited configurations in memory is also the (computationally) limiting constraint on the (exhaustive) lid algorithm274 for finding and studying pockets on metagraphs mentioned earlier; in fact, such lid based explorations usually were stopped when the computer memory was exhausted. Thus minimal memory space encoding of the microstates of the system was a major technical feature of the lid algorithm that needed to be adjusted when a new system was to be studied. Analogously, one would make sure to have an efficient conversion routine between the microstate in the minimal computer representation (in binary if possible) and the actual atom configuration in double precision variables for chemical systems, both for tabu searches and lid-based explorations. 294 Often, one attempts to employ order parameter-like expressions to classify the forbidden regions, but this can lead to problematic restrictions on the way the state space is explored and/or requires a priori knowledge about the landscape. 295 In principle, the deluge algorithm can also be run using MD simulations, using the procedure developed for the MD-threshold algorithm.273 292 293

332 3.11.6.2

Energy landscapes in inorganic chemistry Saddle point techniques

Saddle points, their energy and their width play an important role in the study of energy landscapes and their dynamics.153,309,672–675 However, identifying saddle points requires considerably more complex (and thus somewhat less reliable) search methods than those employed to determine local minima. Intuitively, one can try to identify saddle points by finding all critical points with a mixture of positive and negative eigenvalues of the Hessian. One way would be to find all the minima (with |V E|2 ¼ 0) on the landscape of the help function |VE|2( 0) and then checking which among these candidates are actual saddle points of the original energy surface676.296 As an alternative, one tries to identify the “flat” regions on the landscape that can serve as candidates for saddles, using the slowest slides procedure and closely related methods.123,677–679 At any given point on the energy hyper-surface, one identifies the negative eigenvalues of the Hessian, and then proceeds along the slowest downhill eigenvector direction until only one negative eigenvalue remains. We now move in the opposite direction of this eigenvector up to the saddle. Once the saddle point has been found, we can easily reach the minima (or perhaps other saddle points) that are connected by the saddle we have identified; a slightly related method is the conjugate peak refinement approach680.297 Instead of trying to reach flat regions by starting from a generic microstate of the landscape, we can use the already known local minima as starting points of a search for the nearest or lowest saddle to or between the minima, respectively. One way to do so is the so-called “eigenvector following”681–685: Starting from a local minimum, one moves uphill in the direction of the eigenvector that belongs to the smallest eigenvalue of the local minimum, until a negative eigenvalue is found in the Hessian. Now one proceeds as in the final step of the slowest slides method and follows this direction uphill to the saddle point.298 Another contentious issue is whether one reaches really the lowest accessible saddle points, and thus other methods are often employed, especially if information about expected minima pairs is available. If two minima are known to be in close proximity, one can use (nudged) elastic band methods.687–689 Here, we begin with a more or less educated guess about the path in configuration space between the minima.690 This (usually very short) path in configuration space is then “projected” on the landscape, and we perform many local minimizations along this path (usually starting out orthogonally to the tangent of the path) while not allowing the path to break. This procedure may have to be repeated many times until the path has stabilized and consists of a continuous sequence of partial minima with respect to the directions orthogonal to the tangent along the path. The maximum along the final path is the saddle point. Choosing the appropriate initial path is clearly a major issue, in particular for paths between minima that are separated by several saddle points, with possible side-minima that might need to be avoided or crossed along the “optimal” trajectory.299 As an alternative, one can try to deform the landscape to make the saddle points “stick out” more,691 in a way the opposite procedure to the smoothing methods used to simplify the landscape during global optimizations.300 Finally, we mention a branch-and-bound algorithm (called aBB) for saddle point identification,695 which is inspired by the exhaustive optimization methods,301 and the use of machine learning to assist in the determination of saddle points where the trained neural network is used to efficiently compute forces and energies (compared to the full ab initio energy calculations).696 Once both the minima and the saddle points are known, we can construct a network consisting of (all) the local minima and saddle points.103 Besides the energy barriers at the saddle point, which are given by the energies of the saddle and those of the connected minima, we can also estimate the entropic barrier at the saddle point itself from the positive eigenvalues (corresponding to the curvatures in the uphill directions orthogonal to the reaction coordinate across the saddle) at the saddle point,302 and thus add some entropic barrier term to the energetic barrier represented by the saddle point.698 Much work has been devoted to the study of the dynamics across individual saddles done in this direction,290,698 and also to the investigation of the dynamics based on a minima þ saddle point network picture of the landscape103,318,325,699.303

Note that not every point that fulfills the requirement V|V E|2 ¼ 0 is even a critical point of the energy landscape, not to mention a saddle point. We note that when the steps of sliding down and of following the negative curvature upwards need to alternate, complications can arise. 298 Since the definition of the eigenvector following trajectory is not necessarily unique (in contrast to the downhill movement along a gradient)despecially since it needs to be constructed numerically -, finding the intuitively optimal path to the nearest saddle and the supposedly closest local neighbor minimum is a non-trivial task. The paths will depend on the way they are constructed (based on gradients, second and higher derivative approximations, eigendirections in the harmonic approximation, etc.)das anyone can attest who has used different local minimizers when studying complex landscapes -, and thus a variety of methods686 have been developed to find saddle points in this way. 299 The question arises, what we would call an “optimal” trajectory, i.e., what is the criterion of optimality in this context? One can consider the route along which the probability flow is maximal, but this flow usually is measured as function of temperature, and thus the optimal path would become temperature dependent. On the other hand, we might pick the shortest distance as measured in configuration space, or the path with the minimal saddle point energies (corresponding essentially to a path the system would take at close-to-zero temperature). 300 The activation-relaxation techniques692–694 used to identify saddle regions are known to us already in the context of global optimization algorithms, as they are also often employed to detect neighboring minima starting from a given minimum. 301 This approach is based on finding convex functions (pieces of paraboloids/hyperboloids) that can be used to (locally) approximate the true energy landscape. 302 There are standard methods for determining the curvatures, i.e., the eigenvalues, at critical points,697 and we will not discuss these here. 303 The disadvantage of reducing the dynamical analysis to the saddle þ minima network is the lack of information about the true entropic barriers, i.e., we cannot measure the difficulty to even get close to a saddle point during the time evolution of the system. Similarly, we face the uncertainty, whether the saddle points we find are the only ones connecting two basins, anddif there are many connecting pathsdwhether they are actually the important ones where the main probability flows occur. As a consequence, the entries in the transition matrix describing the dynamics on the network are only of limited validity, and the long-term evolution of the system might take a different route than expected from the minima-saddle point network alone. 296 297

Energy landscapes in inorganic chemistry 3.11.6.3

333

Pockets of the landscape and characteristic regions

To analyze complex energy landscapes from a lid-based pocket point of view (c.f. Section 3.11.2.1.2), one can use the so-called  ! ! lid274 and threshold273,295,296 algorithms to identify the pockets P L; R as function of starting (or reference) point R and energy   ! lid L, measure the local densities of states g E; L; R for a sequence of given lid values L, establish the microstate connectivity of the landscape, and measure the probability flows between the pockets of the landscape. The original lid algorithm82,274 was designed for applications to model systems in physics and mathematics with discrete configuration spaces.82,274,700,701 For discrete systems,  ! ! one exhaustively enumerates all states within the pocket P L; R ; for every starting point R of interest, together with all the  ! connections between the individual microstates, which are defined by the moveclass. Similarly, g E; L; R is completely known, and we can both explicitly construct the metagraph of the system and the various simplified tree structures of the landscape discussed above. For continuous systems or discrete systems with essentially infinitely large numbers of microstates in the regions of interest,273,295,296 the exhaustive approach of the lid algorithm is no longer feasible. Instead of the lid algorithm, we now use a stochastic approach, the threshold algorithm mentioned earlier in the context of global optimization methods (c.f. Section 3.11.6.1.6).304 As mentioned above, the trajectory of the random walker or of the MD simulation remains below the lid, but we periodically stop and perform a set of stochastic local minimizations.305 By using several minimizations at every stopping point we obtain a statistical overview, which local minima are accessible from the stopping point. This procedure yields both the probability flow between the minima and the size of the characteristic regions286 (c.f. Section 3.11.2.1.2) at a given energy level. Furthermore, we can sample the local density of states as function of energy lid. By combining the local DOS for many energy lids, we can estimate the statistical weight of the basin around the local minimum.306 Further information can be extracted from the energies of the attempted moves (including the rejected ones). The degree to which these are correlated with the energies of the points from which the moves were attempted, yields additional insights into the general shape of the energy landscape beyond those provided by the tree structure. The threshold algorithm has been applied to many chemical systems, ranging from crystalline compounds112,255,294,386,387,704–706 to clusters258,259,707,708 and molecules.273,709 Recent extensions of the threshold algorithm are the so-called threshold-minimization algorithm661 that had been originally developed for global optimization purposes, the molecular dynamics threshold algorithm,273 and the threshold-RRT algorithm.339 Moves in the threshold minimization algorithm consist of limited-size jump-moves followed by short (incomplete) local minimizations.307 Besides serving as a global optimization procedure, this algorithm allows us to analyze transitions between minima basin that involve long-time scales. The MD-threshold combines the deterministic aspects of molecular dynamics with stochastic repulsive interactions with the potential energy lid (as described above), and the threshold-RRT explores the landscape via rapidly growing random trees where the nodes and edges of the trees must remain below the given energy lid.

3.11.6.4

Transition path analysis

Studying saddle points, and transition paths in general, yields information about the global barrier structure and the local barriers that surround local minima and locally ergodic regions, and thus allows to estimate the existence, stability and lifetimes of the hypothetical metastable compounds. Similarly, the dynamics and mechanisms of relaxations, chemical reactions and transformations67,690,710 are controlled by the energetic, entropic and kinetic barriers in the system,294,711–716 which implicitly determine the transition paths between the locally ergodic regions representing two metastable phases.717,718 Quite generally, the algorithms that are used for determining barriers and transition paths, fall into two major classes: those where only the starting point of the trajectory is known, and those where both the starting and the end point of (a set of) the trajectories between them are given from the outset, either because they are already known from experiment or have been postulated. As discussed above (c.f. Section 3.11.6.2), another aspect is the distance between the two points or regions in configuration space.

304 Keep in mind that for the barriers to be meaningful, the moveclass should only consist of physically relevant moves on the time scale of an exploration step; jump moves only make sense, if we want to study only the evolution on very long time scales. 305 For molecular dynamics threshold calculations,273 we employ stochastic low-temperature MD-gradient methods in place of the stochastic Monte Carlo quenches. 306 If the local density of states grows exponentially or as a power law with a very large exponent, trapping phenomena might be observed198,274,702,703 and entropic barriers that shield the basin from being entered above the trapping temperature might be present.63,294,701 307 There are some similarities with the genetic-algorithm threshold algorithm,662 where we also explore the landscape below given energy lids using moderately large ensemble moves and keep track of the way the ensemble changes or stays essentially the same, analogous to the jump moves in the threshold-minimization method.661 Note the importance of keeping the jump moves and ensemble moves small when trying to identify basins on the energy landscape and their connectivity.

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Starting with the second category, we note that many studies start with prior chemical information or inspiration to postulate a reaction coordinate in more or less detail, and then to approximately follow this route.719,720 In this case, we usually use local optimization orthogonal to the proposed route and generate a stable path between the two regions of interests. As we have already discussed (c.f. Section 3.11.6.2), this approach is particularly useful if the two minima are quite close on the landscape, such that socalled nudged elastic band (NEB) and related procedures307,687–689,721 allow us to generate a locally optimal path connecting these minima.308 However, in macroscopic chemical systems, both the starting and the end regions, and the transition regions connecting them tend to contain many local minima. Examples are defect structures within the locally ergodic regions describing real crystals, or different conformations of large molecules that contribute to the LER. Analogously, there will be many complex “prenucleation” or intermediary structures inside the transition regions connecting the starting and end phase of the transformation under investigation. Thus, picking a “shortest path” or “ideal” reaction coordinate between the two phases is usually not realistic. To address this problem, alternative algorithms have been developed, such as the string method,722 transition path sampling,723 discrete path sampling,103 the prescribed path method724,309 or the pathopt algorithm.725 Conceptually, transition path sampling,453,586,723,726,727 which works analogously to a Monte Carlo simulation, but where the state space is the space of whole transition paths and not the microstates of the chemical system, and related procedures689,728–731 are very general and powerful. In practice, one matches forward and backward routes between the two minima, which we simulate via molecular dynamics simulations of a single path. Subsequently, we sample the path space via small (random) changes in the initial conditions of the simulations of the forward and backward paths.310 Each path will have a thermodynamic weight, and from the “density of path states”, rate constants for the “local” transition between the minima or LERs can be derived, usually via the transition state theory formalism.103,290,698,732–734 Of course, information about the local density of states is very valuable for such calculations of rate constants. As noted earlier (c.f. Section 3.11.6.2), for continuous landscapes, we can compute the local vibrational DOS in the neighborhood of a critical point within the harmonic approximation. Many approaches to model the dynamics of landscapes via “local” transitions, i.e., without employing global barrier information, such as transition state theory (TST), the closely related RRKM theory103,732 (RRKM), and local random matrix theory456,735 (LRMT) use this local density information as input, in addition to the saddle þ minima network that represents the connectivity among the critical points of the landscape. The basic modeling step in TST and RRKM consists in representing the system’s time evolution by jumps between critical points with certain transition probabilities,311 and that the transitions between neighbor minima are controlled by the energies and shapes of the minima and the connecting saddle point(s) in the harmonic approximation.312 An extension is the LRMT approach, which in some ways is a quantum mechanical version of the TST/ RRKM. The underlying concept of the LRMT is the flow of energy/probability (instead of an ensemble of walkers) via the quantum transitions among harmonic oscillator states (instead of classical thermally activated transitions) between the critical points of the energy landscape. For more details regarding these methods, we refer the reader to the literature on TST,103,698,732–734 RRKM103,732 and LRMT.456 The metadynamics scheme658 mentioned earlier in the context of global optimization techniques (c.f. Section 3.11.6.1.5) and related methods736,737 also can generate candidates for transition paths and transition probabilities between metastable phases. In contrast to transition state sampling, only the starting minimum/phase is provided, and the system is free to identify interesting new phases “on its own”. A return to previously explored parts of the energy landscape738 is prevented by associating penalty type terms with microstates inside the starting region (and in other regions that have been visited already). In this way, the starting region becomes increasingly unfavorable, and the walker gets pushed out of the region. This taboo-like search is not restricted to low temperatures, and it is possible to estimate the local free energy, too.514,739 Of course, this search can be combined with other exploration tools such as quenches along the trajectory, in order to identify neighboring minima (regions), as is frequently done for long MC or MD trajectories (c.f. Section 3.11.6.1). As mentioned earlier, for reasons of efficiency most implementations of metadynamics rely on a good choice of an order parameter, which can efficiently classify the original phase and the new one(s) that are expected

308 These procedures are most suitable at low temperatures when the transition ratesdand the choice of the optimal pathdare dominated by the energy barriers of the landscape. 309 The prescribed path method is a recursive application of the elastic band methods, particularly useful for minima that are far apart, such that many local minima representing possible intermediary states are expected to exist. The algorithm contains a number of ways to explore restricted parts of the landscape via prescribing various kinds of paths combined with local optimizations or short (low-temperature) MC simulations, in the space orthogonal to the prescribed path. Based on the various local minima encountered during those orthogonal explorations, shorter paths connecting these minima are generated, and thus a full network of intermediary minima and connecting (nudged elastic band) paths can be obtained. 310 Of course, each step is much more involved than the simple changes in the atom configurations typically employed during the usual MC-walks in atom configuration space. 311 Such a time evolution can then be mapped into a Master equation approach, discussed earlier in Section 3.11.4.2.1. 312 This assumption of low temperatures is often justified. Furthermore, we assume that the time scale of the steps in the evolution is large enough such that the system is locally equilibrated with respect to the stable degrees of freedom (i.e., the degrees of freedom associated with the stable eigenmodes at the critical points), at the critical points (minima and saddle points). If now the harmonic approximation is sufficient to compute the relevant local densities of states, one can use an Arrhenius-type of argument to define free energy barriers between neighboring minima (basins). Again, note the similarity to the coarsening procedure in the Master equation approach in Section 3.11.4.2.1.

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to appear.313 Similar issues are faced in various versions of coarse molecular dynamics,451,740–744 accelerated molecular dynamics,644 or steered dynamics procedures,745–751 which have been employed for similar studies of transition paths.314 Here, we again mention the use of motion planning methods taken from robotics to design algorithms to simulate the motions and shape changes of proteins.335,669,670 Combined with methods in computational physics such as normal mode analysis,752 or using appropriate multi-scale molecular models,753 robot path-planning algorithms have computed large-amplitude conformational transitions in proteins several orders of magnitude faster than standard simulation methods such as molecular dynamics.

3.11.6.5

Probability flows

The measurement of probability flows on the landscape is closely related to the analysis of transition paths between LERs and the global barrier structure of the energy landscape. For simple transition paths, standard transition state theory or transition path sampling will yield sufficient information to estimate the probability flows. If this is not the case, other methods are required. Probably the most straightforward waydbut not necessarily a very efficient onedto measure such flows is to perform many long constant temperature MC/MD simulations plus quenches. However, at realistic temperatures, the length of the simulations can be extremely large for reasonably sized systems. Instead of measuring the flow as a function of many temperatures, we can obtain it as function of energy slice by performing threshold simulations for many energy lids.154,258,259,294,296 No target is given for the random walker below each energy lid, and by periodically quenching the walkers into local minima, we obtain estimates of the probability flows and generalized barriers between various LERs and landscape pockets, as function of energy slice.315 In contrast, the energy barriers follow from comparisons between the flows for different threshold values starting in the same minimum. We note that these generalized barriers that include kinetic and entropic aspects are only meaningful in a statistical sense.316 This concurs with the fact that we derive such barriers from the probability flows on the energy landscape, which are statistical entities themselves. In essence, the probability flows directly measure the combined entropic and kinetic barriers below the given energy lid154,155; in contrast, the relative energies of the microstates inside the pocket are irrelevant for the dynamics at a given energy lid since every move is accepted as long as we stay below the lid.

3.11.6.6

Locally ergodic regions

A central concept for energy landscapes of chemical systems are the locally ergodic regions defined above (c.f. Sections 3.11.2.1.2, 3.11.4.1.2, 3.11.4.1.3). In principle, the determination of locally ergodic regions consists of three steps: First, we generate candidates for such a regiondat low temperatures, these correspond to individual local minima, while at higher temperatures, several minima def , and finally, we must contribute. Next, we verify that the candidates are locally equilibrated on the time scale of observation tobs ensure that the candidates are kinetically and thermodynamically stable on the time scale of observation.485 At least in the field of (solid) compounds, nearly all investigations have concentrated on the first step, since one implicitly assumes that the check that the candidate is a realistic metastable phase, will happen via the successful synthesis of the predicted modification (which is left to the experimentalist .).317 In practice, most theoretical studies reduce the verification of equilibrium (step 2) and kinetic and thermodynamic stability (step 3) of the candidates to only checking whether the candidate corresponds to a local minimum of the energy. It follows that the vast majority of predictions of new modifications found in the literature are only strictly valid for very low temperatures, T z 0 K. When perusing the literature, one finds three main groups of approaches to generate atom configurations that can serve as representatives or cores of locally ergodic regions. The first one consists of the straightforward determination of local minima and structure families of such minima on the energy landscape.21,101,102,116,120,150,485,754 Next, there are chemically inspired571,755–758 and/ or systematic construction568,759 of hypothetical candidates including database-driven searches577,578 and topological bondnetwork methods541,583,760–764; in this case, we need to perform a local minimization for each candidate. Finally, we can perform simulations that (more or less accurately) reproduce or imitate chemical and physical processes. This procedure by construction suggests a possible synthesis route while proposing candidates; examples are pressure or temperature induced solid-solid phase transitions,765–768 the sol-gel process,769,770 or crystallization from solution771 or the melt.772 To the latter category of approaches belong long Monte Carlo or molecular dynamics simulations (at high temperatures) where one attempts to somehow (visually) identify stable structures about which the system meanders along its trajectory; neural network/ machine learning methods might be useful here, if the network can be trained in some way so as to recognize not-yet-seen patterns 313

Selecting such indicator variables is a non-trivial problem; similar issues arise in the ergodicity search algorithm187 (c.f. Section 3.11.6.6). Note that the determination of saddle points is usually more expensive than an optimization; this is even more true for transition path sampling, metadynamics and similar methods, which are even more complex and can be quite expensive computationally. 315 Since the local densities of states of the individual pockets grow very rapidly in the low- energy range of the landscape, the walker spends about 90% of its time in a thin energy slice right below the current lid value. Thus, one can usually identify the energy slice with the threshold value. Of course, one should check this assumption by sampling the DOS for each energy lid; this is a set of data that are automatically accessible from the threshold run. 316 In fact, even modeling the dynamics in the presence of only energetic barriers assumes multiple attempts to cross the barrier in a quasi-equilibrium situation, leading to an Arrhenius-law type behavior, which is only real in a statistical sense. 317 For clusters and small molecules, the barrier structure has been investigated in a number of studies, but even here many investigations only focus on the determination of the local minima of the potential energy landscape. 314

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in large sets of atom configurations. This can even work for phases that do not correspond to a single, or even any, local minimum of the energy landscape.765,768,773 A systematic version is the ergodicity search algorithm.187 Similar to, e.g., metadynamics, one defines some structurally meaningful indicator variables, e.g., the potential energy and/or the radial distribution function, and analyzes their values along the trajectories at constant pressure and temperature, within time windows of varying size. Now, if the average values of these variables jump by more than the fluctuations within each window when we compare two successive time windows, the sets of microstates visited within each of these two windows are flagged as possibly different locally ergodic regions. We next verify, whether the system is in local equilibrium on the time scale of the sample window by running swarms of short simulations starting from several points along the trajectory in the time window. Finally we run long simulations starting from these microstates for a number of temperatures, in order to measure the probability flow from the region and thus estimate the escape time. Obviously, searching for locally ergodic regions in this fashion is quite expensive computationally, especially as one might want to repeat the analysis for different sizes of time windows along the trajectory. Thus, we would use this procedure at somewhat elevated temperatures, where we suspect that LERs might be present that are stabilized only by entropic barriers,154 such that a standard minima search would not be sufficient to find all relevant candidates.318 As mentioned earlier, many types of locally ergodic regions encompass many local minima, especially at high temperatures. Such multi-minima LERs have a non-trivial contribution of the configurational entropy to the free energy. The corresponding phases often exhibit local or global controlled disorder,118 where we speak of local controlled disorder if the local minima in the LER can equilibrate on the relevant observational time scales, e.g., we might deal with “freely” rotating complex ions in a crystal. In contrast, solid solutions or compounds with partial occupancy can be considered to possess global controlled disorder.319 Finding such LERs without recourse to a priori information about the system requires the identification of very many low-energy minima and a systematic structural comparison based analysis. If we can group some of the minima (of similar energy) into a structure family,88,485 we can consider the LER containing this family of minima as a candidate for a (partly disordered) high-temperature phase.

3.11.6.7 3.11.6.7.1

Free energies and densities of states Densities of states

Computing densities of states is a seemingly straightforward enterprise, especially for discrete energy landscapes where we only face the issue of limits on the computer memory. However, for larger physical or chemical systems, simple enumeration is usually not feasible, and statistical methods are required even for metagraphs and definitely for continuous landscapes, once we go beyond the immediate neighborhood of local minima or saddle points where the harmonic approximation is valid and the vibrational DOS can be analytically calculated.320 We note that anharmonic corrections775,776 and magnetic contributions777–779 can be included to some degree, but even this is usually not sufficient for the complex energy landscapes which one deals with in chemical systems. Clearly, we can sample the (local) DOS via long unbiased random walks,780,781 such as the classical histogram methods.782,783 Since such a sampling only yields relative densities of states, we need to normalize them to, e.g., the vibrational density of states near the critical points. However, this step involves assigning part of the DOS to the various minima (basins) on a complex landscape and is not straightforward. But even if we are only interested in the global DOS, we require very long simulation times. To deal with this issue, one has developed so-called re-weighting methods158,784: By repeating the simulations for many different temperatures, one can re-scale the sampled densities of states according to the acceptance probabilities of the random walk. In this way, the simulation looks like a diffusion on a “flat” landscape. The most popular schemes are weighting of histograms on the fly, e.g., WHAM,785,786 re-scaling of data of constant temperature runs obtained from ensemble simulations,787 and parallel tempering or so-called multi-canonical simulations.158,608,788–795 Conversely, one can modify the landscape itself by (global or local) transformations to make it look “flat”796 and re-weight the sampled DOS afterwards158.321 Of course, the statistical validity of the sampling is a concern, and, for global explorations, our ability to achieve a homogeneous sampling of different parts of the landscape as well. An alternative is to perform the sampling for overlapping energy slices, employing the threshold algorithm,273,295,296 and subsequently match the samples’ DOS.294 Just as for the case of simulations at different temperatures, a serious concern is the quality of the overlap of the distributions taken at different lids or for different temperatures.322 If we cannot normalize the local

318 Stabilization of a locally ergodic region via entropic barriers can lead to quite counterintuitive results. For example, on the energy landscape of CaF2, a stable region centered on a structure exhibiting CaF7 monocapped prisms as coordination polyhedra, was found that showed long life times (ca. 20,000 MC-steps) at very low temperatures, but did not contain any local minima. But when increasing the temperature, not only the fluctuations about the average energy increaseddas one would expect from MC walks at higher temperatured, but also the life time of the region doubled.154 This is just the opposite of what one would expect in the case of a region that is stabilized by energy barriers. 319 Note that the additional local minima corresponding to equilibrium defects in the structure, which appear in all real crystals, are not considered part of the set of minima associated with controlled disorder. In a way, equilibrium defects are a measure of structural noise on top of the ideal crystal, while controlled disorder is part of the definition of the ideal system, where regular ordered crystals would exhibit no controlled disorder, and thus zero configurational entropy. The contribution of defects needs to be added to the free energy after the configurational entropy contribution due to the controlled disorder has been evaluated. 320 Alternatively, one can compute the phonon spectrum from simulations, e.g., via the Fourier transform of the velocity autocorrelation function.774 321 Recall the schemes used to flatten the surface for the purpose of more easily identifying local minima in Section 3.11.6.1. 322 Keep in mind that we are particularly interested in the DOS restricted to locally ergodic regions, which are only defined for finite observational time scales def tobs . Thus, the time scale dependence of the relevant DOS sample can play a role when analyzing and matching the data.

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DOS with respect to the local phonon density of states, then umbrella sampling methods developed for the computation of free energy (differences) can be a possible alternative,797 which allow us to compute the difference in the DOS compared to a reference system, which can then serve as the keystone of the normalization procedure.

3.11.6.7.2

Free energy calculations

Once the local density of states for a locally ergodic region is known, it is straightforward, in principle, to compute the local free energy as function of temperature. Clearly, the quality of the resulting free energy depends on the accuracy of the density of states that is used as input for the calculation. We also note that for most chemical systems, we evaluate the free enthalpy, i.e., we consider the system at constant pressure and not a constant volume. Furthermore, we usually assume that the system is in local equilibrium regarding the metastable phase under consideration.323 If the (local) DOS is not directly available, one usually uses various approximations. If we are dealing, e.g., with a real ordered  !  crystal, then the free enthalpy G p; T; xi ; B ; . can be approximated by the ground state free energy plus independent (!) free energy contributions of various structural and electronic excitations present at the given temperature, pressure, magnetic field, etc., i.e., G ¼ G0 þ Gvib þ Gelectr þ Gdefects þ Gmagn þ .. Each of these contribution can then be evaluated separately, employing standarddexact or model dependentdformulas. Else, statistical mechanical models suitable for solid solution systems, etc., are used to compute the free energy182.324 Another straightforward route to the free energy of a (locally) ergodic region R, F(R), is via p(R), the probability to find the system inside the locally ergodic region R extracted from MC/MD simulations for different thermodynamic parameters. Similarly, so-called entropy sampling Monte Carlo simulations can be used.804,805 Another class of methods aims to compute the free energy by first calculating the free energy difference DF(R / R0) with respect to a reference system R0 with known free energy F0 ¼ F(R0), such that F(R) ¼ F0  DF(R / R0).325 Three related general approaches for calculating free energy differences are thermodynamic integration,806 thermodynamic perturbation807 and computational alchemy469.326 All of these methods follow a given or selected path between the two systems A and B in (thermodynamic and potential) parameter space, along which one smoothly transforms phase A to phase B.808–810 The first two procedures are based on umbrella sampling811 where one evaluates the overlap of probability distributions over the microstates of the system along the path. In contrast, computational alchemy uses thermodynamic inequalities that compare the free energy between two phases with the work needed to move from system A to system B, since this work constitutes upper and lower bounds on the free energy difference between A and B. As far as the accuracy of free energies derived from dynamical simulations is concerned, the major issue is the quality of the sampling process.812 Here, the efficiency of these procedures depend crucially on the path chosen and the allocation of (simulation) time along the path,473,813 although the path does not have to correspond to a dynamically meaningful reaction coordinate.327 We note that if a transition path is given in order parameter space or as a reaction trajectory on the full landscape, we can also compute the free energy along a chosen route via integration over the remaining coordinates.814,815 Assuming that the system stays in local equilibrium along the whole trajectory, such an approach employs the average force calculated along the path to estimate the free energy (difference).816–818 To avoid difficulties regarding the realism of the dynamics along such a path, especially if it is

In the literature, “free energies” are often studies as a function of an order parameter ! m ; and “free energy barriers” between competing phases are defined, !Þ only signifies the probability again, as a function of ! m : As we indicated earlier, as long as one assumes that the system is in global equilibrium, such that Fðm ! ! probðm Þ ¼ expðFðm Þ=kB TÞ=Ztotal that at any given moment one will encounter the system in a microstate with order parameter ! m ; there is no conceptual !Þ has a maximum, can be problem. But dynamical statements about the visually appealing idea that the set of microstates with order parameter ! m ; where Fðm ! Þ are highly problematic. A further discussion regarding various concepts of free energies is given treated as a transition state with a free energy barrier Fðm in.798 324 If such free energy calculations are performed for many different values of the thermodynamic parameters, the outcome is the full phase diagram of the system, for all metastable phases; a number of program libraries are available for this purpose799–803 with most of them having grown out of the CALPHAD project.178 325 Usually, one compares two different chemical systems, A and B, where F(B) is knowndperhaps B is a simple Lennard-Jones system with easily evaluated free energy -, and it is furthermore “easy” to compute F(B)F(A). But if we are only interested in the free energy difference between two modifications A and B in the same chemical system, then the methods discussed are sufficient even without explicit knowledge of F(B), of course. 326 The term “computational alchemy”469 is used because one changes the parameters in the energy function in such a way that, e.g., KCl (in the rocksalt structure) is transformed into a different chemical system NaCl (in the rocksalt structure), which allows us to compute the free energy of KCl if the one for NaCl is known. This procedure is probably preferable, if the goal is the exact free energy of the chemical system, while the thermodynamic integration and perturbation methods that are based on the umbrella potential sampling of distributions constitute a relatively efficient way of determining the free energy difference between two crystalline modifications in a solid or between two isomers of a molecule or a cluster, at least as long as the transition path is reasonably well known. An overview together with discussions of computationally efficient implementations is given in the literature.473 327 Here, one needs to keep the different meanings of “path” distinct. The path between the two systems we choose when transforming one into the other is determined by the changes in parameters of the extended energy landscape, such as thermodynamic parameters, e.g., the applied pressure, or parameters in the potential. In contrast, the ensemble of walkers representing the system as far as its statistical mechanics (possibly restricted to the desired locally ergodic regions) is concerned, “walk” on the energy landscape following their own paths in state space that start and end on the initial and final landscapes, respectively. 323

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defined in order parameter space, such computations are usually performed along a true simulation trajectory, where we integrate over the (local) coordinates orthogonal to the given path, again using umbrella sampling if possible. In this context, we note that a major problem in “realistic” MD simulations on complex energy landscapes is the sometimes exorbitant time spent waiting between two transitions among the minima or LERs, even though the transition itself (once the system has “found” the right route) is very fast. This affects not only free energy (difference) calculations but also the computation of transition rates, and is known as the rare event problem.819–825 One approach for accounting for such rare events in an efficient way is the use of Jarzynski’s inequality,826–828 where one again needs to ensure that the trajectory follows a reasonable reaction coordinate which is much slower than the remaining degrees of freedom (orthogonal to the tangent of the reaction trajectory). Considering the importance of free energies, it is no surprise that a plethora of methods exist beyond those already described. Here, we only mention the grand canonical,829 semi-grand canonical830,831 and extended Gibbs ensemble832 computations: Analogously to the way one can implement a chemical potential by moving atoms from reservoir at certain thermodynamic conditions to the chemical system of interest, we move atoms between two copies of the system,833 one describing phase A and the other phase B, allowing us to compute phase equilibria and differences in free energies. Even though the methods described above should be applicable for all types of chemical systems, efficiently computing the free energies of alloys or solid solutions requires the use of methodologies that take the system-specific aspects into account, in particular the importance of the configurational entropy. Conceptually, many of these methods are related to the simple ideal and regular solution models834 but go far beyond these.176,835–839 At the heart of these approaches, usually called cluster expansion837,839–842 or cluster variation843–845 methods, is the idea that most of the refinement needed for quantitative accuracy can be achieved by studying the properties of local clusters of atoms on top of the global aspects of the free energy incorporated in the solution-like model. One “brute force” approach is the explicit computation of the free energy via summing over the (local) free energies of a large number of microstates. Here, one assumes that these correspond to a representative sample of the set of atom configurations in the alloy system846.328 In all of these approaches, one assumes that the underlying structure (the “reference lattice”) of the alloy, i.e., the atom arrangement seen if all the atoms were identical, e.g., a fcc or bcc lattice, does not change when the distribution of the atoms over the reference lattice or sublattices thereof is varied (except possibly for small relaxations away from the reference positions). Thus, all microstates we consider (and sum over) when estimating the free energy of the alloy or solid solution exhibit the same reference lattice. In that case, every microstate can be mapped isomorphically to an occupation state of the reference lattice176;, mathematically, this corresponds to a n-state (Potts)-spin model847 for n different atom types.329 Usually, it is too expensive to compute the energies on the ab initio level, and thus one approximates the energy by a sum over interaction terms among the Potts-spins. In particular, if our assumption that the atoms only interact with atoms within a small neighborhooddthe clusterdholds, we only need few-body terms (e.g., 1-, 2-, 3-, and 4-body terms) to evaluate the energy function. Furthermore, this implies that we need only to determine a small number of interaction parameters that can be obtained by fits of the approximate energies for a set of atom configurations to the corresponding ab initio energies.330 Once these parameters are known, the approximate energy function can be evaluated very quickly even for large periodic approximants. Thus, the free energies of the alloy phase can be calculated quite accurately via statistical averages over many millions of configurations,848,849 possibly using symmetry adapted selection schemes846 to further reduce the number of energy evaluations involved.

3.11.6.8

Calculation of mesoscopic properties

We close the methods section by shortly considering the issue of the computation of physical properties of materials starting from their energy landscape. Clearly, there are atom-level properties such as the energy or the band structure, which are directly accessible from quantum mechanical calculations of individual atom arrangements, using one or several of the many codes available in the literature.331 Natural extensions are the computation of E(V) curves or H(p) curves, from which one can deduce the bulk modulus of a given modification or the transition pressures between modifications. Similarly, many programs or program packages nowadays allow quasi-automated computations of vibrational spectra (for molecules) or phonon-spectra (for solids), or IR and Raman modes, of a quality that is often comparable with the experiment, and which are suitable to compute the free energy in the 328 This kind of calculation can be performed in a systematic fashion by computing the free energies over all states of the reference lattice defining the basic structure of the alloy, i.e., exhaustively, for a small number of atoms in the periodic approximant. This calculation is repeated for larger numbers of atoms/cell, and we extrapolate to the infinite system limit, analogously to the way we approximate the infinite solid by a sequence of large finite clusters of increasing size. 329 Clearly, we consider only a very limited fraction of the full state space of the chemical system, i.e., we restrict our investigation to a single though very complex locally ergodic region. For the full free energy of this given solid solution phase, we would need to add the defects that violate the reference lattice condition, and similarly, the vibrational energies associated with each arrangement of the atoms on the reference lattice. But usually, these contributions are quite small compared to the main portion of the free energy.88 330 Keep in mind that we are concerned with energy differences between different atom arrangements compared to the “average” energy over all the configurations that takes the long-distance interactions into account. Each microstate has essentially the “same” long-distance term as part of its energy, which can be subtracted (or added) accordingly, while the Potts spin potential deals with the local variation due to the actual atom arrangement. This also often allows us to ignore the variations in the vibrational contributions associated with the individual atom arrangements on the lattice, i.e., with the microstates in this model landscape. 331 As mentioned before, for systems where we do not have experimental data available to tune the functional, basis sets, pseudo-potentials, k-space nets, etc., one should compute the same property with several such combinations, in order to gain some estimate of the validity of the results. This goes beyond the standard convergence procedures in, e.g., the plane wave energy.

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(quasi-)harmonic approximation for low and intermediate temperatures. Similarly, modern computational methods such as density functional perturbation theory or displacement methods can be employed to compute, e.g., dielectric constants,850–852 and also some magnetic properties.853 However, many properties cannot be computed from single point or single basin (in the harmonic approximation) calculations; in particular, transport properties can pose problems. For these quantities, one can find formulas within semi-classical or linear response theory,58,500 which allow us to combine properties such as relaxation or collision times of quasi-particles that serve as carriers, effective quasi-particle velocities, quasi-particle charges, specific heats, etc., as function of wave vector, together with the Fermi- or Bose-statistics, to compute electric, thermal, etc., conductivities. But in practice, these quantities are difficult to compute, and one is often forced to perform MC and/or MD simulations486.332 Analogous problems can appear when trying to compute electromagnetic properties such as dielectric constants or magnetic permeabilities for non-trivial materials, where again averages over mesoscopic size regions are required, in principle,478 or if larger regions with varying dielectric constants need to be investigated.857 Similarly, one is often interested in the behavior of systems on larger spatial scales, which might require extremely large numbers of atoms to simulate and where even an essentially classical treatment with effective energy functions might not be sufficient. Here, analytical modeling and continuum models, or intermediate models susceptible to finite element methods236,237,858 would be required. In such a case, the original energy landscape of the system would be studied to extract local quantities such as transition rates and energy barriers associated with certain re-arrangements of the atoms in the material, which then can be employed in a stepping-stone (multi-scale) modeling approach as input to higher level models suitable for the description of the properties of interest of the mesoscopic or even macroscopic system. Discussing such model approaches takes us beyond the purview of this overview, and we refer to the literature for further information.859

3.11.7

Applications

In this final section, we are going to present examples for many typical applications of energy and cost function landscapes in the context of chemical systems. Here, we have attempted to find case studies that demonstrate the concepts discussed in Sections 3.11.2–5; however, not all of the ideas presented in Sections 3.11.2–5 have already been realized in published work. Considering the large number of investigations that deal with energy landscapes, the references are far from complete, and often we have only been able to mention a small sample from the rich smorgasbord of studies in the literature. Regarding the computational methods employed in the various studies, we refer to the methods Section 3.11.6 for a description of the general features of the different approaches that have been used; more details of the particular implementations can be found in the specific publications.

3.11.7.1

Structure prediction

By far the most common use of energy landscape investigations in chemistry is the prediction of kinetically stable atom arrangements in a given chemical system. Such atom configurations, by definition, correspond to local minima on the potential energy landscape or on one of the extended energy landscapes that incorporate the external, usually thermodynamic but not exclusively,333 constraints due to the environment. Frequently, such local minima, or groups of related minima, are at the heart of the locally ergodic regions on the landscape that constitute thermodynamically (meta)stable polymorphs or isomers for a given temperature and observational time scale.334 Examples of such predictions fall into several categories: predictions for the isolated chemical system, predictions for systems under thermodynamic constraints such as a fixed pressure, predictions in the presence of spatial constraints such as a substrate or a porous matrix on which or within which the atoms are supposed to arrange themselves, respectively, and the predictions of shapes of a given molecule with a fixed bond-network that is assumed to be unchangeable for the purpose of the study. These major categories can be sub-divided. The classification and the methods employed will depend on (i) whether we consider bulk systems or small finite clusters, and whether we (ii) search only in the space of highly symmetric configurations (e.g., look for perfectly periodic crystalline structures) or look for general amorphous arrangements representative for glassy materials or general clusters. Another important aspect is whether (iii) we employ atoms, primary building units862dsuch as CH3 groups, benzene rings or whole side chains in large polymers and proteins,102 complex-cations/anions or coordination polyhedra109,863,864d, larger (secondary) building units such as building blocks of framework compounds,109,540 or whole rigid/flexible molecules as the elementary units 332 Here, it is important to keep the limits of classical molecular dynamics in mind, which does not take quantum mechanical coherence and quantum statistics into account; such quantum effects can dominate the transport at low temperatures, see for example.854–856 333 Examples of non-thermodynamic constraints are, e.g., forcing the atoms to stay inside a monolayer on a substrate or to stay on the surface of a cylinder, or on the sites of a particular lattice. 334 Of course, their theoretically assured existence is no guarantee that there are feasible synthesis routes to access such configurations or compounds, in practice; although the synthetic capabilities in organic and molecular chemistry are quite powerful already (due to the inherent kinetic control of the reactions in this field), this task is much more of a challenge in solid state chemistry. Still, we can always imagine some amazing tools that allow us to place arbitrary atoms next to each other in free space to form just the right kinetically stable arrangements (at zero temperature). In fact, the development of methods based on establishing such starting configurations for the synthesis has been a major enterprise in solid state chemistry in the past 25 years.860,861

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of the arrangements. Finally, there is the issue (iv) of the dimensionality of the system which can range from 3D crystals and glasses over (quasi-)2D monolayers and surfaces, quasi-1D tubes and 1D atom chains, to “quasi-0D” systems like small finite clusters. Providing a comprehensive overview over the prediction of such structures335 would merit its own chapter or book, and will not be attempted here. Nevertheless, we will present a very cursory overview of structure prediction, since a discussion of energy landscapes of chemical systems would be incomplete without mentioning at least some examples and general aspects. Identifying low-energy minima including chemically relevant configurations besides the global minimum is usually achieved via global optimization techniques some of which have been discussed in the previous section. Here, we mention a couple of general issues. For one, using ab initio energies during the global optimization is frequently too time-consuming, and thus many studies employ simple and fast but less accurate energy functions to find minima candidates that are subsequently locally optimized and ranked by energy on ab initio level.336 As mentioned earlier, a new variant of such simple energy functions are the implicit energy functions provided by machine learning techniques409,411,865.337 The main problem is that for many chemical systems the relevant modifications and isomers do not constitute low-energy minima of the landscape of the simple energy function,338 or sometimes are not minima at all.339 Trying to cope with the plethora of local minima on the energy landscape, in particular in the case of crystal structure prediction but also for cluster prediction, one often restricts the search to a subset of the full configuration space that consists of supposedly “chemically reasonable” configurations, and essentially only performs local optimizations of such pre-selected configurations, or global searches on these highly restricted configuration spaces. Such restrictions apply for systems where certain structural elements or local environments of atoms are pre-defined or assumed at the outset based on chemical pre-knowledge, such as primary and secondary building units,109 or where the atoms are assumed to reside on prescribed sublattices usually known from experiment185,831,848,866–872.340 These restricted configurations can be generated via data-mining,577,578,581,582 as bond networks (employing graph theory and mathematical tiling theory) for a given set of allowed bond-connectivities,541,583,760–764,873,874 inspired by chemical information and “experience”571,755–758 and/or by systematic construction employing physical principles like dense sphere packings or symmetry constraints,25,568,572,759,875 and so-called ‘bottom-up’ approaches876–879 where clusters are first optimized and then assembled to crystal structures. We also note that studying a variety of compositions is important for structure prediction in ternary or higher systems, since one often cannot estimate a priori, which composition will be the preferred one, even for ionic systems since charge (or ionization level) conservation gives only one condition on the composition of the system and thus there is still freedom to vary the stoichiometry of the compound.341 In this way, it is relatively straightforward to construct individual structure candidates containing dozens or even hundreds of atoms in a unit cell or a cluster. The main problem when relying on a database or chemical rules is that not all relevant structures

335 The term “structure” is frequently employed rather loosely, generally implying any type of “regular arrangement” of atoms or molecules in space. But while in mathematics a structure (and its symmetries) would refer to just one particular arrangement of atoms in space, i.e., to only one microstate, in chemistry and frequently also in crystallography one implies a statistical average over many microstates associated with a single (usually quite symmetric) minimum configuration, or even with a whole minima basin including all the equilibrium defects of the structure. Thus, when people speak of “structure prediction”, they usually have one configuration in minddtypically the local minimum representing the ideal crystal at zero temperaturedas far as the search itself is concerned, but at the same time assert an agreement with the average structure one would measure in the experiment. 336 Nevertheless, the sheer computer power required makes the systematic determination of all relevant local minima on an energy landscape of a general solid (or cluster) without any direct or indirect input from experiment still extremely challenging for systems with more than about 50 atoms/periodic simulation cell (or 100 atom/cluster), irrespective of the global search algorithm employed. While in the case of clusters, the number of atoms corresponds to the true number of atoms in the cluster, and thus we either find the global minimum or we do not, the situation is more tricky for periodic systems: Here, the true number of atoms is essentially infinite, and we are talking about limits on the number of atoms in the periodic approximant (with a variable unit cell). But since we do not know how many atoms will be in the periodic cell of the true global minimum of the systemdassuming that the global minimum is a crystalline atom configuration in the first place -, we are always left with a feeling of unease, even if we have employed, e.g., periodic approximants with, say, 40 atoms in the desired stoichiometry, during the global search. 337 To a certain extent, this circumvents the use of the energy function by replacing it by a black-box kind of scheme, leading to the typical problems of interpretation and rationalization of the results encountered when using neural network approaches. 338 This can be a major problem in molecular crystals, where the experimentally observed structures are often similar to minimum configurations of the empirical landscape but these are not low enough in energy (at least on the level of the simple energy function) and, thus, may be discarded after the first stage of the search. 339 For “infinite” chemical systems, there can exist a multitude of tiny local minima within each significant minimum basin, due to numerical inaccuracies and cut-off effects of the infinite range interactions, even if we use periodic approximants, leading to many minutely different versions of the same actual minimum. As a consequence, the configurations identified as a minimum are always found in the space group P1, and require a careful analysis of their approximate symmetries, and subsequent idealization (if appropriate) and removal of duplicates, using suitable symmetry search algorithms.314–316 340 In the case of finding the optimal shape of large (bio)molecules, we often use an analogous approximation by treating (some of) the side chains and rings as rigid elements and thus only optimize, e.g., the dihedral angle along the backbone of the molecule.375 341 We have earlier remarked that one can vary the composition and number of atoms within the simulation cell by adding an appropriate chemical potential term to the energy function (c.f. Section 3.11.5.5). However, it is usually not clear what the appropriate value of the chemical potential value would be in practice, i.e., one would need to repeat the calculations for a series of chemical potential values. But one knows from experimental phase diagrams that the locally ergodic region associated with the thermodynamically stable phase of the system for a given composition may consist of a combination of two separate compounds with different compositions. To realize such a situation using periodic approximants requires very large unit cells (if it can be done at all for reasonably sized unit cells), for which the determination of the major locally ergodic regions is greatly impeded by the multitude of amorphous configurations that constitute local minima on the energy landscape. Thus, for most situations it has proven to be more efficient to perform the optimizations for a sequence of fixed compositions; furthermore, for each composition one also repeats the calculation for different numbers of formula units since it is similarly unclear how many formula units should be in the periodic simulation cell, in order to include all relevant modifications in the search. The question, for which compositions a decomposition into separate compounds should take place, can then be addressed using a standard convex-hull construction.880

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on the true full energy landscape of the chemical system are similar to a structure found in a database or obey the chemical rules. The same caveat holds true for similar approaches, where one uses a-priori information about preferred bonding and/or packing restrictions, often including knowledge about the unit cell, to suggest possible structures.881–883 We note that many of these concepts were already tested and explored in the last decade of the 20th century, and one might assume that these issues are no longer of any greater concern. However, this is not true, and both the challenges to circumvent the lack of computer power and to address the problems associated with the various approaches mentioned above are still relevant for today’s structure prediction investigations of chemical systems that employ the energy landscape of the system as its sole criterion.

3.11.7.1.1

3D solids

In the beginning crystal structure prediction was mostly performed for systems where simple approximate energy functions exist, such as ionic compounds. This has changed over the past two decades with the increase in computer power that allows the fast evaluation of ab initio energies. Below, we give a quick overview over some typical systems that have been studied since the beginning of structure prediction using energy landscapes in the early 1990s.342 If the reader wants to know, whether there are structure predictions available for specific chemical systems of interest, we refer to the many reviews21,89,101,102,116,120,147,242–244,884 and the general literature that can nowadays be rather quickly searched using various search engines, and here we just mention a few examples. We also note that while in the early stages of structure prediction there was quite a competition between the developers of algorithms, nowadays the focus is more on the efficient determination of structure candidates in the system of interest, and several different algorithms and combinations thereof, together with database mining and schemes for the generation of candidates based on chemical principles, are frequently employed during the global search. The guiding principle is no longer the methodologypuristic “The Path is the Goal”885 but the “Keep the Eyes on the Prize”886 of a successful prediction in practice, regardless of which algorithm is being employed. Thus, we also mention the recent progress of machine learning887 in the field of crystal structure prediction even though this is only indirectly connected to the topic of energy landscapes.343 3.11.7.1.1.1 Elements, gases and mixtures of gases Originally, only predictions for the structure of noble gas solids889 were feasible since only for these fast empirical potentials were available, but nowadays many elements have been investigated, in particular in the context of high-pressure structures. We just mention work on carbon,538,539,574,575,890 silicon,539,891,892 boron,893 sulfur,538 lithium,894,895 iron,896 and tellurium.897 3.11.7.1.1.2 Ionic and covalent compounds By now, a very large body of work deals with oxides, e.g. the alkali113 and alkaline earth oxides,112 V2O5,898 B2O3,899 SnO221 and SnO,101 NiO,21 ZnO,705 SiO2,21,900 CaTiO3,21 SrTiO3,21 SrTi2O5,21 MgSiO3,538 BaBiO3,901 B6O,902 CoMoO4,903 Ce2ON2,904 SnTiO3905 and TiO2.906 Besides the oxides, nitrides have been investigated, such as the alkali nitrides108,704,907–909,344 Cu3N,912 BN,419 AlN,582 Si3B3N721,571 and other compositions in the Si-B-N-system,487 tantalum nitride,913,914 Cr2SiN4,915 lanthanum pernitride,916,917 and pernitrides917 of Ca, Sr, Ba, and Ti. Sulfides, selenides and silicides have also been studied, e.g. the alkali sulfides108,111, 345 and lead sulfide307,923 (employing an ab initio energy landscape in the latter case), ReSe,924 or WSi2.925 Similarly, there are investigations of the carbonates of the alkali 342 Since high-pressure modifications are of great interest in fields such as planetary science, over the past two decades structure prediction and structure solution of high-pressure phases using global energy landscape exploration methods has grown into a veritable industry employing a variety of landscape exploration algorithms, with many reviews available.242–244 343 In fact, at the beginning of crystal structure prediction efforts in the early 1990’s, such expert systems visualized as a “computer distillation of the experience of the practicing chemist” were seriously discussed but not implemented in the end.888 At the time, the reasons for not doing so were three-fold: 1) The computing power of neural networks at the time was not large enough to build a realistic general prediction network. 2) The ultimate goal being the search for new compounds in chemical systems where no such compound had ever been observeddthe correct or at least viable composition was also envisioned to be part of the predictiondand the expectation that completely new structure types would appear for such a system, made it doubtful, whether a neural network trained on examples drawn from a set of known structures and compounds would allow a successful prediction of structures and structure types that had never been seen before. 3) The structure candidates needed to be capable of existence, i.e., they should correspond to what we now call locally ergodic regions in configuration space, which required the determination of the energy barriers surrounding them on the energy landscape and the local free energies of these candidates, both of which were unlikely to be available for methods that did not make any reference to information about the energy landscape of the system. 344 The question of whether Na3N would be capable of existence had been a long-standing controversy. Combining the global optimization with ab initio calculations for the E(V) curves for the most promising candidates led to the prediction of a sequence of high-pressure phase transitions between the metastable Na3N modifications, starting from the ReO3-type over the Li3N- and Li3P types to the Li3Bi type. After Na3N had been synthesized in the ReO3 structure type,861 ending the controversy about its existence, these high-pressure structures were also found,910,911 validating the earlier predictions. We note that an additional intermediary phase (in the YF3 structure) was obtained in the high-pressure experiments. 345 Experimental high-pressure investigations of the compounds Li2S, Na2S and K2S were also performed at the same time. The theoretical108,111 and experimental918–920 studies identified the same high-pressure structures in these compounds, validating the energy landscape approach for structure prediction at high pressures. Note that several years later, experimental high-pressure investigations yielded for Rb2S921 and Cs2S922 the structures that had been originally predicted.111

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metals926,927,346 magnesium and calcium.538,929,930 Of course, the halides have been of interest, e.g., AlF3541 and gold fluoride,931 the chlorides and fluorides of magnesium and calcium,932 SrCl2,21 and all the binary alkali halides115,379,933,934.347 Furthermore, GaAs and SiC have been explored,891 as well as the quasi-binary system Mg(BH4)2.937 In fact, several binary or quasi-binary systems have been investigated for many different compositions. These include the FeB-system,627 the quasi-binary (Li,Na)-nitrides,908 the quasi-binary Ca2Si-CaBr2387 and MgO-MgF2386 systems,348 and the mixed halides of lanthanum.938 In this context, we remark that for multinary systems, it is not sufficient to study only the enthalpy landscape of the specific compound of interestdespecially if one employs a small periodic approximant that does not allow to realize a decomposition into other phases, because decomposition into, e.g., binary compounds can be thermodynamically preferred if the pressure is varied. As an example, consider the hypothetical alkali metal orthocarbonates M4(CO4)dM ¼ Li, Na, K, Rb, Csd, for which possible modifications were predicted as function of pressure.114 Although some of the orthocarbonates should be stable at high pressures, they are unstable against decomposition into the corresponding oxides M2O and carbonates M2(CO3) at standard pressure114; however, the orthocarbonates might be kinetically stable at low temperatures for finite observation times at low pressures. Here, the computation of the decomposition pressures required the parallel analysis of the enthalpy landscapes of the orthocarbonates, oxides113 and carbonates926 of the alkali metals. 3.11.7.1.1.3 Intermetallics The field of intermetallics is vast and thus it is no surprise that it is not always clear, whether a simple quasi-ionic embedded-atom or Gupta potential is sufficient for successful structure prediction, or whether one has to study the ab initio energy landscape. On the other hand, one can often assume that the different metal atoms approximately occupy sites on some simple underlying lattice and thus the actual structures can be obtained via short local minimizations of very many but easily generated starting configurations for the candidates. This also favors the use of databases577 or various sublattice occupation methods849 in generating either starting structures or providing a restricted configuration space to be searched. Examples of structure predictions are the alloys Au8Pd4,891,939,940 and UCo,941 or the investigation of 80 binary intermetallic alloys942 composed of 4d transition metals, plus several systems containing Al, Au, Mg, Pt, Sc, Na, Ti, and Tc, via a partly database driven approach. Many of these structure prediction studies are inspired by open questions in the phase diagram of the chemical system of interest, in particular due to the wide composition ranges of non-stoichiometric phases. An example is the case of the FeeAl system, where evolutionary algorithms and quenches from MD simulations were employed to predict feasible (metastable) stoichiometric structures for FeAl2, FeAl3 and Fe5Al8.943 3.11.7.1.1.4 Networks as templates Similar to the occupation of (sub)lattices frequently employed for intermetallics, (bond) networks can be used as templates to generate structures, in particular in the context of systems expected to form covalently bonded compounds. As an example, we mention the prediction of crystal structures of group 14 nitrides and phosphides,944 where the new candidates were generated by substituting atoms of different types in known network structures. 3.11.7.1.1.5 Mixed types of interactions If one wants to avoid the use of ab initio energy functions, then systems where many different types of interactions (from the phenomenological modeler’s point of view) compete, pose a particular challenge. In certain situations, where one can identify “fixed” building units, such as complex anions, which are expected to form in the system, one can use these units to recast the compound as a quasi-ionic system. Examples are KNO2,101,485 or MgCN2.101,109,934 Since ab initio energy calculations are no longer exorbitantly expensive for small periodic approximants, ab initio energy functions have also been employed to deal with mixedinteraction compounds; examples are the silicon-carbon system,891,945 H2O-NaCl and carbon oxide compounds,946 CaC2947 or helium-sodium,948 and the BN-system419.349 A related class of systems that either require special building units or ab initio energy functions are compounds where lone electron pairs are expected to be present in some fashion and influence the formation of stable atom arrangements. Examples of structure prediction studies of such systems are, e.g., SnO,101,949 GeF2950 or PbS.307,923,951 3.11.7.1.1.6 Zeolites Zeolites and zeolite-analogues are an important class of chemical systems, and thus have been investigated since the beginning of structure prediction studies using energy landscape approaches.109,116,540,623,760,881,952–955 However, zeolite framework prediction usually requires some pretty drastic restrictions of the state space, because the global minimum and many low-lying minima of the 346

Some of the predicted high-pressure modifications of the alkali metal carbonates have by now been synthesized.928 Subsequently, the predicted wurtzite modifications of LiBr935 and LiCl936 were synthesized using the low-temperature atom beam deposition method. 348 For Ca2Si-CaBr2 and MgO-MgF2, the barrier structure has also been studied, and tree-graph representations of the landscape have been constructed. 349 Note that in the boron nitride system both layered structures (hexagonal BN) and three-dimensional networks (wurtzite-and sphalerite-type) are observed in the experiment, making it very difficult to find all the relevant structure candidates using an empirical potential; thus this global search was performed on the ab initio level. 347

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unconstrained system correspond to high-density phases.350 One way to deal with this issue in the field of structure prediction is the so-called “automated assembly of secondary building blocks” (AASBU)- procedure,540 and another route consists of restrictions varying the atom positions inside the variable cell while enforcing a limit on the overall density of the zeolite structure.485 In the same direction points the use of bubble clusters to generate zeolite candidates877,878,956,957,351 or the introduction of stationary or mobile exclusion zones as “pseudo-atoms” that ensure a low density of the periodic atom configurations.623,958–961 Besides predicting the structure of the “raw” zeolites, the determination of optimal guest arrangements for a given zeolite structure has often been of interest962–964 including studies of transformation paths between the various minimum-atomarrangements.965 3.11.7.1.1.7 Molecular crystals Molecular crystals are a prominent class of crystalline compounds that are targets of structure prediction efforts. Most of these crystals are built up from organic molecules,102,116,117,432,754,953,966–975 although structure prediction of simple inorganic molecules have also been performed, e.g., for hydrogen,539 nitrogen,864,949 silane,539,976 mixtures of noble gases507 or for the system nitrogen þ hydrogen þ oxygen.977 Usually, one proceeds by performing systematic (ideally exhaustive) landscape explorations or by running global optimization searches, possibly with restrictions on the space group symmetries built into the generation of structure candidates.352 Commonly, one employs the whole molecule as a rigid building unit for the purpose of the global search,353 but this is no longer necessary for many molecules of interest, and the molecules can be allowed to change their shape during the search. While simplified model interactions are often used during the global optimization stage of the global search, nowadays ab initio energies can also be employed.981 In contrast to typical inorganic solids, the potential energies of many structure candidates for molecular crystals are nearly the same on ab initio level.354 Thus, the free energy (as function of pressure and temperature) needs to be computed to achieve a ranking of the candidate structures, and, in addition, the kinetic aspects of the synthesis are likely to control the structure of the molecular crystal found in the actual experiment.355

3.11.7.1.2

Low-dimensional solids

Structure predictions for two-dimensional and quasi-two-dimensional systems have also been performed,983 with, e.g., 2D-carbon and silicon systems, and their combinations,119,161,945,984,985 and related systems such as carbon-boron986 being quite popular. In contrast, in the field of quasi-2D materials, in particular in the study of slabs of single layers or stackings of few layers (such as the MoS2 layers consisting of edge-connected MoS6 octahedra), classical energy landscape explorations do not seem to be common. An example that uses an approach reminiscent of database mining starts from (known) layered 3D materials, and generates quasi2D structure candidates by cutting out slabs that are a few atoms thick, which are then locally optimized for various types of atom compositions.987 Going down one dimension further, we can globally explore the energy landscape of one-dimensional systems. Both atom chains of finite length and infinitely long periodic chains have been investigated using density functional tight binding as energy function (c.f. Fig. 8).163 Here, the system was first globally optimized in one dimension, followed by a relaxation in three dimensions, in order to check the stability of the obtained structure. Similarly, global optimizations of quasi-one-dimensional systems such as ZnO nanotubes162 or MgO/BeO nanotubes163 have been performed, using empirical Coulomb þ Lennard-Jones potentials (c.f. Figs. 9–11). Again, the global optimization took place on the prescribed cylinder surface, followed by local optimization in 3D via a small perturbation of the structure into 3D away from the on-cylinder optimum and subsequent local minimization.356 In another study, configurations of carbon nanotubesdwith and without molten salts insidedhave been investigated from an energy landscape point of view.990

350 This is no conceptual issue for the structure determination of zeolites where it is natural to employ restricted cost function landscapes according to the available experimental information.863 351 In this two-step process, one first globally explores the energy landscape of clusters of various sizes for the chemical system of interest, identifying stable (hollow) clusters with a bubble-like structure, besides the usual compact structures. Next, one uses these clusters to construct either densely packed structures or porous frameworks using an analogue of the AASBU procedure. 352 It appears that over 90% of all molecular crystals found so far possess symmetries belonging to only 18 space groups,978 and a large percentage of the crystals contain only one molecule in the asymmetric unit.979 This (statistical) symmetry information has been widely exploited in generating structure candidates for molecular crystals.980 353 Note that treating the molecules as inflexible is a major restriction on the state space of the landscape, and thus important polymorphs can easily be missed. 354 The same similarity of minima energies is also true on the empirical potential level, while the ranking of the structures by energy can vary considerably when switching from empirical potential to ab initio energies. As a consequence, one can easily miss the global minimum or other very low-energy minima if one only minimizes, e.g., the ten local minima with the lowest energy according to an empirical potential. 355 We note that the likelihood of formation of a critical nucleus of a phase depends on the properties of the chemical environment, e.g., the medium (atmosphere, solvent, etc.) or the precursor material (gel, film, etc.), inside which the nucleation takes place. Therefore, this likelihood need not be correlated with the thermodynamic stability of the macroscopic phase that is grown from the nucleus. This is not restricted to molecular crystals; analogous observations have been made during the growth of metastable phases from amorphous precursor films.909,982 356 In this context, we note that many energy landscape studies of low-dimensional system have been performed for physics related (model) systems, since these can serve as testing grounds for algorithms and theoretical modelsdboth on the analytical988 and numerical level989dbeyond the inherent interest in analyzing the behavior of chemical and physical systems as function of dimensionality.

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Fig. 8 Global minimum of a finite one-dimensional atom chain Si6C9, using DFTB energies.163 After global optimization in 1D, the structure was allowed to relax in 3D. Silicon and carbon atoms are shown as blue and brown spheres, respectively.

Fig. 9 Global minimum of an infinite MgO-single wall nanotube with prescribed radius, using empirical Coulomb þ Lennard-Jones potentials.163 Magnesium and oxygen atoms are shown as blue and red spheres, respectively. The atom arrangement resembles a single layer of a (100)-MgO crystal in the rocksalt modification, rolled up into a tube.

Fig. 10 Global minimum of an infinite BeO-single wall nanotube with prescribed radius, using empirical Coulomb þ Lennard-Jones potentials.163 Beryllium and oxygen atoms are shown as green and red spheres, respectively. The atom arrangement resembles a single layer of a (hypothetical112) 5–5 modification of BeO, rolled up into a tube. Taken together, the Be and O atoms occupy the same positions as the carbon atoms on a non-twisted single wall carbon nanotube.

3.11.7.1.3

Atoms, molecules and clusters on surfaces and substrates

Another group of structure prediction studies in (quasi-)low-dimensional systems focus on substrate based structures, where the low-dimensional system is either the surface layer of the bulk material, or the atoms, molecules and clusters that have been deposited on the surface. The difference to the “true” one- or two-dimensional systems are the interactions between the system of interest and the substrate. 3.11.7.1.3.1 Reorganization of surfaces In principle, the surface layer of a material can be treated as a quasi-2D-system, and its energy landscape yields information about the re-organization processes on a surface, and the minimum energy patterns of the atoms in the surface layer. For example, the atom arrangements on a ZnO surface have been investigated160,991 using the KLMC code141 with empirical potentials, and the structure patterns (formed by the surface atoms) with the lowest energies exhibited good agreement with experimental observations. 3.11.7.1.3.2 Particles on closed surfaces Atoms on spherical surfaces constitute prototypical examples of sets of particles located on closed surfaces. Thus, the energy landscape of the classical Thomson problem354dthe optimal distribution of point charges on a spherical surfacedwas studied as function of particle number (with up to 4352 atoms),992 using global optimization techniques. For the minima with the lowest energy,

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Fig. 11 Global minimum of an infinite Mg7Be5O12-single wall nanotube with prescribed radius, using empirical Coulomb þ Lennard-Jones potentials.163 Magnesium, beryllium and oxygen atoms are shown as blue, green and red spheres, respectively. The fourfold coordinated pattern inherited from the MgO structure holds sway in the nanotube. Note that the variable simulation cell contained only 24 atoms, and thus a (possibly preferred) phase separation into two essentially separate MgO and BeO tubes was not possible for the given periodic approximant; but also note the separation into Be-rich and Mg-rich halves of the tube. But there exist many other local minima in the system, where neither the square pattern nor the “separation” according to cation type is observed, and instead also hexagons, octagons, and many combinations of the various structure elements are found.

the number and type of defects in the optimized “lattice” of atoms on the surface was analyzed. Such defects357 cannot be avoided due to Euler’s formula,358 This analysis was repeated for other surfaces with constant mean curvature,994 finding satisfactory agreement with experiments on colloid systems. 3.11.7.1.3.3 Clusters on surfaces In the previous paragraph, we considered idealized particles restricted to given surfaces. Another real type of systems are clusters or single molecules on a (planar) surface or substrate, where a number of energy landscape investigations have been performed. Besides nano-sized Cu clusters on ZnO408 and PdAu and PdPt nanoparticles on MgO,995 the arrangements and stabilities of various intermetallic clusters on oxide surfaces have been studied. These involved structural comparisons of AuCu clusters in the gas phase and on oxides,996 and the analysis of the structure of several other nanoalloys in the gas phase and how the optimal atom configuration changes when the clusters are placed on TiO2997. 359 3.11.7.1.3.4 Thin films on surfaces There are many predictions of possible structures of individual organic and biological molecules on a surface,164 but mostly they are not based on truly global studies of the energy landscape of the system.360 An example of global and local energy landscape investigations of molecules and their monolayers on substrates, complemented by scanning tunneling microscopy (STM) investigations, is the analysis of the depositions of sucrose221 and trehalose999 molecules on Cu-surfaces.361 Similarly, there are global optimization studies of the energy landscape of a limited number of foreign atoms on substrate surfaces, where the goal is the identification of feasible (few layer) atom structures that might form on a substrate.119 Such studies are of particular interest since the structure candidates found often do not correspond to the bulk modifications, at least not to the global minimum structure for the bulk system. However, such unusual substrate-stabilized phases can allow us to realize a metastable bulk modification. 357

These defects are analogous to dislocations in regular crystals. For a simple polyhedron, which is topologically the image of a sphere, the formula V E þ F ¼ 2 must hold, where V is the number of vertices, E the number of edges and F the number of faces, respectively, of this polyhedron.993 Mathematically, this requirement implies, e.g., that a minimum energy fullerene-type isomer must contain 12 pentagons in addition to an arbitrary number of hexagons. All the additional hexagons behave as if they were in a two-dimensional plane, i.e., their additional vertices, faces and edges would fulfill the rule V E þ F ¼ 0. Adding more defects, e.g., heptagons, must necessarily require the addition of another pentagon (and conversely), to balance out the additional distortion. 359 Here specially developed genetic algorithms998 were employed for the global optimizations of clusters on surfaces using ab initio energy functions. 360 Frequently, such studies are complemented by experiments. This allows a direct verification of the predictions, in principle, but also often leads to a more or less strong guidance of the structure search by experimental information. 361 The theoretical part of this study consisted of a multi-stage modeling process. Using the IGLOO code,164,221 single sugar molecules on the Cu-surface were globally optimized with empirical potentials, followed by a local optimization on DFT level. Next, several optimized molecules were placed as rigid building units on an averaged periodic 2D-surface, and this periodic approximant was globally optimized. Comparing the resulting low-energy configurations with the STM data showed good agreement with the experiment. 358

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An example of such a substrate stabilized modification found during energy landscape studies, is the so-called 5–5-structure type362 for AB-systems,379 which has been predicted as a feasible bulk modification in, e.g., the alkali halides115,379 and earth alkaline oxides,112 among other systems,1000 and had first been synthetically realized as a ternary compound Li4SeO5.1001 Furthermore, based on theoretical studies, the 5–5-type structure has been suggested to exist in thin films of the alkali halides on LiNbO31002. 363 Recently, the 5–5-structure was discovered in deposits of NaCl on carbon surfaces (denoted hexagonal NaCl),1005 confirming earlier predictions that NaCl should exhibit the 5–5 type structure as a low-energy minimum (with an energy barrier of about  0.02 eV/atomddepending on choice of potentialdagainst transformation to the rocksalt global minimum structure)21,379; this deposition experiment was complemented by global optimization studies of the energy landscape of Na and Cl atoms placed on a graphite substrate.1005 As mentioned above, in many instances, the theoretical exploration of a deposit on a substrate is combined with an experiment. Another example is the study222 of two amorphous 2D polymers built up from precursor molecules. Starting with a random deposition of ca. 20,000 molecules (as simplified building units to allow the energy computations and MD simulations) on an idealized surface, the time evolution of the two polymers was investigated using molecular dynamics, and the energy of the system was analyzed as function of time. A logarithmically slow decrease in energy with time was observed, indicating a typical aging behavior seen in systems with a tendency towards amorphous structures, as one would expect from a system with such a complex energy landscape.149,484,1006 Concerning the structure of the polymers, a molecule-by-molecule comparison between simulation and experiment does not make much sense because the polymers are amorphous, but local structure properties such as the ring statistics showed satisfactory agreement between simulation and experiment.364

3.11.7.1.4

Clusters and large molecules

We just mention another very large group of chemical systems, for which an enormous number of structure predictions have been performed, and which have served as testing grounds for many exploration algorithms in complex landscape investigations: clusters103,120,124,133,135–137,139–141,255,259,550,624,708,879,998,1010–1021 and large molecules, in particular proteins and polymers,94,102,103,532,1022–1029 where in the case of proteins many studies also aimed at understanding the folding pathways of proteins.365 In the first class of systems, atoms, and possibly small molecules served as the basic elementary units used in describing and modeling the cluster, while in the second group, one usually assumes that the bond structure of the molecule itself does not change and only its shape varies.366 In particular, we note the availability of a database of already predicted clusters,1034 which also is expected to include surface structures in the future. For more information, we refer to the many reviews on cluster103,120,133,139,140,1010,1017,1034 and polymer/protein103,105,1026,1027,1029,1035–1038 structure prediction found in the literature.

3.11.7.1.5

Phase diagram prediction

A logical extension of straightforward structure prediction at zero temperature and pressure for fixed composition is the prediction of whole phase diagrams.89 By identifying the locally ergodic regions for a chemical system for a prescribed set of thermodynamic boundary conditions (pressure p, temperature T, chemical potential mi (i ¼ 1, ., m)dor alternatively many different but fixed P compositions (N1, ., Nm) ¼ (Nx1, ., Nxm) (with N ¼ im¼ 1Ni) for the m different types of atoms in the systemd, .), for the given observation time tobs, and computing their corresponding free energies, we are able to predict the phase diagram of the system, and thus make the final step from atomistic energy landscape to thermodynamic space.367 We note that the observation time tobs 362

The 5–5 structure consists of AB5 and BA5 trigonal bipyramids; topologically, it resembles the hexagonal BN structure, where the hexagonal BN-layers have been moved close to each other such that B and N are five-fold coordinated by N and B, respectively. Few-atom layer thick films of ZnO have been found to exhibit this structure type in the experiment.1003 Similarly, the diffraction pattern of MgO deposited on Al2O31004 agreed with the 5–5-structure type for very thin films of MgO. This interpretation was supported by global optimizations on the energy landscape of few atom layers of MgO on Al2O3,119 where the 5–5-type structure was obtained as a low-energy local minimum besides the standard rocksalt modification of MgO. 364 We note that there exist many simulations dealing with the deposition and/or rearrangement of atoms and molecules on surfaces, providing insights into the experimental observations and results,479,1007–1009 but most of them do not attempt to probe the structure of the energy landscape itself. 365 Note: 20 years ago, ca. 90% of the structure prediction literature dealt with proteins and polymers, ca. 9% with molecular crystals and clusters, and just 1% with inorganic crystal structures. 366 Here we note that recently machine learning algorithms have been employed to predict (folded) structures of long peptides, which was announced as a great breakthrough.1030 But such algorithms have already been employed for 30 years with the goal to predict the structure of proteins and their folding.1031–1033 From the point of view of energy landscapes, we note that the neural network actually cuts out the “middle man” by avoiding the study of the actual energy landscape that controls the structures the peptide can form. Instead, the machine “learns” how to associate an optimal three-dimensional structure directly with the secondary structure sequence of the peptide; this is very reminiscent of the earliest attempts of protein structure prediction by analogy based on the assumption that “similar sequence leads to similar structure”.102 367 So-called solid-solution phases can appear within certain temperature and concentration ranges for many chemical systems, which compete with ordered crystalline modifications. As mentioned earlier (c.f. Section 3.11.6.6), the crucial issue is whether so-called structure families exist among the minima observed for many different compositions which have essentially the same energy for a given composition.88,485 If that is the case, the union of these local minima can be treated as a large locally ergodic region, and the free energy of this alloy phase contains a configurational entropy contributiondessentially an entropy of mixing -, which favors the alloy over ordered crystalline compounds that are lower in energy but correspond to a single minimum basin on the energy landscape; here, vibrational contributions to the free energy are usually of secondary importance since they tend to be similar for ordered and disordered phases. Thus, it is important to insert a structure analysis step in which one compares all the minima and places them into structure families if appropriate. In the case of alloys, the minimum structures belonging to such a structure family can be identified by the fact that, structurally, they possess the same set of sublattices on which the atoms can be more or less randomly placed. In the case of solid solutions, the same overall cation-anion superstructure is present for many compositions in the quasi-binary, ternary, etc. system; the different types of cations and/or anions are randomly distributed over the cation and/or anion sublattices in the superstructure, respectively. 363

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appears as an additional coordinate in such an extended (or generalized or metastable) phase diagram; thus, for a given timescale tobs, one marks for each of the phases P i that can exist as metastable modifications in the system on some time scale, those parts of the thermodynamic (T, p, xi, .) space where P i is locally ergodic on the timescale tobs. Thus, for finite tobs, there will exist several possible metastable phases at every point in the (p, T, mi, .) or more commonly (p, T, xi, .) space, in addition to the thermodynamically stable equilibrium phase. Note that many of these phases will not be present for large observation times that exceed their limits of stability.368 As a consequence, the phase distributions in the cuts through the extended phases diagram will vary with tobs. The standard equilibrium phase diagram corresponds to the (tobs ¼ N)-slice of the extended (p, T, xi, .; tobs)-diagram189.369 Of course, there are also phase diagrams and phase transitions due to magnetic or electric interactions,1056 or quantum phase diagrams1057; we are not discussing these in this chapter, and refer to the literature. We also do not consider liquid crystals,1058,1059 or quasi-crystals,1060 and the prediction of their phase diagrams. Similarly, we do not address the issue of phase diagrams in the context of polymers94,1061,1062 or jammed matter.1063 We remark that we have implicitly assumed that the thermodynamic limit, N, V / N with N/V ¼ constant, applies. As mentioned earlier (c.f. Section 3.11.4.1.3), for chemical systems of finite size, e.g., large molecules or clusters, all regions of the landscape will be visited with a finite probability for sufficiently long experiments or simulations, at essentially all values of the thermodynamic parameters. Nevertheless, ‘phase diagrams’ of finite size systems such as clusters can be constructed although, e.g., the transition temperature between, say, the liquid and the solid phase would not be a precise number but we would define a transition temperature interval using statistical criteria.370 In this context, we note that, quite generally, the melting transition is not directly accessible via only studying the minima of the energy landscape. Here, one needs to perform simulations at high temperatures and compute the free energy difference between the solid and liquid phases.1065 Similarly, one can study the thermodynamics of structurally disordered systems such as glasses.1066 However, systems such as glasses are only marginally ergodic, and thus are never fully in local equilibrium at any time scale tobs beyond the vibrational time scale around individual minima; as a consequence, they cannot be represented in the extended phase diagram. The reason is that the extended phase diagram incorporates all phases that are locally ergodic on some range of timescales tobs and are thus at least metastable thermodynamically, but non-equilibrium states such as the marginally locally ergodic glassy region do not qualify as thermodynamically stable phases. The quasi-equilibrium thermodynamic behavior of such a system leads to, e.g., the afore-mentioned aging phenomena,271,1006,1067 which violate the thermodynamical stability requirements for a phase. 3.11.7.1.5.1 Prediction including compositional variation without experimental input Phase diagrams, with no input except the identity of the participating atoms, have been predicted for about 20 different alkali halide systems.88,188,1068–1071 For the low-temperature part of the phase diagrams below the solidus-liquidus region, the type of phasedsolid solution or ordered crystaldwas correctly obtained from unbiased global energy landscape explorations in all systems studied. Furthermore, for the solid solution phases, good quantitative agreement of the binodal curves with the available experimental data in the literature1072 was found. Similar work was performed for two quasi-ternary semiconductor systems, (Al,In,Ga)eSb90 and (Al,Ga,In)eAs.1073 We note that in several alkali halides metastable solid solution phases were obtained besides the thermodynamically stable one, leading to an extended phase diagram as function of observation time. An example is the LiBreNaBr system, where miscibility gaps for solid solutions were observed that exhibited not only the thermodynamically stable phase with a rocksalt superstructure, but also phases based on a NiAs, wurtzite and 5–5 type of structure were observed.189 Related predictions focused on possible metastable phases in the ZnO1–xSx system, where both the possibility of ordered and solid solution type phases for a variety of stacking polytypes were investigated.875 However, for all polytypes investigated, no ordered crystalline phases for 0 < x < 1 were found as the thermodynamically stable modification, although solid solution type phases are expected to exist at high temperatures. Although experimental input had been available, we also mention, as an example of the class of phase diagram predictions of quasi-2D systems, the theoretical “prediction” of the phase diagram associated with the oxide layers on the Ag(111) surface,1074 where (semi-)grand canonical simulations were employed. Other studies have involved the computation of the phase diagram of superionic ice at very high pressures and temperatures, using ab initio molecular dynamics starting from a bcc-structure,1075 and the prediction of the high-pressure phase diagram of the ternary Ca/B/H system, using an evolutionary search algorithm with DFT energies.1076

368 In Fig. 6, we have given a schematic plot of the time intervals of existence for various types of phases for a given pressure and two different temperatures. For a given tobs, many phases can be present, but they are not in equilibrium with each other, and thus Gibbs’s phase rule52 does not apply. Note that if we consider a region of the landscape that has been equilibrated on a long time scale, and we now pick special thermodynamic conditions where the free energies of two subsystems of the large locally ergodic region are the same, then Gibbs’s rule holds for these phases. In particular, we can derive various co-existence curves for this subset of phases, such as miscibility gaps, and we can use the convex hull construction1039 if necessary. 369 After the phases capable of existence for a given observation time have been identified, standard procedures799–802 can usually be used to compute their free energies and the phase boundaries of each (meta)stable phase for the individual tobs-slices of the extended phase diagram. We also note that we only discuss work that is concerned with the (extended) energy landscape of the chemical system; studies dealing with the computation of free energies and phase diagrams for a given set of phases185,775,1040–1048 or the refinement of experimental phase diagrams844,1049–1055 are not considered. 370 For example, one would declare the cluster to be in the solid and not in the liquid state, if at the given temperature the ratio of the probability to be in the solid state pS exceeds the one of the liquid state pL by some large factor, pS/pL  1012.1064 However, we do not discuss the prediction of such phase diagrams here.

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3.11.7.1.5.2 Prediction including compositional variation restricted to prescribed sub-lattices Many ab initio predictions of phase diagrams found in the literature assumed that all atoms have to be located on the sites of a defined set of sub-lattices usually known from experiment or based on analogies to chemically related systems.185,848,866–872 Frequently, the actual atom positions used in the calculation of, e.g., the free energy were obtained via local optimizations of the sub-lattice based structure candidates. Semi-grand canonical simulations were employed using ionic interatomic model potentials to compute phase diagrams of MgOCaO, MgO-MnO and (Ca,Mg)CO3,831,1077,1078 predicting that these systems should exhibit solid solution phases. Similar results were obtained in a study of the PdeRh system using embedded atom potentials1079 and cluster expansion methods.1080 Furthermore, the phase diagrams of LixCoO21081 and NaxCoO21082 have been successfully investigated, and in other studies that aimed to elucidate the effect of vacancies on the NiAl phase diagram1083 a new phase “NiAl2” stabilized by vacancies was found. A different approach was taken to determine possible binary ground states in the MoeTa system.849 Here, several million atom arrangements with up to 20 atoms/unit cell (on a bcc-lattice) were completely enumerated and their energies and thermodynamic weight computed. Similar investigations were performed for the CueAu and NiePt systems848 and the CuePd system,872 predicting new low-temperature structures for two of the compositions (7:1 and 3:1). 3.11.7.1.5.3 Prediction for fixed composition, as function of pressure and/or temperature Finally, we consider some examples from structure prediction at fixed composition but as a function of pressure and/or temperature.1084–1087 As mentioned earlier, there are many studies investigating phases at high pressures but at zero temperature, such as,106,1086,1088,1089 just to name a few in addition to those listed already earlier in this section. This work essentially corresponds to structure prediction using the potential enthalpy landscape instead of the potential energy landscape. Thus, standard global optimization techniques have been used, and we will not discuss these quasi-one-dimensional phase diagrams any further. An extension of this approach are studies, where the temperature has been varied on top of the sampling of different pressures. For example, in the study of WSe, standard crystal structure prediction for different pressures was supplemented by the computation of free energies at elevated temperatures.1090 In another study, a combination of potential enthalpy landscape exploration and machine learning estimates of free energies was used to generate a phase diagram of TiO2 as function of temperature and pressure.1091 Of more interest is the prediction of high-temperature phases that do not correspond to ordered crystalline modifications, including the investigation of phase transformations as function of temperature. Two main approaches are pursued to identify such phases with controlled disorder. One relies on identifying families of local minima and then verifying that, as a group, they have a noticeably lower free energy than the individual minima with the lowest energies, as discussed above. The alternative are long MC/MD simulations at elevated temperatures where one follows indicator variables along the trajectories, in order to check, whether the system stays within a local minimum region or spreads out over larger subsets of configuration space, similar to the ergodicity search algorithm (c.f. Section 3.11.6.6). An example are MD-simulations for KNO2, CsNO2 and TlNO2 crystals in large supercells containing several hundred atoms.765 The simulations were performed for many temperatures, with the expectation that the nitrite (NO2) group might exhibit unusual behavior. Several phase transition were observed that corresponded to the NO2 group acquiring more freedom of movement. In all systems, the nitrite group could rotate freely at high temperatures, on the time scale of observation. Treating the (NO2) complex anion as a single entity, an average structure of the NaCl- or CsCl-type was observed, in good agreement with the experiment. Similarly, the average structure of molybdenum was studied as function of temperature using ab initio molecular dynamics simulations.768 At elevated temperatures, Mo is observed in a fcc-lattice type phase, although the fcc packing is not a local minimum on the ab initio energy landscape and thus the fcc phase is unstable at zero temperature.371 Another example is the study of the (metastable) phase diagram of the hypothetical compound Si3B3N7 as function of density and temperature covering the solid, liquid and gaseous state.1092 Here, the effects of finite observation time and of the finite size of the (large) periodic approximant were discussed; in particular, the way the observational time scale influences our ability to distinguish between liquid and gaseous states, and the appearance of phase segregation into, e.g., solid þ gas regions in the simulation cell was investigated. Furthermore, the phase diagrams of benzene1093 and silicon409 have been computed as a function of pressure, using metadynamics.658 Finally, we mention the prediction of the phase diagram for crystalline methanol,1094 as an example of phase diagram prediction for molecular crystals.372

3.11.7.1.6

Time crystals

As discussed in Section 3.11.4.3.4, for time-varying energy landscapes, we cannot expect our usual locally ergodic regions on the landscape. On the other hand, it has been suggested1095,1096 that one should be able to define “time crystals”, whose structuredand possibly other propertiesdare periodic in time. These states of the system are not supposed to be periodic excitations (like vibrational modes), but should be the analogues of the ground state of a structural crystal, where the periodicity in R3 is replaced by a periodicity in time. Such a breaking of the time-translation symmetry is not possible for an isolated system or a system in regular 371

The low-temperature modification of Mo exhibits a bcc lattice. The fcc phase of Mo is estimated to be stable only for pressures larger than about 10 GPa. Recall that we only consider phase diagram prediction that involves also the prediction of structuresdthe plethora of predictions in the literature that take the structures as given and “only” compute the free energies, etc. are not discussed. 372

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equilibrium with the environment.1097 But if the system is out of equilibrium, e.g., in a stationary state, this is possible. In particular, one would expect this kind of state to appear for a periodically varying environment, with, e.g., periodic forces or periodically varying currents. If we observe the system at constant intervals, tp, which should be an integer multiple of the time period tvar of the driving force, tp ¼ ntvar, n > 1, then it is conceivable that the system exhibits time-crystallinity. Of course, the periodic interactions with the environment might enforce some periodic behavior synchronized (up to phase) with the external force, but such a periodically driven system1098 is not what is meant by a time-crystal. Instead, we are interested in structures that are not only stable for an essentially infinite number of periods, but also exhibit properties associated with a ground-state such as no net-heating of the material, i.e., no entropy production, caused by the periodically applied forces. Furthermore, the period tp of the time crystal should be larger than tvar, i.e., n > 1, in order to keep the time crystal distinct from a standard driven system.373 Creating such a time crystal in the experiment is non-trivial, and, up to now, examples are small finite systems, where the periodically varying degree(s) of freedom are the spins in, e.g., 1D chains of trapped ions1099 or nitrogen vacancy centers.1100 Nevertheless, complex chemical systems with a time-crystalline ground state with regard to atom positions might be feasible. In particular, possible time-crystalline systems can also be envisioned in clusters or solids. For example, crystalline structures where few degenerate local minima separated by small energy barriers exist as in the case of complex anions that can exhibit different orientations, might be suitable candidates. Usually, these systems are frozen at low temperature into one of the anion orientations while at high temperature these anions occupy all orientations with equal probability on the typical observational time scale. However, if we can force this flipping or rotational motion of the anions to occur already at low temperature via some periodic interaction, a timecrystalline state could be created if the natural transition frequency between the minima can be synchronized with the external periodic force. Similarly, in clusters certain re-arrangements corresponding to back-and-forth jumps between neighboring minima could serve as the template for the construction of a “time isomer cluster”. Clearly, predicting which system would be a suitable candidate for such a state involves both the determination of the relevant local minima on the energy landscape and the study of the energy barriers separating them, as function of time, and the investigation of the probability flows on the time-dependent landscape.

3.11.7.2

Structure determination

A highly important application of cost function landscapes in chemistry is the structure determination from limited experimental information about the synthesized compound. In particular if no single-crystal X-ray data are available, and one needs to solve the structure of the (microcrystalline) solid based on powder diffraction data,374 computational approaches are greatly helpful, and appropriate algorithms and computer programs have been developed.245,1102–1109 In general, determining a structure from powder data consists of a number of individual tasks, ranging from the determination of the cell parameters to the refinement of the atomic positions, using, e.g., the Rietveld method.1110 Each step requires its own (preferably automated) procedure, but usually the most difficult step is the determination of chemically and physically reasonable structure candidates for the final refinement. Regarding the determination of feasible structure candidates, the methods employed differ by the amount of information about the chemical system available from experiment, and by how much information about the potential energy landscape is included in the cost function. The latter can range from full (ab initio or empirical potential) energies to simple penalty terms that only prevent unphysically short atom-atom distances.

3.11.7.2.1

General inverse problem

In the case of the basic inverse problem, no reference is made in the cost function regarding the energy of the chemical system, and just the agreement with the experiment serves as the optimization criterion.375 The most well-known approach is the so-called reverse-Monte-Carlo method (RMC) where the cost function equals the (square of) the difference between measured and computed diffractogram Rdiff.1108,1111,1112, 376 Originally, this method was used to deduce possible structures of amorphous compounds and glasses, but by now many structure solution codes contain such an algorithm, such as the programs ESPOIR1108 or GSAS.245 Another implementation of the RMC-method has been used for lattice and magnetic structure analysis, where RMC is combined with neutron powder diffraction data.1113 Similarly, atom distributions1114 and NMR spectra assignments1115 have been obtained from NMR data using global optimization techniques. 373 Of course, a time-crystallinity with, e.g., n ¼ 2 might correspond to (or be interpreted as) higher order (second harmonic) effects of the driving force, which might appear for a non-linear system-environment interaction. As usual, such concepts and phenomena can be viewed from many perspectives, since they all must rely on the underlying laws of classical and quantum physics. 374 The need for automated procedures for structure solution from limited data is rapidly increasing considering the industrial scale generation of new materials, symbolized, e.g., by the so-called combinatorial (solid state) chemistry.1101 Sadly, this multitude of compounds and modifications in the pipeline of the experimental chemist and material scientist has become a towering tidal wave of structural challenges, and thus has led to presentations of new materials for applications (and perhaps even patents) without actually having solved the structure of the material! 375 There is still quite some freedom in the construction of the cost function. For example, one can assign different weights to the quantitative agreement of the computed and the measured intensities of specific peaks in the spectrum or the diffractogram, where the total cost function is the sum of the peak-by-peak differences multiplied with the corresponding weight factors. 376 Since a pure RMC often produces unphysically short atom-atom distances, one usually supplements it by some restriction on how close two atoms are allowed to be in the proposed microstates, either via penalty terms or restrictions on the space of allowed atom configurations.

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3.11.7.2.2

Reduced search space

3.11.7.2.3

Cost function as combination of potential energy and experimental data

Another general approach first extracts particular chemical and/or crystallographic information from the experiment, which is “directly” accessible from a preliminary analysis of the data, e.g., unit cell parameters, space group symmetries, unit cell content, atom-atom distances (taken, e.g., from similar compounds consisting of the same atoms), etc., and uses this information to greatly restrict the state space of the global search. Thus, the complexity of the cost function landscape is reduced, considerably simplifying the global optimization. Many such studies have been performed.406,447,552,906,1116 Different global optimization methods (simulated annealing, genetic algorithms, etc.) have been employed to solve the structure of crystalline compounds such as NbF4447 and Li3RuO4.622 We note that the difference to the restricted structure predictions mentioned several times in Section 3.11.7.1 is only one of degree, as far as the mathematics is concerned. The main distinction is the fact that for structure determinations one usually has reliable information about the unit cell of the periodic structure. Furthermore, such cell information sometimes allows us to replace the empirical potentials by even more simplified chemically inspired cost functions based, e.g., on the bond-valence potentials.1117–1119

A third method to determine the structure of a chemical system modifies the cost function by adding the potential energy or enthalpy to the term in the cost function representing the agreement between calculated and measured data. A first study in this direction dealt with the structure determination of zeolites.1120,1121 Subsequently, the agreement with the diffractogram Rdiff and the potential energy Epot were combined in the cost function on an equal footing,277 C ¼ lEpot þ (1-l)Rdiff. In such a so-called Pareto-optimization,253 the parameter l (0  l  1) allows us to assign a relative weight to the energy and the diffractogram in the cost function, depending on the quality of each term.377 We note that the value of l can be varied during the global optimization, in order to allow a more efficient access to the deep-lying minima that should correspond to the unknown structures of the crystalline compound.378 Over the past two decades, this approach has been successfully used for the structure determination in a large number of ionic, quasi-ionic, and metallic crystalline systems,277,1122,1123 and, furthermore, for molecular crystals such as a-L-glutamic acid.1123 Here, simulated annealing and genetic algorithms, and modifications thereof were used as global optimization tools. One implementation of such a Pareto optimization is the ENDEAVOUR program,1124 which has been used in the structure determination of many different systems, such as Na3PSO3,1125 Ag2NiO2,1126 Ag2PdO2,1127 K2CN2,1128 sulfur,1129 GaAsO4,1130 ammonium metatungstate,1131 the zeolite-like structure Na1–xGe3 þ z,1132 Tl2CS3,1133 a high-pressure phase of MgSiO3,1134 polymeric carbon dioxide,1135 BaTi2O5,1136 porous molecular crystals1137 and BiB3O6,1138 just to name a few.379

3.11.7.3

Barrier studies

Next to the determination of local minima and the low-lying saddle points that connect them, perhaps the most common energy landscape studies in the literature are the exploration of the structure of the barrier regions connecting two minima or minima basins. As mentioned earlier, in general, these can be split into two types of explorations, those where the starting and end point are very close to each otherdperhaps only separated by a single saddle point thus essentially defining the connecting path implicitly or explicitlyd, and those where the two minima are farther apart and the connecting path is unknown. Regarding applications of such barrier investigations, many studies involve chemical reactions, cluster isomerizations, or phase transformations. The application of the various methods tends to be quite straightforward for the case of finite molecules and clusters, but requires more care for the case of solids. In the case of solid-solid transitions, simple transition paths can usually be identified for second order phase transitions, since these transformations typically involve synchronized movements of the atoms in neighboring cells or small supercells. Such displacive transformations are commonly associated with zero (or nearly zero) frequency modes in the phonon spectrum,380 which appear, e.g., as function of pressure or due to re-arrangements of the electronic structure.381 But for general phase transitions in solids, one must keep in mind that the computational limitations force us to use small periodic approximants when describing the transformation. This usually makes it very difficult to identify the true nucleation and growth mechanism unless soft modes dominate the transformation. While such a movement is usually associated with a second 377 In addition to using the Pareto cost function, we can include additional information from the experiment, such as cell parameters, known positions of some of the atoms (which should be kept fixed during the optimization), or the existence of (rigid or flexible) building units such as molecules or complex ions in the compound under study. This would not affect the cost function but restrict the state space (the set of allowed microstates or atom þ cell configurations) and the moveclass during the global search, compared to free movements of the atoms. 378 This is another example of a variation of the energy landscape with “simulation” or “progress” time. However, since we are not trying to reproduce a physical time evolution, this does not pose any fundamental problems; nevertheless, one must be aware that by such a reweighting many local minima will be modified, eliminated or newly created on the landscape. 379 Clearly, a Pareto optimization can be used for any combination of terms in the cost function, e.g., one could combine the difference between a measured and computed NMR spectrum with the difference between a measured and computed powder diffractogram, and a potential energy term in addition; in this P case, we would have three parameters li ˛ [0, 1] (i ¼ 1, 2, 3), which must fulfill the condition ili ¼ 1. 380 Such very “soft”, i.e., long wave-length, modes usually correspond to shearing-like movements that involve an in-phase shift of equivalent atoms in neighbor cells (repeated over mesoscopic or macroscopic numbers of cells). 381 Examples are possible phase transformations in PbS951 or TiSe2.1139

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order phase transition, we note that this type of transformation route sometimes also can appear during first order phase transitions in bulk crystals if the energy barrier against the soft-mode type transition is small enough: in that case, the barrier in the free energy of activation that needs to be crossed when initiating the “soft-mode” transition is small enough such that this mechanism can be statistically preferred over the usual nucleation-and-growth driven mechanism.

3.11.7.3.1

Nudged elastic band methods

Many variations of the nudged elastic band method (NEB) exist that are used to explore the connectivity of nearby local minima and transitions paths (c.f. Sections 3.11.6.2 and 3.11.6.4), and one general feature of the original NEB is the requirement that both the original configuration and the final configuration of all the participating atoms must be prescribed, before the transition path between them is explored. As we discussed in the context of periodic approximants of crystalline solids, this poses a problem since many approximants describe the same crystal, and, furthermore, while atoms are uniquely defined in the calculations, they are indistinguishable in reality, and thus assigning the “wrong” atom pairs between initial and final configuration can also yield nonsensical or inefficient paths. As a consequence, the dimer method has been developed, which can take care of this problem, by leaving the choice of final path partly open,721 and this has been extended from molecule-molecule to solid-solid transitions,1140 and systems at finite temperature.1141 Since we are not dealing with a global landscape exploration, ab initio energy functions are often employed, in addition to empirical potentials, when using nudged elastic band methods. 3.11.7.3.1.1 Chemical reactions An important application of the dimer method has been to the study of chemical reactions. Several variations of this approach were tested at the example of 24 different reactions1142 on the quantum mechanical level using the so-called growing string method. Other examples of various atom rearrangements deal with small islands on surfaces,1143 the enzyme PHBH,1144 the di-alaninepeptide,1141 interstitial clusters in a-Fe,1145 or the Au/g-Al2O3 model catalyst.1146 3.11.7.3.1.2 Solid-solid transitions An application of the dimer method in solids is the system CdSe, where the wurtzite to rocksalt transition was studied, together with a construction of a disconnectivity graph.1140 Fig. 12 shows two different low-energy transition (soft shear) paths between the wurtzite and rocksalt modification for a small periodic approximant. Similarly, the A15 to bcc transition in Mo was studied with this approach, demonstrating the existence of different transformation mechanisms.1140

wurtzite

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Fig. 12 Two different (soft shear) transition routes between two modifications, wurtzite and rocksalt, in CdSe, using a small periodic approximant, and an empirical Coulomb þ Lennard-Jones potential.1140 Cd atoms and Se atoms are given in red and green, respectively. Note how the saddle structures resemble the 5–5-structure and the GeP structure (shown to the right: grey and red spheres correspond to cation and anion positions in these structure types which consist of trigonal bipyramids and square pyramids of the cations coordinated by the anions, respectively), which had been observed in energy landscape studies724,1147 of the wurtzite-rocksalt transition in ZnO. The figure is adapted from reference Xiao, P.; Sheppard, D.; Rogal, J.; Henkelman, G. Solid State Dimer Method for Calculating Solid-Solid Phase Transitions. J. Chem. Phys. 140 (2014) 174104 with permission by American Institute of Physics 2021.

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Fig. 13 Left: Ab initio (DFT) energy landscape of a single bismuth atom on a Si(100) surface, which exhibits dimerization of the surface Si-atoms (1/4 of the supercell studied is shown), derived from local relaxations and NEB studies.1155,1156 The rows of Si-dimers are indicated by the grey stripes. Dark blue marks regions with minima of the potential energy. Right: Low-energy relaxation paths of movement of single bismuth atoms on this surface. Note that in the region between the rows of silicon surface dimers, the landscape is quite flat and thus the local relaxations can stop on saddle-like regions. The figure is adapted from reference Huang, H. Preparation of Bismuth Single Atom on Silicon (100) Surface, MS thesis, Stuttgart University: Stuttgart, Germany (2017).

3.11.7.3.1.3 Atoms on surfaces A final common application of NEB methods is the study of mobility barriers of atoms, vacancies and molecules on surfaces and in solids. Both the barriers and the specific diffusion mechanism are studied. Examples are vacancy diffusion in Ga2O3,1148 carbon and vacancy diffusion in GaN,1149 surface diffusion barriers controlling Zn-dendrite growth,1150 bulk vacancy migration in 47 different elements that form fcc structures,1151 Fe-adatom diffusion on different Fe-surfaces,1152 molecule diffusion through zeolite pores,1153 diffusion of water-monomers, -dimers and -trimers on Pt(111) surfaces,1154 or the mobility of bismuth atoms on Si(100) surfaces.1155,1156 Fig. 13 shows the ab initio potential energy landscape of a single bismuth atom on a Si(100) surface, which exhibits dimerization of the Si-atoms, which had been deduced from low energy relaxations and NEB calculations.

3.11.7.3.2

Prescribed path method

3.11.7.3.3

RRT based explorations

The prescribed path method485,724 (c.f. Section 3.11.6.4) belongs to the general class of path-oriented exploration methods like the NEB and dimer procedures, insofar as a starting and a final configuration are given. But it can be employed both for nearby local minima and for quite broadly exploring the general landscape between starting and target regions that are far away from each other, where many other local minima possibly exhibiting a wide range of structures can exist in the neighborhood of the path. The prescribed path algorithm has been applied to the study of rotational movement of complex anions in solids, such the NO2 group in KNO2,485 the systematic exchange of atoms in solid solutionsdas one would perform, e.g., in the context of the prediction of solid solution phases88d, and the exploration of the energy landscape of crystalline ZnO concerning possible transformation paths between the major modifications, wurtzite, sphalerite, and rocksalt (c.f. Fig. 14).724

One newly popular approach, especially in the context of biological (chain) molecules but equally applicable to inorganic systems, is the use of the rapidly exploring tree methods (RRT),340 where one fills the whole configuration space using a graph-like structure, often consisting of the union of many trees starting from different initial states (c.f. Sections 3.11.6.1.7 and 3.11.2.2.2). An example is the study of the landscape of the dialanine peptide,336 projected to the Ramachandran coordinates, with the goal to efficiently identify transition paths and local minimum configurations (c.f. Fig. 15).

3.11.7.3.4

Metadynamics and metashooting

In principle, landscape explorations based on metadynamics (c.f. Section 3.11.6.4) and related algorithms such as metashooting1158 where metadynamics is combined with transition path sampling, are “open-ended” in the sense that one usually does not prescribe a particular target structure, and thus they can be also employed to find candidates for new modifications in the chemical system, similar to the lid and threshold algorithm274,295 (c.f. Section 3.11.6.3). However, in many applications, one focusses on one particular target structure or group of such structures. 3.11.7.3.4.1 Metadynamics studies Examples of metadynamics investigations have often dealt with phase transitions,659,1089 e.g., in CdSe1159 or SiO2.900 Another frequent application has been the search for high-pressure modifications, such as in Si,409 or alternative modifications, in general.1093 But there have also been explorations of chemical reactions, of course, using metadynamics.514 And finally, one can try to extract estimates of (local) densities of states1160 and construct free energy landscapes.739,1161

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5-5

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GeP

H.P.

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Fig. 14 Pictorial summary of prescribed path investigations in the ZnO system, where the landscape between the three major experimentally observed (bulk) structures in the ZnO system, wurtzite-, sphalerite- and rock salt-type, were studied. The general direction of the transitions explored are indicated by black arrows.724 The (number of) wavy black lines corresponds to the relative size of the barrier along the prescribed path. The blue arrows represent low-temperature transition routes, and the red arrows represent high-temperature transition routes, respectively. Note that the 5–5 structure type, which has been observed in the experiment for few-layer ZnO films,1003,1157 is encountered as a side-minimum along the low-energy paths emanating from the wurtzite structure, together with the GeP structure type. Furthermore, several transition routes are only observed to be active in one direction, according to the transition path sampling calculations.1147 The figure is adapted from reference Zagorac, D.; Schön, J. C.; Jansen, M. Energy Landscape Investigations Using the Prescribed Path Method in the ZnO System. J. Phys. Chem. C 116 (2012), 16726–16739 with permission by American Chemical Society, 2021 and includes information from reference Boulfelfel, S. E.; Leoni, S. Competing Intermediates in the Pressure-Induced Wurtzite to Rocksalt Phase Transition in ZnO. Phys. Rev. B 78 (2008), 125204.

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Fig. 15 Landscape exploration with the T-RRT algorithm. (Left) Tree generated by the T-RRT algorithm for a fictive 2D energy landscape. It illustrates the ability of the algorithm to explore low-energy and transition areas while avoiding high-energy regions. Note the homogeneous covering of the low-energy part of the configuration space. (Right) Transition paths generated by several executions of the T-RRT algorithm between two local minima of the alanine dipeptide peptide, projected on the Ramachandran angles of the molecule. The routes shown depict a sample of the transition paths from the basin C7ax to the basin C5 (indicated by large white dots), using the AMBER parm96 force field.308 The figure is adapted from reference Jaillet, L.; Corcho, F. J.; Perez, J.-J.; Cortes, J. Randomized Tree Construction Algorithm to Explore Energy Landscapes. J. Comput. Chem. 32 (2011), 3464–3474 with permission by John Wiley and Sons, 2021.

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3.11.7.3.4.2 Metashooting of ZnO Features of the energy/enthalpy landscape of crystalline ZnO were explored using the metashooting approach,1147,1158 to study the pressure-induced wurtzite/sphalerite (B4/B3) to rock-salt (B1) phase transition. Here, the transition path sampling feature of the metashooting procedure1158 was employed to explore the sequence of nucleation and growth events between mixed B4/B3 and B1 structures. Several competing intermediate phases were observed along some of the trajectories, exhibiting, e.g., the 5–5 structure379 and GeP structure types. Interestingly, while the hexagonal 5–5 structure intermediate could indeed be visited during the transition, it was not necessary to form this intermediate structure when transforming from the high-pressure B1 to the standard pressure B3 or B4 structures.382

3.11.7.3.5

Phase transition via free energy landscape investigation

As mentioned in the previous subsections, structural phase transitions can be explored by analyzing the barrier structure of the energy landscape of a system, frequently as function of pressure, and transformations mechanisms can be identified. In most cases, these investigations rely on molecular dynamics simulations, combined with penalty terms in the case of metadynamics or using tools from transition path sampling or nudged elastic band schemes. A different type of study is the detailed investigation of the energy landscape of the displacive phase transition in V2O4 using refined empirical potentials.1162 Here, the free energy was included in the quasi-harmonic approximation, and the mechanism of the transition could be elucidated from the evolution of the free energy landscapedplotted over a simplified configuration space spanned by two soft-phonon modes that served as order parameters. Furthermore, the transition temperature was found in good agreement with the experiment (see Fig. 16).

3.11.7.4

Probability flows

The term “probability flow” is probably most familiar from network dynamics or flows of probability through networks.1163 The connection to chemical systems is twofold: For one, we model probability flows via master equation modeling of the dynamics based on the network nodes, local densities and transition probabilities that have been extracted from global landscape explorations using a coarsening procedure as discussed earlier (c.f. Section 3.11.6.5). The second connection is the direct “measurement” of the probability flows on the energy landscape extracted from dynamical simulations on the landscape, using MC or MD simulations, under certain boundary conditions, such as temperature, or allowed energy ranges. Here, we mention some examples of probability flows that have been derived from simulations; master equation approaches are covered in other subsections (c.f. Sections 3.11.4.2.1 and 3.11.7.7.1). Perhaps the earliest systematic study of probability flows on complex landscapes studied the multi-basin dynamics of a protein in aqueous solution1164 and subsequently in a crystalline environment.291 From this, it was possible to derive transition probabilities between the basins and derive tree graph representations analogous to those tree graphs derived from the threshold295 and lid274 algorithms, which subsequently served as prototypes of the popular disconnectivity diagrams320 and disconnectivity trees.321 In practice, the simulations that allow us to measure the probability flow are performed for some constant temperature. By performing them for many temperatures, one can deduce transition probabilities as function of temperature and thus extract, e.g., effective (free) energies of activation, if the transition probabilities exhibit an Arrhenius-like behavior as function of temperature. However, there is a serious problem of time scales, since for the low realistic temperatures the transition rates are very small: we are dealing with so-called rare-event problems (c.f. Section 3.11.6.7.2). If one circumvents these sampling rate problems by steering the dynamics via reaction coordinates of some kind or by accelerating the dynamics in some fashion, one might still be able to deduce the energetic part of the generalized barrier between the starting and the end point of the transition, but one nearly invariably loses the entropic aspect of the barrier that describes the many possible ways the system can take off into the wrong directions.383 As an alternative, one can employ the threshold algorithm (c.f. Section 3.11.6.1.6) to study the connectivity of the landscape within a sequence of energy ranges. In this fashion, the probability flows measure only the entropic barrier; however, as a consequence, one needs to combine the explorations for many energy lids when one constructs the Markov matrices that describe the probability flows as function of temperature. This method has been employed for many systems, such as clusters258,259,421,707,708,1018 where in some instances comparisons between the empirical potential and the corresponding ab initio landscape were performed, various disaccharide molecules,273,709 and solids.112,294 But probability flows on landscapes have also been studied for biomolecules, e.g., the probability to reach certain parts of the kinase energy landscape in enzyme controlled reactions.1165 However, in most instances, the probability flows are studied on the level of master equation models,328,1166–1168 which are sometimes called Markov state models in this community.

382 On the other hand, the transition to the high pressure rocksalt type structure appeared to always involve the generation of local motifs of the hexagonal 5–5 structure minimum on the landscape, before transforming to the final rock-salt (B1) modification. 383 We remark that while steered dynamics essentially always underestimates the size of the entropic barrier, an accelerated dynamics can lead to an overestimate since, e.g., simply applying a very high temperature can open up so many more paths to the system that the entropic barrier we measure is greatly increased. Of course, one will try to adjust the results to take such issues into account, but the general problem usually cannot be eliminated.

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Fig. 16 Free energy landscape (including phonon contribution in the quasi-harmonic approximation) of V2O4.1162 First row: Potential energy (“static lattice energy”) landscape (left) and free energy landscape at T ¼ 0 K (right); the (free) energy is given in eV per (monoclinic) unit cell.1162 A refined empirical potential combining a Buckingham potential with a shell model and a Morse-potential was employed. Note the strong effect of the zeropoint vibrations on the free energy, which eliminate the small local minimum representing the tetragonal phase on top of the center hill of the landscape. Furthermore, they lead to a splitting of the four equivalent global minima representing the monoclinic phase into two slightly different pairs of two equivalent minima, and the discontinuity between the locally ergodic region corresponding to the tetragonal phase and the four subregions together constituting the locally ergodic region of the monoclinic phase indicates that the displacive transition is as first order phase transition. Rows 2–4: Free energy landscapes at various temperatures, T ˛ [10, 500] K, with contours shown beneath, as a function of order parameters L1 and L2 of V2O4 (the frequency cut-off in the computation of the phonon free energy differs from the one in the T ¼ 0 K plot). The order parameters indicate the deformation between the monoclinic and tetragonal (rutile) phase of V2O4. The estimated transition temperature from the monoclinic low-temperature phase to the tetragonal high-temperature phase (Tsim z 300 K) is in good agreement with the experiment (Texp z 340 K). The figure is adapted from reference Woodley, S. M. The Mechanism of the Displacive Phase Transition in Vanadium Dioxide. Chem. Phys. Lett. 453 (2008), 167–172 with permission by Elsevier, 2021.

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3.11.7.5

Graph representations of energy landscapes

Due to the high complexity of multi-minima energy landscapes, two-dimensional cuts through the landscape, projections into two dimensions, or using two “characteristic” degrees of freedomdfor example, two cell parameters, or the unit cell volume together with the average coordination number of (some of) the atomsd, i.e., visual depictions in essentially two dimensions, can only provide a very limited impression of the true complexity of the landscape. Still, having only two (or three) dimensions available for a model of the landscape, many approaches have focused on certain features of the landscape at the neglect of others. As discussed earlier (c.f. Section 3.11.2.2.2), disconnectivity type or lumped tree graphs consider the energy barriers associated with the successive lumping of larger and larger basins, while probability flow graphs concentrate on the likelihood of reaching neighbor basins using the edges between the nodes to represent the relative sizes of the flows. Such graphs are often modified by adding other features, e.g., return barriers or free energies, on top of the energies of the minima, etc. Free energy graphs yield some information about the thermodynamic stability of the system, but we can also follow the way the system equilibrates via mergers of basins into superbasins, by generating so-called equilibration trees274 (c.f. Section 3.11.4.2.2). Quite generally, however, one must note that the vast majority of energy landscape studies in chemical systems has only focused on the minima themselves since they provide insights into possible metastable compounds and modifications atdnowadaysdrelatively small computational effort, while the generation of even approximate graph representations usually requires at least one order of magnitude more computer time. As a consequence of this computational cost, most of the studies that include barriers and minima in a visual representation have been performed with empirical potentials for small systems, such as clusters or crystalline solids with few atoms in the periodic approximant.

3.11.7.5.1

Tree/disconnectivity graphs

3.11.7.5.1.1 3D solids For three-dimensional solids, most of the work on tree graph representations has been performed for simple crystalline systems, such as the alkali metal fluorides MgF2 and CaF2,294 the alkaline earth oxides BeO, MgO, SrO and BaO,111 Lennard-Jones crystals,1169,1170 Na3N,704 Mg2OF2386 or Ca3SiBr2.387 Other investigations have led to tree graphs of amorphous materials and liquids, in particular for (binary) Lennard-Jones systems62,482,1171 and silica.80 3.11.7.5.1.2 Clusters and molecules Compared to the three-dimensional bulk systems, the number of tree graph representations of clusters and molecules is considerably larger, and landscapes of clusters or molecules have often been employed to test and develop new exploration algorithms and landscape representations. Since the energy calculations for clusters are not as expensive as for solids, it has also been possible to generate tree graphs on the ab initio level for small ionic258,707 and intermetallic clusters.259 For the Cu4Ag4 cluster discussed in Ref. 259 it is interesting to see the comparison between the DFT and Gupta potential based tree graphs. Apart from the fact that not all DFT minima were minima on the empirical potential landscape, and conversely, the typical barrier heights were also different by a factor of nearly four (c.f. Fig. 17). But most of the studies still employ empirical or semi-empirical potentials.292,320,321,326,708 Recently, it has become possible to generate such tree graphs using DFTB (density functional tight binding) as was shown in a study of the Au20 cluster421 (c.f. Fig. 18). For more examples, we refer to the literature.103,133,159

3.11.7.5.2

Free energy landscape

Both the energies and the barriers between locally ergodic regions should be replaced by the free energies and the so-called free energy barriers, once we consider the landscape on longer observation times tobs. Ideally, one would depict the landscape at a sequence of observation times, with a different shape of the tree graph for every observation time. However, one often just plots the energy barrier structure and associates with each minimum or basin its corresponding free energy, and employs, e.g., return probabilities294 as representative of the entropic barriers surrounding each minimum (basin). 3.11.7.5.2.1 3D solids An example for such a simplified free energy landscape for a crystalline solid is the study of SrO111,187 as function of pressure and temperature. For several different pressures, the minima of the potential enthalpy landscape were obtained, and subsequently, the ergodicity search algorithm187 was employed, in order to identify possible high-temperature phases that were not associated with individual local minima. The free energies were computed for each minimum structure in the quasi-harmonic approximation using DFT with a B3LYP functional (see Fig. 19). 3.11.7.5.2.2 Clusters and molecules For clusters and molecules, several such free-energy modified tree graphs can be found in the literature, such as for small LennardJones clusters and alanine,1172 but also for proteins.322 However, such trees usually only present the energy barriers, connecting the various regions that are locally ergodic at a given observation time. A greater focus on free energy barriers and their observation time dependence is a study of the lumped free energy representation for the so-called Holliday junctions300; however, in that work no tree graph was derived.

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Fig. 17 Tree graphs and important low energy structures of a Cu4Ag4 cluster.259 (a) Tree graph of the Gupta-potential energy landscape of the system, (b) Tree graph of the DFT energy landscape of the system. Structurally related local minima are depicted with the same color; the energy is in units of eV/atom. Note that the structural variety of the DFT landscape is larger than the one of the Gupta landscape; furthermore, the energy barriers of the DFT landscape are larger on average than those on the Gupta landscape, (c) Global minimum of the Gupta landscape (type: dodec); (d) Lowest energy minimum of the McPB class of structures on the Gupta landscape; (e) Global minimum of the DFT landscape (type: TcTd); (f) Second lowest minimum on the DFT landscape (type: Buck-P). Note that the Buck-P-type structure does not constitute a minimum on the Gupta landscape; and the global minima for the two energy landscapes are also different. Cu and Ag atoms correspond to reddish and grey spheres, respectively. The figure is adapted from reference Heard, C. J.; Schön, J. C.; Johnston, R. L. Energy Landscape Exploration of Sub-Nanometre Copper-Silver Clusters. Chem. Phys. Chem. 16 (2015), 1461–1469 with permission by John Wiley and Sons, 2021).

421 Fig. 18 Tree graph of a Au() The different labels refer to groups of 20 cluster, using DFTB energies, together with five structures of local minima. structurally closely related individual minima; the ground state (G1) exhibits a tetrahedral arrangement of the 20 Au atoms, and the other local minima correspond to various deviations from this arrangement. Note that grouping structurally related local minima into multi-minima basins for the purpose of analyzing the probability flows on a complex energy landscape where a chemically useful characterization requires going beyond individual minima to “order-parameter-like” classifications, and developing the conceptual and practical tools do deal with such systems, is an active area of research in the field. The figure is adapted from reference Rapacioli, M.; Tarrat, N.; Schön, J. C. Exploring Energy Landscapes at the DFTB Quantum Level Using the Threshold Algorithm: The Case of the Anionic Metal Cluster. Theor. Chem. Acc. 140 (2021), 85 with permission by Springer Nature, 2021.

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Fig. 19 Free energy tree of SrO at standard pressure, using empirical potentials for the global explorations112 and the high-temperature ESA simulations, and ab initio energies for the structure refinement and the computation of the phonon contributions to the free energy.187 Note that for very low temperatures, the free energy is higher than the energy of the electronic ground state (indicated by the white circle) for the various modifications, because we need to add the zero-point energy of the vibrations. The barriers indicated in the figure are only the energetic barriers indicated by the white circles where different branches of the tree merge; adding the entropic barriers would have made the figure too difficult to read. No high-temperature modifications were observed in long Monte Carlo simulations using the ergodicity search algorithm (ESA) up to the melting temperature; thus, all the modifications shown correspond to local minima of the potential energy. The figure is adapted from reference  carevic, Z. P.; Hannemann, A.; Jansen, M. Free Enthalpy Landscape of SrO. J. Chem. Phys. 128 (2008), 194712 with permission by Schön, J. C.; Can American Institute of Physics, 2021.

3.11.7.5.3

Equilibration trees

Finally, we mention an example for the so-called equilibration trees, the 2D neon system in a periodic approximant with varying cell parameters, using a Lennard-Jones potential with parameters fitted to neon.296 Besides the ground state consisting of a close packing of spheres, the side-minima with the lowest energies were determined, which exhibited one or two defects (corresponding to missing atoms in the 2D close packing) at different distances per simulation cell. After determining their lumped tree graph representation, the local densities of states of each basin up to and beyond the energies where they merged were evaluated. From this information, a master equation level description of the system was derived, and the time evolution of the system, for given temperatures, was followed, where the time was registered, when the occupancy ratio for pairs of the basins (and pairs of already equilibrated sets of basins) became equal to their equilibrium Boltzmann ratio. In this fashion, an equilibration tree for the system was constructed (c.f. Fig. 20).

3.11.7.6

Liquids and glasses

We recall that the original motivation to study complex energy landscapes in chemistry was the non-equilibrium thermodynamics and statistical mechanics of glasses, their structure, the glass transition and their aging behavior, and the related issue of the dynamics and structure of liquids, where time scales are of importance when trying to distinguish between a liquid and a gas.1092,1173 Trying to address these issues from a conceptual point of view instead of just running extremely long MD and MC simulations, led to the use of energy landscapes, starting with the local minima accessible to the system as function of

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temperaturedthe so-called inherent structures1174d, over the size and growth of (locally) equilibrated but not locally ergodic regions, to the barrier structure and its influence on the dynamics.

3.11.7.6.1

Structural liquids and glasses

3.11.7.6.2

Aging phenomena

Inherent structure studies287 where one explores the range of accessible local minima via periodic minimizations from stopping points along some simulation trajectory or from otherwise generated (random) starting points have been performed for over 40 years by now, and many such studies can be found in the literature. The systems investigated range from artificial potential landscapes,260 solid He4,1175 the ubiquitous Lennard Jones system72,1176,384 water,1181,1182 molten salts,1183 hydrocarbons,1184 proteins,1185 to crystalline systems such as pentacene1186 or tetracene.1187 Many of these inherent structure analyses are performed in the context of studying the glass transition1188,1189 or, more recently, the jamming transition1190.385

Besides the glass transition, one important aspect of studying the energy landscape of various glassy systems is the so-called aging phenomenon associated with nested marginally ergodic regions (c.f. Sections 3.11.4.1.1 and 3.11.4.1.2). To study aging, one quenches the system out of its equilibrium state, usually the liquid state in the case of structural or polymer glasses, quite deep into the glassy state. Due to the fast quench, the system is only in a very small locally ergodic region with a very short escape time after the quench has taken place, and thus the system very quickly leaves the region, explores larger parts of the landscape but without ever reaching local ergodicity. Instead, the system is marginally ergodic, i.e., only on a time scale tobs smaller and more or less proportional to the aging time sage (or waiting time tw)386 the system has been allowed to relax before the measurements begin, tobs < sage, does it appear as if the glass were in equilibrium during the measurements. But once the length of the measurement exceeds the earlier relaxation time, tobs > sage, one notices non-equilibrium behavior such as drifts in the time average of the energy, sudden peaks in the specific heat, deviations of the two-time correlation functions from their equilibrium value, etc.387 Many studies of aging have been performed for polymers,270,1006,1192,1193 glasses1194–1197 and spin

384 Quite generally, Lennard-Jones systems in, e.g., two dimensions are a popular system for studying rare events that require very long simulation times, since the interaction potential is not only fast to evaluate but can serve as an effective potential to represent more complex interactions in materials.1177–1180 Furthermore, being a two-dimensional system, visualization of the minima and of the trajectories among them is much more lucid than for three-dimensional systems, although the variety is greatly reduced compared to the analogous 3D-system. 385 Note that in some of the studies of inherent structures, the focus is on the structure prediction aspectsdafter all, the inherent structures can also be obtained via global optimization disassociated from the dynamicsdand less on the dynamical features. 386 The notation tw is frequently used in simulations instead of sage. 387 This phenomenon has also been called “weak ergodicity breaking”,1191 inspired from the classical “broken ergodicity” point of view.460

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w ÞE ðtw Þi Fig. 21 Aging in the amorphous ceramics Si3B3N7.484 On the left, the normalized two-time energy-energy correlation f ¼ ðE ðtðEobsðtþt as function w ÞE ðtw Þi of observation time tobs is shown, for three different waiting times tw (¼3 $ 103, 104, 105 MCC) in red, green and blue, respectively, and for three different temperatures. Here, time is measured in Monte Carlo cycles (MCC): during 1 cycle each of the ca. 1000 atoms in the simulation attempted one move, on average. The averages h.i are taken over ensembles of independent simulation trajectories. We find that deviations from the equilibrium value (one) appear for tobs > tw for temperatures below the glass transition temperature (ca. 2000 K), i.e., for 1250 K and 1750 K, while the value fluctuates around one in the liquid state at 2500 K. (The greater fluctuation about the average curves in the simulations at 1750 K and 2500 K than for those at 1250 K is due to the use of a smaller ensemble of walkers at the higher temperatures compared to a large ensemble of ca. 100 walkers at 1250 K.) On the right, the logarithmic decrease in the average potential energy as function of time is shown for different temperatures below the glass transition temperature of the system; note how the energy decreases faster for higher simulation temperatures. The figure is adapted from reference Hannemann, A.; Schön, J. C.; Jansen, M.; Sibani, P. Non-equilibrium Dynamics in Amorphous Si3B3N7. J. Phys. Chem. B 109 (2005), 11770–11776 with permission by American Chemical Society, 2021.

glasses,149,263,271,700,1198,1199 both in the experiment and in simulations, and for the standard Lennard-Jones and related softsphere systems266,1200–1204 in simulations, but also for realistic materials such as amorphous Si3B3N7484 (see Fig. 21).

3.11.7.6.3

Spin glasses

Besides the Lennard-Jones systems (clusters, crystalline and amorphous solids and liquids), spin glasses1205 are one of the most important prototypical systems with complex energy landscapes, which have been employed to develop and test many algorithms and concepts concerning energy landscapes, such as simulated annealing547 or models for aging,149 and thus we shortly mention them.388 With an essentially random interaction among neighboring spins, spin glasses exhibit a multitude of local minima with nearly the same energy in contrast to the well-developed global minimum of a spin system with, e.g., ferromagnetic interactions. This leads to highly complex multi-minima landscapes with aging behavior,271 which have been studied82,84,276 using a variety of exploration algorithms. The local densities of states were found to grow approximately exponentiallydat least in the low-energy region of the landscaped, and equilibration trees were computed, which showed a noticeable hierarchy on logarithmic time scales, even though the corresponding tree graphs did not exhibit a multiple-level-branching structure.82 In this context, we also note the new developments concerning the so-called spin-liquids1206,1207; here, however, one stresses the importance of coherent states, which are not easily captured by the classical energy landscape approach, and thus tools from energy landscape theory are only rarely applied, unless a “classical” spin-based model can be employed.1208

388 Spin glasses are usually more of interest in the physics community, but the physical realizations of spin glasses are inorganic compounds, of course!271 In practice, the “random” interaction is often established by employing a dilute spin system with essentially random spin-spin distances.

Energy landscapes in inorganic chemistry 3.11.7.7 3.11.7.7.1

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Optimal control Master equation dynamics

As mentioned earlier, once lumped models of the energy landscape have been constructed, together with information about the probability flows between the nodes of the model as function of thermodynamic parameters such as temperature or pressure, we can describe the dynamics of the system via a master equation approach, where we multiply the probability distribution vector of the system by an appropriate Markov matrix for given values of the thermodynamic parameters (c.f. Section 3.11.4.2.1). Typically, this requires numerical simulations, but due to the great reduction in complexity of the original microstate based landscape, such simulations can usually be easily performed. This allows us to develop optimal temperature and/or pressure, etc. schedules, which can be used to steer the time evolution of the system such that, e.g., various modifications of the system can be reached with high probability. A simple demonstration of such an optimal control of a chemical system is the optimal control of the probability flows on the energy landscape of a small periodic approximant of MgF2.143,198,301 Using results from global energy landscape explora! tion,286,294,932 a transition probability matrix G ðT; pÞ between four of the major basins on the energy landscape, representing the rutile, anatase, a half-occupied rocksalt, and a monocapped prism structure, respectively, as function of temperature and pressure, was constructed. Subsequently, application of optimal control methods1209,1210 to the task of maximizing the final probability ! to be in one of these basins after Nsim time steps, i.e., Nsim multiplications by the Markov matrix G ðT; pÞ, it was shown how different temperature and temperature þ pressure schedules could guide the system into preferentially reaching each of these modifications (c.f. Fig. 22).

3.11.7.7.2

Finite-time thermodynamics for chemical processes

3.11.7.7.2.1 Computational alchemy Another important application of optimal control of energy landscapes of chemical systems is the computation of free energy differences between compounds via the smooth change of the energy function, sometimes called computational alchemy.469–472,1211 As mentioned earlier (c.f. Section 3.11.6.7.2), we change the underlying energy function of the system, while we use thermodynamic integration methods806–809 to compute the free energy difference that is accumulated along the “path” connecting the two compounds or modifications of interest. Applications where computational alchemy has been employed to compute free energies or free energy differences are, e.g., in the field of drug discovery,1212,1213 free energies of defects,1214 free energies of sublimation,1215 phase stability of solid solutions,1216 or catalyst discovery.1217 Similarly, there are applications that do not focus on free energies but attempt to compute or predict new materials and their properties, such as band structures, using computational alchemy.1218,1219 3.11.7.7.2.2 Output maximization using mesoscopic models Another important application of optimal control for finite-time processes is the maximization of the output of certain chemical reactions202,1220 or distillation processes1221–1223 for finite time and a prescribed allowed range of temperatures and pressures. However, such studies usually employ only single phases, in the sense, that the system is treated completely macroscopically on the level of thermodynamics. An example study, which deals with the chemical system at the mesoscopic level of optimizing nucleation and growth processes, in order to maximize the output of competing crystalline modifications, considers the optimal synthesis of glycerol crystals.144 Here, classical time evolution models for nucleation rates and growth rates as function of temperature and concentrations are employed to set up an objective function that describes the yield as function of temperature schedule. This simplification actually allows us to derive analytical solutions to the problem, as long as we are given experimentally derived values for the parameters in the model descriptions. However, a consequence of such simplifications, i.e., the use of continuum models and thermodynamic descriptions of the processes involved, is that the connection to the description of the system on the level of atom configuration based energy landscapes is only tenuous since the model parameters are difficult to derive from the energy landscape, and thus usually taken from experiment. This contrasts with the master equation approach, where the parameters of the coarsened landscape model are extracted from global energy landscape explorations on the microstate level, as we had seen in the example of the MgF2 optimal control discussed above.

3.11.7.8

Synthesis routes and materials design

In some fashion, the holy grail of the use of energy landscapes in chemistry is the prediction of feasible synthesis routes, i.e., we not only enumerate and rank the many possible phases that are kinetically stable and thermodynamically metastable in the system on given observational time scales using various structure prediction methods, but we also lay out a plan to achieve their synthesis, preferably in an effective and efficient way. But just as the exploration of the generalized barrier structure of an energy landscape is at least one order of magnitude more expensive computationally than the determination of the local minima, trying to follow the chemical system through its synthesis route on the landscape is clearly even more demanding, especially since we need to

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Fig. 22 Optimal control of temperature and pressure in the MgF2 system. In each row, the left plot shows the optimal temperature and pressure schedule, while the right plot depicts the evolution of the probabilities; rows 1, 2, 3 and 4 maximize the production of rutile, anatase, 1/2Occp and MP1, respectively. Starting point was the anatase minimum on the empirical potential energy landscape of the periodic approximant Mg2F4, and the paths were set up such that the maximal amount of the four major local minima considered, exhibiting the rutile structure (red circles), anatase structure (blue circles), 1/2 filled NaCl structure (called 1/2Occp) (black filled squares), and a mono capped prism structure (called MP1) (hollow squares), can be produced within a finite number of steps (corresponding to a finite time available).198,301 Details on the structures and the energy landscape of the system are given in.294 The temperature (green triangles) along the y-axis is given in terms of x ¼ exp.(0.1/T) with 0.0  x  0.6 where the temperature T is in units of energy, such that x ¼ 0.6 corresponds to ca. 2300 K. Similarly, the pressure (black triangles) along the y-axis is normalized with an allowed range of 0  p*  1.0, such that p* ¼ 1.0 corresponds to 128 GPa or ca. 0.8 eV/Å3. Along the x-axis, the time is given in terms of the number of applications of the Markov transition matrix of the discrete master equation. The figure is adapted from reference Hoffmann, K. H.; Schön, J. C. Combining Pressure and Temperature Control in Dynamics on Energy Landscapes. Eur. Phys. J. B. 90 (2017), 84 with permission by Springer under Creative Commons, 2021.

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deal with an extended landscape that incorporates varying thermodynamic boundary conditions and exposure to sources and sinks of environment atoms in form of substrates, catalysts, educts and side-products, just to name a few. To achieve such a goal, we need to model, analyze and optimize many families of classical synthesis procedures that, hopefully, can be put together in a step-by-step fashion to form the full synthesis route to a desired, predicted or already known, modification of a compound. All the procedures that are going to serve as building blocks of the complete synthesis route must be analyzed by following the system’s trajectory on the appropriate extended energy landscape. This should lead to a library of such basic elements of syntheses, that can be employeddpossibly in an appropriately modified fashiondto construct an optimal synthesis route for a given product148.389 However, just as it took 30 years from the first simple structure determination and structure prediction studies to today’s efficient combinations of many global landscape exploration methods, this grand vision will take time to be realized in a systematic fashion, e.g., via master equation approaches on time-varying extended energy landscapes as discussed above.198 At the moment, the energy landscape usually enters the design of synthesis routes and complex materials as such by providing possible targets, and general guidance by elucidating the barrier structure of the chemical system of interest for different thermodynamic boundary conditionsdincluding varying pressure and chemical potentials, and exchange of atoms with the environment. This general information about the system then can inspire the selection of possible candidates for synthesis routes; a procedure one could call “rational synthesis route selection”. Below, we discuss a couple of examples, where energy landscape concepts have entered the design of materials and the selection of synthesis routes.390

3.11.7.8.1

Materials design

3.11.7.8.1.1 Zeolites A class of materials, where design has been built into their DNA from the beginning, are zeolites and zeolite-like materials, since, as has been mentioned earlier (c.f. Section 3.11.7.1.1), they constitute energetically high-lying minima on the energy landscape of the participating atoms952,955,1224.391 Thus, their synthesis usually has to employ various kinds of templates, which can be used to guide the atoms and possible metastable building blocks into forming the desired structure.15,1225,1226 Such templates have been the target of systematic theoretical design in the past for given zeolite target structures.954,1227,1228 3.11.7.8.1.2 Artificial drugs Another class of materials where energy landscape concepts implicitly enter the design are artificial drugs. Of course, here structure prediction methods have been employed to predict structures, both the structure of molecular crystals117,120,970,975 and the secondary and tertiary structure of biomolecules,105,1029 and also to predict feasible ligands647,1229,1230 and docking sites.1231–1234 Since this area properly belongs to the world of organic or biochemistry, we just mention a couple of reviews for the interested reader.1235–1237 3.11.7.8.1.3 Property driven design One particular class of materials design is concerned with the final properties of the compound. In the past, a tricky issue has been the question of kinetic and thermodynamic stability of the suggested compound,1238 and there have been debates about the issue, to what extent one can speak of “design” in chemistry in the first place,32 but it is by now agreed that only compounds correspondexp , should be included when discussing ing to thermodynamically metastable configurations, for a given observational time scale tobs 392 the issue of design. Much of the work in property driven computational design relies on the exploitation of databases, which provide candidate structures in the chemical system of interest or in related chemical systems,1239 where a first step is to verify the kinetic stability of the suggested compound. Examples are, e.g., the computation of elastic properties of over 1000 inorganic compounds,1240 with the goal to identify promising candidates for materials, or the open quantum materials database.1241 Similar energy landscape inspired strategies have been developed for the design of inorganic materials, such as solid oxides,1242 or the design of modified metal-organic frameworks for the efficient uptake of CO2.1243

389 One should always keep in mind that the individual processes involved need to proceed spontaneously, and the system thus follows a descending trajectory on the (currently visited region of the) hypersurface of the (extended) free energy. 390 In the past, many practicing synthetic chemists would often point out that he or she employed his/her experience and chemical intuition to design the new material or pick the synthesis route, even though, in many cases, this intuition reflects information about the energy landscape of the system, even if this is not explicitly recognized or acknowledged. But this has greatly changed by now, with many experimentalists actually visualizing and discussing their synthesis in an energy landscape picture. 391 In this fashion, zeolites show similarities to organic molecules, where also a multitude of kinetically stable but thermodynamically only metastable molecules can be constructed from the same set of, e.g., carbon, hydrogen, oxygen, nitrogen, etc. atoms, but for which the “ground state” would probably be a mixture of molecules like CO2, H2O, etc., that constitute the “thermodynamic sink” of the system. 392 Of course, one must not forget systems that are only marginally ergodic, such as glasses, which are nevertheless highly valued in applications. Such systems exhibit aging behavior and are not in thermodynamic equilibrium, and thus their “design” is not reproducible from the energy landscape point of view, where reproducibility implies local ergodicity on the time scale of interest. Nevertheless, many mesoscopic and macroscopic properties of interest, such as mechanical stability, transparency, conductivity, etc., are constant for a long enough time, and the material can be produced in a sufficiently reproducible fashion with respect to these properties, to justify the use of the term “design” when talking about the tuning of the material’s properties.

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3.11.7.8.2

Synthesis routes

The materials design discussed in the preceding subsection is still focused on the energy landscape of the isolated system. Now, the energy landscape of an isolated system contains only the atoms of the target compound. Thus, we can only describe syntheses that take place by spontaneous and complete reactions from, e.g., the gas phase or from solid educts, without any side products; a special case would be a (spontaneous) phase transformation. But most actual syntheses involve additional atoms that are present in the system, e.g., the atoms in the educts which form side products, the atoms or molecules of the solvent, or the atoms that constitute (the surface of) a substrate. Furthermore, in many cases, we need to guide the synthesis. For example, we might want to vary the pressure or the temperature, in order to force the system into a high-pressure or high-temperature modification (which subsequently might be quenched thus becoming a metastable modification at standard pressure and temperature), or to allow the system to overcome barriers on the energy landscape that impede the synthesis process. Thus, in practice, all syntheses correspond to manipulations of the (instantaneous) extended energy landscape of the chemical system of interest. In the past, this use of the energy landscape has been mostly implicit; a famous example would be the so-called reverse synthesis in organic chemistry.18,19 The great success of this concept rests on the fact that the synthesis of the molecule from its educts can be split into many (intuitively sensible) reaction stepsdand subsequently put together again in (possibly) modified formd,1244 each of which can be quantitatively, or at least qualitatively, analyzed on the atomic level using the appropriate energy landscape. This systematic visualization of the individual reaction step as a movement on an energy landscapedusually in a very simplified fashiond, feeds the intuition of the experimentalist about how to design an efficient reaction sequence. Together with the availability of a wide variety of experimental tools that allows changing the energy landscape as desired, this yields an exquisite kinetic control of the synthesis. Trying to understand such reactions in detail on a quantitative leveldand thus to justify the intuitiondhas led to many studies of energy landscapes for known chemical reactions.309,515,1245 However, we can turn this around: Starting from the energy landscape of the chemical system, we can try to systematically identify many hypothetical reactions by following the evolution of the chemical system on its energy landscape for given values or time schedules of the external parameters. Such “designed” reactions, transformations or other synthesis processes can then be validated and implemented in the experiment.393 Quite generally, one should try to extract the energy landscape features responsible for various prototypical types of reactions, such as redox reactions in biochemical systems,1247 or valence bond delocalization.516,1248 As mentioned above, this should lead to a whole library of analytical or numerical model descriptions of synthesis processes that could be combined into more-or-less optimal synthesis procedures148; an example for a step in this direction is the computational compilation of a database of transition states of elementary chemical reactions from the study of their energy landscape on the quantum mechanical level.517

3.11.7.9

Optimal parameters for chemical modeling

Quite generally, the extraction of model parameters for the description of chemical of physical systems in a simplified fashion always involves some kind of optimization, ranging from the least-square fit of simple empirical models to complicated weighting procedures and error analyses when designing multi-scale models of materials. Since the cost functions involved in such problems are rather removed from the question of complex energy landscapes of chemical systems, we only shortly discuss two examples of complex cost functions, dealing with the parameters of neural networks that have become popular for the analysis of (classes of) chemical systems, and the optimal selection of parameters in empirical potentials.

3.11.7.9.1

Neural networks

We recall that a neural network can be viewed as a mapping of a set of inputs to a selection of possible outputs (e.g., different classifications for the system described by the input), where the mapping function usually consists of many levels of convolutions of many simple “neuron”-type functions that can be described by some simple parameters. However, the number of the “neuron” functions and the many possible connections among them results in hundreds, thousands, or millions of parameters, whose values determine the success rate of the neural network mapping. The cost function landscape associated with these parameters corresponds to the success ratedhowever measureddof the neural network when applied to large sets of training and test inputs, as function of the parameter values. Since many choices of parameters can yield decent or good classifications while slight changes of each of these locally “optimal” choices leads to a worsening of the outcome, we are dealing with a multi-minima cost function.394 Thus, researchers have taken to employing global optimization techniques previously applied for potential energy landscapes, for the search of optimal neural network parameter combinations.257 One such study considers a comparatively simple problem, the so-called XOR (exclusive OR), and studies its cost function landscape as function of network complexity (number of hidden 393 Clearly, there will have to be much feedback between theory and experiment in the realization of such a systematic energy landscape based approach to synthesis planning, in practice.1246 394 An example would be the classification of points as being inside or outside of a circle, using six neurons in a single hidden layer, where each neuron effectively splits space into two parts by introducing a planar divider surface, and yields a 1 if the point is to the right and a 1 if the point lies to the left of the surface. It was found that one locally optimized choice of parameters approximates the circle by a hexagon (each side of the hexagon was coded by one neuron), while another choice is an approximation as a square, where three of the neurons code one of the sides each, and the other three neurons code the remaining side of the square (where two of these neurons just cancel each other).1249

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nodes, etc.), deriving minima, disconnectivity graphs and transition states on the network-parameter landscape.1250 While the XOR is not directly related to chemical system neural networks, the analysis yielded insights into generic properties of such networks and could be used to analyze the stability of the neural network to perturbations of the input data.

3.11.7.9.2

Empirical potentials

The optimal selection of parameters for empirical potential functions is not straightforward and often an art in itself, since many choices of parameters can be reasonable but not perfect. The reason is that we are dealing again with a multi-minima cost function landscape, where the agreement, with the experiment, of various computed quantities, such as energies, atom-atom-distances, but also bulk moduli or vibrational spectra, serves as the cost function. We note that in such cases, weighting of various properties is unavoidable, i.e., we are actually dealing with Pareto optimization problems. Thus, one often attempts to use global optimization techniques to identify high-quality force field parameters in an efficient fashion. Examples of such approaches are the use of a genetic algorithm to identify good force field parameters for the description of coarse-grained models of RNA,1251 or the annealing optimization of parameters for the description of proteins,1252 or for molecule interactions in crystals.1253 Similarly, the so-called genetic programming method has been employed to find parameters in empirical potentials that yield good agreement with ab initio energy calculations,1254 where both the functional form and the parameters of the potential are optimized. Recently, methods have been described that optimize the agreement between computed and target data surfacesdinstead of a point-by-point comparisondas the optimization criterion when fitting parameters to empirical potentials.1255 We note that it would be expected that using such an agreement between whole surfaces or pieces thereof for ab initio and empirical potential surfaces should also be a feasible choice as the optimization criterion for fitting empirical potentials. Finally, we point to the use of databases of ab initio energy functions as the input for the optimization of empirical potentials; while such databases have been used for a long time when fitting empirical potentials, the issue of an optimal selection of database entries for generating a globally robust empirical energy function is not yet fully resolved.1256

3.11.8

Outlook

Over the past 30 years, the energy landscape of chemical systems has changed from an esoteric tool at the fringe of the field to a fundamental entity at the heart of chemistry. And not only theorists but also synthetic chemists now think and speak about their research in the language of energy landscapes, and employ energy landscape concepts when visualizing their syntheses, both regarding the choice of targets and the synthesis routes to reach them. These concepts have become invaluable for understanding the reaction mechanisms and processes involved in the synthesis of molecules and solids, and when considering the kinetic and thermodynamic stabilities of the compounds one deals with in practice. This holds true for chemists from all sub-disciplines, whether they are dealing with organic and inorganic molecules, biomolecules, clusters, surfaces, crystals, porous materials, or amorphous compounds. In this chapter, I have tried to provide a comprehensive overview over the field of energy landscapes in chemistry. The focus has been on inorganic materials, ranging from general concepts for energy landscapes of isolated chemical systems and systems in interaction with the environment, including time variation, to cost functions for inverse problems such as structure determination or parameters for machine learning and simplified potentials. We have given a short introduction to the basic elements of the most popular computational tools to study energy landscapes locally and globally, together with a sampling of the most common practical applications of energy landscapes in inorganic chemistry. We have only mentioned in passing the vast field of energy landscapes of molecules and biomoleculesdthe topics of protein folding or docking alone deserve separate chapters, and the same holds true for the study of molecular reactions. But the same landscape concepts that address, e.g., the existence of metastable crystalline compounds, also apply in cluster isomers, and the aging of an ionic or metallic glass is based on the same general features of the energy landscape as the aging of an organic polymer. As a consequence, the same tools can be applieddwith obvious modifications, of coursedwhen studying the transformations in the shape of a large molecule and the vacancy diffusion in a solid, when predicting the feasible existence and structure of crystalline modifications, inorganic molecules, and proteins, when including an interaction with the environment for various chemical systems, or when optimally controlling a chemical reaction or a crystallization process. However, realizing that the extended energy landscape controls the time evolution of a chemical system and its interaction with the environment, in principle, only corresponds to an “existence” theorem, and there are a multitude of open questions both regarding fundamental aspects of energy landscapes and practical realizations and applications.395 From the practical point of view, we still face the enormous computational effort needed to globally explore such a landscape and simulate or model the time evolution of the system. This holds true even for energy landscapes that do not vary in time, but it is a much more formidable problem if the interaction with the environment depends on time, e.g., when we try to control a chemical process in detaildwhich is one of the major goals of chemistry, of course. As ever, the various time scales in play, such as observation times, equilibration times, 395 An analogy from mathematics would be that knowing that a unique solution to a differential equation exists does not tell us the solution for the specific problem at hand.

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escape times, time scales on which the environment changes, and their relative sizes, are central to understanding the behavior of the chemical system. Another practical issue is the spatial variation of the influence of the environment, and the associated need to compute properties of the material (in the continuum approximation) on the material point level involving thousands of atoms for each such point, if we want to include effects like enforced steady-state currents, etc., in a compact fashion via, e.g., conductivities or permeabilities. Regarding the spatial variation, this is no problem, “in principle”, if we can deal with so many atoms that we do not need to use periodic approximants for describing the bulk solid.396 In practice, this is still not possible, although, e.g., molecular dynamics simulations involving several million atoms using empirical potentials are becoming quite common in studying various processes in materials on time scales of nanoseconds,1257,1258 and simulations involving 109 atoms1259 have been performed for biomolecules.397 More critical is the computation of material properties that are fed into the extended energy or cost function, since these are often not associated with a single microstate but require ensemble or time averages. Here, the expectation would be that, e.g., linear response models can be employed that use information about the relevant microstates and minima basins of the system, but it might also be possible that short MD or MC simulations can be used to compute such propertiesdanalogous to the way one would compute, e.g., chemical potentials, etc., in an implicit fashion via short-time simulations of an external reservoir.508,1260 One possible way to address the issue of extreme numbers of atoms is to work with hierarchies of configuration spaces and energy landscapes.398 We have seen such an approach already in the construction of master equations based on information about the locally ergodic regions of the landscape. Similarly, we can construct stepping-stone models of the system, analogous to multiscale simulations,1261–1263 where we derive an effective configuration space on, e.g., the material point level: each microstate of the mesoscopic system is now defined by the properties associated with all the material points such as density, local structural order, local permeability and conductivity, etc.danalogous to the way the coordinates for all the atoms in the original atom level configuration space characterized a given microstated, together with effective cost or energy functions that now represent the state of the system. Of course, now the moveclass must reflect the dynamics of the spatially distributed system where the properties at each material point can change according to some probabilistic rules, analogous to the movement of atoms or the change of spin states on the potential energy landscape.399 We note that at this level of spatial (and implicit temporal) resolution reside the free energy landscapes that are defined as function of spatially varying order parameters, which are commonly used to analyze the statistical mechanics and thermodynamics of phase transitions. Such a coarsening would also be a first step in moving from the atomistic energy landscape to the “landscape of compounds”dboth for bulk systems or moleculesd, where the microstates now correspond to (spatially distributed) metastable compounds and isomers, which can evolve in time and transform into different modifications. This “landscape of compounds” is often the level of description employed when drawing a qualitative energy landscape of a bulk solid for illustration purposes of an experimental synthesis or phase transformation. This “landscape of compounds” could be considered a “macroscopic” energy landscape, while the material point landscape might be denoted a “mesoscopic” energy landscape. Further coarsening would then lead to the generalized phase diagrams mentioned earlier,189 where all metastable phases are incorporated as function of observational time scale, but without any spatial resolution, i.e., the system is considered to be spatially homogeneous in each of its metastable phases. Of course, these two possible intermediary-scale landscapesdthe actual number of intermediary scale landscapes would depend on the chemical system of interest and its inherent complexitydwill only be sensible for appropriately chosen observational time scales, such that the properties that characterize the system at the material point level or at the “compound” level are stable, and thus meaningful, in the first place. This also implies that changes in temperature result in changes of the properties of the mesoscopic or macroscopic “microstates” and consequently in the cost associated with the microstate. This is in contrast to the atom level energy landscape, where the configuration space and the landscape itself are temperature independent400 and only the locally ergodic regions, their equilibration and escape times, and the generalized barriers separating them, depended on temperature. Another step beyond the classical atom based energy landscape with the microstates defined solely by the atom coordinates, will be required when the Born-Oppenheimer approximation does not hold, and the ionic and electronic degrees of freedom can no longer be decoupled. Similarly, there are problems, if we no longer can separate the spin degrees of freedom from the movement of the atoms, which had allowed us to, e.g., focus solely on the spin Hamiltonian of the system,401 or if we are interested in the energy landscape of electronic phases resulting in, e.g., metal-insulator transitions,1264,1265 charge-density waves,1266,1267 or spin liquids.1206 In that case, the quantum mechanical aspects of the landscape cannot be hidden in the energy function and the

396 However, even then we might need to perform property simulations over short simulation times, and we must consider the time scales on which an externally applied field can establish itself throughout the system. 397 Still, the availability of bigger computers is not the magic bullet, since the difficulty of globally exploring a complex landscape grows exponentially with system size, i.e., with the number of atoms per molecule or periodic approximant. 398 Divide-and-conquer approaches that work by restricting the allowed configuration space during some stages of the exploration, e.g., during the search for local minima or estimates of barriers, (by combining certain atoms into rigid building groups, for example, which can then be “packed” as if they were simple spheres) often appear to be able to deal with large simulation cells or molecules. However, one always runs a substantial risk of overlooking important structure candidates or saddle points, or identifying the wrong barriers or locally ergodic regions. 399 Regarding some features of the time evolution at the level of this material point landscape, we might actually draw upon the well-established finite-element methods236,237,858 (which would need to be adapted to the requirements of this and other intermediary energy landscapes, however). 400 An exception would be the use of temperature-dependent statistical potentials, but these are only valid for relatively large observational time scales. 401 We have seen such landscapes in the context of spin glasses, for example.82

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moveclass, e.g., as proton tunneling or spin flips. Instead, we need to find an efficient way to define microstates of the quantum mechanical system, together with a moveclass that reflects both the probabilistic nature of the quantum mechanical time evolution and the possible presence of coherent or entangled states. For the fundamental state space, we might need to employ the Hilbert space of the quantum mechanical system; similarly, the question of the introduction of density matrices instead of pure states or eigenstates of the Hamiltonian H0 as the appropriate microstates in the context of an energy landscape description needs to be discussed. While the use of eigenstates of the Hamiltonian H0 as the underlying state space allows a straightforward definition of the energy function of the landscape as the energy of the eigenstates, the “natural” generalization of the energy for a general quantum mechanical state as the expectation value of the Hamiltonian H0 for this state is much more fraught with potential problems. For example, can we define a simple time average of this energy (expectation value) along the time evolution trajectory of the quantum system without confronting the issue of measurements and the reduction of the quantum state to one of the eigenstates of the Hamiltonian upon such a measurement, and what consequences does this have for our standard definition of ergodicity as the equality of time and ensemble averages? And what is the appropriate ensemble to perform an average over? Or how do we implement the interactions with the environment via an interaction Hamiltonian Hint that allows a quantum mechanical time evolution of the system beyond the “trivial” Schrödinger time evolution due to the Hamiltonian H0 that defines our energy landscape? Some work on quantum mechanical systems has been done, such as modeling the energy flow in such a system,456 or studies on eigenstate thermalization in quantum systems, but this can only be the beginning.402 Clearly, construction and proper handling of a quantum mechanical energy landscape for physical and chemical systems will be one of the great challenges in the field of energy landscapes. A completely different challenge will be how to deal with the every-increasing amount of information we are able to gain about an energy landscape of a chemical system, such as the millions of local minima and saddle-points, or the probability flows between them as function of temperature or energy slice, which are determined using various exploration algorithms. Classifying these states, computing their properties, such as ab initio energies (if the search did not take place on the ab initio energy landscape from the beginning), symmetries, curvatures, magnetic moment, etc., and grouping them into locally ergodic regions or, at least, minima basins with their local free energies, together with the barriers separating them, is starting to overwhelm the user. Whereas 30 years ago each minimum was studied in depth, in order to extract all possible information about the system, one nowadays tries to deal with the excess of data by using automated systems. But since such classification algorithms are programmed and trained with certain preformed expectations and categories in mind, it can easily happen that we will only see what we trivially expect to see, and overlook the exciting exceptions from the rule which will yield the qualitative instead of just the quantitative advances in the field. But personal inspection of every structure candidate or probability flow cannot be the answer, either.403 Thus, next to automated data generation, we need to develop tools for automated data analysis that can spot what is new. One could imagine that the highly developed neural networks which classify according to explicit rules and/or implicit rules acquired via training, can be taught to extract the unusual but not crazy examples, which do not fall into the known categories. However, this will clearly be a major challenge, by definition. Conversely, we are currently facing the issue that the data-mining and machine learning based data analyses produce results, which are exceedingly difficult to rationalize, such that one must worry that if one accepts the classification by the network, one no longer can explain, e.g., why a certain compound has a certain property, such as a high conductivity or elasticity. This is especially true if the input is very generic such that not only one intermediary step of a model calculation but several such steps have been dropped. This cutting-out-of-the-middleman can also happen in the context of the energy landscape itself. Neural network based empirical potentials have become quite popular409,411; while these are usually still based on known functional forms and only the parameters in the potential are adjusted from microstate to microstate, we can easily imagine that at some point the empirical potential will be gone completely, and the network directly generates the energy associated with the given microstate after having been trained on a gigantic set of ab initio energies. In that case, we have achieved a great improvement in speed compared to the ab initio calculation with, hopefully, a robust black-box neural network, but it will become very difficult to understand the meaning behind the numbers.404 But what is perhaps more critical is the use of machine learning tools to completely ignore the energy landscape, at least as far as, e.g., the prediction of feasible structures is concerned.887,1038,1273,1274, 405 Here, the great concern is that we miss out on the richness of the landscape, where many structures are actually capable of existence, and thus the generation of only one candidate greatly narrows our view as far as synthesis targets and synthesis routes are concerned. Quite likely the most efficient way to deal with

402 We have already mentioned the use of excited state Born-Oppenheimer surfaces, which are often needed to properly describe details of chemical reactions or interactions with external fields such as electromagnetic radiation. However, this approach side-steps the actual challenge, and really only works best if the excited state surface shadows the ground state Born-Oppenheimer surface. 403 Even in a perfectly automated world where every human does what humans do bestdoptimistically speakingdi.e., be creative and flexible, and recognize patterns that follow new principles not included in one’s training, while machines do all the routine work from agriculture to consumer goods production and bureaucracy, we would not have enough humans to review the data we can produce with our global landscape explorations. This has been called “In Schönheit sterben” (“dying in a state of beauty”).1268 404 Already ab initio energy calculations are often difficult to rationalize compared to the results of empirical potential calculations, but at least with ab initio calculations we have some idea what happens to the physics (or mathematics) of the system, if we enlarge/reduce/modify the basis set or adjust some parameters in a pseudo-potential or functional. But what do we learn from adjusting some generic parameters in the neural network? It is quite interesting to note that one nowadays has started to write additional code to clarify what happens inside the neural network, in order to understand, whether the decision making process correlates well with our understanding of the situation (here: the physics and chemistry of the system).1269–1272 405 This is actually not a new development but goes back to the last time neural networks were highly popular, in the late 1980’s and early 1990’s.1273,1275

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this situation is to pursue combined searches, where candidates are generated from data-mining of known crystal, cluster, protein, etc., structures, from machine learning suggestions, and via global energy landscape exploration. As has been noted earlier, puristic approaches that focus on only one methodology are needed when developing and fine-tuning new algorithms,406 but, in practice, combinations of several approaches are most likely to be the ones that will efficiently generate feasible candidates in not-yet experimentally explored systems. And, furthermore, we have seen that the local minima only scratch the “surface” of the energy landscape. We want to identify complex locally ergodic regions that are important at elevated temperatures, and estimate the kinetic stability, equilibration times and life times of the possible modifications and isomersdideally as function of the environmentd, which requires landscape information beyond the local minima. Similarly, grouping many local minima according to some (chemically inspired) order parameter(s) into multi-minima basins, which may or may not correspond to locally ergodic regions, will become an important tool for both the classification of chemical systems and the modeling of their equilibrium properties and dynamics. In turn, such classifications will hopefully enable the practicing chemist to establish a useful intuition about a chemical system of interest that can guide the development of synthesis routes to new materials and provide a detailed understanding of their potential for applications in devices. Clearly, developing new, and improving available, exploration tools that can, e.g., identify chemically and physically relevant (multi-minima) basins and efficiently determine generalized barriers and probability flows between these regions will constitute a major enterprise in the future. While computational short-cuts offered by the machine learning approach should therefore be employed with a big grain of salt, they nevertheless open up new possibilities regarding general aims like the optimal control of chemical reactions and syntheses. If such neural networks and various expert systems that can incorporate and organize a large amount of experimental and theoretical data can be properly combined with a judicious use of explorations of the energy landscape at the atomic and the mesoscopic level, and with analytical modeling of basic elements of chemical reactions such as nucleation phenomena1277,1278,407 we should be able to build up a library containing both feasible target and possible intermediary compounds, together with models of chemical processes,148 which then can be employed to optimally guide the chemical system’s movement on the extended energy landscape to the desired target compound, and to inform us about the stability and behavior of the material in various application environments.

Acknowledgments I would like to thank the many colleagues and students over the past 33 years, with whom I have had the pleasure to discuss and analyze questions concerning energy landscapesdfrom basic concepts over exploration algorithms and numerous applicationsd, starting with the mathematics department at SDSU in San Diego, the physics departments at the universities in Odense and Copenhagen, the chemistry department at Bonn university, the solid state chemistry and nanoscale science departments at the Max-Planck-Institute for Solid State Research in Stuttgart, my many collaborators at universities and research institutes around the world, and, last but not least, the many participants of the energy landscape workshops that have taken place on a nearly yearly basis over the past 30 years. Furthermore, I would like to thank H. Huang for providing Fig. 13, J. Cortes for providing Fig. 15, D. Zagorac for drawing Figs. 8–11, and R. Noack for drawing Figs. 4, 7, 12, 14, 16 and 19–22, in this chapter.

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Scaling Molecular Dynamics beyond 100,000 Processor Cores for Large-Scale Biophysical Simulations. J. Comput. Chem. 2019, 40, 1919–1930. 1260. Hörrmann, N. G.; Andreussi, O.; Marzari, N. Grand Canonical Simulations of Electrochemical Interfaces in Implicit Solvation Models. J. Chem. Phys. 2019, 50, 041730. 1261. Horstemeyer, H. Multiscale Modeling: A Review. In Practical Aspects of Computational Chemistry; Leszcynski, J., Sjukla, M. K., Eds., Springer: New York, 2009; pp 87–135. 1262. Bruix, A.; Margraf, J. T.; Andersen, M.; Reuter, K. First-Principles-Based Multiscale Modelling of Heterogeneous Catalysis. Nat. Catal. 2019, 2, 659–670. 1263. Radhakrishnan, R. A Survey of Multiscale Modeling: Foundations, Historical Milestones, Current Status, and Future Prospects. AIChE J. 2021, 67, e17026. 1264. Imada, M.; Fujimori, A.; Tokura, Y. Metal-Insulator Transitions. Rev. Mod. Phys. 1998, 70, 1039–1263. 1265. Shachkin, A. A.; Kravchenko, S. V. Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions. Sppl. Sci. 2019, 9, 1169. 1266. Grüner, G. The Dynamics of Charge-Density Waves. Rev. Mod. Phys. 1988, 60, 1129–1181. 1267. Zhu, X.; Guo, J.; Zhang, J.; Plummer, E. W. Misconceptions Associated with the Origin of Charge Density Waves. Adv. Phys.: X 2017, 2, 622–640. 1268. Breu, J. In Scho¨nheit Sterben. Private Communication, 1998. 1269. Hochuli, J.; Helbling, A.; Skaist, T.; Ragoza, M.; Koes, D. R. Visualizing Convolutional Neural Network Protein-Ligand Scoring. J. Mol. Graph. Model. 2018, 84, 96–108. 1270. Alishani A.; Chrupala C.; Linzen T. Analyzing and Interpreting Neural Networks for NLP: A Report on the First Blackbox NLP Workshop, arXiv:1904.04063 (2019). 1271. Bouwmans, T.; Joved, S.; Sultana, M.; Kung, S. K. Deep Neural Network Concepts for Background Subtractions Systematic Review and Comparative Evaluation. Neural Netw. 2019, 117, 8–66. 1272. Alber, M.; Lapuschkin, S.; Seegerer, P.; Hägele, M.; SChütt, K. T.; Montavon, G.; Samek, W.; Müller, K.-R.; Dähne, S.; Kindermans, P.-J. iNNvestigate Neural Networks!. J. Machine Learn. Res. 2019, 20, 1–8. 1273. Holley, L. H.; Karplus, M. Protein Secondary Structure Prediction With a Neural Network. PNAS 1989, 86, 152–156. 1274. Liang, H.; Stanev, V.; Kusne, A. G.; Takeuchi, I. CRYSPNet: Crystal Structure Predictions Via Neural Networks. Phys. Rev. Mater. 2020, 4, 123802. 1275. Rost, B.; Sander, C. Combining Evolutionary Information and Neural Networks to Predict Protein Secondary Structure. Proteins 1994, 19, 55–72. 1276. Salamon, P. Beware the Claims of the Producer!. Private Communication, 1988. 1277. Catlow, C. R. A., DeLeeuw, N. H., Anwar, J., Davey, R. J., Roberts, K. J., Unwin, P. R., Eds.; Faraday Discussions 136: Crystal Growth and Nucleation, The Royal Society of Chemistry: London, 2007. 1278. Anwar, J.; Zahn, D. Uncovering Molecular Processes in Crystal Nucleation and Growth by Using Molecular Simulation. Angew. Chem. Int. Ed. 2011, 50, 1996–2013. 1279. Santoro, M.; Schön, J. C.; Jansen, M. Finite-Time Thermodynamics and the Gas-Liquid Phase Transition. Phys. Rev. E 2007, 76, 061120.

3.12

First principles crystal structure prediction

Lewis J. Conwaya, Chris J. Pickarda,b, and Andreas Hermannc, a Department of Materials Science & Metallurgy, University of Cambridge, Cambridge, United Kingdom; b Advanced Institute for Materials Research, Tohoku University, Sendai, Japan; and c School of Physics and Astronomy, The University of Edinburgh, Edinburgh, United Kingdom © 2023 Elsevier Ltd. All rights reserved.

3.12.1 3.12.2 3.12.3 3.12.4 3.12.4.1 3.12.4.2 3.12.4.3 3.12.4.4 3.12.4.5 3.12.4.5.1 3.12.4.5.2 3.12.4.5.3 3.12.4.5.4 3.12.5 3.12.5.1 3.12.5.2 3.12.5.3 3.12.6 3.12.6.1 3.12.6.2 3.12.6.3 3.12.6.4 3.12.7 3.12.8 3.12.8.1 3.12.8.2 3.12.8.2.1 3.12.8.2.2 3.12.8.2.3 3.12.8.3 3.12.9 Acknowledgments References

Introduction Crystalline configuration space First principles calculations in solids Modern approaches Random structure searching Particle swarm optimization Genetic and evolutionary algorithms Database informed methods Local approaches Metadynamics Minima hopping Basin hopping Following soft modes Fitness functions Compositional stability Structural and mechanical properties Electronic properties Accelerating the search process Hard sphere potentials and volumes Modular decomposition Orbital free DFT Machine learned potentials Visualizing the PES Examples Compositional complexity High pressure Elements Hydrogen bonding Polyhydride superconductors Target properties Challenges and conclusions

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Abstract First principles calculations, usually in the shape of density functional theory, are an indispensable component of modern day solid-state inorganic chemistry. Over the past 15 years, those types of calculations have successfully evolved from explanatory to predictive tools. At the heart of this development is the now widely recognized ability of density functional theory to predict, without any input from experiment or recourse to chemical intuition, the crystal structures of inorganic compounds. In this chapter, we begin by presenting arguments why crystal structure prediction works, seemingly against the combinatorial odds, due to the shapes of the potential energy landscapes in crystalline configuration space. We then review the modern approaches used in the field that sample this space, either on a global scale aimed at finding global and metastable minima, or on a local scale aimed at finding nearby minima and transition states. We include an overview of attempts to tailor searches toward desired materials properties, and discuss ongoing work to make structure prediction faster and more efficient. Finally, we give examples from fields where crystal structure prediction is, or could be, a successful driving force for new research avenues, and conclude by pointing toward some of the remaining challenges.

3.12.1

Introduction

Determining the structure of a material has been a problem faced by computational scientists for decades. Without this information, it is difficult to predict or explain material properties. As the rise in computational resources matched the development of accurate

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first principles calculation methods, high-throughput approaches have become more feasible routes to material design and can be used to study a wider variety of systems. This approach allows efficient screening of material properties, particularly in unusual chemical environments. The study of a material by experiment is a labor intensive process. To use a trial-and-error approach experimentally to design and study new materials is prohibitively slow. Whereas, a computer d by its nature d can perform repetitive tasks and screen large numbers of potential compounds quite straightforwardly. Scientists who previously spent years studying a handful of materials, can now uncover insights into novel materials with relative ease using crystal structure prediction (CSP). That is not to say that high-throughput experiments are not possible. For instance, brilliant x-ray radiation from x-ray free electron lasers can be used in ‘high-throughput’ approaches to crystal structure determination.1 Even if such high-throughput experiments were more common, the results are not as definitive as those obtained in calculations. Typically, the crystal structure is inferred from diffraction based methods. Due to restrictions caused by noise, low symmetry, and sample synthesis, diffraction may only provide partial or inconclusive information about the crystal structure. This process can be helped significantly if guided by initial calculations which can provide likely stable stoichiometries, material properties, and diffraction patterns. It would have been difficult to conceive of a novel high-Tc superconducting sulfur hydride as a decomposition product of H2 S, without CSP.2 Recent machine learned high-throughput methods can be used to quickly estimate superconducting temperatures3 of a wide variety of hydrides, extending the predictive power of CSP; promising metastable polymorphs with desirable Tc’s can be quickly identified. It would also be difficult to sample a wide range of complex stoichiometric environments without CSP. Cathode materials can consist of a wide variety of quaternary or even quinary stoichiometries. The space associated with conceivable structures of these types is very large, but CSP can help identify low energy polymorphs4,5 with promising properties. However, the methods used to obtain total energies and material properties are always approximate solutions to the problem, typically a trade-off between speed and accuracy. For example, in some cases, an ionic pair potential or force field is sufficient to calculate energies and is computationally very fast. In other cases, e.g. where electronic properties are relevant, such a potential might be insufficient to describe the behavior. The latter typically employs a density functional based approach, which in turn might be problematic for instance for strongly correlated materials. In either case, the search results should not be considered the final answer; temperature effects are difficult to consider in structure searches, and the choice of level of theory can influence the results. For these reasons, both experimental and theoretical methods should be used together to uncover the accurate picture; CSP should motivate experiments and vice versa. This chapter will introduce the methods used to perform effective structure prediction from first principles. To appreciate the power of CSP, we begin by discussing historical approaches to determine molecular and crystal structures before introducing modern approaches to the global minimization problem. An early example of theoretical structure prediction preceding an experimental discovery is that of van’t Hoff, who postulated d to account for the optical activity of carbon-containing compounds d that a central carbon atom should be bonded to its neighbors such as to form a tetrahedron.6 Van’t Hoff’s idea of this asymmetric carbon atom led to the idea that an allene d which takes the form of RR’C ¼ C ¼ CRR’ where R represents a functional group d should be chiral. Many years later, this was confirmed experimentally.7 Following the development of x-ray diffraction and the discovery of atom-level periodicity in crystals, more systematic predictions were sought. One approach is to predict the likely coordination number for a given stoichiometry based on ionic radii ratios; a concept first applied to extended lattices by Victor Goldschmidt in 19268 and 1927.9 For example, a 1:1 ionic binary compound with an ionic ratio of around 0.3 will likely be tetrahedrally coordinated in a cubic sphalerite structure or hexagonal wurtzite structure. For ratios around 0.6, it will likely form octahedrally coordinated structures such as rock-salt, and if it is close to 1 (i.e. the ions are similar in size), it will likely be 8-fold coordinated like CsCl. Similar arguments can be made for the stability of perovskite structures (ABO3). Intuition about the expected coordination of the atoms and the stoichiometry d in other words, to appeal to conventional wisdom d may be enough to generate some likely candidates from previously known structures. However, there are many ways to stack structures tetrahedrally or to stack perovskite units, and the conventional behavior of chemical bonding is different when pressure is applied. Additionally, stoichiometric variability is a concern. The ionic radii rules will not generate stable structures of the type Na3Cl, Na2Cl, Na3Cl2, NaCl3 forming under pressure,2 or the aforementioned H3 S10 forming from the decomposition of H2 S. With the advancement of density functional theory techniques, it became feasible to calculate the stable configurations of crystal structures, including the predictions mentioned in the previous paragraph. While computational limitations would mean initially limiting the candidates to small primitive unit cells with high symmetry, computational resources sufficient for unrestricted (and importantly, unsupervised) structure searches have been available for close to 20 years now. While the determination of the global energy minima is a daunting task, the nature of the potential energy surface means it is surmountable. There are many crystal structure prediction methods, including evolutionary algorithms11,12 and machine learning methods,13 all of which have been successful in predicting novel crystal structures, particularly at high pressures14–21 including the structure of phase III of solid hydrogen,16 a transparent phase of sodium,22 and superhard tungsten nitride.23,24

3.12.2

Crystalline configuration space

We begin by making a case for why crystal structure prediction works, despite the apparently overwhelming number of degrees of freedom on offer.

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Long-range translational symmetry sets crystals apart from localized or extended amorphous systems. This symmetry is captured by defining three lattice vectors (a1, a2, a3) (in three-dimensional space) such that the crystal is invariant under any translation vector R ¼ n1a1 þ n2a2 þ n3a3 that is a linear combination of the lattice vectors. The atomic basis, the set of atomic positions within the parallelepiped spanned by the lattice vectors (the unit cell), completes the definition of a specific crystal structure. The unit cell is not uniquely defined; an infinite number of choices of lattice vectors are possible that describe the same crystal. A primitive unit cell is the smallest possible unit cell (by volume or number of atoms), but is also not uniquely defined; several approaches exist that construct uniquely defined reduced cells for a given lattice, which involve finding the shortest possible primitive unit cell edges.25–28 Every lattice (in real space) has a corresponding reciprocal lattice (in reciprocal space), with lattice vectors (b1, b2, b3) defined via the identities ai $ bj ¼ 2pdij. The reciprocal lattice (with units of inverse length) is important whenever phenomena are studied that are modulated by the crystal lattice, such as harmonic lattice vibrations, phonons, with specific wave vectors q that reside in the reciprocal unit cell; see Section 3.12.4.5.4. More importantly, Bloch’s theorem shows that single-electron states in a periodic potential have wave functions of the type jk(r) ¼ uk(r) exp ikr, with a lattice-periodic component uk(r) and a phase factor determined by wave vector k, which resides in the first Brillouin zone, the region of reciprocal space closer to the origin than to any other reciprocal lattice vector; this is important in any electronic structure calculation, see Section 3.12.3. The configurational space for an N-atom primitive unit cell is 3 N þ 3 dimensional (3 N – 3 dimensions for atomic positions, 6 dimensions for the unit cell). A specific unit cell volume V ¼ NVat or, equivalently, number density n ¼ 1/Vat, which introduce a characteristic atomic volume Vat, reduces the dimension by one. A combinatorial approach shows29,30 that the remaining space for atomic positions alone is incredibly vast. If atoms are placed within the unit cell on a discretized grid with spacing d, i.e. S ¼ Vat/ d3 sites per atom, the number of configurations C is ! NS C¼ zð2pNÞ1=2 eN SN ; N utilizing Stirling’s approximation. The exponential growth of the number of possible configurations makes exhaustive exploration of the space impossible. For compounds, where not all atom types are equal, an additional binomial factor increases C even further, but the largest contribution remains the freedom of placing atoms in space. However, since the potential energy surface (PES), the energy associated with any point in the configurational space, is the result of physical interactions, it is, in most cases, smooth. Unphysical configurations, such as those with overlapping atoms, can be considered irrelevant for materials synthesis processes and ruled out altogether from detailed exploration. Most other configurations do not correspond to energy minima. Nonetheless, the number of local minima Cmin still scales exponentially with system size,31,32 Cmin  eaN. However, the topology of the potential energy surface and the energy spectrum of these local minima have been subject to several studies that yield promising insights. Doye and co-workers noted that lower energy minima on the PES tend to have larger basins of attraction.33,34 They show that topologically, the PES resembles an Apollonian packing of spheres, in that the largest (and deepest) basins have the highest number of connections to adjacent basins, just like the largest Apollonian spheres have the most touching neighbors,35 see Fig. 1. One consequence is that by performing geometry optimizations to find the local minimum from a given starting configuration, the likelihood of finding the global minimum is significantly increased. In fact, taken to its extreme, this basin size distribution implies that the global minimum has the highest probability of any minimum to be found with a random starting structure; however, due to the sheer number of local minima the absolute probability can still be very low, requiring many initial attempts. Another consequence of the expected high connectivity of the global minimum basin is that local exploration of the PES from some initial local minimum is a reasonable strategy to find more favorable minima. The Bell-Evans-Polanyi principle provides some arguments for the basin size distribution and correlation with the minimum energy. It states that the smaller the energy barrier from one basin to a neighboring basin, the lower the neighboring basin’s energy. In other words, low energy minima are close together. In a similar manner to the Marcus equation for electron transfer, this principle can be shown by assuming identical quadratic forms for neighboring basins with minima A and B, along a reaction coordinate x, at x ¼ 0 and x ¼ 1. The result is that the transition coordinate is x ¼ 12 for a thermoneutral reaction (that is to say both minima are equal) and that the transition coordinate and energy barrier is lower if minimum B is lower in energy than minimum A.36 An alternative schematic is shown in Fig. 1: lowering the minimum energy of the central basin will both lower the transition barriers to access it from its neighbors, and increase its size. The Bell-Evans-Polanyi principle relies on the assumption that basins have comparable shape; we assume equal curvatures in the previous paragraph. This is not unreasonable: a reaction coordinate connecting two local minima across a transition state represents coupled atomic displacements and lattice strains, and the restoring forces and stresses, in the harmonic/linear elastic regime, can be expected to be similar, if comparable bond breaking and bond formation processes occur around both minima. Another assist for CSP is symmetry. Crystal structures can be highly symmetric, and high symmetry atomic configurations are often very low or very high energy minima.37 In organic crystals, the uneven distribution of space groups (the 230 symmetry groups for three-dimensional crystals) has been documented for a long time38,39 and is related to close-packing of unevenly shaped objects.40–42 That restriction means space groups that combine inversion centers, 2-fold screw axes, and glide planes, are preferred43 and indeed the six most dominant space groups in the Cambridge Structural Database (CSD), which are of that type, make up almost 83% of the over 1.1 m organic crystal structures in the CSD.44 (Note that Kitaigorodskii41,42 built and described a mechanical

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(A)

(B)

(D) (C)

Fig. 1 Crystalline PES characteristics. (A) Basin size distribution as Apollonian sphere packing, energy axis indicated; (B) Bell-Evans-Polanyi principle connecting a basin’s energy minimum to its size (black dotted bars) and transition barriers to adjacent basins (blue dots); (C) crystal system distribution in the ICSD database; (D) space group frequency of the most prominent space groups in the ICSD database.

‘structure seeker’ to explore and in effect predict close packed molecular crystals). In inorganic systems, the symmetry requirements and therefore their statistical distribution are markedly different. Fig. 1 shows crystal system and space group frequencies extracted from the over 200,000 entries of the Inorganic Crystal Structure Database (ICSD).45 The different crystal systems are relatively evenly distributed, but within them particular space groups stand out. Those tend to be of the highest symmetries that each crystal system offers and suggest a symmetry principle in the formation of inorganic crystals.39 The 20 most prominent space group account for just over 63% of all ICSD crystal structures. For structure prediction, this could mean that symmetry-constraint searches, or at least initial guesses, could be a fruitful avenue to explore. But note that four of the 20 most relevant space groups are triclinic or monoclinic; insisting on high symmetry for a particular compound might lead only to metastable states.

3.12.3

First principles calculations in solids

The exploration of the crystalline configurational space can, in principle, be done with any type of interatomic potential that attaches total energies to specific configurations and generates forces to optimize said configurations. Using off-the-shelf interatomic potentials or force fields is problematic if one aims to explore different conditions, e.g. high pressure, but even at ambient conditions every structure prediction approach (see next section for details) involves creating structural candidates that are initially far from structural equilibrium (database mining approaches would be an exception), where force field parameterization might be poor. Even so, in organic crystal structure prediction the use of force fields remains common, due to otherwise prohibitive computational cost,46 but is most often combined with very sophisticated and specialized algorithms to generate structural candidates that avoid unphysical configurations. For inorganic systems, an additional challenge is their potentially non-trivial electronic structure, so metallic and insulating configurations need to be described with similar accuracy. Taken together, using parameter free ab initio methods has therefore become the default approach in inorganic crystal structure prediction, helped by advances in computational power that have made high-throughput first-principles calculations feasible about 20 years ago. However, note that recent advances in machine learning of interatomic potentials has meant that computationally cheap, transferrable potentials can be generated for a wide variety of systems, and we will discuss further below how this is used in crystal structure prediction. Here, we give a very brief overview of total energy calculations for crystalline systems in conjunction with density functional theory, the standard method for first-principles calculations in extended systems. A wide selection of textbooks is available for more detailed reading in this area, see for example Refs. 47–50. For any independent particle approach to electronic structure in crystals, Bloch’s theorem means that the many-electron problem only needs to be solved for the number of electrons Ne per unit cell, instead of the whole crystal. For every wave

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vector k, the single-electron wavefunctions jnk and their eigenvalues εnk are labelled by a band index n; in an insulator, the lowest Ne/ 2 bands are occupied, all others unoccupied. In a macroscopic crystal, electrons occupy states on a dense mesh of k-vectors; the mutual orthogonality of states at two wave vectors k1 and k2 satisfies Pauli’s exclusion principle. Many properties, such as total electronic energy, or electron density, need to be added up or averaged over all occupied electronic states. It turns out that the dense mesh of k-vectors (with on the order of Avogadro’s number of so-called k-points) can be replaced for those purposes by a much coarser regular grid of k-vectors, with on the order of 10–1000 k-points.51 This, then, makes electronic structure calculations for crystals feasible: a relatively small number (101000) of eigenvalue problems of order Ne needs to be solved to obtain the electronic structure and hence information about total energy, Coulombic forces on the atoms, etc. In standard density functional theory (DFT), the eigenvalue problems can be solved independently for different k-vectors, which is ideal for numerical parallelization. In those flavors of DFT, the exchange-correlation energy is approximated by local or semilocal density functionals.52,53 The Hamiltonian, as in standard SCF (Hartree-Fock) theory, depends via the density on its own solutions, and is solved self-consistently. Due to Bloch’s theorem it is advantageous to use plane waves as basis functions to expand all electronic states. The latticeperiodic component uk(r) of any Bloch state can be expanded in a Fourier series of plane waves with periodic lattice wave vectors, P unk(r) ¼ Gcnk(G) exp iGr, where all G ¼ m1b1 þ m2b2 þ m3b3. The basis set quality can be captured by a single quantity, the cutoff radius kGmaxk of the G-vector expansion; most commonly expressed in terms of the “plane wave [kinetic] energy cutoff”, Ec h12kGmax k2 . In plane wave representation, solving the DFT eigenvalue equations then becomes a matrix diagonalization problem for the coefficients cnk(G), replicated over the finite set of k-vectors. However, the choice of a plane-wave basis set is not well suited to the core regions of atoms where the spatial variation in the wavefunction is rapid and would require a very high value of Ec d and hence a prohibitively slow calculation d to give accurate results. However, core electron orbitals are unlikely to depend on the atomic environment and so electrons within a given cut-off radius of the nucleus are pre-calculated such that electronic structure calculations need only consider the remaining valence electrons. This is akin to the “effective core potential” approach, which in plane-wave DFT is extended to the whole periodic table. The pseudopotential replaces the hard divergent potential close to the core with a softer potential. This is typically done by solving the radial Schrödinger equation as an ‘all-electron’ problem and generating a pseudo-wavefunction that satisfies a number of constraints.54,55 First principles structure prediction relies on robust pseudopotentials that deliver accurate results with relatively low values of Ec. Note that including the Fock operator explicitly in the exchange-correlation energy expression severely increases computational cost of plane wave calculations (unlike those based on localized basis sets). Hartree-Fock or hybrid functional DFT calculations for crystal structure prediction are therefore prohibitively expensive, as are any other wave-function based approaches for CSP. This can be a challenge in systems where so-called exact exchange is crucial to obtain the correct electronic structure. Once the electronic structure is obtained, forces on all atoms and stresses on the unit cell are obtained using the HellmannFeynman approach. Those forces and stresses are then minimized using standard iterative techniques, until residuals are below acceptable thresholds; at which stage the final crystal structure (lattice and atomic positions) together with its energy and, optionally, other observables, are available for further processing. An efficient implementation of the geometry optimization is key for CSP, and the DFT packages typically chosen for CSP are those that have fast yet robust algorithms to minimize forces and stresses.

3.12.4

Modern approaches

A variety of different algorithms have been implemented in modern CSP approaches, and are described in detail below. They universally employ local geometry optimizations as described above. This distinguishes them from one of the most intuitive early structure searching approaches, simulated annealing.56 In that method, Monte Carlo or molecular dynamics steps are applied to modify an initial structure, with an initially very high temperature that is continuously reduced during the simulation, leading ideally to the global minimum. No local geometry optimizations are involved during the individual simulation steps, which arguably restricts the method’s efficiency.

3.12.4.1

Random structure searching

As we will discuss, all approaches in ab inito structure searching necessarily utilize some level of randomness. The AIRSS code (Ab initio Random Structure Searching)57 uses random structure search exclusively. While this approach may seem simplistic, it is remarkably effective thanks to the generation of sensible random structures, an emphasis on trivial parallelism and effective postprocessing tools. The speed of ab initio crystal structure searches depends on how many geometry optimization steps are required. This can be reduced by generating ‘sensible’ starting structures. This means the geometry optimization spends less time in unphysical configurations, far from the local minima. AIRSS incorporates a powerful set of keywords to generate sensible random structures by invoking a useful program, buildcell. Important structure features such as the target volume and minimum atomic separations can be chosen manually or automatically based on initial searches. Symmetry can be imposed by specifying a range of symmetry operations or a specific space group.

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For molecular crystals, buildcell can place molecules in random positions in the cell while applying symmetry operations. Constraints can be placed on the unit cell, and on positions of sets of atoms. This is useful when only partial information about the crystal structure is available. A technique used in AIRSS is ‘relax and shake’ (RASH), which generates new structures by shaking structures around a local minimum. This can result in new structures with lower symmetry, or similar structures in distorted super-cells. This process can be performed recursively as a minimal input ‘learning’ approach. Unlike more sophisticated approaches, random structure searching does not require any communication between different computations. This means calculations can be trivially parallelized so that thousands of structures can be generated and calculated simultaneously. For the same reason, AIRSS does not require special treatment for variable stoichiometry searching. The user can specify an arbitrarily complex composition space over which AIRSS can sample. 58 AIRSS uses the SHELX .res file format for crystal structures, supplemented with metadata. These files can be read by visualization tools such as Vesta59 and directly by AIRSS’ analysis tool, cryan, which can rank structures, build convex hulls and provide geometrical analyses.

3.12.4.2

Particle swarm optimization

A general approach to go beyond purely random structure generation is to use information from already completed geometry optimizations to map the configurational space and build new structures. Particle swarm optimization (PSO) is a global minimization technique used in a wide range of applications with non-convex minimization problems, such as modelling the electrical power market.60 The technique allows individual agents to move around configurational space (where particle i has coordinate xijt at timestep t with velocity vijt). The positions and velocities are initialized randomly and allowed to relax to a global minimum with position yijt. The new velocities are calculated from a weighted sum of;

• • • •

the difference between the initial and relaxed geometry, the difference between the starting configuration, the best known minimum energy configuration, zijt, and the velocity at the previous timestep.

The update equation used by the CALYPSO code61 is given by,     vijtþ1 ¼ A1 yijt  xtij þ A2 ztij  xtij þ uvijt :

(1)

Where A1 and A2 are randomly generated numbers between 0 and some value A1max and A2max. The ‘memory’ factor u decreases linearly between generations. A schematic of the algorithm and its traversal of the potential energy surface is shown in Fig. 2. Several extra constraints are required to avoid stagnating in local minima: a penalty function is applied such that high energy structures are not used in the PSO, further exploiting the Bell-Evans-Polanyi principle; and each generation includes a certain (user-specified) percentage of new randomly generated structures to retain the population diversity.

3.12.4.3

Genetic and evolutionary algorithms

An alternative approach to exploit information during a structure search is to use genetic algorithms (GA’s) or evolutionary algorithms (EA’s). These borrow ideas from biological evolution, of mutation, breeding and natural selection, to build upon promising intermediate structural candidates. Ideally, that allows the search to focus on the relevant sections of the PES, generate more competitive guesses faster, and find the global minimum with less attempts. Their use for structure prediction was pioneered in the early 1990s for clusters and nanoparticles62–64 and shortly thereafter for inorganic solids.65 GA’s encode all properties of a particular structure in a one-dimensional alphanumeric or binary string, the “genome”. All modifications (described below) are done by direct manipulation of “genes”. The projection from structure to genome (and vice versa) removes GA’s from the structure prediction realm, into the wider area where GA’s are applied. The advantage is that a large set of GA operators developed in different research fields can be applied straightforwardly; the disadvantage, that any links to crystalline configurational space, and to physico-chemical intuition as relates to structure formation, are severed. EA’s on the other hand describe algorithms where structural information (lattice vectors and atomic positions) is stored and manipulated directly, typically with the use of evolutionary operators that are designed for structural modifications. The majority of software packages in the field (see Table 1) are of the EA type. Evolution of the Darwinian type implies that changes to the genome only occur in the procreation process, combining parental features with a small amount of random changes (mutations). For structure prediction this implies generating a new structure and determining the enthalpy for its given (fixed) geometry. This often has a very low success rate. However, allowing local optimization of the structure (i.e., relaxing to the bottom of its local basin in the PES) results in a much more relevant structural candidate for further consideration. This local optimization corresponds to changes to the genome during the “lifetime” of the structure, and allowing those changes to be passed on to later generations follows the Lamarckian view of evolution, that traits acquired during

First principles crystal structure prediction

Fig. 2 Flow chart (left) outlining the particle swarm optimization technique (equation is specific to energy surface (right). Table 1

CALYPSO)

to find the global minimum of a potential

A selection of CSP packages for inorganic systems available for academic use (possibly following registration), and the methodology they are based on. Equivalent packages that specialize on organic CSP include DMACRYS, GAtor, GRACE, MOLPACK, and UPACK.66–70

Package

Method

References

AIRSS

Random structure searching Particle Swarm Optimization Bayesian Optimization Genetic Algorithm Evolutionary Algorithm Evolutionary Algorithm Evolutionary Algorithm

17,57 61 71 72,73 74 11 75

CALYPSO CRYSPY GASP MAISE USPEX

XTALOPT

399

an organism’s lifetime can be passed on to offspring. Such is the success of this approach, that local structural optimization has been identified as a key ingredient of GA or EA structure prediction.11 The typical workflow of a GA/EA structure search is shown in Fig. 3. Often a “generation” of structures is optimized, followed by a selection process of the best candidates, which influence the next generation of structures. The latter features various genetic or evolutionary operators. In another (incidental) nod to biology, EA implementations show great diversity in choice and weighting of particular evolutionary operators, but the following operations are commonly used:

• • •

Mutation: manipulating lattice vectors and/or atomic positions of a single structure. Can include lattice strains, atom exchanges between sites, individual or collective atomic displacements. Breeding: combining two “parent” structures into one “offspring”. Most often a real-space ‘cut and splice’ operation. Random generation: adding completely random new structures to the dataset to retain diversity of the “gene pool” and avoid premature focus on particular parts of the PES.

Structural candidates, whether created at random or via specific operations, might occupy the same basin within the PES, and are then (within numerical noise) identical following the geometry optimizations. This is not an issue for random structure searching – re-discovering the same structure several times in fact contains information about relative basin sizes – but is a challenge for algorithms with built-in memory: GA/EA and also PSO implementations need to be able to identify identical structures to remove duplicates from the pool of structures considered for the next generation. Given that crystal lattices and atomic positions are not uniquely defined, this is a non-trivial task. Duplicates are usually identified by having very similar enthalpies, and by either matching fingerprints based on local atomic connectivity,76 pairwise separations of atoms77 or successful mapping of suitably normalized unit cells.75 Lyakhov et al.78 show that fingerprints, and the ‘distance’ between these fingerprints, can be used to perform more effective EA structure prediction. To maintain diversity, structures with ‘nearby’ and hence similar fingerprints are identified and only the

400

First principles crystal structure prediction

Fig. 3 Top: sketch outlining the evolutionary algorithms technique to find competitive minima of a potential energy surface. Bottom: visualizations of typical EA operations: mutations that involve lattice strain, collective displacements, or atom swapping; and hereditary ‘cut and splice’ operation.

lower energy structure is carried forward. They also note that structures which are very dissimilar should not ‘breed’ as the “offspring” structures will often be high energy.

3.12.4.4

Database informed methods

Yet another approach is closer to the aforementioned ‘appeal to conventional wisdom’ approach. With high-throughput calculations easily accessible, it is feasible to build all candidate structures using many geometrically valid structures or structures from an extensive database (typically 1000s of structures) of known geometries. The ICSD has over 200,000 experimental inorganic structures. The majority of these fall into a given structure type; a set of structures with identical lattice shapes and (relative) atomic positions.79 For example, the cubic ZnS wurtzite structure type (labelled B4 in Strukturbericht notation) is shared by AgI, BeO, ZnO, GaN and many others. This suggests that a lot can be learned from these known structure types with different stoichiometries. As more new structures are discovered, new structure types may also emerge. Before global optimization-based CSP methods, some effort had been applied to combine ab initio calculations with highthroughput approaches, building new structures from known structure types80 and generating databases of electronic structure information. Since then, tools have been developed to assist these types of calculations such as high-throughput pseudopotentials81 and band structures.82 This technique has been applied to otherwise compositionally complex systems such as ternary and quaternary oxide compounds.83,84 This method is similar to, but distinct from, database screening methods which aim to screen for material properties based on data from a database, without input from ab initio calculations.85,86 A related approach has an emphasis on local connectivity, or bond network topology, achieved by representing crystal structures as graphs. A graph is a set of nodes connected by edges. Just as a crystal, a graph can be infinite. Placing nodes in space (such as a plane or 3-dimensional space) is called embedding. A crystal can be considered a three-dimensional embedding of a graph where nodes are atoms and edges are bonds (or connections between nearest neighbors). The infinite graph of a crystal will have the same 3-periodicity. This can be simplified to a finite labelled graph by a process called the vector method.87 In short, this involves drawing N nodes for the N atoms in the unit cell. An edge can be drawn between the nodes for bonded atoms where both are within the unit cell. Where atom i is inside the unit cell and is bonded to an atom j in the neighboring unit cell, an edge can be drawn from atom i to atom j’s image in the unit cell and labelled with the lattice vector describing the shift in unit cell.

First principles crystal structure prediction

401

Some examples of this method include considering all possible three-fold coordinated structures to find sp2 carbon polymorphs (restricted by the number of atoms in the unit cell),88,89 or tetrahedral networks to find sp3 carbon polymorphs.90,91 Further sp3 carbon polymorphs have been generated by using databases of zeolite structures.92 A similar approach has been used to great effect to reproduce a priori all known phases of ice.93 Structures generated using these methods may have much larger primitive unit cells and require more computational resources per structure, however each optimization starts from a chemically sensible configuration. Graph theoretical methods have also been invoked in the study of alumino-silicates94,95 where particular care can be taken to build structures obeying the Löwenstein avoidance rule,94,96 which suggests alumino-silicates will be unlikely to have AleOeAl bonds, an assertion that has kinetic97 and thermodynamic arguments.98 Topological methods can be extremely efficient, if the underlying assumptions about relevant bond network topologies hold true. Conversely, they quickly fail if those assumptions break down. For instance, studying silica polymorphs based on connected SiO4 tetrahedra will fail to identify the high-pressure stishovite phase, which is based on SiO6 octahedra.99

3.12.4.5

Local approaches

The prediction methods discussed so far are “global” approaches in the sense that they survey, at least initially, the full crystalline configuration space. Approaches that retain memories during the searches, such as the PSO and GA/EA algorithms, should eventually focus on the most relevant sections of this space. However, the location of this section can not be directed by the user, nor do the results give any indications about the local connectivity of the PES, i.e. which local minima are connected by transition states and the size of transition barriers. Further below we shall introduce recent approaches to extract such connectivity information from global structure prediction surveys, but in this section want to briefly discuss “local” approaches to predict crystal structures. By this we mean approaches that explore the local topology and connectivity of the PES, usually starting from a single user-defined configuration, and not always concerned with identifying the global minimum. There are a multitude of scenarios where such information is relevant. Fig. 4 depicts an idealized potential energy surface of carbon, which is dominated by graphite and diamond but features a myriad of other local minima that can be seen as intermediates in the sense that they combine sp2- and sp3-hybridized carbon (the x-axis in the figure should not be taken too literally; there are multitudes of purely sp2- and sp3-bonded carbon polymorphs other than graphite and diamond100). At ambient conditions graphite has the lowest free energy of all carbon polymorphs101 but diamond is stabilized at moderate pressures. However, the cold (i.e., non-heated) compression of graphite does not result in diamond102–107 because the potential barrier toward its formation can not be overcome, at least not on the experiments’ time scale. Instead, another metastable carbon allotrope forms that is configurationally “closer” to graphite but features sp3-bonded carbon. Various crystal structure prediction approaches have yielded a variety of candidate allotropes very close in enthalpy, such as M-, bct-C4-, W-, and Z-carbon.29,108–112 Of those, M-carbon seems the most likely candidate due to the lowest kinetic barriers to transformation from graphite.113 Hence, knowledge of the local topology of the crystalline PES can be invaluable to interpret experimental findings or explore whether new materials are accessible from specific precursors. Most of the approaches described below are not exclusive or tailored for crystal structure prediction, they instead represent quite general methods for sampling free energy surfaces, often aiming to accelerate rare events such as phase transitions where substantial energy barriers need to be overcome.

3.12.4.5.1

Metadynamics

In metadynamics,114–116 a molecular dynamics simulation is performed, beginning from a local minimum on the PES. During the simulation, a bias potential is constructed that penalizes previously visited configurations, thus driving the system away from the initial and toward new metastable or stable configurations, see Fig. 5. One key ingredient is the concept of “collective variables”

Fig. 4 Schematic of carbon’s potential energy surface at some moderate pressure where diamond (“dia”) has a lower enthalpy than graphite (“gra”) but other metastable structures are more readily accessed.

402

First principles crystal structure prediction (A)

(B)

(C)

Fig. 5 Local approaches’ explorations of potential energy surfaces (red) with structural candidates (blue). (A) Metadynamics simulation early on (left) and at a later stage (right). Green indicates the bias potential. (B) Basin hopping. (C) Minima hopping.

(CV’s) {s(R, a)}, a set of coarse-grained order parameters much smaller than the full set of atomic positions R and lattice vectors a, that can distinguish the different metastable states. The bias potential V(s) is built successively as a series of Gaussians G(s, s’) ¼ W exp  (s  s’)2 in s-space that are centered around the CV’s present at regular intervals during the simulation, si    1 bVn1 ðsn Þ : Vn ðsÞ ¼ Vn1 ðsÞ þ G s; sn Þexp  g1 The exponential scaling term, introduced as well-tempered metadynamics,115 ensures that the changes to the bias potential decay over time, and features the inverse temperature b ¼ 1/kBT and a positive constant g > 1. Those, together with the width of the Gaussians W are the main computational variables in a metadynamics simulation. The x-axis of the metadynamics schematics in Fig. 5A should therefore be considered as one of these CV’s. Ongoing efforts in developing CV-based approaches are often centered around building better iterative bias potentials, to further accelerate rates of transitions between metastable minima.117–121 However, the main physical choice is around appropriate CV’s themselves. For proteins, these might be certain torsional angles that distinguish relevant foldings. For crystal structures (and many other systems) there are no obvious choices. Moreover, in exploratory searches, where the target structure is not known, it is nigh on impossible to guess the most efficient CV’s that link initial and target structure. However, noting that space group symmetry and more generally the crystal lattice often sufficiently distinguish different polymorphs within a given system, metadynamics simulations in inorganic solids most often use the lattice vectors as CV’s, s ¼ (a1, a2, a3).122,123 Strictly speaking, the six independent lattice lengths and angles are used.

3.12.4.5.2

Minima hopping

3.12.4.5.3

Basin hopping

Minima hopping124,125 is another approach to survey metastable minima around an initial configuration that are accessible via increasingly larger energy barriers. It also uses molecular dynamics to explore the PES, but uses more explicit bookkeeping to retain memory of the explorations. After a relatively short MD run a full geometry optimization is performed, see Fig. 5. If the system has reached a new configuration, it will be added to the list of known configurations if it is less than a threshold Ediff higher in energy than the previous configuration. The threshold Ediff and the MD temperature T are then decreased. On the other hand, if the energy of a newly found configuration is more than Ediff above the previous configuration, Ediff will be increased; and if the optimization ended in a previously visited configuration, T will be raised. With those adjustments, repeated visits to known configurations are penalized, the exploration will always escape from local trappings, while locating new relevant minima slows the exploration down.

The basin hopping approach was first introduced for clusters and is still mostly applied to localized systems126; it is similar to a prior “Monte Carlo minimization method” suggested to explore protein folding.127 In this method, changes to the current configuration are made discontinuously, using random update steps such as moving or swapping atoms. Importantly, those updates are followed by full geometry optimizations, and the optimized configuration is accepted or rejected based on a Metropolis criterion, see Fig. 5.

First principles crystal structure prediction

403

By attempting to sample the space of local energy minima rather than the full PES, a high-dimensional problem is vastly reduced. But not only is the information from each PES basin reduced to its minimum energy value, this approach also loses information about barriers between adjacent basins. By scaling the overall update step size the method can be tuned to explore configurationally close to or farther away from the initial structure; if it is too large, the method essentially becomes random structure searching.

3.12.4.5.4

Following soft modes

A method specific for crystal structure prediction is to “follow” soft phonons. Phonons are the elementary quasiparticles of lattice vibrations and span the space of all possible atomic displacements within a crystalline system. They are harmonic and monochromatic, characterized by a wave vector q, frequency ui(q), and displacement vector ei(q) (index i labels the phonon “branch”, of which there are 3 N  3 for N atoms in the unit cell). Phonon frequencies and displacement vectors are the eigenvalues and -vectors of the dynamical matrix D(q), which itself is the Fourier transform of the real-space force constant matrix. The appearance of nonzero q is due to translational symmetry in the crystal; in molecular systems, the equivalent of the dynamical matrix is the Hessian. Solving for phonon frequencies is equivalent to diagonalizing the Hessian many times over, at each q-vector of interest. More precisely, the eigenvalues of D(q) are the squared phonon frequencies u2i (q), proportional to the curvature of the PES along the displacement vector ei(q). A dynamical instability is characterized by a negative curvature, and therefore u2i (q) < 0. This is referred to as an “imaginary phonon mode”, though it isn’t actually a phonon (there is no lattice vibration without a restoring force), rather a saddle point on the PES. When plotting the phonon dispersion relations ui(q) along special directions of q, imaginary modes are commonly drawn as negative frequencies, as sketched in Fig. 6. The corresponding atomic displacement vector ei(q) points toward a lower-energy structure on the PES, and this can be exploited to find low-lying minima in configurational space. This method is similar in principle to the RASH technique used in AIRSS. The entire procedure is sketched in Fig. 6. Starting from simple, high-symmetry initial structures for the system of interest (e.g., known structure types), geometries are relaxed and phonon dispersions calculated. If clear imaginary modes appear, the displacement vector is added to the atomic positions. Note that for non-zero q vector, the displacement always has a wave length that spans multiple crystalline unit cells; it needs to be applied to the atomic positions in a suitable supercell. That can make the resulting structures very large, and it is usually advisable to follow imaginary modes at q ¼ 0 (the G-point in the Brillouin zone), or otherwise modes at special points at the Brillouin zone boundary. Once a suitably scaled displacement is added, structures are relaxed, and the process repeated until dynamically stable structures have been generated. The absence of any imaginary modes means a minimum on the PES has been reached. This approach has been used to predict new perovskite compounds129–131 (starting from the ideal cubic perovskite structure), and new transition metal monoxides128,132 (starting from the rocksalt structure). There are no limits to its applicability, but it will only discover structures that are connected to the initial guess via a series of saddle points on the PES. A somewhat flipped approach that acknowledges the spirit of dynamical instabilities has been used in random structure searching to explicitly discover high-symmetry structures with one or two imaginary modes.133 Just like many perovskites have a hightemperature phase with the cubic structure,134 those saddle point structures are candidates for high-temperature phases of materials, where thermal excitations result in the averaged high-symmetry structure.

3.12.5

Fitness functions

So far, discussions have implicitly focused on the target to minimize the total energy (or enthalpy) during the structure prediction process. This is reasonable, as the global minimum on the potential energy surface for a given chemical composition is surely (A)

(B)

(C)

(D)

Fig. 6 (A) Illustration of following soft phonon modes. (B) Phonon dispersion curve of rocksalt HgO, as per reference 128. Points indicated by crosses are explicitly calculated phonon frequencies, red lines are Fourier interpolations. (C) Primitive unit cell of HgO in a rock-salt structure. (D) Imaginary phonon mode with wave vector q ¼ (0.5, 0.5, 0.5) of rock-salt HgO described in a 2  2  2 supercell of the primitive cell. Blue atoms are moved along the displacement vector. The perturbed structure is lower in energy.

404

First principles crystal structure prediction

the most likely to be synthesized. However, local search approaches have already been motivated by the need to identify low-barrier transitions from given initial structures to competitive metastable alternatives, rather than the global minimum. Furthermore, it is often desirable to identify and rank crystal structures by specific physical or electronic properties. Finally, compounds are often compared across composition ranges, so the total energy needs to be evaluated against all possible ‘escape’ or decomposition reactions. All these considerations have led to the development of more generalized ‘fitness functions’ F to attach to a given crystal structure S, X F ðSÞ ¼ ð1  aÞEðSÞ þ a bi Pi ðSÞ; i

P where Pi are appropriately normalized properties other than the compound’s total energy/enthalpy E, with weights ibi ¼ 1 chosen to reflect the user’s priorities. The properties Pi can reflect a desirable absolute maximization or minimization of a property (e.g. hardness or mass density) or closeness to a target value (e.g. optical band gap or lattice constant). Once F is chosen, structural candidates can be ranked according to this more general criterion, which influences (in search approaches with memory) choice of future structural candidates accordingly. To be fully utilizable, the properties Pi must be calculable with little effort; any property calculations that are computationally more demanding than a structure’s geometry optimization are unsuitable for unsupervised structure prediction. Some properties are trivially derived from a crystal structure, such as density or atomic coordination numbers; for others, for example Vickers hardness, estimators exist based on atomic configurations; yet others, such as electronic band gap or electride character, come at the cost of a single SCF calculation – which is a defensible cost if the structure prediction is done at DFT level of theory. For structure-property relationships that are hard or impossible to calculate, correlations can be explored using machine-learning techniques.135 Originally used to mine materials database information, machine-learned models can also be used in crystal structure prediction, provided the structural candidates are not too far removed from the model’s training dataset. Pareto analysis is an economics-inspired alternative to specifying a fixed multi-component fitness function. It produces solutions that as best as possible optimize all desired properties (E, Pi) simultaneously, ranking as best structures that are optimal in at least one property, and where every alternative includes a trade-off of properties.136–138 The Pareto analysis is an example of how desirable properties can be included in a structure prediction without designing a specific fitness function. A possible workflow could be to determine a full set of structural candidates, e.g. by random structure searching, then determine all properties of interest either for the full set or a sub-set of low-enthalpy candidates, depending on computational cost. The latter, in particular, is a very popular approach to find materials with specific properties among lowenthalpy (and therefore presumably synthesizable) structural candidates. Below, we discuss some of the property estimators that can be employed directly as part of a fitness function. Before that, we introduce the Maxwell stability criteria for multi-component systems, also known as convex hull or tieline constructions.

3.12.5.1

Compositional stability

Where the system of interest contains a range of possible compositions or stoichiometries, the thermodynamic stability of each structure against a decomposition into a mixture of other stoichiometries should be calculated. A Maxwell, or convex hull, or tieline, construction can determine this. If H(F) is the enthalpy of a structure with composition F comprising a set of species, {Ai}¼{A,B,...,X}, the relative enthalpy of an n-species mixture containing elements A, B to X compared against decomposition is 



DHðxÞ ¼ H Ax1 Bx2 .Xxnn 

n X   H Ai xi :

(2)

i¼1

An n-species composition space can be described by normalized barycentric coordinates in n  1 dimensions by imposing the constraint n X

xi ¼ 1;

(3)

i¼1

where xi are the composition weights. The mechanism for calculating stability against decomposition to any composition is to construct the n-dimensional convex hull of points in a composition-enthalpy space. Points that form the (lower portion of the) convex hull are stable against decomposition. In a binary system this looks like

DHðxÞ ¼ HðAx B1x Þ  Hð AÞx  HðBÞð1  xÞ:

(4)

The composition can be described by one parameter, x, and the convex hull is a polygon where each corner represents a stable composition, as in Fig. 7A). In a ternary system this looks like

DHðxÞ ¼ HðAx1 Bx2 C1x1 x2 Þ  Hð AÞx1  HðBÞx2  HðCÞð1  x1  x2 Þ

(5)

First principles crystal structure prediction

405

Fig. 7 Examples of binary and ternary convex hulls with A, B and C end members, binary material (BM ¼ Ax B1  x) and ternary material (TM ¼ Ax1 Bx2C1  x1  x2). In the binary case, the data is represented in two dimensions; one for relative enthalpy and one for composition. In the ternary case, two dimensions are used to represent composition and the convex hull is projected onto the surface, dividing composition space into different (triangular) regions labeled by the relevant decomposition reactions.

where the pure A, B, C end members are corners of an equilateral triangle. Explicitly describing the ternary convex hull in Fig. 7B, the composition space is divided into irregular triangles, each of which represents a linear mixture of the surrounding stable structures. Points on the ridges connecting points represent binary mixtures of the two endpoints. In a quaternary system, with a fourth species, D, this looks like

DHðxÞ ¼ HðAx1 Bx2 Cx3 D1x1 x2 x3 Þ Hð AÞx1  HðBÞx2  HðxÞx3  HðDÞð1  x1  x2  x3 Þ:

(6)

where the end members are corners of a regular tetrahedron. In this case, the composition space is represented in three dimensions. The composition space is divided into irregular tetrahedra which contain compositions that form linear mixtures of the four surrounding compounds. Points on the faces of the tetrahedra represent ternary mixtures of the face corners and points on the edges of the tetrahedra represent binary mixtures of the two endpoints.

3.12.5.2

Structural and mechanical properties

Often, partial structural information about an unknown compound exists. In particular, partial powder x-ray diffraction (XRD) data might have been collected, but in insufficient detail or quality to fully solve the compound’s structure. In such a case, the fitness function could be augmented by a term P  k1  dfgk that contains the difference dfg between the measured and simulated XRD patterns for predicted structures.139–141 This approach resembles a new type of XRD refinement procedure, based on firstprinciples structure optimizations. In practice, the use case for this approach has so far been limited. More often, the partial information from XRD (such as unit cell size or space group) is used to constrain conventional structure searches to the relevant regions of configurational space142,143; the interactions between experimental and computational collaborators tend to beat explicit fitness functions in those cases. For a given structure, the response to external stresses via collective distortions away from equilibrium determines its mechanical properties. The mechanical properties of materials are extremely important for potential technological applications. Superhard and incompressible materials, for example, are sought after in various industrial and aerospace settings. While the bulk modulus, the inverse of compressibility, determines a material’s resistance against compression, its shear modulus is a better descriptor for its hardness, the resistance against indentation.144,145 Conventionally, elastic moduli of crystals are determined from a series of finite strain calculations, but this is too computationally demanding for high-throughput calculations. The bulk modulus can be estimated via an empirical relation to local bond lengths.146,147 For the hardness, several different microscopic models have been developed148–151 that (intrinsically or with small modifications) rely only on the atomic configurations and no additional electronic structure information. These computationally cheap empirical models have been implemented in several crystal structure prediction packages, and allow to include hardness in the fitness function.78,152 As an alternative,153 the shear modulus of the predicted structures has been estimated by a machine-learned model based on the AFLOW materials database,85 from which the hardness itself can be estimated.145 We expect that machine-learned structureproperty relationships will become increasingly more important to quickly assess newly predicted structures with respect to mechanical or other properties.

3.12.5.3

Electronic properties

For many applications of inorganic materials, their electronic properties are at least as important as mechanical or elastic properties – are they insulators, semiconductors, or conductors? Is their fundamental band gap (the solid-state equivalent of the

406

First principles crystal structure prediction

HOMO-LUMO gap) suitable for matter-light interaction, e.g. sensing or photovoltaics? Are their transport properties amenable to thermoelectric applications? Are they promising superconductors? Those properties are accessible in electronic structure calculations, but usually at (comparatively) large computational cost. For high-throughput scenarios such as crystal structure prediction, fast approximations are required. The optical band gap of materials determines their ability to absorb light of given wave lengths. For solar cell materials, the optimal gap value is around 1.4 eV.154 Hence, a fitness function that uses the property P  (Eg  1.4 eV)2 would be straightforward to implement in a search process, where Eg is the band gap obtained from DFT. There are several issues that need to be considered in such an approach. Firstly, the optical gap is different from the single-particle energy gap; it is the smallest direct band gap including excitonic states, which are bound electron-hole pairs.155,156 Calculations of exciton binding energies are feasible but computationally very demanding.157 Secondly, all DFT implementations systematically underestimate even the fundamental band gap.158 Exchange-correlation functionals can be chosen to closely mimic experimental data159 but again can be computationally expensive (usually if they include exact exchange) and their appropriateness for new structure types discovered in a search is not always clear. Finally, for photovoltaic applications the band gap needs to be direct, and that assessment requires full band structure calculations instead of relying on the electronic density of states (DOS), which would be readily available within the structure search. For metals, the potential of superconductivity is of interest. In fact, the prediction of conventional high-pressure polyhydride superconductors is one of the most impressive examples of the successes of crystal structure prediction. The superconducting transition temperature Tc can be calculated via the electron-phonon coupling strength l, which is again computationally expensive.160,161 Hence, the preferred modus operandi has been to sample a given system, and determine Tc for the most stable or metastable structural candidates only. However, the electron-phonon coupling can be re-written in terms of single-site Hopfield parameters hj,162 which in turn can be estimated from scattering theory using only single-particle wave function information around each atom.163,164 A fitness function to maximize the Hopfield parameters, P ¼ maxihi, could be used for prediction of high-Tc candidate materials. In a recent alternative,165 the Hopfield parameters were used as a secondary filter (after closeness to the convex hulls) to identify high–Tc candidates, for which full calculations of l and Tc were then performed; the amount of structural candidates that could afforded to be tested (in this approximate way) for superconducting behavior was several orders of magnitude higher than before. Materials where electrons localize in the interstitial spaces between atoms are called electrides; they are stoichiometric pseudoionic compounds where the localized electrons take up the role of the anions.166 The electron sites can be occupied by electron pairs or single electrons; the latter giving rise to interesting magnetic properties.167–169 Either way, the high electron mobility and low work function make electrides valuable catalysts and reducing agents.170,171 They are readily identified from electronic structure calculations by topological analyses of the electron density or derived measures, such as the electron localization function (ELF)172,173: ELF maxima not associated with ionic cores or covalent bonds, or density maxima away from atomic positions, are indicators for electride character that are obtained at very little computational cost during structure searching. A fitness function that maximizes such regions in the unit cell, P ¼ Vinter/Vunit, has been used successfully to predict new electride materials.174 Thermoelectrics are energy materials where the thermoelectric effect, a voltage difference due to a thermal gradient, is exploited, for instance for refrigeration or waste heat recovery. Thermoelectrics are most often characterized by their Seebeck coefficient S ¼ DV/DT and their figure of merit ZT ¼ sST/k, which combines S with their electrical conductivity s and thermal conductivity k. Accurately determining those transport coefficients is computationally expensive but reasonable approximations, such as the constant relaxation time approximation, have been implemented175,176 that allow estimates of S and ZT based on standard DFT band structure information. This allows for high-throughput screening for thermoelectric properties.177,178 For actual structure prediction, both S and ZT have been used (next to total energy) as part of multi-objective optimization searches.179,180

3.12.6

Accelerating the search process

The techniques describes in Section 3.12.4 all require a series of first principles geometry optimisations, each of which consists of tens of energy and force calculations. This presents a bottleneck for CSP. Here, we describe some techniques that can reduce the number of expensive calculations and speed up the structure search.

3.12.6.1

Hard sphere potentials and volumes

Considering the vast size of the potential energy surface, it should be appreciated that a significant portion of the potential energy landscape consists of unphysically dispersed or dense structures. Care should be taken to generate structures that are physically reasonable before invoking the chosen energy calculator. Therefore, candidate structures should be generated based on realistic estimates of the density and typical bond lengths. Any initial structures that deviate significantly from those should be discounted or adjusted prior to calculation. A related, albeit extreme case is that pseudopotentials have a finite cut-off radius separating the frozen core region and the Kohn-Sham valence electrons. When the core regions overlap too much, the ab initio calculation will often be nonsense and should be avoided. Reasonable atomic separations should be considered pair-wise in multispecies systems; for example, LiCoO2, which is used as a cathode material, has crystal structures which are likely to have close LieO contacts but not LieLi or CoeLi.5 Each pair

First principles crystal structure prediction

407

combination can have a different minimal allowed separation. This is sometimes input as a matrix or list of pair distances. In Fig. 8, an example is shown for BaTiO3. One approach, as implemented in AIRSS, is to apply hard-sphere pre-optimizations d with sphere radii determined by the expected bond length d to all proposed structures prior to first principles calculation. With a determined unit cell chosen, the atoms can be placed inside and optimized with the simple potential. In some cases, the ideal density and bond lengths are unknown a priori. In this case, while the search can begin with no restrictions, after some time ‘sensible’ low energy structures will emerge. From those, hard sphere radii and volumes can be determined and used throughout the remaining searches. In a sense, when used to construct hard-sphere potentials, this is a basic d and perhaps trivial d example of training a potential on ab initio data.

3.12.6.2

Modular decomposition

Complex crystal structures can often be decomposed into recognizable building blocks. A trivial example is that a random search of stoichiometric H2O would find the low energy structures are networks of H2O molecules. The conclusion from this would be that the dimensionality of the search space could be reduced by building structures with H2O molecules from the beginning; the composition space of OHand Hþis less relevant and should be explored less. On a larger scale, complex structures like the allotropes of Boron can be decomposed into building blocks of clusters. Alpha Boron, a 12 atom rhombohedral structure is known to be formed of connected B12 icosohedra. Gamma Boron, a 28 atom orthorhombic structure, found at high pressure, consists of B12 icosohedra and B2 dimers.181 Ahnert et al. used this concept, conceiving a smart decomposition of structures with an emphasis on modularity, to perform structure searches for other allotropes of Boron.182 The configuration space is reduced by building structures only containing the expected ‘building blocks’. This modular decomposition was also used to accelerate searches to rediscover complex fibrous Phosphorus crystal structures with a machine learned potential.183 In the Gamma Boron example, decomposing the structure into 12 atom icosahedra, the 3 N degrees of freedom reduce to just 5N 12 2N degrees of freedom containing 3N 12 translational degrees of freedom and 12 rotational degrees of freedom.

3.12.6.3

Orbital free DFT

Another approach is to use faster implementations of DFT to perform the geometry optimizations. The Kohn-Sham formalism of DFT described in Section 3.12.3 uses orbitals to calculate the kinetic energy and the electron density, which is in turn used to calculate the remaining energy contributions. In practice, this process requires a mesh sampling of the Brillouin zone to capture the orbitals as they vary with wave vector, k. There is a trade-off between computational cost and accuracy depending on this sampling. In the case of metals, this becomes problematic as poor sampling leads to overly rugged landscapes, and as such the smoothness of the PES and hence surmountability of CSP described in Section 3.12.2 becomes less applicable. Orbital free methods use a kinetic energy functional to determine the total energy using electronic density alone and as a result should scale in theory linearly with the number of atoms in the unit cell. This has been demonstrated to recover low energy structures of Li, Na, Mg, and Al184 but requires a good functional to estimate the kinetic energy. Development of suitable kinetic energy functionals is ongoing.185

3.12.6.4

Machine learned potentials

Interatomic potentials have been used widely for a range of purposes since calculations are fast and can be used to study large systems. However, they will not describe well the novel environments seen in the candidate crystal structures. Hence, there is

Fig. 8 The crystal structure of BaTiO3. Pair distances are shown in the matrix. Sensible low-energy structures will have similar pair separations. In AIRSS, this is given with the command, #MINSEP ¼ 1.0 O-O ¼ 2.81 O-Ti ¼ 1.88 O-Ba ¼ 2.80 Ti-Ti ¼ 4.06 Ti-Ba ¼ 3.44 Ba-Ba ¼ 4.06.

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a practical upper limit to the number of atoms used in first principles CSP. Machine learned (ML) potentials aim to describe the ab initio PES but calculated at a fraction of the cost. The concept of fitting interatomic potentials to ab initio data is not new, but historically restricted to datasets of a few atoms in a small number of configurations.186 The increased availability of high-throughput ab initio data and the renewed interest in nonlinear fitting techniques such as artificial neural networks187 and Gaussian regression188 has garnered much interest in recent years. Typically, ML potentials are effective because the model can “learn” local energies εi for each atomic environment, taking advanP tage of the ‘shortsightedness’ of atomic interactions. The total energy is then a sum of each atoms contribution, E ¼ iεi. This means the ab initio training data can use relatively low numbers of atoms, remaining computationally manageable but still capturing a diverse set of local environments. The potential can then be used on much larger systems, previously inaccessible with DFT. ML potentials generally consist of three components; a reference dataset, a descriptor, and a regression tool. The reference dataset consists of several atomic configurations with known energies and forces calculated with high accuracy such as with DFT. The transferability of the resulting potential depends heavily on this dataset. A descriptor is selected to accurately represent each atomic environment, and eventually return a local energy and forces. Hence, the descriptor must be invariant to permutations, translations and rotations of atoms.189 Meaning, swapping two identical atoms, should give an identical descriptor and hence an identical energy. For practicality, the descriptor only describes the environment within a sphere with radius of a few Angstrom. Finally, the regression tool is trained to predict total energies for previously unseen crystal structures. This can be a linear regression,190 artificial neural network187 or kernel methods.188 One example descriptor is the ‘Smooth Overlap of Atomic Positions’, or SOAP,191 which consists of an atomic density field formed of gaussians placed at each neighboring atom site, expanded in a basis of spherical harmonics and radial basis functions. This descriptor lends itself well to the Gaussian Approximation Potential (GAP) framework,188 which uses Gaussian process regression to estimate the atomic energy and forces of a given environment by comparison with known energies for environments present in the reference dataset. It can take a long time to generate potentials that are sufficiently transferable and accurate, often as a result of intuitive curation of reference datasets. These potentials can then be used on their own going forward in a variety of applications. An alternative approach is to routinely generate new potentials ‘on-the-fly’ with minimal human input. The latter potentials will often be less accurate but specific to the desired use. As an example, while silicon has been well studied and has very accurate machine learned interatomic potentials,187,192 these would be of little use to study a less ubiquitous silicon-containing compound such as SiH4 or SiC for which new potentials are required. Crystal structure prediction, accelerated by MLPs, uses the ‘on-the-fly’ approach.193–195 The potential need only be accurate enough to generate low energy structures, which can ultimately be refined using DFT. Generating potentials suitable for global optimization requires a reference dataset that spans low energy and high energy configurations to capture the atomic environments seen throughout the optimization. Training data for these potentials can be generated from single point energy calculations on randomly generated crystal structures. It can also be effective to iteratively train the model by using the potential to optimize structures to local minima and calculating ab initio energies for structures close to the predicted local minima. This enables structure discovery with significantly fewer explicit ab initio calculations and can search unit cells with significantly more atoms. Recent applications of MLPs with unit cells containing large numbers of atoms include a priori ‘rediscovering’ allotropes of Boron,13196 prediction of allotropes of phosphorus197 and prediction of a novel 60 atom phase of SiH4 at high pressure.196

3.12.7

Visualizing the PES

The PES exists in a high dimensional space of all atomic configurations and so is inherently difficult to visualize. This represents a well known task in data science, to project the high dimensional data into three dimensions or less. A few approaches have been introduced to do this. However, as a first step the crystal structures must be ‘vectorized’ to define a consistent high dimensional basis. This means rather than choosing a local descriptor to describe atomic environments, we seek a global descriptor that describes the entire crystal structure. This is similar to the concept of collective variables discussed in Section 3.12.4.5.1. However, these global descriptors are not constrained to just a few variables. Earlier, we discussed the crystal fingerprints d or global descriptors d suggested by Valle and Oganov,77 which are histograms of pair-wise atomic separations. These histograms are a 1  N vector where N is the number of histogram bins. It is useful to define a measure of how separated the structures are from each other based on this descriptor. Valle and Oganov use a cosine distance measure d meaning the dot product of the descriptor vectors d and find that plotting energy against this distance measure gives information about the potential energy landscape. In particular they show how local minima can appear as ‘funnels’ toward the lower energy minima.198,199 An alternative vectorization is to, in some way, combine the local atomic environment descriptors in the unit cell to give a global descriptor. The aforementioned SOAP descriptor vectors, for example, can be simply averaged over atoms in the unit cell to give a global descriptor vector.200 Of course, averaging the descriptors in this way loses some information; two distinct crystal structures could give similar averaged global fingerprints. Alternative but more complex global descriptors calculated from local environments are described by De et al.200

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Several approaches have been used to reduce the dimensionality of this vector space, such as sketch maps201 and disconnectivity graphs.202 However, the specific task relevant to crystal structure prediction of representing a range of local minima d as opposed to a series of molecular dynamics snapshots d involves capturing information about the basin sizes, connectivity and separation. A recent approach by Shires and Pickard203 does this by applying state-of-the-art dimension reduction techniques, such as t-SNE and UMAP,204,205 to project the vectors on to three or fewer dimensions. The size of the basins are considered by combining similar structures, where more repetitions are represented by larger symbols. The connectivity of the basins is retained by imposing hard-sphere potentials to points in the reduced dimension projection. An example of this is shown in Fig. 9.

3.12.8

Examples

Thanks to the ubiquity of high-performance computing resources that can efficiently run multiple parallel instances of DFT codes, CSP has become a burgeoning field. Below we collect examples from the literature from different fields where first principles CSP is (or could be) a successful driving force of research.

3.12.8.1

Compositional complexity

In chapter 2, we introduced a combinatorial estimation of the number of atomic configurations for a single element system. Naturally, introducing multi-element variation increases this number dramatically, making CSP more difficult. It is a challenge that is of interest, since superconducting cuprate materials typically contain between four and five element types,206 and d as will be discussed d ternary superconducting hydrides are showing theoretical and experimental evidence for high temperature superconductivity.207–211 An application where compositional complexity dominates is in cathode materials. Particularly, with a desire to design nextgeneration Lithium-ion batteries for electric vehicles or ubiquitous energy storage, featuring longer lifespans and storage capacities.212 A prominent material and the first commercialized Lithium-ion battery is LiCoO2.213 Since then, efforts have been made to improve the capacity by substituting transition metals into the LiMO2 framework, again presenting a combinatorial problem. This was explored initially through ab-initio calculations by Ceder et al.,214 with manual construction of LiMAlO2 derivates (where M ¼ Ti, V, Fe, Co). Then, more ambitiously, using high-throughput screening of ternary oxide compounds using known crystal structures from a database,83 and calculations of AxM(YO3)(XO4) (with A ¼ Na, Li; X ¼ Si, As, P; Y ¼ C, B; M ¼ a redox active metal;) in the sidorenkite crystal structure.84

Fig. 9 SHEAP map of LiCoO2 structures generated from a random structure search. Circle sizes correspond to the number of structure repetitions (hence the size of the local basins), colors represent energy scale from purple (most stable) to yellow (most unstable). Reproduced with permission from Lu et al. Lu, Z.; Zhu, B.; Shires, B.W.B.; Scanlon, D.O.; Pickard, C.J. Ab Initio Random Structure Searching for Battery Cathode Materials. J. Chem. Phys. 2021, 154(17): 174111, doi:10.1063/5.0049309.

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However, perhaps the next-generation material does not conform to a known structure type. Recently, calculations have attempted to generate novel structures for a wide range of stoichiometries using AIRSS,4,5 reproducing known polymorphs and suggesting new stable and promising new materials in LiCoO2, LiFePO4, and the Lix1Cux2 Fx3 ternary. Due to the sheer scale of the computational problem, these efforts may have only scratched the surface and perhaps the best material is a quaternary or quinary structure in an unknown structure type. Cathode materials will be at the cutting-edge of CSP in the coming years. An entirely different proof-of-concept structure search by Pickard provides an unbiased sampling of the quaternary HeCeNeO chemical space at 1 GPa pressure.215 While the pressure removes the dominance of dispersion interactions in the structures formed, the dataset reflects an ambitious survey of the most dominant organic chemical space (the PubChem database lists close to 45 million compounds within this composition space216). The dataset contains many known thermodynamic sinks, and the quaternary nature of the space is well reflected if visualized using the SHEAP mapping introduced above203; it remains available for further investigations. Unrelated studies of planetary ice mixtures, motivated by the unknown makeup of deep planetary interiors, presented quaternary HeCeNeO phase diagram at several Mbar pressures,217,218 implying that compositionally complex systems can now be studied using first principles CSP.

3.12.8.2

High pressure

Energy landscapes of elements and compounds at high pressures often deviate from chemical intuition attuned to ambient conditions.219,220 Due to the non-intuitive effects, the exploration of high pressure energy landscapes with crystal structure prediction methods has proven extremely useful to reveal fundamental new physics and chemistry that governs materials’ stability and properties at high densities.221

3.12.8.2.1

Elements

The lightest element, hydrogen, retains immense interest due to its complex high-pressure phase diagram and the promise of hightemperature superconductivity following its eventual metallization.222 One of the most intriguing effects is the appearance of a strongly infrared-active vibron mode in its phase III above 150 GPa, implying the presence of polarized H2 molecules in an anisotropic matrix of their peers.223 An early success of random structure searching was the discovery of a structure that fits this description16; while free energy calculations of the same dataset revealed a candidate structure for the high-temperature phase IV.224 Computational work on hydrogen continues, and its unique challenges due to the low nuclear mass drive methodological developments in structure prediction.133 Boron tends to form very complex crystal structures, with polyhedral motifs borne out of an electron deficiency that favors multicenter bonding. This tests the ability of structure prediction methods to successfully perform searches with large numbers of atoms. The g - B28 phase was found in evolutionary algorithm searches by exploiting experimental unit cell information.181 Boron provides motivation for continued methodology development to scale up structure predictions, such as automated detection of atomic motifs (making polyhedra instead of atoms the units of the search),182 or on-the-fly training of interatomic potentials to vastly increase structure throughput.13196 Sodium is a textbook simple metal at ambient conditions but is also exemplar for non-trivial electronic transitions at high pressure, most strikingly by taking up an insulating phase above 200 GPa that was found in evolutionary algorithm structure searches.22 The appearance of bad metals or insulators among most group I and II elements correlates with their tendencies to form highpressure electrides, including in complex incommensurate host-guest phases.225 Random structure searches predict that trivalent aluminium will also take up such host-guest phases in the TPa pressure range.226

3.12.8.2.2

Hydrogen bonding

The evolution of hydrogen bonds under pressure is relevant to many fields beyond inorganic chemistry, including biology (boundaries of life at extremes), geosciences (water storage inside minerals) and planetary sciences (icy planet interiors). Among hydrogen-bonded ices the pressure response varies significantly. Ammonia was predicted from random structure searching to self-ionize into (NH2)(NH4)þ,15 which was later confirmed in experiment.227,228 Water is predicted to undergo a series of phase transitions beyond hydrogen bond symmetrisation,21,229–231 ultimately decomposing in the TPa pressure range.232 Experiments have not yet reached those pressure conditions, instead new ice phases are found at high temperatures, in the superionic regime of hot dense water.233–235 Ammonia-water mixtures are again predicted to form ionic high pressure compounds,20,236 which has been confirmed by experiment.237 A cautionary tale is hydrogen chloride, where (post hydrogen bond symmetrization) a series of low-symmetry ground state phases was predicted,238–240 whereas experiments and molecular dynamics simulations found a high-symmetry, room-temperature superionic phase.241 The emergence of high-symmetry phases at elevated temperatures, whether caused by quantum or entropic effects, continues to present challenges for crystal structure prediction methods that rely on full structural optimizations toward local minima. Metal hydroxides of formula MOH (for group I metals) or M(OH)2 (for group II metals) are another class of hydrogen-bonded materials with rich structural variety under pressure. Crystal structure prediction was instrumental in both enabling and correcting previous experimental structure assignments in dense alkali hydroxides.242–244 Hydrous minerals are compounds that store hydrogen at deep Earth conditions; the hydrogen bonding influences individual phases’ stabilities and elastic properties. Computational predictions have successfully preceded experimental syntheses in this

First principles crystal structure prediction

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area,245,246 while crystal structure prediction has been applied to scenarios that range from Earth’s mantle transition zone to the primordial core.247–250

3.12.8.2.3

Polyhydride superconductors

The theory of metallic and superconducting dense hydrogen evolved toward the concept of ‘chemically precompressed’ hydrogen in hydride compounds.252 The very first published random structure search results were in this area, on silane,17 and since then the field has seen an extraordinary sequence of crystal structure prediction followed by experimental synthesis and characterization. The structure predictions usually follow the sequence of identifying low-enthalpy structures, nowadays across variable composition ranges, before calculating superconducting properties for the best candidates. As a general theme, high pressure favors the formation of polyhydrides with higher hydrogen content than at ambient or low pressure conditions, hence searches need to screen a wide range of hydrogen compositions. Polyhydrides that have been predicted (all with superconducting Tc above 200 K) and subsequently synthesized (all above 150 GPa) include CaH6,253–255 H3 S,18,256 LaH10,257–260 and YH6  9.261–263 Full screening of ternary composition spaces is now possible, and ternary polyhydrides have been predicted with even higher Tc, or with lower formation pressures.207–210

3.12.8.3

Target properties

In Section 3.12.5 we discussed different types of fitness functions (other than total energy or enthalpy) that can be used to direct structure searches toward specific properties, using multi-objective optimizations. In truth, those targeted approaches have not been as prominent as one might expect – despite the need for new materials with specific properties. That is not to say that applications do not exist: multi-objective searches have yielded superhard materials,264–266 electrides,174 and thermoelectrics.179,180 However, a much more common approach is to search for low-energy or low-enthalpy structures first, and then investigate those for properties of interest. This often includes expensive follow-up calculations, and is therefore restricted to a few candidate structures. There is therefore merit in developing or adapting computationally cheaper predictive models, to enable pre-screening for interesting properties across a wider range of candidates – ideally the entire dataset of structures that emerge from a CSP search. The surrogate model for superconducting Tc, based on single-electron wave functions, is an example, as it allows rapid qualitative statements on prospects of superconductivity.165 Across materials informatics, a plethora of similar structure-property relationships have been codified for all kinds of materials properties in recent years, often using machine learning techniques, which could be used in conjunction with CSP.135,267–269

3.12.9

Challenges and conclusions

While first principles crystal structure prediction has been shown to be successful in materials discovery, challenges remain. Those challenges can stem from deficiencies in the DFT calculations to describe particular compounds; or from inadequacy of total energy calculations to deduce stability. The latter relates for example to the fact that DFT calculations of a material require well defined atomic positions; disorder is not usually handled by CSP. However, real materials can be intrinsically disordered. In fact, a large number of structures on the ICSD contain partial site occupancy (SOF < 1), implying some disorder. This could be a structural disorder such as in H2O, where the molecules appear to be tetrahedra with 0.5H atoms on each corner (see Fig. 10A). Crystal structure prediction will only discover the ordered approximation of these structures. In this example, it can discover the ordered structure, ice XI, but not the isostructural disordered ice Ih. Alloys can have well defined crystal structures with non-integer occupancies that would not be described well using DFT. The stoichiometries accessible with CSP are limited by the number of atoms used in the search. Fig. 10B) shows an experimental determination of the crystal structure of an Mn0.95  dFe1.05  x þ dCox B alloy. However, the exact stoichiometry is not easily accessed in calculations. Furthermore, it is not clear that CSP is the right tool for alloys: an analysis of metallic compounds within the AFLOW materials library suggested that their formation is dominated by entropy rather than energy gains if they contain four or more elements.270,271 Configurational entropy is but one contribution to a compound’s Gibbs free energy that DFT total energy calculations struggle to quantify. The aim of first principles CSP, to find the global minimum of the potential energy landscape, does not give the complete (or often, sufficient) picture. At finite temperatures, dynamic effects can change relative stabilities of different structures, which has two major consequences. Firstly, CSP methods need to find and catalog the spectrum of metastable states, in addition to the global minimum, and do this in an ergodic way, i.e. representing ensemble averages. For search methods with ‘memory’ this is not a priori clear. For the ‘local’ approaches discussed in Section 3.12.4.5 the challenge is the opposite, as they need to ensure to sample the local environment of the PES wide enough to encounter all relevant metastable states. For the ‘global’ survey approaches introduced in Section 3.12.4 additional benefit would emerge from information on the connectivity of different basins within the PES. Some progress is currently being made by visualizing the PES based on structural similarities, but identifying reaction pathways and transitions states remains a big challenge. The second consequence of dynamic effects is the stabilization of phases that are not even local minima on the PES, as they are intrinsically stabilized by entropy. The prototypical cubic perovskite structure will not usually emerge from CSP of ABO3 compounds, because it often features a dynamical instability toward lower symmetry structures with rotated and tilted octahedra.

412

First principles crystal structure prediction

Molecular crystals may have rotational (‘plastic’) states at elevated temperatures,272 which will not be uncovered from CSP alone. Some materials display superionicity, ionic diffusion, or chain melting (in each case a subset of atoms becomes mobile against a fixed lattice of the remaining atoms). Far below the characteristic temperature scales for these phenomena, free energies can be estimated using vibrational entropies alone. This has led to a candidate structure for hydrogen’s phase IV.224 Those calculations require expensive phonon calculations and are therefore not routinely accessible to CSP at the moment – but may be via structure-property relationships or efficient interatomic potentials, if those can readily deliver phonon densities of states. An entirely different issue with hydrogen, due to its very low mass, is the large contribution of nuclear quantum effects, which arguably require much more computationally intensive approaches than standard DFT to capture phase stabilities and transitions properly. For some systems the periodic setup of DFT calculations for solids is insufficient. Some compounds contain periodicities that are incommensurate with the lattice. This can manifest as a host-guest structure, where the periodicity of the ‘host atom’ lattice is incommensurate with the ‘guest atom’ lattice. Known examples include the alkali metals, the heavier pnictogens, and predictions for Al, all under pressure,225,226,273 while LiBx and Hg3  dAsF6 have linear incommensurate chains at ambient conditions.274–276 Incommensurate structures can also manifest as charge density waves (CDW) or modulated crystal structures, where atoms are displaced along a wave vector incommensurate with the average crystal structure. Examples of these include uranium at ambient and the chalcogenides SeTe under pressure.277 Finally, there are compounds, such as NaNO2 and CaAu-II that display occupational modulation where the site occupancies vary along an incommensurate wave vector. Those phases cannot be modelled in periodic DFT, nor discovered directly using CSP. However, commensurate or ordered approximants of host-guest and CDW structures can be constructed around a feasible superstructure that captures the two periodicities.226,278 Incommensurate structures may have an ordered variant, stable above a transition temperature, where either the CDW amplitude vanishes (as in the high pressure SeIV to SeV transition), or the periodicity becomes commensurate. An excellent review of incommensurate structures and the related nomenclature has been given by Sander Van Smaalen.279 First principles CSP is also undermined in cases where DFT fails to capture significant portions of the electronic interactions. This used to include weakly bound systems dominated by long-range dispersion interactions but the development of dispersioncorrected exchange-correlation functionals in the last 10–15 years now allows accurate calculations of those. However, strongly correlated electron systems still suffer from the inadequate formalism of electron correlation within DFT. This includes openshell systems (where DFT can struggle to reproduce magnetism) and Mott insulators (which tend to be metallic within vanilla DFT), among others. In addition, the systematic underestimation of electronic band gaps within DFT means that even if the electronic structure is qualitatively correct, quantitative predictions, e.g. of metallization pressures or optical band gaps, remain difficult. Corrections for these issues exist for specific systems, for example on-site Hubbard repulsion terms that penalize metallic states or material-specific functionals to reproduce experimental band gaps, but introducing system-specific parameters terms is not ideal in the spirit of automated CSP. One main purpose of CSP is to uncover materials with desirable properties in silico, thereby bypassing expensive trial-and-error synthesis efforts. As pointed out in Section 3.12.8.3, the direct search for specific properties has probably not reached its full potential. Partly, this might be due to the difficulty in capturing relevant physicochemical properties based on structural information alone: catalytic activity, for instance, often relates to defects and surfaces of materials, rather than their bulk crystal structure. Nonetheless, ongoing exploits in machine-learned structure-property relationships can hopefully be interfaced with CSP more often and thus lead to more relevant predictions in materials science.

(A)

(B)

(C)

Fig. 10 Crystal structure prediction can be challenging on systems with partial occupancy (A, B) or incommensurate structures (C). (A) Ice Ih with partial H occupancy and ordered ice XI. O/H atoms in red/blue. Partial occupancies given by partially filled spheres. (B) Mn0.95  dFe1.05  x þ dCox B alloy with occupancies measured experimentally by XRD, and three ordered approximations in the same unit cell that may be found by CSP. Mn/Fe/ Co/B atoms in purple/bronze/blue/green. Partial

occupancies given by partially filled spheres. (C) Incommensurate host-guest phase of K-IIIa at high pressure, along the [001] (left) and the 110 directions. Reproduced with permission from Woolman, G.; Naden Robinson, V.; Marqués, M.; Loa, I.; Ackland, G.J.; Hermann, A. Structural and Eelectronic Properties of the Alkali Metal Incommensurate Phases. Phys. Rev. Mater. 2018, 2(5), 053604, doi:10.1103/PhysRevMaterials.2.053604.

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A final fundamental challenge is the combinatorial problem stated at the outset of this chapter. CSP (whether from first principles or not) will always be limited in unit cell size and compositional variety that can be searched by the computational resources necessary to sample configurational space. Even if the ground state minimum is the most likely structure to be found, in absolute terms it might still take many thousands or millions of attempts to locate it. The modern approaches in CSP have had remarkable success yet ongoing methodology developments aimed to speed up the searches, as discussed in this chapter, remain necessary to allow even more relevant searches to take place.

Acknowledgments We are grateful for support by the UK’s Engineering and Physical Sciences Research Council over many years, in particular for financial support for L.J.C. (EP/L015110/1 and EP/S021981/1).

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