Atomic Clusters with Unusual Structure, Bonding and Reactivity: Theoretical Approaches, Computational Assessment and Applications 0128229438, 9780128229439

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Atomic Clusters with Unusual Structure, Bonding and Reactivity

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Atomic Clusters with Unusual Structure, Bonding and Reactivity Theoretical Approaches, Computational Assessment and Applications

Edited by

Pratim Kumar Chattaraj Indian Institute of Technology Kharagpur, Kharagpur, India

Sudip Pan Institute of Atomic and Molecular Physics, Jilin University, Changchun, China

Gabriel Merino Universidad de Merida, Merida, Mexico

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 © 2023 Elsevier Inc. 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 must 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. ISBN: 978-0-12-822943-9 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

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Contents Contributors

xi

M. Molayem and M. Springborg

1. Describing chemical bonding in exotic systems through AdNDP analysis Edison Osorio 1. Introduction 1.1 AdNDP implementation 2. Boron hydrides 2.1 Chemical bonding scheme in B3H
y complexes 2.2 Isostructural relationships in BnHn series 2.3 Electronic transmutation 2.4 Chemical bonding in deltahedral BnH2
n systems 3. Boron nanowheels 3.1 Dynamic behavior in small boron clusters 3.2 Boron wheels members of Wankel motor family 3.3 Design of sandwich structures 3.4 Dynamic behavior of B36 cluster 4. Summary References

1 2 3 3 3 4 5 7 7 7 9 13 13 15

2. Electron delocalization in clusters Jose M. Mercero and Jesus M. Ugalde 1. Introduction 2. Electron delocalization in flat clusters 3. Electron delocalization in (pseudo) spherical clusters 4. Conclusions Acknowledgments References

3. Bimetallic clusters

19 20 30 35 36 36

1. Introduction 2. Computational methods 2.1 Energy calculator 2.2 Global structure optimization methods 3. Structural properties of bimetallic clusters 4. Conclusions Acknowledgments References

41 41 42 43 44 57 58 58

4. Unusual bonding between second row main group elements Gurudutt Dubey and Prasad V. Bharatam 1. Introduction 2. Low-valent state of main group elements 3. L ! E L complexes 3.1 Carbones: Divalent C(0) systems 3.2 Borylenes: Monovalent B(I) systems 3.3 Nitreones: Divalent N(I) systems 4. Debates on the bond representation 5. Summary References

61 61 63 63 68 71 80 82 82

5. Conceptual density functional theory and all metal aromaticity Debolina Paul and Utpal Sarkar 1. Introduction 2. History and descriptors of aromaticity 2.1 Relative aromaticity indices (4X) 3. Aromaticity in the context of metallic systems

87 89 90 91

v

vi

Contents

3.1 Alkali and alkaline earth metal 3.2 Transition metal 4. Conclusion Acknowledgments References

91 93 94 94 94

6. Structural evolution, stability, and spectra of small silver and gold clusters: A view from the electron shell model Pham Vu Nhat, Nguyen Thanh Si, and Minh Tho Nguyen 1. Introduction 2. Equilibrium structures and growth mechanism 3. Thermodynamic stabilities 4. Phenomenological shell model (PSM) 5. Electronic absorption spectra 6. Concluding remarks Funding information References

99 100 107 110 113 117 118 118

7. Optical response properties of some metal cluster supported host-guest systems 123 124 125

125 127 130 132 134 135 136

8. Group III–V hexagonal pnictide clusters and their promise for graphene-like materials Esha V. Shah and Debesh R. Roy 1. Introduction 2. Computational details

140 140 142 143 143 143 145 146 148 151 152 152 153

9. M(L)8 complexes (M = Ca, Sr, Ba; L = PH3, PF3, N2, CO): Act of an alkaline-earth metal as a conventional transition metal Hai-Xia Li, Zhong-Hua Cui, Dandan Jiang, Lili Zhao, and Sudip Pan

Arpita Poddar and Debdutta Chakraborty 1. Introduction 2. Computational details 3. Results and discussion 3.1 Geometrical structures and thermodynamic feasibility of obtaining the corresponding hostguest moieties 3.2 Optical and electronic properties of the selected metal cluster-host complexes 3.3 AIM analysis 3.4 EDA study 3.5 TDDFT analysis of the guest@OA complexes 4. Conclusion References

3. Benzene and its group III–V pnictide cluster analogues 3.1 Structural properties 3.2 Electronic properties 4. Polymeric growth of benzene and its III–V analogues 4.1 Structural properties 4.2 Electronic properties 5. Group III–V graphene-like materials from potential cluster units 5.1 Monolayer indium nitride for thermoelectrics 5.2 Mono- and multilayer thallium nitride for thermoelectrics 5.3 Other two-dimensional group III–V materials 6. Conclusions Acknowledgments References

139 140

1. Introduction 2. Computational details 3. Structure and stability of M(L)8 complex 4. MOs and correlation diagram 5. Energy decomposition analysis 6. M(Bz)3: 20-electron complex 7. Conclusions Acknowledgments References

157 158 159 161 162 167 170 170 171

10. Structures, reactivity, and properties of low ionization energy species doped fullerenes and their complexes with superhalogen Abhishek Kumar, Ambrish Kumar Srivastava, Gargi Tiwari, and Neeraj Misra 1. Introduction 2. Computational techniques 3. Low IE species doped endofullerenes 3.1 Li@C60 vs SA@C60 endofullerene (SA = FLi2, OLi3, and NLi4)

173 174 175 175

Contents vii

3.2 Li@C60 vs Lr@C60 endofullerene 4. Endofullerene-superhalogen complexes 4.1 Li@C60
PF6 endofullerene complex 4.2 SA@C60dBF4 endofullerene complex 5. Conclusions and perspectives Acknowledgments Conflict of interests References

176 177 177 179 180 181 181 181

Gourhari Jana and Ranita Pal 185 187 187 188 190 191 191 192 192 192 200 202 206 206 206 206

12. Studies on hydrogen storage in molecules, cages, clusters, and materials: A DFT study K.R. Maiyelvaganan, M. Janani, K. Gopalsamy, M.K. Ravva, M. Prakash, and V. Subramanian 1. Introduction 2. H-storage in various motifs—The road map representation 2.1 H-storage in small molecules 2.2 Hydrogen storage in molecular cages

222 226 232 232 232

13. A density functional theory study of H3+ and Li3+ clusters: Similar structures with different bonding, aromaticity, and reactivity properties Dongbo Zhao, Xin He, Meng Li, Chunna Guo, Chunying Rong, Pratim Kumar Chattaraj, and Shubin Liu

11. Generation of global minimum energy structures of small molecular clusters using machine learning technique 1. Introduction 2. Our proposed methodology and algorithm (parallel implementation) 2.1 Particle swarm optimization 2.2 Firefly algorithm 2.3 ADMP-CNN-PSO approach 3. Computational details 4. Experimental setup 4.1 PSO, FA, and ADMP-CNN-PSO 5. Results and discussion 5.1 PSO: Boron clusters, Bn (n = 5, 6) 4
5.2 CNN and PSO: N2
4 , N6 , Aun (n = 2– 8) and AunAgm (2  n + m  8) clusters 5.3 Firefly algorithm with density functional theory 6. Conclusion Acknowledgments Conflict of interest References

2.3 H-storage in molecular clusters 2.4 H-storage in materials 3. Conclusions Acknowledgments References

213 214 215 220

1. Introduction 2. Methodology 3. Results and discussion 4. Conclusions Acknowledgments References

237 238 240 243 243 243

14. Designing nanoclusters for catalytic activation of small molecules: A theoretical endeavor Anup Pramanik, Sourav Ghoshal, and Pranab Sarkar 1. 2. 3. 4.

Introduction N2 activation H2 activation Activation and reduction of CO2 4.1 Specific role of metal hydride for the reduction of CO2 5. Activation of O2 and oxidation of CO on Aun nanoclusters 5.1 Effect of doping in Aun nanoclusters 5.2 Al
n anionic nanoclusters: Effect of electron spin 6. H2O activation 7. C–X and C–H bonds activation 7.1 C–X bond activation on Aln nanoclusters 7.2 Competitive H–X elimination on alumina nanoclusters 7.3 Selectivity of alumina nanoclusters during elimination 7.4 Selective C–H bond activation 8. Summary and future outlook Acknowledgments References

247 248 251 253 254 255 256 257 258 260 260 260 262 262 264 265 265

viii

Contents

15. Molecular electrides: An overview of their structure, bonding, and reactivity

6. Conclusion Acknowledgments References

Ranajit Saha and Prasenjit Das 1. Introduction 1.1 Electrides 1.2 Confinement of the electron 1.3 Development of organic electrides 1.4 Development of inorganic electrides 1.5 Toward the molecular electride 2. Norms and conditions of being a molecular electride 3. Computational methodology 4. Examples of molecular electrides 4.1 Alkali metal-doped electrides 4.2 Mg2EP, molecular electride and small molecule activation L)2]2
complex 4.3 Bonding in [Mg4(Dipp H and its electride nature 4.4 Mg2@C60 and its electride characteristics 4.5 Binuclear Sandwich complexes of alkaline earth metals as electrides 4.6 Li3@Cg (Cg = B40 and C60) and their electride nature 5. Conclusion Acknowledgments Authors note References

275 275 275 275 276 277 278 279 281 281 283 285 286 287 288 289 290 290 290

16. Hydrogen trapping potential of a few novel molecular clusters and ions Sukanta Mondal, Prasenjit Das, and Santanab Giri 1. 2. 3. 4.

Introduction Theoretical background Computational details Atomic and molecular clusters 4.1 Mg and Ca clusters 4.2 B2Li and B2Li2 moieties 4.3 C12N12 cage 5. Ionic clusters 5.1 N4Li2 and N6Ca2 clusters 5.2 Li+3 and Na+3 ions 5.3 B2Li+ and B2Li+2 ions 5.4 M5Li+7 (M = C, Si, Ge) clusters

297 299 301 302 302 302 303 305 305 306 307 308

308 309 309

17. Polarizability of atoms and atomic clusters Swapan K. Ghosh 1. Introduction 2. Basics of response properties and polarizability 3. DFT-based approach to calculation of polarizability 4. Polarizability of spherically symmetric systems: Atoms and atomic clusters within the jellium model 5. Chemical reactivity indices-based route to polarizability 6. Discussion on polarizability values of atomic clusters 7. Concluding remarks Acknowledgments References

313 314 314 317 318 318 319 319 319

18. Advances in cluster bonding: Bridging superatomic building blocks via intercluster bonds Nikolay V. Tkachenko, Zhong-Ming Sun, Alexander I. Boldyrev, and Alvaro Mun˜oz-Castro 1. Introduction 2. Intercluster bonding of gold clusters 3. Intercluster bonding of Zintl clusters 4. Extended networks 5. Conclusions Acknowledgments References

321 322 323 328 329 330 330

19. Zintl cluster as a building block of superalkali, superhalogen, and superatom Swapan Sinha, Ruchi Jha, Subhra Das, and Santanab Giri 1. Introduction 2. Computational details 3. Zintl superalkali 4. Zintl superhalogens 5. Zintl superatom 6. Concluding remarks Acknowledgments References

333 333 334 335 339 342 342 342

Contents

20. Metallic clusters for realizing planar hypercoordinate secondrow main group elements and multiple bonded species

2.4 Temperature-induced transformation 3. Perspectives and conclusions References

Amlan J. Kalita, Shahnaz S. Rohman, Chayanika Kashyap, Lakhya J. Mazumder, Indrani Baruah, Ritam Raj Borah, Farnaz Yashmin, Kangkan Sarmah, and Ankur K. Guha 1. Introduction 2. Planar hypercoordinate main group elements 3. Planar pentacoordinate nitrogen 4. Metal cluster supported multiple bonded second-row main group element 5. Conclusions and future aspects Acknowledgment References

Dandan Jiang, Manas Ghara, Sudip Pan, Lili Zhao, and Pratim Kumar Chattaraj

345

1. 2.

Introduction The chemistry of Lewis acids and bases 3. Identification of FLP reactivity 4. Mechanism of H2 activation by FLPs 5. Thermodynamics on H2 activation by FLP 6. Activation of other small molecules 7. Aromaticity-enhanced small molecule activation 8. Catalytic hydrogenation 9. Boron-ligand cooperation 10. Polymerization reaction 11. Summary and outlook References

345 348 349 353 353 353

Prasenjit Das, Sudip Pan, and Pratim Kumar Chattaraj 357 357 361 365 365 368 369 369

387 389 389 392 393 397 398 401 403 407 407

Yukatsu Shichibu and Katsuaki Konishi 1. Introduction 2. Representative examples of theoretical studies 3. Diphosphine-ligated gold clusters 3.1 Jellium models and core shapes 3.2 Geometric studies 3.3 Electronic studies 3.4 Effects of ligands on geometric and electronic structures 4. Conclusion References

Saniya Gratious, Sayani Mukherjee, and Sukhendu Mandal 373 373 373 375 379

387

24. Ligand-protected clusters

22. Transformation of nanoclusters without co-reagent

1. Introduction 2. Co-reactant-free transformations 2.1 pH-induced transformation 2.2 Solvent-induced transformation 2.3 Photo-induced transformation

381 384 385

23. Application of frustrated Lewis pairs in small molecule activation and associated transformations

21. Planar hypercoordinate carbon

1. Introduction 2. Planar tetracoordinate carbon (ptC) 3. Planar pentacoordinate carbon (ppC) 4. Planar hexacoordinate carbon (phC) 5. Higher coordinate carbon 6. Conclusion Acknowledgments References

ix

Index

411 411 411 411 413 414 416 420 420

423

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Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

Sourav Ghoshal (247), Department of Chemistry, VisvaBharati University, Santiniketan, India

Indrani Baruah (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India

Santanab Giri (297, 333), School of Applied Sciences and Humanities, Haldia Institute of Technology, Haldia, India

Prasad V. Bharatam (61), Department of Medicinal Chemistry, National Institute of Pharmaceutical Education and Research, S.A.S. Nagar, Punjab, India

K. Gopalsamy (213), Center for High Computing and Inorganic Physical Chemistry Laboratory, Central Leather Research Institute, Council of Scientific and Industrial Research, Chennai, Tamil Nadu, India

Alexander I. Boldyrev (321), Department of Chemistry and Biochemistry, Utah State University, Logan, UT, United States Ritam Raj Borah (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India Debdutta Chakraborty (123), Department of Chemistry, Birla Institute of Technology, Mesra, Ranchi, Jharkhand, India Pratim Kumar Chattaraj (237, 357, 387), Department of Chemistry, Indian Institute of Technology, Kharagpur, India Zhong-Hua Cui (157), Institute of Atomic and Molecular Physics, Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), Jilin University, Changchun, China Prasenjit Das (275, 297, 357), Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur, India Subhra Das (333), School of Applied Sciences and Humanities, Haldia Institute of Technology, Haldia; Department of Chemistry, Cooch Behar Panchanan Barma University, Cooch Behar, West Bengal, India Gurudutt Dubey (61), Department of Medicinal Chemistry, National Institute of Pharmaceutical Education and Research, S.A.S. Nagar, Punjab, India

Saniya Gratious (373), School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram, Trivandrum, Kerala, India Ankur K. Guha (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India Chunna Guo (237), Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), Hunan Normal University, Changsha, Hunan, PR China Xin He (237), Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), Hunan Normal University, Changsha, Hunan, PR China Gourhari Jana (185), Department of Chemistry, Indian Institute of Technology Bombay, Mumbai, India M. Janani (213), Department of Chemistry, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Chengalpattu, Tamil Nadu, India Ruchi Jha (333), Advanced Technology Development Center (ATDC), Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India

Manas Ghara (387), Department of Chemistry and Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur, India

Dandan Jiang (157, 387), Institute of Advanced Synthesis, School of Chemistry and Molecular Engineering, Jiangsu National Synergetic Innovation Center for Advanced Materials, Nanjing Tech University, Nanjing, China

Swapan K. Ghosh (313), UM-DAE-Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai, India

Amlan J. Kalita (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India xi

xii Contributors

Chayanika Kashyap (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India Katsuaki Konishi (411), Graduate School of Environmental Science, Hokkaido University, Sapporo, Japan Abhishek Kumar (173), Department of Physics, University of Lucknow, Lucknow, Uttar Pradesh, India Hai-Xia Li (157), Institute of Atomic and Molecular Physics, Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), Jilin University, Changchun, China Meng Li (237), Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), Hunan Normal University, Changsha, Hunan, PR China Shubin Liu (237), Research Computing Center; Department of Chemistry, University of North Carolina, Chapel Hill, NC, United States K.R. Maiyelvaganan (213), Department of Chemistry, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Chengalpattu, Tamil Nadu, India

Pham Vu Nhat (99), Department of Chemistry, Can Tho University, Can Tho, Vietnam Edison Osorio (1), Faculty of Natural Sciences and Mathematics, University of Ibague, Ibague, Colombia Ranita Pal (185), Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, India Sudip Pan (157, 357, 387), Institute of Atomic and Molecular Physics, Jilin University, Changchun, China Debolina Paul (87), Department of Physics, Assam University, Silchar, India Arpita Poddar (123), Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India M. Prakash (213), Department of Chemistry, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Chengalpattu, Tamil Nadu, India Anup Pramanik (247), Department of Chemistry, SidhoKanho-Birsha University, Purulia, India M.K. Ravva (213), Department of Chemistry, SRM University—AP, Amaravati, Andhra Pradesh, India

Sukhendu Mandal (373), School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram, Trivandrum, Kerala, India

Shahnaz S. Rohman (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India

Lakhya J. Mazumder (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India

Chunying Rong (237), Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), Hunan Normal University, Changsha, Hunan, PR China

Jose M. Mercero (19), Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), Donostia, Euskadi, Spain Neeraj Misra (173), Department of Physics, University of Lucknow, Lucknow, Uttar Pradesh, India M. Molayem (41), Physical and Theoretical Chemistry, Saarland University, Saarbr€ ucken, Germany Sukanta Mondal (297), Department of Education, Ashutosh Mukhopadhyay School of Educational Sciences, Assam University, Silchar, Assam, India Sayani Mukherjee (373), School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram, Trivandrum, Kerala, India Alvaro Mun˜oz-Castro (321), Grupo de Quı´mica Inorga´nica y Materiales Moleculares, Facultad de Ingenierı´a, Universidad Autonoma de Chile, El Llano Subercaseaux, Santiago, Chile Minh Tho Nguyen (99), Institute for Computational Science and Technology (ICST), Quang Trung Software City, Ho Chi Minh City, Vietnam

Debesh R. Roy (139), Materials and Biophysics Group, Department of Physics, Sardar Vallabhbhai National Institute of Technology, Surat, India Ranajit Saha (275), Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Sapporo, Japan; Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur, India Pranab Sarkar (247), Department of Chemistry, VisvaBharati University, Santiniketan, India Utpal Sarkar (87), Department of Physics, Assam University, Silchar, India Kangkan Sarmah (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India Esha V. Shah (139), Materials and Biophysics Group, Department of Physics, Sardar Vallabhbhai National Institute of Technology, Surat, India Yukatsu Shichibu (411), Graduate School of Environmental Science, Hokkaido University, Sapporo, Japan

Contributors

xiii

Nguyen Thanh Si (99), Department of Chemistry, Can Tho University, Can Tho, Vietnam

Gargi Tiwari (173), Department of Physics, Patna University, Patna, Bihar, India

Swapan Sinha (333), School of Applied Sciences and Humanities, Haldia Institute of Technology, Haldia, India

Nikolay V. Tkachenko (321), Department of Chemistry and Biochemistry, Utah State University, Logan, UT, United States

M. Springborg (41), Physical and Theoretical Chemistry, Saarland University, Saarbr€ ucken, Germany

Jesus M. Ugalde (19), Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), Donostia, Euskadi, Spain

Ambrish Kumar Srivastava (173), Department of Physics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, Uttar Pradesh, India V. Subramanian (213), Center for High Computing and Inorganic Physical Chemistry Laboratory, Central Leather Research Institute, Council of Scientific and Industrial Research; Academy of Scientific and Innovative Research (AcSIR), Chennai, Tamil Nadu, India Zhong-Ming Sun (321), State Key Laboratory of Elemento-Organic Chemistry, Tianjin Key Lab of Rare Earth Materials and Applications, School of Materials Science and Engineering, Nankai University, Tianjin, China

Farnaz Yashmin (345), Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India Dongbo Zhao (237), Institute of Biomedical Research, Yunnan University, Kunming, Yunnan, PR China Lili Zhao (157, 387), Institute of Advanced Synthesis, School of Chemistry and Molecular Engineering, Jiangsu National Synergetic Innovation Center for Advanced Materials, Nanjing Tech University, Nanjing, China

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Chapter 1

Describing chemical bonding in exotic systems through AdNDP analysis Edison Osorio Faculty of Natural Sciences and Mathematics, University of Ibagu
e, Ibagu
e, Colombia

1. Introduction The chemical bonding theory is one of the most important concepts in chemistry and its objective is to explain the stability of thousands of compounds present in nature. However, these concepts are constantly evolving in order to satisfy the major challenges of research through the formulation of new and increasingly sophisticated binding models. The Lewis chemical bonding theory, formulated through lone pairs and two-center two-electron (2c-2e) covalent bonds, has been used to explain the nature and behavior of chemical bonding at the undergraduate and research levels for more than 90 years. The success is the simplicity of model, the availability and simplicity of rules for building structures, and the graphical representation of chemical bonding pattern. Successfully, these models could be associated with the properties of species and their reactivity, thus providing a descriptive and predictive model. [1,2] Nevertheless, there are situations in which the chemical bond and properties of certain species cannot be satisfactorily described by this theory. Within these groups, we emphasize the clusters, aggregates of atoms or molecules bonded together by different types of interactions. Studies of these systems began in the 1950s and are often synthesized using mass spectrometer ion sources or laser vaporization techniques. The latter technique allows the researcher to “assemble” species of any composition and to go beyond studies of aggregates of volatile materials [3–5]. Structurally, clusters are an intermediate form of matter between the atomic level and the solid phase, and their properties are extremely sensitive to composition and charge and can be drastically altered by the addition or abstraction of as little as a single atom or electron, so creating species homologous to those observed in molecular beams is extremely difficult [6,7]. One of the chemical systems in which chemical bonding cannot be explained by Lewis bond theory are boron hydrides, systems characterized by atypical 3c-2e, 4c-2e, etc., chemical bonds [8–10]. The Adaptive Density Natural Partitioning (AdNDP) is a theoretical tool developed in 2008 by Zubarev and Boldyrev to determine the chemical bonding patterns in different systems of interest. [11] This approach has been successfully applied for more than a decade to describe the chemical bonding in aromatic and antiaromatic organic molecules [12] and different atomic clusters, including boron clusters and combinations of different chemical elements such as C, Si, Ge, Sn, Mg, Ca, Sr Ba, Be, and among others [13–22]. AdNDP is based on the concept of electron pair as the main element of chemical bonding model and allows representing the electronic structure in terms of nc-2e bonds, where n includes the interval of total number of atoms in a particular atomic ensemble. This approach recovers the Lewis chemical bonding model and delocalized bonding elements associated to the concepts of aromaticity and antiaromaticity. In this perspective, AdNDP provides a perfect description of systems with localized and delocalized bonds, without involving the resonance concept. Essentially, AdNDP is a powerful visual approach for interpretate of wave functions based on molecular orbitals (MOs); nevertheless, MOs will not be considered properly as a chemical bonding model, unless they are used as a part of aromaticity/antiaromaticity concept for delocalized bonds. Nevertheless, in systems where not involve the sharing of electrons such as hydrogen bridges, electrostatic interactions and Van der Waals forces are prevalent, there is no electron density sharing in the target region and AdNDP tool fails to describe these types of interactions correctly. However, this methodology allows describing a correct localization of core electrons and lone pairs responsible of stability for certain molecules, validating the results obtained by other methodologies, which are better to describe this type of interactions. An example of this alliance was applied in the study chemical bonding scheme in EC+3 , EC+4 , EC+5 , and EC+6 species (E ¼ Sc, Y, and La), work published by Osorio et al., in depth using a combination of different theoretical strategies [23]. In a first instance, the exhaustive explorations of relevant potential energy surfaces (PESs) provided a fan-like structures as the most energetically stable configurations. The chemical bonding

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00016-4 Copyright © 2023 Elsevier Inc. All rights reserved.

1

2

Atomic clusters with unusual structure, bonding and reactivity

analysis using the natural bond orbital (NBO) analysis indicated that the metal-carbon interaction has strong ionic character, increasing when going from Sc to La. Besides, NBO predicted the presence of some degree of covalent metal-carbon interaction, result verified by means of energy decomposition analysis (EDA) [24]. The EDA results showed that in all studied cases, both electrostatic and covalent components significantly contribute to bonding interaction between the 2+ carbon fragment (C1 n ) and metal (E ). Additionally, the topological analysis of electron density showed that metal-carbon interactions are mainly of a closed-shell nature (ionic-like interactions). However, they also have a degree of covalent character. Finally, the AdNDP results support the covalent component in these interactions and, in turn, describe metal-carbon bonds as delocalized forms. In organic molecules area, an example where a combination of different theoretical methodologies is used to explain chemical bonding can be found in the theoretical description of mechanism for the walk rearrangement in Dewar thiophenes clarified by Restrepo et al. [25], where the results obtained by NBO and AdNDP tools showed a interesting evolution picture of bonding during the rearrangements, fully consistent and complementary to the Bader’s theory [26–28], where the nature of bonding interactions and evolution of bonding using descriptors calculated at the bond critical points (BCPs) showed an expected increase (decrease) in the electron density at the BCPs associated to chemical bonds in the process of being formed (broken). Another example is the theoretical study of reaction steps during the biosynthesis of suicidal clavulanic acid (coformulated with b-lactam antibiotics and used to fight bacterial infections) [29]. In this work, Restrepo et al., provided evidence of a reaction channel for the double inversion of configuration that involves a total of six reaction steps. The molecular geometries and electronic structures calculations showed a substantial reorganization of electron density right at the onset of reaction, mostly involving a cyclic evolution/involution of large regions of p delocalization used to stabilize the excess charge left after the initial proton abstraction. A number of bonding descriptors derived from analysis of electron topology distributions showed the evolution of bond orders and are quite consistent with the plots of evolution of electron density and bonding orbitals localized by AdNDP analysis.

1.1 AdNDP implementation The AdNDP methodology is a NBO analysis generalization based on the optimal transformation of a multielectron wave function to a localized form, consistent with the theoretical Lewis chemical bonding model. The first order reduced matrix operator for a closed shell system, independent of spin, is defined as: Z gð1j10 Þ ¼ N Cð1, 2, …N ÞC∗ ð10 , 20 , …N Þd1d2…dN where 1 and 10 are the abbreviations for w1 and w10 , respectively, and the matrix element is Z Pkl ¼ N wk ð1Þgð1j10 Þwl ∗ ð10 Þd 1 d 1 0 So, g(1 j 10 ) can be expressed as an orthonormal basis set of atomic orbitals {wk} X gð1j10 Þ ¼ Pkl wk ð1Þwl ∗ ð1Þ kl

The diagonal elements Pkl of matrix density P ¼ {Pkl} correspond to occupancy number (ON) of wk orbitals. If wk are the bonding orbitals with a maximum occupancy, the set of hybrid orbitals should be considered as optimal, in the sense in which the approximate wave function constructed using the wk orbitals will have a better overlap with the original wave function. It is necessary to perform some approximations in the search for these hybrid orbitals with maximum occupancy, since this procedure is computationally demanding. The density matrix P is represented in block form as follows: 22 33 P11 ⋯ P1N 66 77 ⋱ ⋮ 55 P ¼ 44 ⋮ PN1 ⋯ PNN where block Pjj corresponds to the jth atomic center. The natural spin orbitals with maximum occupancy are eigenvectors of complete density matrix P. It is possible to obtain hybrid orbitals maximizing the occupation over an atomic center, this means, diagonalizing the subblocks P which involve this atomic center. The procedure corresponds to solving the following eigenvalue problem:

Describing chemical bonding in exotic systems Chapter

ðjÞ

ðjÞ

1

3

ðjÞ

Pij hl ¼ nl Sij hl

(j) where Pij is the density matrix of subblock on the jth center, Sij is the overlap matrix, and h(j) l and nl correspond to the first (j) eigenvector and eigenvalue of Pij, respectively, where nl is close to 2.00. The algorithm implementation is called adaptive natural density partitioning (AdNDP) and is based on the diagonalization of n-atom subblocks of density matrix for an n-atomic molecular system written in the bases of natural atomic orbitals (NAO). NAOs are 1-center orbitals of maximum occupancy for a given molecular wave function derived from the atomic subblocks of density matrix. The goal of algorithm is to reveal the most probable regions in which localized electron pairs exist [11,12]. In the next sections, a review of some application of AdNDP methodology to boron chemistry and the combination of these with other elements of the periodic table will be presented. With this knowledge, it is expected that the scientific community will be motivated to explore systems that differ in a significant way from classical chemistry and understand the chemical bonding concepts presented in them.

2. Boron hydrides Boron is one of the lightest chemical elements in the periodic table which can form covalent bonds with hydrogen and, therefore, can be useful in forming units or building blocks for designing and constructing hydrogen storage materials, or in other areas such as catalysis [9]. Understanding the electronic structure, chemical bonding and stability of different conformations or boron hydrides series with different stoichiometries will allow the establishment of boron-hydrogen ratios useful for different kinds of applications.

2.1 Chemical bonding scheme in B3Hy2 complexes Although the structure and properties of a large number of boron hydride compounds are well known [10,30–32], there are still hundreds of unexplored systems that could be theoretically designed from the BxHy00/n+/n general formula. In order to understand the structural relationship between three boron atoms and n hydrogens, the Boldyrev’s group explored the PES of neutral and anionic clusters of B3Hy series (y ¼ 4–7) using the Gradient Embedded Genetic Algorithm (GEGA) program [33]. The chemical bonding scheme revealed by AdNDP analysis allowed to explain correctly the geometrical and not very particular shape of these systems: presence of 2c-2e BdB and BdH, 3c-2e BdHdB bonds, and finally one BdBdB 3c2e sigma bond on B3 triangle [34]. In relation to the importance of B3Hn systems, it has been experimentally demonstrated that the octahydrotriborane anion B3H8 occupies an intermediate position in the ranking of boron hydride compounds of lower complexity, and due to the triangular (deltahedral) geometry, this anion can be considered as a building block for the preparation of polyhedral boron hydrides [35]. Information related to synthesis of these compounds can be reviewed and consulted in the literature [36–39]. In order to understand and establish the binding scheme and stability of B3 aggregates, the Boldyrev’s group conducted an exploration on the PES to understand the reversible dehydrogenation of Mg(B3H8)2 system, which occurs experimentally under certain special conditions [40]. The study was performed using the Coalescence Kick (CK) algorithm [41,42] and localized the most stable conformation for B3H8 anion and an additional structure very close in energy which explains the fluxional behavior of this anion. The AdNDP analysis revealed the presence of classical BdH and BdB bonds (2c-2e), BdHdB 3c-2e bonds and one BdBdB 3c-2e delocated bond on B3 [40].

2.2 Isostructural relationships in BnHn series Is there a relationship between carbon and boron chemistry? To answer this question, it is necessary to review the established concepts taught in undergraduate chemistry courses. Carbon and boron are neighbors on periodic table, but their chemical bonding is different. Molecules that possess carbon can form 2c-2e bonds with other carbon atoms or with other elements such as hydrogen. A specific example is the saturated hydrocarbons with CnH2n+2 stoichiometry, compounds characterized by the presence of 2c-2e bonds formed by hybrid orbitals denominated as sp3 [2]. According to the periodicity trends and the number of valence electrons of chemical elements, the BnHn+2 series would be expected to consist of classical 2c-2e bonds, which should be formed by sp2 hybrid orbitals. An exploration of PES on BnHn+2 (n ¼ 2–5) series, using the CK program, revealed that the classical structures composed of sp2 hybrid bonds become progressively less stable as the series becomes larger, i.e., geometrically more compact structures are created and different chemical bonds like 3c-2e, 4c-2e, etc. appear [43]. The conformations and chemical bonding analysis are shown in Fig. 1.

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 1 Global minima and chemical bonds identified by AdNDP analysis. (Picture obtained from E. Osorio, J.K. Olson, W. Tiznado, A.I. Boldyrev, Analysis of why boron avoids sp2 hybridization and classical structures in the BnHn+2 series, Chem. A Eur. J. 18 (2012) 9677–9681. https://doi.org/ 10.1002/chem.201200506 with the permission of Chemistry A European Journal.)

This performance occurs because the boron atoms in the molecules studied try to avoid sp2 hybridization, since an empty 2p atomic orbital would be highly unfavorable. This affinity of boron to have a certain electron density in all 2p atomic orbitals is one of the main reasons why classical structures are not the most stable configurations [43]. On the other hand, Tiznado et al. performed a PES scan on LinBnH2n series (n ¼ 3–6) and showed that boron avoids adopting structures similar to those of organic cycloalkanes (CnH2n), where cyclopentane (C5H10) and cyclohexane (C6H12) are the most stable systems. However, the authors reported the design of smallest analog of aromatic carbocations (C3H3+), the Li3B3H3+ system, where the global minimum has a triangular B3H32 shape with structural features and chemical bonding patterns similar to its organic counterpart. The authors conclude that aromaticity is a key factor for designing analogs of cyclic organic compounds based on lithium boron hydrides [44].

2.3 Electronic transmutation One of the most important objectives of alchemists at the beginning of history was the transmutation (transformation) of base metals into gold or silver. To achieve this goal, the scientists of time learned to extract certain metals from minerals and to produce different types of inorganic acids and bases, which established the fundamentals of modern chemistry [45,46]. Already in the 20th century, scientists demonstrated that although nuclear transmutation is possible and one element can be transformed into another only by a nuclear reaction, such reactions require significantly high energies compared to a normal chemical transformation. Recent investigations show that the old alchemist’s idea of chemical transmutation is not completely dead. Theoretical analyses show that, in particular systems, when a boron atom acquires an additional electron a kind of electronic transmutation occurs, and the chemical bond and geometrical structure of resulting specie behave like a carbon atom [47]. The work reported by Olson et al. showed that the most stable geometric shape of Li2B2H6 system contains a Li2B2H6 nucleus which is isostructural to C2H6 ethane molecule. The authors propose that this concept may have a significant effect on prediction of new chemical compounds [47]. According to the electronic transmutation concepts, it could be generalized to different systems. The exploration on PESs of Si5-n(BH)n2 and Na(Si5n(BH)n)  systems, with n ¼ 0–5, showed that an isoelectronic substitution of a Si atom for a BdH unit along the transformation from Si52 to B5H52 is possible and the deltahedral shape of global minima is not affected as one moves up the series [48]. The chemical bonding scheme for the Si5n(BH)n2 series is presented in Fig. 2. The AdNDP analyses showed that the conservation of structure is due to valence electrons combining to form six 2c-2e Xeq-Xax bonds (X ¼ Si or B), 5-n lone pairs on the silicon atom and n BdH 2c-2e bonds; a particular fact is the closeness in electronegativity values of boron and silicon atoms (2.04 vs. 1.90, on the Pauling scale respectively), which explains the prevalence of structures at several levels of isoelectronic substitution.

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2 2 2 2 FIG. 2 Chemical bonding picture of Si2 (E), and B5H2 5 (A), BHSi4 (B), B2H2Si3 (C), B3H3Si2 (D), B4H4Si 5 (F) revealed by the AdNDP analysis. (Picture obtained from reference E. Osorio, A.P. Sergeeva, J.C. Santos, W. Tiznado, Theoretical study of the Si5-n(BH)n2- and Na(Si5-n(BH)n) - (n ¼ 0-5) systems, Phys. Chem. Chem. Phys. 14 (2012) 16326–16330. https://doi.org/10.1039/c2cp42674a with the permission of Physical Chemistry Chemical Physics Journal.)

Electronic transmutation on Si52 system can be explored with other elements of periodic table such as aluminum, which has the same electron number as boron. Osorio et al. investigated the transformation of Si52 to Al5H52 through the successive substitution of silicon atoms by AldH units, exploring the PESs for Si5n(AlH)n2 (n ¼ 0–5) systems [49]. The results showed how the global minima, with n ¼ 1–3, keep the same deltahedral structure of Si52 cluster and the same chemical bonding scheme. The chemical bonding analysis for Si5n(AlH)n2 (n ¼ 1–3) is presented in Fig. 3. Nevertheless, in the case of n ¼ 4 (Al4H4Si2) the deltahedral conformation is completely destroyed and Al4Si fragment adopts a planar conformation with C2V symmetry. This result shows how this concept of electronic transmutation is not always applicable to a given system [49]. Regarding the nature of aluminum, it is well known that aluminum-hydrogen atomic clusters are stabilized by the conventional AldAl bonds and AldHdAl multicentric bonds, however, information about the existence of double Al]Al or triple Al^Al bonds is limited. Olson et al. reported through combined studies of photoelectron spectroscopy and ab initio simulations the presence of an Al]Al double bond within the LiAl2H4 cluster, which was proposed through the theoretical model of electronic transmutation [50]. Exhaustive searches for the most stable structures of LiAl2H4 cluster showed that the global minimum possesses a geometrical structure similar to Si2H4, thus demonstrating that an electronic transmutation phenomenon occurs from Al to Si through electronic donation. Theoretical simulations of photoelectron spectrum allowed to establish the coexistence of two isomers and confirmed the presence of a Al]Al double bond [50]. Unfortunately, the electronic transmutation model is only feasible for some particular systems, i.e., it cannot be extended to all atoms of the periodic table. This behavior was described by Olson and et al. who performed ab initio studies for electronic transmutation of beryllium atom into boron. Exhaustive searches of lowest energy structures for LinBen and Bn (n ¼ 3–5) showed how the structure corresponding to the global minimum of Li3Be3 possesses a chemical bonding scheme and geometrical structure similar to B3 system. However, in the case of series with n ¼ 4 and 5, the minimum energy structures do not resemble their Bn counterparts [51].

2.4 Chemical bonding in deltahedral BnHn22 systems Within the enormous existing information on boron-hydrogen systems, the BnHn2 are particularly interesting since they are characterized by their high 3D symmetry, by being stable aromatic species and also because they possess n + 1 valence electron pairs, according to Wade’s rules [52–55]. However, the electronic structures cannot be explained by the classical Lewis structure picture due to the electron deficient character. The multicentric bonding scheme for BnHn2 systems, with n ¼ 2–17, has been revealed through AdNDP analysis in order to obtain information on stability and aromaticity.

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 3 Chemical bonding scheme for Si52, AlHSi42 (A1), Al2H2Si32 (B1), and Al3H3Si22 (C1) revealed by the AdNDP analysis. (Picture obtained from I. Fuenzalida-Valdivia, M.J. Beltran, F. Ferraro, A. Vasquez-Espinal, W. Tiznado, E. Osorio, Isoelectronic substitution from Si52 to Al5H52 : exploration of the series Si5n(AlH)n2  (n ¼ 0–5), Chem. Phys. Lett. 647 (2016) 150–156. https://doi.org/10.1016/j.cplett.2016.01.062 with the permission of Chemical Physics Letters Journal.)

Different theoretical analyses used to describe chemical bonding show how all s and p valence electrons in BnHn2 are involved in the formation of multicentric bonds on the cage surface, which coincides with different symmetric configurations. In this sense, AdNDP is able to detect five different types of multicentric bonds, including an open 3c-2e BBB bond, a triangle-shaped 3c-2e BBB bond, a diamond-shaped 4c-2e bond, an 8c-2e bond in the form of a double ring, and a bond totally delocalized over the entire surface of boron cage [52]. The AdNDP tool has also been useful for studying the stability of larger clusters, such as the B80H20, C80H20 and Al80H20 systems, which contain tetrahedral B4H, C4H, and Al4H fragments in place of CdH fragments at the vertices of dodecahedral scaffold. The AdNDP, NBO and ELF analyses demonstrate the chemical bonding in C80H20 can be described in terms of classical 2c-2e CdC s bonds, while the electron-deficient B80H20 and Al80H20 analogs show the presence of 2c-2e and 3c-2e s bonds as responsible for the bonding between and within the tetrahedral fragments, respectively [56].

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3. Boron nanowheels This section presents an analysis of chemical bonding in boron wheels to understand their behavior and how the incorporation of certain metals into their structure proceeds.

3.1 Dynamic behavior in small boron clusters At present, it is possible to find in the literature innumerable information about the lower energy structures of small and medium-sized boron clusters, most of which are made up of a pair of concentric rings and present two-dimensional shapes [41,57,58]. However, the current research needs to add efforts to understand the dynamic behavior of these systems and to find applications for them. A study carried out by Merino’s group using Born-Oppenheimer molecular dynamics (BO-MD) simulations tools [59,60] showed how in clusters B11, B13+, B15, and B19 the rotation barriers of one of these rings with respect to other are remarkably low [61]. In these systems, the outer ring subunits surround the inner fragment, which in a sequence of random motions makes its way around the main axis in one direction or the other, allowing partial rotation of inner fragment. The AdNDP analysis presented in Fig. 4 shows how these systems possess a network of 2c-2e single bonds linking the boron atoms in the outer rings. In the case of inner rings, the bonding is due to a network of multicenter s bonds involving almost exclusively the inner atoms, and a separate set of bonds between three and, in some cases, more centers linking the inner and outer rings. As a conclusion of this work, the authors argued that a combination of electronic and geometric factors is necessary for a decrease in rotational barriers to occur in these two-dimensional clusters. These factors can be summarized in three items: (i) a sufficiently large outer ring; (ii) a s-skeleton of individual rings that remains essentially intact during rotation; and (iii) a transition state for inner ring rotation that involves the transformation from a square geometric shape to a diamond, a rule that may be related to a mechanism suggested decades ago for the isomerization of carboranes and boranes [61].

3.2 Boron wheels members of Wankel motor family The research carried out by Boldyrev and Wang’s groups on the existence of a boron wheel formed by 19 boron atoms (B19) [41], which contains an internal pentagonal fragment of six boron atoms and surrounded by 13 other atoms, inspired different authors to understand the motion of these two concentric rings. Merino’s group determined through BO-MD simulations how in the B19-cluster the pentagonal fragment and the outer boron ring can rotate almost freely in opposite directions, similar to a Wankel motor [62]. Inspired by these results, the same research group reported the theoretical design of B182 cluster, a system that has a dynamic behavior similar to B19 and would therefore behave like a Wankel motor [63]. This dianionic and electronically stable cluster has a double aromatic concentric system, s and p. The internal unit, composed of six boron atoms, undergoes an almost free rotation within the perimeter of B12 ring. The AdNDP analysis presented in Fig. 5 shows the absence of 2c-2e sigma bonds located between the outer ring B12 and inner unit B6; additionally, AdNDP only detects the presence of multicentric delocalized bonds as 3c-2e, 4c-2e, etc., which migrate easily from one position to another during the rotation of B6 unit. Similarly, the AdNDP tool located a set of 4c-2e delocalized p bonds between the inner and outer ring, and one 6c-2e bond located on the inner ring. The absence of any localized s bond between the inner ring and peripheral boron atoms makes the B182 system show a fluxional behavior. The understanding of dynamic performance of these systems has allowed to Merino’s group establish how to stop or prevent one fragment from rotating around another. Their theoretical investigations shows how the substitution in the B19 system of a carbon atom for a boron atom can give a neutral species, CB18, and cancel the fluxionality of this anion [64]. The AdNDP analysis reported in Fig. 6 shows how CB18 cluster has a pi electron distribution analogous to B19; however, the sigma electron distribution is considerably different: eleven peripheral BdB 2c-2e bonds, two CdB 2c-2e bonds and one extra 2c-2e CdB bond, which connects the peripheral and internal pentagonal rings. This last localized bond is the main reason why the CB18 cluster has a radically different dynamic behavior than the B19 cluster. Returning to small 2D boron clusters, theoretical studies show how the transformation of a rigid system such as the B12 cluster, which has a high rotational energy barrier in the inner ring, to a dynamic Wankel motor system is possible through the incorporation of certain transition metals, mainly Ir, which lowers the rotational barrier significantly, transforming MB12 clusters into Wankel Motor [65]. The global minimum of IrB12 is a symmetric bowl-shaped structure in which the Ir atom is located on the concave side of bowl, similar to its lighter congeners CoB12 and RhB12. Although all these MB12 clusters show a dynamical behavior

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 4 Sigma bonds detected by AdNDP for ground states and transition states of B+13 and B 15 systems. (Picture obtained from S. Jalife, L. Liu, S. Pan, J.L. Cabellos, E. Osorio, C. Lu, et al., Dynamical behavior of boron clusters, Nanoscale 8 (2016) 17639–17644. https://doi.org/10.1039/c6nr06383g with the permission of Royal Society of Chemistry.)

analogous to the so-called “Wankel motors,” rotation of inner B3 ring around of peripheral B9 ring, the energy barrier is lower for IrB12 system (5.0 kcal mol1). A low interaction energy between B3 and MB9 fragments is the main reason why the rotational energy barrier is lower for IrB12 than for CoB12 and RhB12 clusters. The chemical bonding scheme for IrB12 obtained by AdNDP analysis is presented in Fig. 7 and locates two lone pairs on Ir atom, nine 2c-2e localized s bonds in the peripheral B9 ring, and one 3c-2e delocalized s bond in inner B3 ring. One of the most important results is that inner B3 and outer B9 rings are connected by three delocalized s bonds of 3c-2e type and there are no localized bonds between the

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FIG. 5 AdNDP analysis for B182. (Picture obtained from D. Moreno, S. Pan, L.L. Zeonjuk, R. Islas, E. Osorio, G.M. Guajardo, et al., B182: ax quasiplanar bowl member of the Wankel motor family, Chem. Commun. 50 (2014) 8140–8143. https://doi.org/10.1039/c4cc02225d with the permission of Royal Society of Chemistry.)

metal Ir and inner B3 ring. Additionally, AdNDP analysis shows interactions between Ir and the B12 moiety through three s and one p bonds. Furthermore, four fully delocalized orbitals also contribute to the interaction between M atom and B12 moiety.

3.3 Design of sandwich structures The discovery of ferrocene Fe(C5H5)2 in 1951 attracted countless interest in fundamental research at the time due to its unusual structure and bonding characteristics, results that led to numerous applications in materials science such as dissolution of metal ions, catalysis, and biological response; among others [66,67]. One of rules for structural design of these systems involves ligands that can coordinate with transition metal atoms through interactions between the delocalized p MOs of ligands and partially occupied d orbitals of transition metals.

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 6 Chemical bonding scheme for CB18 reported by AdNDP. (Picture obtained from F. Cervantes-Navarro, G. Martinez-Guajardo, E. Osorio, D. Moreno, W. Tiznado, R. Islas, et al., Stop rotating! One substitution halts the B19  motor, Chem. Commun. 50 (2014) 10680–10682. https://doi. org/10.1039/c4cc03698k with the permission of Royal Society of Chemistry.)

FIG. 7 AdNDP analysis for IrB12. (Picture obtained from L. Liu, D. Moreno, E. Osorio, A.C. Castro, S. Pan, P.K. Chattaraj, et al., Structure and bonding of IrB12-: converting a rigid boron B12 platelet to a Wankel motor, RSC Adv. 6 (2016) 27177–27182. https://doi.org/10.1039/c6ra02992b with the permission of Royal Society of Chemistry.)

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FIG. 8 AdNDP analysis of endohedral structure (I) of CrB24. (Picture obtained from L. Liu, D. Moreno, E. Osorio, A.C. Castro, S. Pan, P.K. Chattaraj, et al., Structure and bonding of IrB12-: Converting a rigid boron B12 platelet to a Wankel motor, RSC Adv. 6 (2016) 27177–27182. https://doi.org/10. 1039/c6ra02992b and L. Liu, E. Osorio, T. Heine, The importance of dynamics studies on the design of sandwich structures: a CrB24 case, Phys. Chem. Chem. Phys. 18 (2016) 18336–18341. https://doi.org/10.1039/c6cp02445a with the permission of Royal Society of Chemistry.)

Studies reported by Li’s group show how boron clusters can be used for the design of sandwich structures using certain transition metal atoms. In this light, Li et al. reported how CrB24 cluster can be constituted by a kind of sandwich-type structure in which the chromium atom is located between two B12 sheets, parallel to each other [68]. Intuitively, the CrB24 sandwich complex might not necessarily be the most thermodynamically stable structure, since two B12 units are weakly coordinated through a chromium atom. Investigations focusing on BO-MD simulations showed how the sandwich-like CrB24 structure has extremely poor dynamic stability: the conformation collapses resulting in a highly symmetric endohedral structure with a chromium atom in the center of a B24 cage [69]. The AdNDP analysis for CrB24 system is presented in Fig. 8 and shows how the endohedral CrB24 complex is stabilized due to presence of six 3c-2e s bonds delocalized between boron atoms in the central part of box and the central chromium atom. These bonds correspond to a mixing between pz orbitals of boron atoms and d orbitals of chromium atom. Finally, the whole system is stabilized by presence of three 25c-2e p bonds delocalized over the whole box. Although the small boron clusters prefer planar (2D) conformations to maximize the network of multicenter twoelectron bonds, the presence of a metal with suitable characteristics can reshape the morphology of cluster and form systems of boron like Nanowheel, in which the metal is located in the center of it. Theoretical and photoelectron spectroscopy

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 9 The bonding elements recovered by the AdNDP analysis for Ng2M©B10 complexes. (Picture obtained from S. Pan, S. Kar, R. Saha, E. Osorio, X. Zarate, L. Zhao, et al., Boron nanowheels with axles containing noble gas atoms: viable noble gas bound M©B10  clusters (M ¼Nb, Ta). Chem. - A Eur. J. 24 (2018) 3590–3598. https://doi.org/10.1002/chem.201705790 with the permission of Wiley.)

studies have allowed the design and experimental detection of a series of boron wheels with transition metals in the center, systems as M©Bn/0, among which the following stand out: Co©B8 and Ru©B9 [70], M©B9 and M©B9 (M ¼ Rh, Ir) [71] and Fe©B8 and Fe©B9 [72]. The continuing search for new boron wheels has determined how 10 the highest coordination number in a planar system, and occurs in Ta@B10 and Nb@B10 clusters [73]. Sudip et al., explored by ab initio calculations the feasibility of introducing an axle on both sides of wheel, something like an inverted sandwich of boron atoms where the axles are formed with noble gas atoms, NgnM©B10 (Ng ¼ Ar-Rn; M ¼ Nb, Ta; n ¼ 1, 2) [74]. Free energy calculations indicate how these systems, especially the Xe and Rn complexes, may be good candidates for experimental synthesis in a low temperature matrix. The AdNDP analysis for these systems is reported in Fig. 9 and reveals how the Ng-M bonds are covalent in nature, formation of a 2c-2e s orbital between the Ng and M centers. Likewise, the EDA and natural valence orbital analysis showed how the orbital and electrostatic interactions at the Ng-M contact are nearly identical. Finally, AdNDP analysis indicates how the doubly aromatic character (both s and p) in the M@B10 clusters is not perturbed by the interaction with the Ng atoms.

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3.4 Dynamic behavior of B36 cluster Recent research has shown both B36 and B36 to have bowl-shaped structures with C6v and C2v symmetries, respectively (Fig. 10), both with a hexagonal hole in the center [75,76]. Shortly thereafter, Li et al. found that the most stable forms of B350/ clusters are quasiplanar structures with an overall hexagonal shape possessing two adjacent hexagonal holes (Fig. 10) [77]. These two hexagonal holes are adjacent to each other sharing a BdB bond. The presence of a hexagonal hole is quite common in small boron clusters since it acts as an additional electron acceptor and is the reason for 2D boron clusters stability. There are two possible routes for B35 cluster can be convert to B36: the first corresponds to the addition of extra boron atom to the outer hexagonal hole of B35 cluster, which forces the central hexagonal hole to be retained; while the other path is add extra boron atom to central hexagonal hole of B35 cluster and outer hexagonal hole to remain. Piazza and Chen et al. determined how the most stable form of B36 cluster is the one with a central hexagonal hole, i.e., the first case is the most likely [76]. Liu and et el., computationally designed a pathway to determine why the most stable form of B36 cluster prefers to have a central hexagonal hole [78]. The authors analyzed and compared the chemical bonding properties of structures along the path from I to III, which are shown in Fig. 10. The chemical bonding scheme obtained by AdNDP for structures I, II, and III presented in Figs. 11 and 12, show the high stability of B36 cluster. The stability of central hexagonal hole is due to several factors: (1) the outer B18, central B12 and inner B6 rings are connected through a maximum number of delocalized 3c-2e or 4c-2e s bonds, and these “islands” cover the entire boron sheet creating a complete network that stabilizes the system; (2) there are three delocalized 12c-2e p bonds around the hexagonal hole that reinforce connections between central B12 and inner B6 rings; and (3) three delocalized 36c-2e type p bonds are located throughout the boron sheet without contributing electron density around the central hexagonal hole, stabilizing the connections between three boron rings. Finally, the findings published in this work complement the belief that fully delocalized global 36c-2e p-type bonds are the only reason for the stability and planarity of B36 cluster. Understanding the concept of chemical bonding through the dynamical motion of these systems provides important information for the future design of planar boron clusters and twodimensional boron sheets [78].

4. Summary Understanding the electronic structure, chemical bonding and stability of different conformations allow to establish stoichiometric relationships useful for different types of applications. The chemical bonds revealed by AdNDP allow to explain correctly the geometrical and unparticular shape of small and medium boron hydrides, which mostly contain 2c-2e BdB, BdH, 3c-2e B-H-B, and B-B-B bonds. Regarding the concept of electronic transmutation, the AdNDP revealed how the chemical bonding scheme the global minima is not affected as one advances in the series; unfortunately, this concept is only valid for some particular systems. On the other hand, the AdNDP analysis shows that the dynamic behavior of small boron

0/ FIG. 10 Global minima for B0/ 36 [75,76] and B35 [77]. (Picture obtained from L. Liu, E. Osorio, T. Heine, Understanding the central location of a hexagonal hole in a B 36 cluster, Chem. Asian J. 11 (2016) 3220–3224. https://doi.org/10.1002/asia.201601106 with the permission of Wiley.)

FIG. 11 AdNDP analysis for s bonds of structure I, II, and III. (Picture obtained from L. Liu, E. Osorio, T. Heine, Understanding the central location of a hexagonal hole in a B 36 cluster, Chem. Asian J. 11 (2016) 3220–3224. https://doi.org/10.1002/asia.201601106 with the permission of Wiley.)

FIG. 12 AdNDP analysis for p bonds of structure I, II, and III. (Picture obtained from L. Liu, E. Osorio, T. Heine, Understanding the central location of a hexagonal hole in a B 36 cluster, Chem. Asian J. 11 (2016) 3220–3224. https://doi.org/10.1002/asia.201601106 with the permission of Wiley.)

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clusters depends strongly on how the inner and outer ring atoms are bonded, and how this motion is truncated when there are localized 2c-2e bonds between these two rings. Finally, the theoretical design of systems with fluxional behavior is possible if the formation of localized bonds between these outer and inner rings is modulated or restricted.

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Describing chemical bonding in exotic systems Chapter

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Chapter 2

Electron delocalization in clusters Jose M. Mercero and Jesus M. Ugalde Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), Donostia, Euskadi, Spain

1. Introduction Clusters are “ensembles of like objects or individuals,” according to the dictionary. In chemistry, it refers to aggregations of atoms or molecules that appear close together and are chemically interrelated. Clusters are made of a number of atoms or molecules, not too small and not too big, loosely referred to as “intermediate” between the molecular regime and the nanoparticle regime. Normally, the size of clusters ranges within the nanometer scale and consists of a hundred or so of atoms. Smaller ones are referred to as subnanometer clusters consisting of a few tens of atoms. All in all, clusters have a definite number and type of chemically interacting atoms in a specific geometrical arrangement, and consequently represent precise chemical species and studied as such. As a natural extension of this idea of intermediacy between molecular and nanoscale regimes, alluded to above, clusters are customarily thought of as being intermediate “in character” between molecules and solids. However, the truth of the matter is that the chemistry and physics of clusters represent an entirely unique new field of research. The themes of interest in cluster science expand over a wide range of questions that remain unshared even among disciplines close by. For instance, how atomic constituents bind together in aggregates by “stepwise addition” of the building units, and how do their properties change during such a process are crucial issues in cluster science. The atomicity of clusters is also an important issue, namely, the characterization of the clusters requires determining how much matters have the cluster and how much electric charge it bears. Since clusters are often, not always, synthesized in the gas phase at elevated temperatures, cooling results inevitable. Clusters formed in a high-temperature nonequilibrium regime become more ordered and as energy is removed by the cooling, as they approach to thermal equilibrium, which is not necessarily always reached. The balance between the quantum nature of the cooling and the quantum nature of the electronic structure of the clusters themselves dictates the resulting order. One salient feature of the electronic structure of clusters is its electron delocalization. All quantum treatments of the clusters’ electronic structure give a delocalized description for their valence electrons, which remain confined within the cluster due to the attractive potential exerted by its ionic core, as determined by its geometrical structure. Consequently, understanding the features of the electron delocalization in clusters is a key goal that enables rationalization of geometrical structures and electronic properties, and provides predictive power to sustain further advances. Herein, we shall focus on homoatomic clusters in order to keep this chapter as simple as possible in order to deliver concepts and insight in the more understandable manner. For an updated revision of the state of the art in heteroatomic cluster science, the reader is referred to the following recent publications [1–4]. This selection might not be the best one, and it is not even intended to be exhaustive. The lecture delivered by Feynman at the annual meeting of the American Physical Society on December 29, 1959, entitled “There’s plenty of room at the bottom: an invitation to enter a new field of physics,” is cited by many as the milestone which established the scientific credibility of nanoscience, which advocates for the concept that “small is different” [5]. Clusters are smaller than nanoparticles, despite we call them “intermediate.” Consequently, we might be well legitimated to proclaim that “intermediate is different and amazing,” and that there is “even more” room at the intermediate. Thus, exploring this “intermediate” regime of matter should be worth doing, for cluster science offers fundamental knowledge to be discovered and the possibility to transfer that knowledge to industry. Namely, cluster science offers opportunities for all sorts of scientists and engineers. Herein, we shall concentrate in unveiling the close relationship between the electron delocalization of the valence electrons and the overall geometric structure of the clusters. We will treat separately two-dimensional (2D), flat clusters for which electron delocalization and aromaticity are found to be closely related concepts. Then, we shall move on to threedimensional (3D), (pseudo)spherical clusters, where the jellium model will be utilized to put forward the physical basis of Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00013-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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the valence electrons’ delocalization. In both cases, we shall rely on specific examples with the sole aim of best illustrate the concepts introduced.

2.

Electron delocalization in flat clusters

Let us commence with the hydrogen trimer cation, H+3 , which by stretching the concept of cluster arguably beyond reasonably acceptable limits [6], can be seen as the smallest cluster with chemical entity. This cluster contains one electron-pair delocalized over the molecular frame. Recall that H+3 admits two structural isomers, that is, the linear one and the equilateral-shape triangular one. The former isomer is not considered to be an aromatic species, because it is not cyclic, but the latter it is. The reason claimed at this stage is that aromaticity requires “cyclic” delocalization. This makes the connection with the “traditional” concept of aromaticity as it was earlier conceived for rationalizing the “smell” of cyclic unsaturated hydrocarbon species. But, at variance with the venerable organic chemistry tradition, the delocalized electrons in H+3 are of s-type, rather than of p-type. The most outstanding example of this class of aromaticity is the s

aromaticity characterized for the H
5 moiety of the PtZnH5 anionic cluster. The H5 ring contains 4n + 2 with n ¼ 1, namely, six s-type delocalized electrons around its pentagon-like frame [7]. Let us now consider [C3H3]+, the cyclopropenyl cation. Common organic chemistry knowledge tells us that [C3H3]+ bears one electron pair of the p-type (the carbon atomic orbitals that combine to form the molecular orbital which describes such electron pair are of p-type, vide supra) delocalized over the equilateral-like carbon ring structure. Hybridization of the atomic 2s and 2p carbon atomic orbitals lies the conceptual basis for rationalizing the bonding in [C3H3]+. Thus, in this case, the sp2 hybridization results in three orbitals for each carbon of which two combine with their corresponding hybrids of the adjacent carbon atoms, and the third one with its terminal hydrogen’s 1s orbitals. The so-formed molecular orbitals constitute the skeletal s molecular orbitals. Each of them localized on the two atoms over which that s molecular orbital expands and occupied with one spin-entangled electron pair. The remaining three 2p carbon atomic orbitals (one per each carbon atom) are not hybridized, but they combine linearly to yield three p-type molecular orbitals. Notice that since these 2p orbitals lie perpendicular to the molecular plane, they have a node on the molecule’s plane with different signs up and down the plane. Consequently, they are of p-type symmetry. Since, there are only two electrons but three molecular orbitals in this p-type molecular orbital system, we have now to decide the occupation pattern of the three molecular orbitals. The Aufbau principle states that orbitals are filled starting with the lowest-energy and energy degenerate orbitals should be equally filled. For our p molecular orbital system, this means that the lowest-energy molecular orbital will be filled and the two higher-energy-degenerate ones will remain empty. Consequently, the cyclopropenyl cation does qualify as a H€uckel aromatic molecule, that is, fulfills the 4n + 2 electron counting rule, with n ¼ 0, for two cyclically delocalized electrons over the entire molecular frame. This is the type of aromaticity ordinarily referred to in organic chemistry. However, there are atoms for which the s/p atomic orbital energy gap is large enough as to prevent or at least to make unfavorable substantial hybridization of s- and p-type orbitals. This is the case of aluminum, which has an 3s/3p atomic orbital energy gap as large as 4.99 eV [8]. Let us consider the [Al3R3]2
cluster having an equilateral-like aluminum ring with each of the three R terminal substituents bonded to each of the aluminums. This cluster is isostructural to the cyclopropenyl cation, that is, both structures have D3h symmetry. However, its molecular orbitals are rationalized in a rather different way. First, the 3s, and the 3px, y atomic orbitals of aluminum will yield combinations resulting in molecular orbitals of s symmetry, no nodes on the molecular plane, while the 3pz atomic orbitals will lead to p-symmetry molecular orbitals, the molecular plane is a nodal plane. Consequently, the latter orbitals will not mix with the s-type orbitals because of their different symmetry. Second, the s-type orbitals (s) resulting from the 3s atomic orbitals will not mix with the s-type orbitals resulting from the 3px, y atomic orbitals because of their large energy splitting. Third, the 3px, y atomic orbitals will adapt their symmetry to yield molecular orbitals of s radial (r) and tangential (t) symmetry, which transforms in accordance with the irreducible representations of the D3h symmetry point group. Finally, we end up with four sets of molecular orbitals, which will be denoted as ss, sr, st, and p, respectively. The energy ordering of the molecular orbitals in each set is determined by the dimension of the irreducible representations of the D3h symmetry point group, namely, the a0,00 1,2 of dimension 1, and the e0 , 00 of dimension 2. Consequently, the molecular orbitals which transform like a0,00 1,2 will not be degenerate, but those which transform in accordance with e0 , 00 will be doubly degenerate. It turns out that for n membered ring, with n even of the four above-mentioned sets, arrange as one nondegenerate molecular orbital below (n
2)/2 pairs of degenerate orbitals capped with one nondegenerate molecular orbital. On the other hand, when n is odd, all the molecular orbitals, of ss, sr, and p sets, arrange as one nondegenerate molecular orbital below (n
1)/2 pairs of degenerate orbitals, and those of the st set arrange as (n
1)/2 pairs of degenerate orbitals capped with one nondegenerate molecular orbital [9, 10].

Electron delocalization in clusters Chapter

2

21

The filling of these molecular orbitals will be done in accordance with Aufbau principle. Thus, the three ss orbitals will be filled first. This leaves eight valence electrons to be accommodated in the sr, st, and p sets. It turns out that the lowestenergy orbitals filled correspond to lowest-energy nondegenerate orbitals of the sr and p sets, and the lowest-energy doubly degenerate two orbitals of the st set. The resulting picture is that this cluster shows substantial electron delocalization, namely, eight electrons delocalized over the three equilateral-shape triangular structure of the three aluminum atoms. However, our analysis provides a deeper view of this overall electron delocalization. It arises from the delocalization of the three distinct sets of valence orbitals, of these two are of s type and hence, they delocalize on the molecular plane, and the third one is of p type, which delocalizes above and below the molecular plane. Deciphering the aromaticity of this cluster is an interesting issue. Naturally, one would say that the cluster has “two” aromatic systems in the H€ uckel sense, the sr and p systems both having 4n + 2, with n ¼ 1, that is, two electrons cyclically delocalized over the equilateral-shape triangular aluminum ring. This brings about the much debated concept of “multiple” aromaticity [9, 11–16], which refers to the existence of multiple sets of molecular orbitals’ systems which do not mix with each other, and each of them fulfilling H€ uckel’s rule of aromaticity. Finally, notice that although the st set does not fulfill the H€ uckel’s rule of aromaticity, for it has four delocalized electrons, it could also be taken as an aromatic set for it has the lowest-energy molecular orbitals fully and equally occupied. All this renders [Al3R3]2
cluster as a doubly s and single p, triple aromatic cluster. One such type of cluster compound has been synthesized recently by Power et al., namely, the Na2[Al3R3], R ¼ 2,6-dimesitylphenyl, complex (see Fig. 1 and [17]). A preliminary inspection of the calculated valence molecular orbitals of this complex by the same authors revealed that its second highest occupied molecular (HOMO-2) corresponds to a p-type orbital delocalized over the three aluminum atoms, which show that Na atoms above and below the aluminum molecular planar act an electron donor rendering a [Al3R3]2
cluster at the core of the compound. In view of this fact, the authors stated “… Na2[Al3R3] ‘is aromatic’, in accordance with H€uckel’s (4n + 2) rule.” Nevertheless, the aromaticity of the [Al3R3]2
cluster was examined in deeper detail by Mercero et al. [18], who established that its 14 valence electrons are arranged as shown in Fig. 2 and consequently, the cluster is s- and p-aromatic in accordance with H€ uckel’s rule as applied to each of the valence molecular orbitals’ sets. This interpretation comes along

(A)

(B)

C(7) Na(1) C(2) C(1)

Al(1A)

Al(1)

Al(1B) C(2E)

C(7E)

Na(1A) ˚ ) and angles FIG. 1 Left: Thermal ellipsoid plot (30% probability) of Na2[Al3R3], R ¼ 2,6-dimesitylphenyl without H atoms. Selected bond lengths (A (degrees): Al(1)–Al(1A) 2.520(2), Al(1)–C(1) 2.021(3), Al(1)–Na(1) 3.285(2), Na(1)–C(7) 3.066(2), Na–Cring 3.066(2)–3.808(2) [av. 3.459(2)], Mes(centroid)–Na(1A) 3.177(2); A1(1)–Al(1A)–Al(1B) 60.0, Al(1A)–Na(1)–Al(1B) 45.12(3), C(1)–Al(1)–Al(1A) 142.8(1), C(1)–Al(1)–Al(1B) 157.2 (1). Dihedral angle between Ali3 plane and Na(1)–Al3(centroid)–Na(1A) plane: 90.0. Mes ¼C6H2-2,4,6-Me3. Right: Kohn-Sham orbital representation for the delocalized HOMO-2 of Na2[(AlAr)3] (Ar ¼C6H3-2,6-Ph2). (Reproduced with permission from R.J. Wright, M. Brynda, P.P. Power, Synthesis and structure of the dialuminyne Na2[Ar0 AlAlAr0 ] and Na2[(Ar00 Al)3]: AlAl bonding in Al2Na2 and Al3Na2 clusters, Angew. Chem. Int. Ed. 45 (2006) 5953.)

22

Atomic clusters with unusual structure, bonding and reactivity

˚ above the ring center (bottom number of each of the FIG. 2 CMO-NICS, in ppm, analysis at the ring center (top number of each of the pair) and at 1 A pair) of [Al3H3]2
and of [Al3F3]2
, in parenthesis. (Reproduced with permission from J.M. Mercero, M. Piris, J.M. Matxain, X. Lopez, J.M. Ugalde, Sandwich complexes of the metalloaromatic Z3Al3R3 ligand, J. Am. Chem. Soc. 131 (2004) 6949–6951.)

˚ above the center of the ring NICS(1) with the calculated NICS at the center of the ring, NICS(0) ¼
13.04 ppm, and at 1 A ¼
11.02 ppm. The former is an indicator of s-aromaticity and the later of p-aromaticity. Furthermore, an analysis of the magnetic responses of the valence molecular orbitals through the inspection of the NICS values of the each of the canonical molecular orbitals, CMO-NICS, revealed that the st system is antiaromatic in [Al3H3]2
, as shown by their positive CMO-NICS values reported in Fig. 2. However, it was also found in the same research that the aromaticity of the [Al3R3]2
cluster depends markedly on the nature of the R substituent. Thus, it was found that both p-acceptors, like
C ≡N, and s donors, like
CH3, increase the aromaticity of cyclotrialane ring, relative to that of [Al3H3]2
. But the largest enhancement of the aromaticity of the ring occurs for halides. In particular, [Al3F3]2
was predicted to be highly aromatic as suggested by its large negative NICS values, NICS(0) ¼
45.14 ppm NICS(1) ¼
27.61 ppm. Observe, see Fig. 2, that for [Al3F3]2
, even the st degenerate canonical molecular orbitals are slightly aromatic, opposite to their noticeable antiaromaticity in [Al3H3]2
. However, not all [Al3R3] equilateral-shape triangular fragments are aromatic. Thus, Schn€ockel et al. have succeeded to crystallize the [Al(
3Al3R3)2]
, R ¼ N(SiMe3)2 [19] and the radical [Al(
3Al3R3)2], R ¼ N(SiMe2Ph)2 [20], sandwich complexes, see Fig. 3. Voluminous substituents, R, at the Al3R3 rings, have, indeed, been used to protect aluminum atoms from collapsing and to provide isolation from the environment. The latter compound falls short by one electron to yield two [Al3{N(SiMe2Ph)2}3]2
capping ligands above and below the central Al3+ ion. As shown in Fig. 3, the singly occupied molecular orbital (SOMO) corresponds to the lowest-lying ptype molecular orbital of the ligands. The single electron occupation of this molecular orbital prevents the setting up of the aromaticity for this compound. Since the former of the above-mentioned Schn€ockel compounds has this one more electron, it could be aromatic. However, a thorough investigation of the electronic structure of the [Al(
3Al3H3)2]
model compound, concluded that the [Al3R3]2
ligands should not be regarded as aromatic systems because of the lack of a ring-current-induced high field shift for the central Al. Namely, the calculated ring-current-induced field shift at the central Al is d(Al) ¼ +798 ppm in [Al(
3Al3H3)2]
, which should be compared with the value of d(Al) ¼
114 ppm induced by real aromatic rings, like ˚ above the plane of in the aluminoceniun [Al(
5Cp*)2]+ cation [21]. The calculated the NICSs at the center and at 1 A 2
3
the [Al3R3] rings in [Al(
Al3H3)2] , NICS(0/1) ¼
1.34 ppm/
6.47 ppm, indicate that the aromaticity of the [Al3H3]2
ligands decreases substantially upon complexation, in accordance with the prediction of Schn€ockel et al. [20]. Although, given the strong dependence of the aromaticity of the [Al3R3]2
ligands with respect to the nature of the substituents [18], it sounds plausible to find out substituents R that protect the aluminum atoms from collapsing, and at the same time retain the aromaticity of the ligands upon complexation. It is anticipated that finding such ligands will be a challenging task [22]. Bare equilateral-shape triangular clusters also show electron delocalization. The Al
3 cluster anion, and in general the X
(X ¼ B, Al, Ga) clusters, resembles the simplest all-metal ring-like aromatic clusters. These clusters have been 3 carefully studied recently by Kuznetsov and Boldyrev [23] who have carried out both density functional theory B3LYP/6-311+G* and single-reference CCSD(T)/6-311+G* calculations for geometry optimization and frequency calculations. They found that in their electronic ground states, all X
3 cluster anions are equilateral triangles, with
the 1A01 state being the lowest-energy state. The point is, however, that Al
3 , along with the other X3 cluster anions,

Electron delocalization in clusters Chapter

23

FIG. 3 Left: The molecular X-ray crystal structure of [Al7{N(SiMe2Ph)2}6]. The Al1–Al2 bond length (the distance between the central and each of the six symmetry˚ . The Al–Al bond lengths in equivalent Al atoms) is 2.73 A ˚ . All Al–N bond lengths the Al3 rings (Al2–Al20 ) is 2.61 A ˚ . The environment of each N atom is planar are 1.81 A (sum of angles ¼ 360 degrees). The N–Si bond lengths are ˚ . Right: (a) The Kohn-Sham spin-density and 1.75 A (b) the Kohn-Sham SOMO (a2u) of the [Al7{NH2}6] model compound. ( Reproduced with permission from P. Yang, € R. Koppe, T. Duan, J. Hartig, G. Hadiprono, B. Pilawa, € I. Keilhauer, H. Schnockel, [Al(Al3R3)2]: prototype of a metalloid Al cluster or a sandwich-stabilized Al atom?, Angew. Chem. Int. Ed. 46 (2007) 3579.)

(A)

N1 A12

A12¢

2

A11

(B)

(a)

(b)

has only four electrons to be placed at its four lowest-lying near-degenerate molecular orbitals, namely, the two lowest-lying st degenerate molecular orbitals which belong to the e0 irreducible representation of the D3h point group, and the lowest-lying molecular orbitals of the sr and sp systems which belong to the a01 and the a002 irreducible representations, respectively (see Fig. 4), in which case it is very likely that the resulting electronic states bear substantial multireference character.

0.2 0.1

′ ′ ′

′

0.0 –0.1 –0.2

″ ′ ′

′

′ ″

′

′

′

′

–0.3 –0.4 –0.5

′

′

′

′ ′

FIG. 4 Valence molecular orbitals of D3h Al
3 ordered in increasing energy (left) and grouped into its four independent aromatic systems, ss, sr, st, and p (right). Notice the same energy scale in both representations. (From J.M. Mercero, E. Matito, F. Ruip erez, I. Infante, X. Lopez, J.M. Ugalde, The electronic structure of the Al
3 anion: is it aromatic?, Chem. Eur. J. 21 (2015) 9610–9614.)

24

Atomic clusters with unusual structure, bonding and reactivity

TABLE 1 Lowest-lying MCQDPT/MCSCF(10,12)/aug-cc-PVTZ singlet, triplet, and quintet electronic states of the Al 2 3 cluster anion. State

DE

R

n

1 0

0.00

2.509

262.9 (e0 )

A1 (D3h)

383.6 (a01 ) 3

A1(C2v)

9.40

2.632/2.534

101.4 295.4/351.3

5 00

A2 (D3h)

12.46

2.763

225.4 (e0 ) 286.8 (a01 )

DE stands for the relative energy, in kcal/mol, including the zero-point vibrational corrections, with respect to the 1A01 state whose energy is
726.2986479 hartree. R stands for the Al–Al bond length, in A˚. n stands for the harmonic vibrational frequencies, in cm
1, with their symmetries shown in parenthesis. Source: Taken from J.M. Mercero, E. Matito, F. Ruiperez, I. Infante, X. Lopez, J.M. Ugalde, The electronic structure of the Al
3 anion: is it aromatic?, Chem. Eur. J. 21 (2015) 9610–9614.

Thus, when electron correlation is properly taken into account the resulting picture is grossly at variance with that described by single-reference calculations [23]. Mercero et al. [24] optimized and evaluated the harmonic vibrational frequencies of the lowest-lying singlet, triplet, and quintet states of Al
3 at the multiconfigurational self-consistent field (MCSCF) level of theory [25] with 10 active electrons in 12 orbitals using the augmented correlation-consistent polarized valence triple-z basis set [26], hereafter denoted as MCSCF(10,12)/aug-cc-PVTZ. The selected 12 orbitals are the ones depicted in Fig. 4. The obtained geometrical data and the harmonic vibrational frequencies are shown in Table 1. Subsequently, in order to account for the dynamic electron correlation missing by the MCSCF procedure, multiconfigurational quasidegenerate perturbation [27] (MCQDPT) calculations were carried out on the MCSCF(10,12)/aug-ccPVTZ optimized geometries. All valence and virtual orbitals have been correlated in the MCQDPT calculations. Inconsistencies caused by the so-called intrude states, which appear when the perturbation expansion of the reference MCSCF wave function has vanishingly small energy denominators, were remedied by shifting them 0.02 a.u., as recommended earlier [28]. The singlet 1A01 state has a noticeable multireference character as evidenced by the composition of its MCSCF(10,12)/ aug-cc-PVTZ wave function, namely, 0:877477  ða01 Þ2 ðe0 Þ0 ðe0 Þ0 (a002 )2
0:114000  ða01 Þ0 ðe0 Þ2 ðe0 Þ0 (a002 )2
0:114057  ða01 Þ0 ðe0 Þ0 ðe0 Þ2 (a002 )2. The reference MCSCF(10,12)/aug-cc-PVTZ for the C2v-symmetric 3A1 state is 0:9673000  ða01 Þ2 ðe0 Þ1 ðe0 Þ1 (a002 )0
0:2259669  ða01 Þ0 ðe0 Þ1 ðe0 Þ1 (a002 )2, and finally the wave function of the lowest-lying quintet state is dominated by the (a01)1(e0 )1(e0 )1(a002)1 configuration with has an optimum coefficient of 0.917093. The configuration with the next largest coefficient,
0.078707, is 1(a01)1(e0 )1(e0 )1(a002)2(e0 ) with the last molecular orbital being one of the two degenerate highest-lying orbitals of the sr system, see Fig. 4. Therefore, these calculations suggest that the electronic structures of the lowest-lying singlet, triplet, and quintet states of Al
3 are highly correlated. Consequently, the independent particle approximation breaks downs rendering the single-reference molecular orbital representation of the electronic structure incomplete. This impedes deciphering electron delocalization by inspecting the delocalization of the valence molecular orbitals over the molecular frame. Likewise, it also impedes deciphering aromaticity by counting the number of electrons that occupy delocalized valence molecular orbitals and comparing the resulting number with either H€uckel’s or Baird’s rules, or the extensions of the latter. Several theoretical approaches, that go beyond the single-reference approximation, have been developed to build insight into electron delocalization in strongly correlated systems. Most of them [29, 30] rely on the one-particle density matrix, Z Gðr1 , r01 Þ ¼ N dr2 …drN Cðr1 , r2 , …, rN ÞC* ðr01 , r2 , …, rN Þ (1) which unlike molecular orbitals, it is uniquely defined in many-electron systems, irrespective of their multiconfigurationality. One particularly useful such approach for highly correlated electronic structures is the multicenter normalized Giambiagi ring-current index ING [31],

Electron delocalization in clusters Chapter

ING ¼

p2 ½IG ðAÞ1=N 4 NdðAÞ

2

25

(2)

where A is an ordered set fAk gN k¼1 of the N atoms of the ring, and Z Z Z Y N ^ ÞGðr ,r Þ ⋯ ^ ðr ÞGðr , r Þdr IG ðAÞ ¼ dr1 Aðr A 1 1 1 k k k k+1 k

(3)

k¼2

is the Giambiagi electron delocalization multicenter index [32]. It is worth mentioning that in Eq. (2), the numerical factor accounting for the straight linear correlation between ING and the topological resonance energy per p-electron has been not included for convenience. The projector operator Z 0 0 ^ ðr Þ ¼ A dðr1
r Þdr (4) k 1 OðAk Þ

restricts the integrals in Eq. (3) to the atomic basins, O(Ak), of atoms Ak, which are defined by using the fuzzy-atom partition method of Mayer and Salvador [33, 34]. Finally, dðAÞ stands for the total delocalization index [35], which is the sum of all the atom-pair delocalization indices, Z Z ^ ^ ðr ÞGðr , r Þ dðAj ,Ak Þ ¼ 2 dr1 Aj ðr1 ÞGðr1 , r1 Þ  dr1 A k 1 1 1 Z Z (5) ^ ðr Þ dr A ^
2 dr1 A j 1 2 k ðr2 ÞDðr1 , r2 ;r1 ,r2 Þ of the ring. D(r1, r2;r1, r2) is the diagonal element of the spinless two-electron density matrix [36]. The ING index is known to capture the extent of the delocalization of the electron density [37]. In addition, it produces the proper quantitative ordering of aromaticities of both mono- and hetero-cyclic compounds [31]. Table 2 shows the total, 0 dðAl
3 Þ , and adjacent atom-pair delocalization indexes, d(Al, Al ), along with the multicenter normalized Giambiagi electron delocalization multicenter indices, ING, for the three spin states of the Al
3 cluster calculated from their corresponding MCSCF(10,12) multiconfigurational wave functions. Inspection of the data shown in Table 2 reveals a substantial electron delocalization in the 1A01 state of the Al
3 cluster as inferred by the large value, d(Al, Al0 ) ¼ 1.13, of the delocalization index between a pair of adjacent atoms. This is to be compared with the typical value of the delocalization index of single-bonded atom pairs of aromatic cyclic hydrocarbons, 0.75, calculated with comparable correlated wave functions [35]. Excitation to the triplet 3A1 state causes a D3h !C2v symmetry reduction that pushes electron density toward the two equivalent aluminum atoms, resulting in two long bonds with 1.04 delocalized electrons each and one short bond with 1.20 electrons. Remarkably, the total number of delocalized electrons remains similar to the lowest-energy 1A01 singlet state. In sharp contrast, as seen in Table 2, the quintet 5A002 state delocalizes nearly one less electron than its lower-spin singlet and triplet counterparts. Conversely, the electron delocalization sharing between adjacent aluminum atoms decreases significantly, as reflected by the lower value, d(Al, Al0 ) ¼ 0.82, of its delocalization index. The ING values are consistent with the mentioned electron distribution picture, attributing a similar and high aromatic character to singlet and triplet states, and a yet significant but much lower aromatic nature to the quintet species. Notice that

0 TABLE 2 Total, dðAl2 3 Þ, and adjacent atom-pair delocalization indices, d(Al, Al ), along with the normalized Giambiagi 3 electron delocalization multicenter indices, ING (multiplied by 10 ), for the Al2 3 cluster.

ING

dðAl
3Þ

d(Al, Al0 )

1

108.3

3.38

1.13

3

A1(C2v)

103.8

3.29

1.04/1.20

A002 (D3h)

68.2

2.46

0.82

State A01 (D3h)

5

All quantities are expressed in electrons. Source: Taken from J.M. Mercero, E. Matito, F. Ruiperez, I. Infante, X. Lopez, J.M. Ugalde, The electronic structure of the Al
3 anion: is it aromatic?, Chem. Eur. J. 21 (2015) 9610–9614.

26

Atomic clusters with unusual structure, bonding and reactivity

for the 1A1g ground state of Al2
4 , the MSCSF(8,8)/6-311G** normalized Giambiagi electron delocalization multicenter 2
index, ING, is 52.4 [38]. This suggests that Al
3 is even more aromatic than Al4 . The most prominent example of a planar bare metal cluster showing strong electron delocalization is the abovementioned squared Al2
cluster [39, 40]. Fortunately, the electronic structure of Al2
can precisely be described by 4 4 single-reference methods, which greatly simplify the analysis of the features of the electron delocalization.
The tetra-aluminum dianion, Al2
4 , was isolated as a bimetallic charge-compensated system of composition MAl4 ,


with M ¼ Li, Na, or Cu. Wang and coworkers reported the photoelectron spectra of bare CuAl4 , LiAl4 , and NaAl4 clusters claiming that the planar square structure of the Al2
4 cluster, a building block of all these clusters, is aromatic [39, 41]. It was

2
found computationally that CuAl
, LiAl , and NaAl 4 4 4 clusters have pyramidal structures with the planar square of Al4 being a base of these pyramids. Comparison of calculated results and experimental photoelectron spectra confirmed these theoretical findings. Furthermore, the search for the global minimum of the metastable Al2
4 cluster revealed that the planar square structure was actually the lowest one in energy. It is not stable with respect to an electron detachment [42], but when a compact basis set [43, 44] is used, the obtained electronic structure is consistent with its electronic structure in the singly

2
charged CuAl
4 , LiAl4 , and NaAl4 clusters. The question is why the Al4 cluster adopts this high symmetry structure? The 2
answer is: because in this square planar structure results that Al4 has two s- and one p-like delocalized molecular orbital systems over the four aluminum atoms, as shown in Fig. 5. Indeed, as mentioned earlier, single-reference electronic structure methods suffice to show that the four lowest canonical MOs form four lone pairs each one located on every aluminum atom and do not participate in chemical bonding. The three remaining MOs are responsible for bonding in this cluster. The HOMO is the bonding lowest-energy molecular orbital of the p system. The two electrons on that MO delocalize around the cluster. The HOMO-1 is the bonding lowest-energy molecular orbital of the sr system. The two electrons on that MO are also delocalized around the cluster. Finally, the HOMO-2 is the bonding lowest-energy molecular orbital of the tangential st system. Its two electrons are also delocalized around the cluster. Thus, this is an example of a system with threefold (sr, st, and p) electron delocalization. The three molecular orbitals, that contribute to the chemical bonding in Al2
4 , are orthogonal to each other since they are formed from linear combinations of atomic orbitals of different symmetry. Conversely they can also be expressed as valence-bond resonant structures, as elegantly put forward by Dixon et al. [45]. Thus, each of three delocalized bonding molecular orbitals has four independent resonant structures. Consequently, the ` -like structures. valence-bond representation of the chemical bonding in Al2
4 involves 4  4  4 ¼ 64 resonating Kekule Naturally, not all of them have the same weight. In particular, it was anticipated [45] that a full valence-bond calculation

FIG. 5 Valence molecular orbitals of the 1A1g state of the D4h symmetry of the isolated Al2
4 cluster. The topmost 1a2u orbital, the HOMO orbital, corresponds to the p system, the topmost 1b2g to the s-tangential system, and the topmost 2a1g to the s-radial system. All orbitals shown have a population of two electrons.

Electron delocalization in clusters Chapter

2

27

with all these 64 resonating structures will show that the resonating structures associated with triple Al–Al bonds will have a very small weight. Kuznetsow et al. [41] eliminated also the resonant structures featuring p Al–Al bonds with no s bonds between the same pair of atoms, resulting all together in 12 resonant structures. Finally, Havenith and van Lenthe [46] carried out high-level ab initio valence-bond calculations and found that the bonding structure of Al2
4 can be described with six main resonant structures, four Kekule` like, and two Dewar like (diagonal bonding). Surprisingly, the Dewar ones have the largest weights. The large number of resonance structures of Al2
4 accounts for its large resonance energy, RE, 2

1 REðAl2
4 Þ ¼ DEðAl4 ! 4Al + 2e Þ
3  DEðAl2 ð Sg Þ ! 2AlÞ

(6)

estimated as the difference of the atomization energy of Al2
4 and the dissociation energy of three localized Al–Al bonds, because Al2
4 has three bonding electron pairs. High-level ab initio calculations of Dixon et al. [45], based on extrapolating the computed CCSD(T)/aug-cc-pVxZ (x ¼ D, T, and Q) resonance energies to the complete basis set limit, 1 yielded REðAl2
4 Þ ¼ 72:7 kcal/mol. Notice that in Eq. (6), the lowest-lying singlet Sg of Al2 has been taken as the ref3 erence state for the localized Al–Al bonds. However, when the Pu ground state of Al2 is taken as the reference state, the resonance energy of Al2
4 turns out to be 52.5 kcal/mol. This latter estimate is closer the average resonance energy of 48 kcal/mol calculated by Boldyrev and Kuznetsov [47] from the atomization energy of the Al4Na2 cluster referred to a system with two Na–Al interactions and three Al–Al bonds. Finally, Havenith and van Lenthe [46] were able to calculate the resonance energies of the s and p systems of Al2
4 by means of their ab initio valence-bond calculations. They found that the s-system, which is composed by the two independent radial and tangential systems each containing two delocalized electrons, has a resonance energy significantly higher than that of the p-system (123 vs. 40 kcal/mol). Note2+ worthy, the p resonance energy of Al2
4 is substantially lower than that of its p-isoelectronic hydrocarbon [C4H4] (167 kcal/mol). Therefore, all together, both molecular orbital and valence-bond theories describe the electronic structure of the Al2
4 cluster as an electron delocalized electronic structure, with three electron pairs, each of them of different symmetry, delocalized over the square-like aluminum four-membered ring. Recall that the equilateral-shape triangular Al
3 cluster has four
delocalized electron pairs (vide infra). This comes along with the fact the Al2
4 is less aromatic than Al3 , as indicated earlier. The B+5 bare cluster was first proved to be an interesting species by Anderson et al., who found [48] that the mass distribution of B+n clusters generated in laser ablation experiments, showed numerous “magic numbers” in the range n ¼ 1
20. Furthermore, the systematic analysis of the appearance potentials of B+ and B+n
1 , as obtained from their additional collision-induced dissociation experiments, for the B+n clusters, revealed that of the noted “magic” clusters, only B+5 and B+13 were especially stable. A straightforward electron counting reveals that the B+5 cluster is valence isoelectronic to Al
3 , that is, both have four delocalized valence electrons. Thus, it is anticipated that its electronic structure will require a multiconfigurational approach. The results of an extensive search of the singlet, triplet, and quintet potential energy surfaces at the MCSCF(4,15)/aug-cc-pTZVP level of theory supplemented with subsequent single-point MCQDPT calculations, including all inactive electrons in the perturbational space, show that the minimum energy structure of B+5 corresponds to the D5h symmetry 1A01 singlet state. For the triplet spin, a 3B2 state of C2v symmetry bears the minimum energy, while for the quintet spin the C2v symmetry 5A1 state is found to be the minimum energy isomer. The obtained geometrical data, the harmonic vibrational frequencies and the resulting relative energies are shown in Table 3. Given the large energy gaps between the ground-state singlet and the low-lying triplet and quintet states, it can be safely concluded that the structure synthesized by Anderson et al. corresponds to the singlet 1A01 state of the D5h symmetry isomer. Thus, we will restrict our electron delocalization analysis to this isomer. As for the Al
3 , the large multiconfigurational character of B+5 prevents an unambiguous assessment of its electron delocalization features. Nonetheless, analysis of the adjacent atom-pair delocalization indices, d(B, B0 ), along with the normalized Giambiagi electron delocalization multicenter indices, ING, enables to state the electronic density of B+5 is substantially delocalized around the five nuclei (Table 4). The d(B, B0 ) value of 1.27 for the 1A01(D5h) ground state of B+5 is indicative of a substantial electron delocalization among adjacent boron atoms. This is consistent with the total electron-pair delocalization dðB+5 Þ, which shows that the 7.50 delocalized electrons are delocalized among the five boron nuclei in two ways, namely, 6.35 (5  d(B, B0 )) on the periphery, and the remaining 1.15, radially. The calculated value of the normalized Giambiagi index, ING ¼ 15.6  10
3, reveals that the peripheral circular electron delocalization in the ground state of B+5 is about seven times smaller than that of Al
3 (see Table 2), a fact that should be ascribed to the larger diameter of the boron pentamer cation cluster.

28

Atomic clusters with unusual structure, bonding and reactivity

TABLE 3 Lowest-lying MCQDPT/MCSCF(4,15)/aug-cc-pVTZ singlet, triplet, and quintet electronic states of the B+5 cluster cation. State

DE

R

n

1 0

0.00

1.550

151.0 (e02 ), 317.3 (e002 ), 1004.2 (a01 )

A1 (D5h)

1117.9 (e01 ), 1425.4 (e02 ) 3

B2(C2v)

21.49

See Fig. 6

294.0 (b1), 333.3 (a2), 341.2 (a1) 460.9 (b2), 785.2 (a1), 862.9 (a1) 927.5 (b2), 1073.0 (b2), 1321.1 (a1)

5

A1(C2v)

37.17

See Fig. 6

166.8 (a2), 216.1 (b1), 244.1 (a1) 344.6 (b2), 726.6 (a1), 735.5 (b2) 1005.7 (b2), 1018.1 (a1), 1527.7 (a1)

DE stands for the relative energy, in kcal/mol, including the zero-point vibrational energy corrections with respect to the 1A01 state, whose energy is
123.354588 hartree. R stands for the B–B bond length, in A˚. n stands for the harmonic vibrational frequencies, in cm
1, with their symmetries shown in parenthesis.

TABLE 4 The normalized Giambiagi electron delocalization multicenter, ING (multiplied by 103), along with the total, dðB+5 Þ, and adjacent atom-pair delocalization, d(B, B0 ), indices for the lowest-energy isomer of B+5 cluster. State

ING

dðB+5 Þ

d(B, B0 )

1 0

15.6

7.50

1.27

A1 (D5h)

All quantities are expressed in electrons.

One salient feature of the geometries of the isomers shown in Fig. 6 is the lack of intraannular bonds for the lowestenergy isomer, the singlet state 1A01 (D5h). However, inspection of the geometries of the lowest-energy isomers of the triplet and quintet spin potential energy surfaces (recall their reduced C2v symmetry relative to the D5h of the minimum energy isomer) shows indisputable evidence of intraannular B–B bonding, rendering isosceles-shape boron triangles as the basic geometrical feature of these structures. Remarkably, it is this feature that results to be the dominant structural motif for the larger quasiplanar boron clusters, as conclusively posed by Boustani [49]. Thus, the second stable “magic” number cluster discovered by Anderson et al., namely, the B+13 is made of this isoceles-shape boron triangles. The lowest-energy geometry of the B+13 is the so-called Ricca structure, which is estimated to be 10 kcal/mol more stable than it closest contender, the so-called Boustani structure [50]. The former resembles a circularly rounded pseudoplanar coin, and the latter an ovally rounded one. Fortunately, the electronic structures of these isomers of the B+13 cationic clusters are amenable to single-reference electronic structure methods, as shown by Fowler and Ugalde [50]. Their B3LYP density functional calculations revealed a remarkable feature though. Namely, they show substantial electron delocalization for the Ricca isomer of B+13 , with a FIG. 6 MCSCF(4,15)/aug-cc-pVTZ optimum geometries for the 1A01 state of the D5h symmetry, and for the 3B2 and the 5A1 (in parenthesis) states of the C2v symmetry isomers ˚. of B+5 cluster. Bond lengths in A

1.618 (1.972) 2.025 (1.662)

1.550 Symm: D5h State: 1A′1

1.593 (1.494) Symm: C2v State: 3B2 (5A1)

Electron delocalization in clusters Chapter

2

29

structure very much reminiscent of that of benzene, featuring one molecular orbital with no nodes and then next in energy two pseudodegenerate molecular orbitals of p-symmetry delocalized over the whole cluster. This interpretation is reinforced when the parent B
13 anion is considered. Notice that for the singlet-spin state, the twonode pseudodegenerate molecular orbitals of the Ricca structure will be unequally populated, and consequently will undergo a Jahn-Teller distortion toward the oval-shape Boustani structure. This is indeed what takes place yielding the 1 0
Boustani B
13 anion’s Cs symmetry A singlet state 10 kcal/mol stable than the Ricca B13 anion’s C2v symmetry 3 B2 state. This distortion is very reminiscent of that of cyclobutadiene as it converts the triplet spin state to a more stable singlet-spin state [51]. All of these data support the argument that the B+13 cationic cluster is especially stable because of its large electron delocalization, which shares its mean features with that of the benzene to the point that both can be seen as 6p aromatic molecules [50]. Electron delocalization and aromaticity are concepts that have proven to be very useful to experimentalists [52] and very much debated by theoreticians, even to this day [11, 15]. Both concepts have been profusely applied to small boron clusters resulting in a remarkable increase of understanding and predictive ability. A remarkable such prediction has been the experimental characterization of the B36 planar cluster featuring one empty hexagon in the middle [53]. Earlier theoretical calculations [54] have shown that hexagonal vacancy should enhance the stability of triangular lattice boron monolayer sheets. Furthermore, the role of the empty hexagon feature has been rationalized in terms of bonding analysis as carried out within the adaptive natural density partitioning (AdNDP) method [44]. AdNDP analyzes the one-particle density matrix, Eq (1), and searches for core and lone electron pairs, denoted as 1c-2e, and then extends the search to electron pairs (de)localized over n-centers, which will be denoted as nc-2e. Thus, AdNDP retrieves the localized Lewis core and lone-pair orbitals and the delocalized bonding orbitals nc-2e (n > 2) from molecular orbital-based wave functions. All in all, AdNDP, for our purposes, can be seen as a very efficient method to visualize the electron delocalization. The AdNDP analysis of D6h structure of B36 can be seen in figure 3 of Ref. [53]. Notice that the D6h structure is just the transition state between the two symmetrically equivalent minimum energy bowl-shaped C6v structures, which lie 9 kcal/mol below the D6h structure. However, it gives very similar, though a bit nicer, AdNDP bonding orbitals which are more suitable for visualization. Inspection of figure 3 of Ref. [53] readily suggests that B36 can be seen as a lattice formed by the 12 lattice’s triangular structural motifs subtending the 12 3c-2e s-type bonds, which form six hexagonal B7 building blocks over which delocalize the six 36c-2e p-type bonds. Furthermore, due to their stability, B36 clusters can easily stick together, like Lego pieces, to build larger 2D flat structures. Wang et al. [53] hypothesized that this cluster could constitute the structural motif of extended one-atom thick boron sheets. This prediction was soon confirmed experimentally. Thus, inspired by an earlier suggestion [55] that planar boron sheets could be synthesized on coinage metal surfaces, Feng et al. [56] grew boron sheets on an Ag(111) surface by molecular epitaxy under ultrahigh vacuum conditions. They identified two distinct types of boron sheets, consisting of a triangular boron lattice with different arrangements of the hexagonal holes. By setting the silver substrate temperature at 227°C, they were able to grow islands of perfectly ordered structures which subsequent observation under highresolution scanning tunneling microscope revealed to correspond with the so-called b12 sheet structure [57]. By annealing the b12 phase to 277°C, it was transformed into another phase which was identified to be the so-called w3 phase [57]. Both phases can be seen to be made by sticking together B36 units side by side. The main difference between these phases being the distinct arrangement of the hexagon holes. An additional type of boron sheet, which grows under specific experimental conditions, like precise control of boron flux during the epitaxy, and high substrate temperatures, 727°C, was synthesized by Mannix et al. [58]. This phase which is less stable than the formers shows no hexagon holes and lacks planarity due to a large degree of out-of-plane buckling. Finally, using the same ultrahigh vacuum molecular epitaxy technique of Feng et al. [56], Zhong et al. [59] have identified one more boron planar phase on the Ag(111) surface, which has turn out to be the socalled a-sheet. This a-sheet phase of hexagonal symmetry does present hexagonal holes each sustaining a remarkable 6c-2e p-type electron delocalization [60]. The physical and chemical properties of boron sheets, collectively known as borophene, have been extensively studied. Thus, its highly anisotropic metallicity, its stiffness, also anisotropic, which rivals graphene along one axis, have been well documented. In addition, bare samples of boron sheets have been found to oxidize only partially under ambient conditions. Capping with an amorphous silicon/silicon oxide layer fully prevents oxidation. These properties [61–63] of the boron sheets have spurred interest and substantial advance in their synthesis. However, as pointed by Yakobson [61], the separation of these films from their metallic supports and the controlled synthesis of the various phases jeopardizes their way toward realizing practical applications. More amenable to experimental observation is the newly discovered graphene [64–66]. Even more, graphene has already made its way toward practical applications [67]. Like borophene, graphene is a one-atom thick sheet made of, in this case, carbon. Graphene is densely packed carbon in a honeycomb planar lattice expanded by three two-center

30

Atomic clusters with unusual structure, bonding and reactivity

two-electron localized nodeless s-type C–C bonds for each carbon atom. Then, each carbon atom contributes its fourth valence electron to the delocalized p-type manifold. The resulting structure resembles that of an infinite lattice of laterally fused benzene molecules, with the hydrogen atoms at the edges stripped off. This similarity has been considered in many studies on the delocalization of the p-electrons. Indeed, a number of recent studies conclude that all properties of graphene stem from the coherent combination of the aromatic properties of benzene-type p-electrons [68]. However, the six-membered rings of graphene do not hold six p-type delocalized electrons as benzene does. Recall that every carbon atom belongs to three adjacent such rings, then only 1/3 of its p electrons belong to a particular ring. Namely, since each ring has six p electrons, we have 6 1/3 ¼ 2 electrons per ring. Popov et al. [69] carried out a thorough and full 10+ AdNDP [44] bonding analysis of three fragments of graphene, namely, C6H4+ 6 (1 six-membered ring), C24H12 (7 six16+ membered rings), and C54H18 (19 six-membered rings). Notice that the charges were fixed in order to preserve two p electrons per six-membered ring for each of the fragments. The calculations show that C6H4+ 6 has one six-center twoelectron (6c-2e) p bond fully delocalized around the six carbon atoms. C24H10+ has seven (6c-2e) p bonds each delocalized 12 around each of the seven six-membered rings. Finally, C54H16+ has 19 (6c-2e) p bonds each delocalized around each of the 18 19 six-membered rings. Consequently, if this analysis can be extrapolated from these fragments to graphene, the resulting picture is that electron delocalization in graphene is local. The p electrons are not fully delocalized around the whole structure, but they are partitioned in as many delocalization domains as there are six-membered rings, each holding one delocalized electron pair.

3.

Electron delocalization in (pseudo)spherical clusters

When planar clusters are wrapped as spherical clusters, along with the gained third coordinate dimension, they acquire (pseudo)spherical symmetry. Consequently, the delocalized valence electrons are now confined by a (pseudo)spherically symmetric potential set by the ions of the cluster. This confinement potential is very reminiscent of the potential exerted by the nuclei on the electrons in the atomic systems. Though, in this latter case, the potential is perfectly spherically symmetric. Given this analogy, it cannot come as a surprise that the electronic structures of spherical clusters and atoms share a great deal of their main features. Indeed, the idea that the electronic properties of spherical clusters result from the quantum confinement of a nearly free electron gas, which stands for the valence electrons, in a spherically symmetric potential has evolved into the rich concept of the superatom [70]. These superatoms can then be used as the building blocks for a new class of solids with unique structural, electronic, optical, magnetic, and thermodynamic properties. Indeed, recent advances in the assembling techniques for stable superatom-like clusters with atomically precise structures have opened a new promising chapter in materials design [71]. The model which put on physical grounds these ideas is the so-called “jellium” model [72], which refers to a “jelly” made of the cluster’s ionic core within which the valence electrons are confined. For the description of the valence electrons, density functional theory within the Kohn-Sham formalism is adopted. Now, due to the spherical symmetry of the problem (the jelly’s confinement potential is obviously spherical symmetric), angular momentum is conserved, and consequently, the obtained Kohn-Sham orbitals can be classified in accordance with the eigenfunctions of the angular momentum operators L^ and L^z. Let the cluster’s ionic core to be represented by a sphere of smeared out positive background charge density represented by a Heaviside function of height no, namely, n+(r) ¼ noY(R
r), being R the radius of the ionic core, such that the total positive charge of the ionic core produces the total positive jellium charge that neutralizes the total negative charge of the valence electrons, N, Z 4 no  pR3 ≡ N ¼ rðrÞdr (7) 3 where r(r) stands for the electron density of the valence electrons. Then, the energy functional for the electron density of the valence electrons can be cast as E½r ¼ Ekin ½r+Exc ½r+Ees ½r

(8)

where the electrostatic energy functional for the interaction between electrons and the core ions (the jelly) is simply given by a Coulombic term, Z (9) Ees ½r ¼ drvcore ðrÞrðrÞ

Electron delocalization in clusters Chapter

where vcore(r) is the spherically symmetric potential generated by the n+(r) positive charge distribution, 8   r 2  N > > 3
rR

> :
N r>R r

2

31

(10)

and, the electron density of the N valence electrons is given by their spectral summation expanded over the Kohn-Sham orbitals, XN rðrÞ ¼ c* ðrÞci ðrÞ (11) i¼1 i which are obtained by iteratively solving their corresponding Kohn-Sham equations:  
r2 + vef f ðr; rÞ ci ðrÞ ¼ Ei ci ðrÞ 2 with the effective potential given by

Z vef f ðr; rÞ ¼ vcore ðrÞ+

dr0

rðr0 Þ dExc ½r + 0 jr
r j drðrÞ

(12)

(13)

The scheme is closed when a choice for the exchange-correlation functional, Exc[r], is made. Fig. 7 shows the results of above-sketched calculation procedure for 20 valence electrons, when the local density approximation (LDA) is used in Exc[r]. The resulting Kohn-Sham orbitals are fully delocalized around the cluster represented by the jellium of positive charges. One can observe two kinds of shells, from the inspection of the shape of electron density of Fig. 7, as reflected by the two maxima of shown in r(r). In addition, also observed from Fig. 7, some of the electron density of the valence electrons is seen to spill out the cluster, which gives further support to the delocalization of the valence electrons. Inspection of Fig. 7 reveals that in accordance with the Aufbau principle, N ¼ 2, 8, 18, and 20 valence electron will achieve shell closure and thus, attain enhanced stability. This agrees with the experimental evidence relative to the Cu8 cluster [73], which not only shows strong evidence of the enhanced (chemical) stability, but also shows that the anionic Cu
8 cluster behaves as an alkali toward chlorine.

FIG. 7 The electron density of the N ¼ 20 valence electrons in units of no, the ionic core’s density, r(r), in units of no, the effective potential, veff(r;r), and the eigen-energies of the occupied Kohn-Sham orbitals labeled but their corresponding orbital momentum eigenvalues. The occupation numbers are shown as superscripts. (Modified from W. Ekardt, Work function of small metal particles: self-consistent spherical jellium-background mode, Phys. Rev. B 29 (1984) 1558–1564.)

32

Atomic clusters with unusual structure, bonding and reactivity

Even more interesting is the case of the Cu
18 cluster. This cluster with its 19 valence electrons falls either one electron
far off from the 18-electron shell closure of Cu
17 or one electron short for achieving the 20-electron shell closure of Cu19.
However, experimental evidence demonstrates that this open-shell Cu18 cluster is remarkably stable [74]. The explanation resides in the structure of the cluster, which can be viewed as Cu
@Cu17 atom-in-cage cluster. Indeed, natural population analysis reveals that the central Cu atoms bear a charge of
1.06 jej, and subsequent energy decomposition analysis of the natural orbitals for chemical valence (EDA-NOCV) shows that the interaction energy between the Cu
and Cu17 fragments is grossly dominated by the attractive electrostatic interaction term. Consequently, the resulting picture is that the unpaired electron is localized on the 2S orbital of the endohedral Cu atom, which is wrapped by 18 electrons, delocalized on the 17 Cu atoms of the cluster’s surface, which form the 1S2P61D10 shell closure. The latter imparts electronic stability and deprives the inner unpaired electron of its innate chemical reactivity. Another cluster that it is worth to be examined by the jellium model is Al
13. This cluster was observed first as the inner core of the larger Al77 cluster synthesized by Ecker et al. [75]. Subsequent experimental studies [76] suggested
that Al
13 is highly symmetric and very unreactive, for instance, it cannot be etched by oxygen. The structure of Al13 is widely accepted to be icosahedral, as proposed earlier [8, 77], having two five-membered atom rings that are staggered with respect to one another. The eclipsed isomer was found to be the transition state for the simultaneous rotations of the two halves of the cluster in opposite directions. Al
13 cluster owes its stability to the jellium model shell closure of 40 electrons, 1S21P61D102S21F142P6. Its neutral counterpart Al13 cluster behaves as a halogen, because it lacks one electron to attain the closed shell of Al
13 . The theoretically estimated electron affinity [78] of the Al13 cluster, 3.57 eV, suggests that it could form stable salts with alkali atoms. Indeed, KAl13 was characterized [79] as a stable
ionic cluster bound by the electrostatic interaction between K+ and Al
13 . However, Al13 clusters are not stable when put in close proximity of each other [80], for they coalesce into bulk-type aggregates. Same behavior has been found + recently when the gas-phase stable (CH3)4N+Al
13 species is tried to crystallize. While the (CH3)4N cations keep their
structure, the Al13 clusters fuse together into more compact units, at least for the three crystal structures tried, bodycentered cubic, rock salt, and zinc blende [81]. We have considered clusters made of metallic elements only, so far. But, clusters made of nonmetallic elements are ubiquitous. In this regard, the fullerene C60 has gained notoriety in recent years as “the cluster” of main group elements. C60 is a hollow spherical football-shaped cluster that has opened a whole new vibrant research area [82]. The electronic structure of C60 can also be addressed by the jellium model. We shall consider that the two 1s2 electrons of the 60 carbon atoms will make up the ionic core, and their four 2s22p2 valence electrons will constitute the cluster’s valence shell of 240 electrons. Since the cluster is hollow, the jellium will be constrained into a spherical layer of thickness ˚ , for this is the approximate diameter of the carbon atom. Thus, the jellium potential may be cast as [83] DR ¼ 1.5 A 8 3NðR22
R21 Þ > > r < R1
> > > 2ðR32
R31 Þ > > <    N 2R31 2 2 (14) vcore ðrÞ ¼
3R2
r 1 + 3 R1  r  R2 > > r 2ðR32
R31 Þ > > > > > :
N r > R2 r ˚ the radius of C60. The resulting molecular orbitals, once classified where N ¼ 240, and R1(2) ¼ R  DR/2, being R ¼ 3.54 A in accordance with their corresponding angular momentum eigenvalues, are 2 6 14 e 18 6H e 10 4e e 22 7eI26 8K e 30 9L e 34 10M e 38 2S2 3P6 4D10 5F14 6G18 7H10 1e S 2e P 3D F 5G

(15)

The tilde stands for nodeless orbitals, while the remaining orbitals have one radial node at R. The chemical interpretation is straightforward, namely, the 180 electrons occupying the tilded orbitals correspond to the localized s C–C bonds, and remaining 60 electrons in the untilded orbitals are the surface delocalized p-type electrons. Clearly, the electronic shell closure is the key feature behind the remarkable stability of this iconic cluster. Similar conclusions can also be reached based on the so-called spherical aromaticity rules [84–86]. In this vein, it is worth mentioning that Poater and Sola` have recently extended the jellium model to open-shell half-filled systems [87], which show promising perspectives for investigations of stable single high-spin molecules. For chemically oriented research, though, the jellium model should be extended for ionic core densities that have symmetries other than spherical. In particular, polyhedral-type symmetries are highly regarded by solid-state chemists and for those interested in the electronic structure of ligand-stabilized clusters, for the constrains due to the ligand binding often forces the cluster to adopt such polyhedral shapes. The computational efficiency of modern density functional theory

Electron delocalization in clusters Chapter

2

33

implementations makes it feasible nowadays to computationally address any polyhedral shape [88]. The price paid is that the obtained Kohn-Sham orbitals are no longer eigenfunctions of the angular momentum. However, the orbital momentumlike analysis of the Kohn-Sham orbitals can be recovered by inspecting their projections over the various spherical harmonics, Yl, m(f, y), as estimated by Z 2p Z p 2 Xm¼l Z Ro 2 ci,l ðRo Þ ¼ dr r df dy Y ðf, yÞc ðr, f, yÞ (16) l,m i m¼
l 0

0

0

where Ro is the radius of a large enough sphere as to enclose the cluster, and (r, f, y) are the spherical coordinates of r. The polyhedral distortions to the spherical symmetry do not alter the energy shell ordering for both the occupied and the virtual Kohn-Sham orbitals. But, as expected, the polyhedral potential splits some of the angular momentum shells. Thus, the 1F and 1G shells are split in the icosahedral symmetry, the dodecahedral symmetry splits the 1F, 1G, and 1H shells, etc. (see Fig. 8). This puts forward the important role played by the geometry in the overall structure and reactivity of the clusters [89]. Clusters are hardly perfectly spherically symmetric, and since the energy ordering, symmetry and degeneracy of the valence electrons’ orbitals are determined by the geometry of the confinement potential, the orbitals’ degeneracy splittings due to the distortions (lowering the symmetry) from sphericity of the confinement potential can yield subshells separated by sizeable energy gaps. When the lower-energy subshell gets filled up the cluster can acquire a stability on par of those with a magic number of valence electrons. One illustrative example of this phenomenon is the unexpected little reactivity of the

DOS

Sphere

10 8 6 4 2 0 10 8 6 4 2 0 10 8 6 4 2 0 10 8 6 4 2 0 10 8 6 4 2 0 10 8 6 4 2 0

Icosahedron

Dodecahedron

Decahedron

Cuboctahedron

Octahedron

s p d f g h

Spherical

Icosahedral

i Total Dodecahedral

Decahedral

Cuboctahedral

Octahedral

–5

–4

–3

–2

–1

Energy difference from the lowest unoccupied orbital (eV)

0

1

FIG. 8 Electron shells obtained in a fixed-background DFT/LDA jellium calculation for various shapes (shown on top) of the background density. The systems contain 58 electrons corresponding to the 6s-electron density of bulk gold. The LUMO state (59th electron) is marked by a dashed vertical line. € (Reproduced from H. Hakkinen, Electronic shell structures in bare and protected metal nanoclusters, Adv. Phys. X 1 (2016) 467–491.)

34

Atomic clusters with unusual structure, bonding and reactivity

2 6 10 Ag
13 cluster [90]. This cluster has 14 valence electrons, which are 4 electrons short to fill up the 1S 1P 1D , 18 electron closed-shell configuration of the spherical jellium. However, the geometrical distortion to its D2h symmetry optimum geometry has two remarkable consequences, the former is the swap of the virtual 2S orbital into the occupied orbital space, and the latter is the splitting of the 1D shell, such that two of its five orbitals are stabilized while three are destabilized and pushed up to the virtual orbital space, opening a large energy gap at the D shell. The delocalized Kohn-Sham orbitals of 2 4 2 2 4 0 Ag
13 are now best described as: 1S {1P 1P }2S {1D k1D }, where k stands for the HOMO-LUMO gap. The passivation and functionalization of self-assembled ligand-stabilized clusters, one long-standing goal in materials design nanoscience [71], has recently been addressed [91] by means of extensive DFT calculations on a number of liganddecorated clusters of precisely known composition and structure. The picture emerging from the analysis of explicit large-scale DFT calculations is that the jellium model can also be applied to the core moiety of the cluster, ligands excluded, but the count of its delocalized valence electrons should be made using the following recipe:

N * ¼ N
WN W + DN D
q

(17)

where N is the number of valence electrons of the core cluster in the absence of the ligands, NW is the number of electron withdrawing ligands each withdrawing W electrons, ND is the number of electron-donating ligands each donating D electrons, and q is the signed overall charged of the ligand-protected cluster. For instance, consider the closododecaborate [(BH)12]2
anion, which has Ih symmetry, and consequently splits the F shell as shown in Fig. 8. The electron counting as formulated in Eq. (17) renders 12  3
12  1
(
2) ¼ 26 valence electrons, which are arranged as 1S21P61D10{1F8k1F6}, as it can be appreciated in Fig. 9, which correspond with the (ag)2(t1u)6(hg)10(gu)8 symmetryadapted valence molecular orbitals of the 1Ag (Ih) ground state [92]. This puts forward that 26 jellium valence electrons result in shell closure for [(BH)12]2
, which is accordance with its high stability. However, our calculations are at variance with those of Tsukamoto et al. [93], who have reported a 1S21P61D102S2{1F6k1F8} jellium shell-structure, despite their and our calculations have been carried at the same level of theory, namely, B3LYP/6-31g(d,p). Recall that adhering to the criteria laid down in Eq. (17), electrons in molecular orbitals featuring a substantial contribution from B–H bonds are not counted as jellium electrons [94]. In accordance with Eq. (17), the ligand-protected Au102(SPhCOOH)44 cluster has N* ¼ 58 (102–44) valence electrons. Its precise atomic structure has been elucidated by Jadzinsky et al. [95] by means of and X-ray crystal structure determi˚ resolution, and it is shown on the left panel of Fig. 10. nation down to 1.1 A Concurrently, Lopez-Acevedo et al. [96] investigated its electronic structure by means of large-scale DFT calculations using a grid-based implementation of the projector-augmented wave method. Furthermore, they used a minimum frozencore approximation with H(1s), C(2s2p), S(3s3p), O(2s2p), and Au(5d1s) treated as valence orbitals. FIG. 9 The 1S2, 1P6, 1D10, and 1F8 (arranged from bottom to top) occupied jellium orbitals of the closo-dodecaborate anion [(BH)12]2
, arranged from bottom to top.

Electron delocalization in clusters Chapter

2

35

FIG. 10 Left panel: Ligand-protected Au102(SPhCOOH)44 cluster. Right panel: Angular momentum projection coefficients ci, l, from Eq. (16), of the € Kohn-Sham orbitals, as a function of the eigenenergy. (Reproduced with permission from from O. Lopez-Acevedo, P.A. Clayborne, H. Hakkinen, Elec€ tronic structure of gold, aluminum, and gallium superatom complexes, Phys. Rev. B 84 (2011) 035434. Modified from H. Hakkinen, Electronic shell structures in bare and protected metal nanoclusters, Adv. Phys. X 1 (2016) 467–491.)

The spherical jellium predicts a shell closure for this cluster with a fully occupied ninefold degenerated 1G symmetry HOMO and a fivefold degenerated 1D symmetry LUMO (see Fig. 8). The explicit DFT calculations of Lopez-Acebedo et al. confirm the prediction of the symmetry of the HOMO orbitals, as shown in the right panel of Fig. 10. However, for the LUMO orbitals a 1H symmetry, with a tiny 1D character, is found. In addition, the calculated large HOMO-LUMO gap of 0.53 eV agrees with the experimentally observed high stability of this cluster, and with the spherical jellium model prediction of a substantial HOMO-LUMO gap due the shell closure at the HOMO. Thus, in spite of its simplicity, the spherical jellium model accounts for most of the electronic structure features of this large ligand-protected cluster, giving full support to idea that valence electron delocalization of the symmetric, compact, ligand-protected cores acquire enhanced stability when electronic shell closure is attained. Ligand-protected silver clusters are singular due to their tendency to form multiply charge anions. This has been
ascribed to the formation on Ag(SR)3, Ag2(SR)
3 , and Ag4(S2R)2 thiolates of the surface of the silver core of the cluster. 4
In this vein, the recently synthesized Ag44(SC6H3F2)30 cluster [97] is remarkable for both its high charge and its core-ligand structure, which has been established to be best represented as [Ag12@Ag20]6Ag2(SC6H3F2)5, namely a silver Ag12@Ag20 core protected with six Ag2(SC6H3F2)5 tetradentate ligands. The cluster has 32
(6  3)
4 ¼ 18 valence electrons, in accordance with Eq. (17), and consequently it should be electronically stable for it accomplishes the 18 electron 1S21P61D10 closed-shell configuration of the spherical jellium model. However, it is remarkable that in spite of its large charge of
4 jej, the cluster is stable toward electron autoionization [98]. The reason has been traced to the formation of a repulsive Coulomb barrier (RCB) [99] due the combined effect of the high stability imparted by the electron delocalized shell closure of the core of the 4-charge anion, and the electrostatic repulsion between the detached electron and the resulting parent 3-charge anion. The RCB barrier which impedes the electron ionization from the core of the 4-charge cluster turns out to be larger than the barrier for cluster fragmentation [98].

4. Conclusions We have reviewed the features of electron delocalization in both flat and (pseudo)spherical (homoatomic) clusters. For the flat clusters, electron delocalization and aromaticity often run in parallel, a fact that has enabled to extent the concept of aromaticity to structures other than polycyclic hydrocarbons, and has brought remarkable predictive power into the field. In the early years, aromaticity was based on the so-called s
p separation. Namely, the decoupling of the skeletal localized s electrons from ring-delocalized p electrons, and this theory was vastly applied to ring-like hydrocarbons. However, the discovery of very stable s-electron deficient metallic clusters has extended the concept of electron delocalization and aromaticity to these electrons too. In such a way that s-aromaticity, p-aromaticity, etc., have become nowadays part of the modern chemist’s toolbox in order to describe general electron delocalization in ring-like electron deficient structures. The application of the theory to a vast number of clusters has shown that the growth of stable planar clusters, their geometries, and electronic properties stem from the details of their valence electrons’ delocalization, for small and large clusters alike, including both boron (borophene) and carbon (graphene) sheets. In (pseudo)spherical clusters, the separation, which customarily is made, is between core electrons and valence electrons. The former make the ionic core of the cluster that set up the confinement potential for the valence electrons, which are normally treated as a delocalized free electron gas. This is the basics of the simply jellium model that in spite of its

36

Atomic clusters with unusual structure, bonding and reactivity

simplicity has accounted for the salient experimentally observed oscillatory stability of many clusters as a function of the number of the atoms. Although the jellium model is very attractive, its validity has been criticized severely. Modern extensions of the model to include polyhedral symmetry confinement potentials and ways to analyze the outputs of explicit largescale Kohn-Sham DFT calculations have given the method a rebirth, which has allowed to study also the important ligandprotected cluster family. The insightful description of the delicate balance at play between the clusters’ atomistic geometries and the delocalization of the valence electrons should provide a fertile playground for the interaction between theoreticians and experimentalists. In this vein, the jellium model provides a straight language very convenient to communicate easy ideas and concepts that pertain to a number of different fields. Cluster research is a truly multidisciplinary endeavor, which brings in methods from atomic and molecular physics, solid-state physics, thermodynamics, quantum chemistry, and crystallography, just to name a few. However, the jellium model is not a panacea. The key assumption of the jellium model is that the detailed positions of the nuclei do not play a significant role. However, this is an approximation which neglects the fact that in reality, a substantial number of isomers may lie within a narrow energy range. Indeed, it is well known that many clusters are fluxional, their structures fluctuate among several isomers of similar energy. The experimental control of the isomeric forms and the proper statistical treatment of this phenomenon is still in its infancy. Another problem that requires further attention refers to the metallization of clusters. A cluster becomes metallic when its valence electrons are delocalized, the density of states at both the HOMO and LUMO levels is high, and they are close enough in energy for the LUMO orbitals to become thermally accessible. Concomitantly, the accessibility of these excited LUMO orbitals triggers the competition between closed-shell singlet-spin states and open-shell triplet-, or higher, spin states for the ground state. In such cases, the determination of the cluster’s ground state could require extensive inclusion of electron correlation effects, well beyond those brought in by the jellium model. One salient consequence of the electron delocalization in clusters is that they can absorb light through the excitation of a collective mode of the valence electrons in the field of the ionic core charges. This is reminiscent of the collective oscillations of the valence electrons in metals, which are called plasmons. Due to their collective nature, the plasmonic oscillations cannot be described in terms of single-electron excitations. Since plasmon excitations have been observed in both large and small clusters, the rationalization of the plasmonic properties of (metallic) clusters [100] requires incorporating methods from solid-state physics to handle properly the collective excitational behavior of delocalized valence electrons [101]. Efforts to define precisely what a plasmon is in the few-atom clusters, where the character of the optical excitations cannot be unambiguously assigned, are currently being pursued [102]. Self-assembling of atomically well-defined ligand-protected clusters is expected to receive considerable attention in the near future due to its potential implications for the emerging field of nanomolecular electronics [103]. Recent theoretical [104] and experimental [105] advances are very suggestive of the possibility to reduce the size of nanomolecular electronic devices, with application-specific electronic properties, which can be placed in between metallic nanocontacts [106], from the current 20 to 1–3 nm. This remarkable miniaturization could open massive opportunities for the fabrication of nanomolecular devices for a wide range of applications.

Acknowledgments We thank Professor Elixabete Rezabal for her help during the preparation of this chapter. Our work has been supported over the years by Eusko Jaurlaritza (Basque Government) and the Spanish Office for Scientific Research. Financial support comes from MCIU/AEI/FEDER, UE (PGC2018-097529-B-100) and Eusko Jaurlaritza (Ref. IT1254-19).

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13 cluster: quantification and origin of exciton collectivity, J. Phys. Chem. C 120 (2016) 12742–12750. [101] L. Bursi, A. Calzolari, S. Corni, E. Molinari, Quantifying the plasmonic character of optical excitations in nanostructures, ACS Photonics 3 (2016) 520–525. [102] K.D. Chapkin, L. Bursi, B.D. Clark, G. Wu, A. Lauchner, A.-L. Tsai, P. Nordlander, N.J. Halas, Effects of electronic structure on molecular plasmon dynamics, J. Phys. Chem. C 124 (2020) 20450–20457. [103] T. Nakamura, T. Matsumoto, H. Tada, K.-I. Sugiura, Chemistry of Nanomolecular Systems. Towards the Realization of Molecular Devices, vol. 70 of Springer Series in Chemical Physics, Springer-Verlag, Berlin, Heidelberg, 2003. [104] S. Malola, L. Lehtovaara, S. Knoppe, K.-J. Hu, R.E. Palmer, T. B€urgi, H. H€akkinen, Au40(SR)24 cluster as a chiral dimer of 8-electron superatoms: structure and optical properties, J. Am. Chem. Soc. 134 (2012) 19560–19563. [105] S. Jin, X. Zou, L. Xiong, W. Du, S. Wang, Y. Pei, M. Zhu, Bonding of two 8-electron superatom clusters, Angew. Chem. Int. Ed. 57 (2018) 16768–16772. [106] S.M. Jafri, A. Hayat, A. Wallner, O. Sher, A. Orthaber, H. Ottosson, K. Leifer, Nanomolecular electronic devices based on AuNP molecule nanoelectrodes using molecular place-exchange process, Nanotechnology 31 (2020) 225207.

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Chapter 3

Bimetallic clusters M. Molayem and M. Springborg € Physical and Theoretical Chemistry, Saarland University, Saarbrucken, Germany

1. Introduction Clusters have sizes below those of the thermodynamic limit. This means that their properties depend irregularly on their size and that simple scaling laws do no longer apply. Exactly this behavior is the reason for the large interest in “nanomaterials,” including clusters. Systems with specific, desirable properties can be obtained for the properly chosen cluster sizes. On the other hand, this behavior also represents the largest challenge in cluster science. It is hardly possible to predict the materials properties for clusters of a given size, implying that the identification of systems with optimal properties becomes very challenging. Theoretical studies can provide useful information in the understanding of how the properties of the clusters depend on their size. However, the properties depend critically on the structure of the clusters, so an identification of the lowest-energy structures for a given cluster becomes important if one wants to make qualified predictions on the properties. Thereby, the number of metastable structures is known to scale nonpolynomial with the number of atoms in the cluster. As a consequence, theoretical studies of the properties of clusters have to involve one of the following two approximations. One may apply highly accurate methods but then limit the number of structures and/or cluster sizes. Alternative, with more approximate methods, it becomes possible to study more structures and/or cluster sizes. In the present contribution, we shall represent the results of our own work based on the second approach. Thus, we have applied approximate total-energy methods that allow for, in principle, unbiased structure optimizations also for larger ranges of cluster sizes. Thereby, we hope that it becomes possible to identify general trends, although we also have to accept that the results for specific cluster sizes may be less accurate than what more accurate electronic-structure methods could provide. With more than one type of atoms in the clusters, the complexity—and accordingly the challenge—grows significantly. The existence of so-called homotops, that is, systems with the same geometric arrangement of the atoms but with one or more atoms of different types being interchanged, leads to a dramatic increase in the number of possible metastable structures. On the other hand, for a given cluster size, that is, total number of atoms, stoichiometry offers an additional parameter to vary the cluster properties, so also theoretical studies of such systems are highly relevant. Here, we shall summarize our studies on such systems, whereby we shall concentrate on clusters with two (and in one case, three) types of atoms of which at least one is a metal. In all cases, our goal has been to study a larger class of clusters, so that approximate methods have had to be employed. We shall concentrate on the structural and energetic properties and thereby make use of various descriptors that are devised to capture the main features of the obtained results. In the following section, we shall describe briefly our computational methods. Subsequently, we shall present our results for a larger class of system. The presentation is closed by a summary of our findings.

2. Computational methods Studies of the structural and energetic properties of clusters involve the use of a combination of two computational tools: an energy calculator for a given structure and a global structure optimizer. If one wants to conduct a global optimization survey on clusters, such an optimizer contains in turn two parts, that is, a local optimizer, such as a steepest-descent method, and a methodology for (theoretically) sampling all possible geometries of a cluster. In this section, we will provide a brief overview of the different tools we have used in our studies.

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00006-1 Copyright © 2023 Elsevier Inc. All rights reserved.

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Atomic clusters with unusual structure, bonding and reactivity

2.1 Energy calculator There are various approaches for calculating the interactions between atoms in a system. Some of these approaches consider the interactions either at the electronic level such as density functional theory (DFT) methods or at the atomic level as given by the embedded-atom or Gupta potentials. The energy calculations at the electronic level are more accurate, but they demand significant computational resources if the systems are not very small. Therefore, most global optimization studies of clusters have employed simpler descriptions of the interatomic interactions, and just in recent years studies at DFT or other more accurate methods have started to be applied for clusters. In almost all of our studies on cluster properties, we have employed one out of two methods, that is, the embedded-atom model (EAM) and the density-functional tight-binding (DFTB) method.

2.1.1 Density-functional tight binding The DFTB method is based on the density-functional theory formulation of Kohn and Sham [1] and has been developed by Seifert and coworkers [2, 3]. The DFTB method expresses the total energy of a given system relative to that of the isolated, noninteracting atoms according to Xocc XX 1X Etot ¼ E  E + U ðjR  Rj0 jÞ: (1) i jm i 2 j6¼j0 ij j m j In Eq. (1), Ei are the single-particle energies of the system of interest, Ejm are those of the isolated atoms (i.e., jth eigenvalue of the mth atom). The third term in Eq. (1) is a pair potential description of short-range interactions. In DFTB, one considers only the valence electrons, while all other electrons are treated as frozen core. In DFTB, the orbital energies are calculated by expanding the orbital wave functions (Kohn-Sham orbitals ci) in a set of atom-centered basis functions, wjm, that in practical calculations are written as linear combinations of larger number of Slater-type orbitals. For wjm, m is the atom index and j distinguishes different functions centered at the mth atom. Accordingly, the Kohn-Sham orbitals are written as X ci ðrÞ ¼ Cijm wjm ðrÞ: (2) jm

In addition, one assumes that the potential experienced by the electrons can be described as a superposition of potentials of the individual atoms, X ! ! ! Vð r Þ ¼ V m ð r  R m Þ: (3) m

Subsequently, in setting up the secular equation, hwj1 m1 jwj2 m2 i and hwj1 m1 jh^eff jwj2 m2 i (with h^eff being the sum of the kineticenergy operator and the potential of Eq. 3) are extracted from accurate calculations on diatomic molecules as a function of interatomic distance. These matrix elements are then stored as parameters in tables and used on the fly during the DFTB calculations. The third term on the r.h.s. of Eq. (1) represents the repulsive interactions and one determines it such that the total energy as a function of interatomic distance for the diatomic molecule as obtained from parameter-free density-functional calculations is reproduced accurately. Accordingly, the common approach is based on parameterizing results from the diatomic molecules and, subsequently, using those for larger systems. The transferability of the parameters/approach can be checked for instance by comparing calculated bulk properties with experimental values. Although one might expect that the DFTB method as described earlier can provide accurate results only for dimers and crystals, our studies (cf. the following sections) and those performed by other groups have revealed that accurate results will be found also for intermediate-sized systems such as clusters.

2.1.2 Embedded-atom model The EAM was originally developed by Daw and Baskes [4, 5] and has been applied successfully to many metallic systems including clusters [6–11]. Within the EAM, any atom is considered as an impurity embedded in a host comprising all the other atoms. The energy of this atom is written as a functional of the electron density (rhi) due to the host atoms. The approximate total energy of the system of interest contains as one term the sum of the embedding energies of the individual atoms. In addition to this, a correction due to the core-core interactions is included. This takes the form of short-ranged pair

Bimetallic clusters Chapter

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potentials. Accordingly, the functional form of the total energy for an N-atomic system relative to the noninteracting atoms is given by  XN  1 XN h F ðr Þ+ F ðr Þ : (4) Etot ¼ i i i¼1 j¼1, j6¼i ij ij 2 In Eq. (4), Fi ðrhi Þis the embedding energy and Fij(rij) is the pair potential between atoms i and j with an interatomic distance of rij. The values of the parameters of the embedding functions and pair potentials are determined by fitting to experimental data of the bulk system, such as heat of solution, elastic constants, and sublimation and vacancy-formation energies. Eventually, also properties of lower-dimensional systems like smaller molecules and surfaces can be included in the fitting process. One important advantage of the EAM is that the embedding functions depend only on the local electron densities but not on the source of those densities. If the system of interest contains two different types of atoms, it is common to use a geometric mean of the pure pair potentials in order to obtain the heteroatomic ones [12], that is, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fAB ðrÞ ¼ fAA ðrÞ  fBB ðrÞ: (5) As constraints on the pair potentials, it is required that they decay monotonically and vanish beyond certain cutoff distances. By studying the performance of the resulting method, it turns out that smaller cutoff distance for the homoatomic interactions is a good choice for the heteronuclear ones.

2.2 Global structure optimization methods Among the global optimization methods, two have been used exhaustively for cluster studies, that is, genetic algorithms (GA) and the basin-hopping (BH) methods. For both of these methods, the goal is to avoid exploring the complete potential energy surface (PES) of the assumed clusters but instead being able to identify efficiently optimal candidate structures for the global total-energy minimum. Here, it suffices to have a brief overview of the two methods and the interested reader is referred to the large number of available publications for more details.

2.2.1 Basin-hopping algorithm The BH algorithm is a stochastic global optimization method based on Monte Carlo simulations [13–15]. The BH method has successfully identified the structures of the global total-energy minima for many different types of systems such as pure [13] and bimetallic clusters [16–18]. The BH method maps the complex PES of the system under study onto a simpler surface for which the energy is stepwise constant. Thus, the value of the energy for a given structure is taken as the total energy that is obtained after relaxing the structure to its nearest local total-energy minimum structure. Subsequently, a Monte Carlo simulation is performed on this transformed total-energy surface. In each Monte Carlo step, a new structure is accepted if its energy, Enew, is lower than the previous one, Eold. Otherwise, it is accepted with a probability given by exp ½ðEold  Enew Þ=kB T . The nonphysical/virtual temperature T is an hyperparameter, not to be confused with an annealing temperature. The value of T that is used in the Monte Carlo search should be such that the BH algorithm can identify the (known) structures of the global total-energy minima with the lowest computational time. In addition, an acceptance ratio is used that defines the number of accepted trials. The commonly used value is 1/2. In all runs of BH global optimization, the starting configuration can be completely random. This is the strategy that we followed in all of our cluster studies to keep the global optimization unbiased [16, 17].

2.2.2 Genetic algorithm GA is an evolutionary algorithm based on a cut-and-splice procedure [19, 20]. In our studies, we mainly combined GA with the DFTB method for the determination of the total energies for a given cluster structure. There are many different versions of the GA, and we shall not discuss them all here, but focus on a single example that we have used in some, but not all, of our own studies. In that version of the GA, an initial population of structures is created from more sets of randomly placed atoms inside a box. Subsequently, these are relaxed to their nearest local total-energy minima. Each cluster is ascribed a simple fitness value, that is, its total energy. In each generation, the two fittest clusters are allowed to form offspring clusters with every other population member by cutting two parent clusters along a randomly defined plane and reassembling them randomly making sure that the stoichiometry is kept fixed (the so-called crossover process). One might also apply mutation operators

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Atomic clusters with unusual structure, bonding and reactivity

to all or some of the current population members and then, for instance, move all the structures (the current population members, the offsprings, and the mutants) into a large pool. A selection process based on a tournament is then used to build the next generation. Possible mutation operators include the following: random displacements of all atoms, permutation of two atomic species to explicitly include homotops of the same structural motif in the search process for bimetallic clusters, twinning mutation in which two halves of a cluster are rotated by a small random angle in the opposite directions, etch-add process in which a randomly picked atom is removed and placed next to another atom. Since there is no strict stop criterion defined or known in evolutionary algorithms of this kind, we stop the algorithm simply after it has not led to clusters with a lower energy during some specific number of generations, for example, 150.

3. Structural properties of bimetallic clusters Among the clusters we have studied theoretically, we have also considered different metallic clusters, including monometallic, bimetallic, and other metal complexes. Here, we will concentrate on our results on bimetallic clusters although for the sake of completeness a brief review on the binary clusters made of metal + nonmetal element will be provided as well. Titanium-carbon. As the first metallic cluster, we studied titanium-carbon (metallocarbohedrene, i.e., metcar) clusters using DFTB and GA [21]. A stable structure/stoichiometry was found to be Ti8C12 with a tetrahedral geometry that, however, was not found to be particularly stable although it plays a prominent role in experimental studies. The radial distribution of the atoms reveals that the Ti atoms form the cores of the structures consisting of two tetrahedra, while the C atoms are mostly found on the surface layer. In addition, we noticed the tetrahedron shows a distortion interpretable as being due to a Jahn-Teller effect. Cadmium-selenium. We used DFTB to study CdSe clusters with initial structures as cuts out of various crystals. Local relaxations of different sizes n of CdnSen clusters revealed that clusters with initial structures cut from rock salt are less stable than those derived from zinc blende or wurtzite [22] (cf. Fig. 1). As a function of the size of the cluster in the range of 2  n  108, the most stable structure at each size is sometimes the wurtzite structure and sometimes the zinc-blende structure, in agreement with experimental results [23]. The radial distributions of atoms (for definition cf. Ref. [22]) in different cluster sizes reveals that for the optimized structures the largest changes compared with the initial structure occur ˚ from the outer most atoms. Fig. 2 shows the strong in the outer region of the cluster (cf. Fig. 2), that is, up to around 3 A relaxation effects, whereby in each panel the distribution of atoms in the unrelaxed structure (lower part) is compared to that of the relaxed structure. Moreover, the radial distribution of the Cd and Se atoms shows that in the surface region, the selenium atoms move outward whereas the cadmium atoms move inward. This different behavior of chalcogen and metal atoms has also been found by other authors for CdSe clusters [24], as well as in CdS [25] and InP clusters [26]. Nonstoichiometric CdSe have been experimentally synthesized more often than the stoichiometric ones. The radial distribution of zinc-blende-derived clusters for different CdSe cluster sizes is shown in Fig. 3. In contrast to the stoichiometric clusters, the radial distribution of both types of atoms on the clusters surface layer is more scattered. Moreover, the structural relaxations are observed in the complete cluster. For these nonstoichiometric clusters where the outermost atoms are of only one type, the tendency of the chalcogen atoms to move outward and of the metal atoms to move inwards upon structural relaxation, as found for the stoichiometric clusters, is not recovered here. Aluminum-hydrogen-oxygen. Global optimization of (AlHO)n clusters with n  26 using DFTB and GA revealed that in these clusters the Al and O atoms form the inner part of the structures, whereas the H atoms are only found on the

Energy (eV/pair)

–0.8

–1.2

–1.6

–2

0

12

24

36

48

60

72

84

96

108

n FIG. 1 The total-energy variation per CdSe pair for stoichiometric CdnSen clusters as a function of n for (crosses) zinc-blende, (circles) wurtzite, and (squares) rock-salt-derived clusters.

Bimetallic clusters Chapter

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No. of atoms

10 0 –10

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FIG. 2 Radial distribution of cadmium and selenium atoms for zinc blende (left column), wurtzite (middle column), and rock-salt-derived clusters (right column) of different sizes: (a), (e) Cd16Se16; (i) Cd35Se35; (b), (f) Cd37Se37; (c), (g), (j) Cd58Se58; and (d), (h), (k) Cd83Se83. The upper part in each panel represents the relaxed and the lower part the unrelaxed structure, respectively.

No. of atoms

10

(b)

(c)

(d)

FIG. 3 Radial distribution of Cd and Se atoms for nonstoichiometric Cd-rich (left column) and Se-rich (right column) clusters of different sizes: (a) Cd28Se19, (b) Cd19Se28, (c) Cd79Se68, and (d) Cd68Se79. The upper part in each panel represents the relaxed and the lower part the unrelaxed structure, respectively.

0 –10 10

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surface (Fig. 4) [27]. Moreover, in these cores, there are mainly Al–O bonds and essentially no Al–Al or O–O bonds. The H atoms on the surface bind only to Al atoms. These results are supported by DFT calculations on clusters with n ¼ 13. More insight into the geometrical properties of (AlHO)n, clusters can be extracted by looking at the pair-correlation function gAB(R), that is, the number of A-B pairs with an interatomic distance of R as a function of cluster size n [27]. Fig. 5 confirms that there is a strong preference for Al–O and Al–H nearest-neighbor bonds, both for the small and for the large clusters. By a further analysis of the nearest surroundings of the Al atoms, we notice that the coordination number of Al is between 3 and 6 with the value 6 for Al atoms that are in the central part of the clusters.

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Atomic clusters with unusual structure, bonding and reactivity

13

Al

12

O

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11 Radial distance

10 9 8 7 6 5 4 3 2

0

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FIG. 4 Radial distances for Al, O, and H atoms as functions of the size of the cluster n for (HAlO)n clusters. Each small horizontal line indicates the presence of at least one atom of the corresponding type at that given distance to the center of the cluster for a given value of n.

gAB(R)

80 60 40

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FIG. 5 Pair-correlation functions gAB(R) for (HAlO)n clusters for (upper part) n ¼ 2 and (lower part) n ¼ 26 with R in a.u.

Zinc-selenium. Spherical cuts from zinc-blende and wurtzite crystals were relaxed using DFTB to analyze the structural and energetic properties of ZnmSen clusters for cases m ¼ n and m6¼n (m + n  200) [28]. The distribution of atoms in a given ZnmSen cluster, before and after energy optimization, reveals that the structural relaxations are almost throughout the complete cluster with a somewhat larger relaxation in the outer region. We notice that in the outer parts only selenium atoms move outward (away from the cluster center), whereas the zinc atoms move inward. Zinc, as a typical metal atom, prefers a high coordination, whereas selenium atom prefers a low coordination. This difference in behavior of chalcogen and metal atoms is also found for CdS, CdSe, and ZnS clusters. In case of nonstoichiometric ZnmSen clusters, the relaxation effects extend throughout the major parts of the clusters. Almost all atoms change their positions significantly. For these nonstoichiometric clusters, where the outermost atoms are of only one type, the tendency of the chalcogen atoms to move outward and of the metal atoms to move inwards upon structural relaxation are also recovered here. The Zn-rich clusters are simultaneously the clusters with Zn atoms as the

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Radial distance

Radial distance

outermost atoms before relaxation. We find that these clusters possess an overall contraction. On the other hand, the Se-rich clusters with Se atoms as the outermost atoms expand. Potassium/rubidium-cesium. In a study of (KCs)n and (RbCs)n clusters using BH and EAM, we established that the high stability for the magic K-Cs and Rb-Cs binary clusters is strongly correlated with drastic changes in structure toward a higher symmetry, compared to their monometallic counterparts [29]. A cluster is defined as being magic if its second difference of energy with respect to neighboring sizes possesses a strong maximum at this size. As an example, clusters with a total number of N ¼ 34 atoms are magic for pure as well as for binary clusters, but it displays a different symmetry in the two cases. For the pure cluster, it has a T symmetry, but for the bimetallic cluster the heteroatomic interactions lead to the formation of a fivefold so-called “pancake” and to an increase in the symmetry to D5h. This fivefold pancake is the main structural motif for different magic sizes of (KCs)n and (RbCs)n clusters. The high stability of these polyicosahedron (pIh) binary clusters can be explained by the decrease in internal strain when the inner atoms of a pure pIh cluster are substituted by smaller ones. Moreover, the larger atoms have a strong tendency toward segregation, whereby core-shell pIh clusters become favored. One notices that K and Rb in K-Cs and Rb-Cs systems are 16% and 9% smaller than Cs atoms, respectively. We considered the radial distances of the K and Cs, Rb and Cs atoms separately, in order to study the possibility of a segregation in the nanoalloys as shown in Fig. 6. Clearly, the Cs atoms segregate preferentially toward the surface in both types of bimetallic clusters, whereas the K and Rb atoms are primarily located in the core. This is also consistent with the differences in the surface energies of Cs, Rb, and K, that is, 95, 117, and 145 erg cm2, respectively, as well as in the atomic ˚ , respectively. Moreover, the tendency to phase separation in a shell-like segregation is radii, that is, 2.72, 2.50, and 2.35 A predicted for both K-Cs and Rb-Cs clusters with up to at least 1000 atoms. In our study on (KCs)n and (RbCs)n, we used a similarity function to make a quantitative comparison between different cluster structures. The similarity function is so constructed that it approaches the value 1 if two objects are structurally very similar and 0 when they are very different. Our experience is that values below 0.7 imply significant structural differences. We shall repeatedly make use of this function. Comparing K-Cs to K clusters and Rb-Cs to Rb clusters, we noticed that there is a structural agreement between the bimetallic and the pure clusters for clusters with less than N ¼ 26 atoms in total. When we compared both types of binary clusters to pure Cs clusters, we observed structural differences for exactly the cluster sizes 16 and 24 for which the pure clusters of K and Cs or Rb and Cs differ from each other. A reduction in the similarity functions at N ¼ 28 and from N ¼ 34 upwards implies the emergence of novel structures different from those of the pure K, Rb, and Cs clusters [29]. Overall, these bimetallic clusters show a much more regular growth behavior than their monoatomic counterparts. Nickel-copper. We studied structural and energetic properties of NinCum clusters for all combinations of m and n, and sizes N ¼ m + n ¼ 220, 23, 34 using GA and EAM [11]. Whereas in their crystalline phase, both Cu and Ni have the same structures (i.e., fcc), pure copper and nickel clusters, CuN and NiN, have different structures for certain values of N. This behavior is even more complex in the binary nanoalloys. For example, for many sizes, including N ¼ 13 and 23, the geometry of the clusters is the same as that found for both of the pure clusters. Almost exclusively, in all these alloy clusters the Ni atoms tend to occupy the central parts of the clusters, whereas Cu atoms are often found in the outer parts. On the other hand, the structure of NinCu38n clusters with a total of 38 atoms dramatically changes as n varies. For n < 5 and n > 25, the nanoalloys have the same structure as the pure one, that is, the truncated octahedron. But for n ¼ 525, structures with pentagonal symmetry (C5V), presenting an icosahedral fragment are formed [11]. This structural change can be understood from the fact that the nearest Ni–Ni distance is 3% shorter in icosahedral than in the octahedral

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FIG. 6 The radial distances of the K and Cs, Rb, and Cs atoms in their nanoalloys, that is, (KCs)n and (RbCs)n.

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Atomic clusters with unusual structure, bonding and reactivity

Similarity function

1.3

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–2 0 2 4 6 8 10 12 14 16 18 20 22 24 n

FIG. 7 The similarity function versus the number of Ni atoms n. In the left panel, the structures of the bimetallic clusters of the sizes N ¼ 15, 17, and 23 are compared to those of the pure Ni15, Ni17, and Ni23 clusters, respectively. The panel to the right shows the comparison with the corresponding pure Cu clusters. Cu structure. Thus, for the Ni atoms, which possess the higher cohesive energy (ENi coh ¼ 4.44 eV, Ecoh ¼ 3.49 eV), it becomes possible to form stronger bonds and consequently lowering the total energy. The similarity function could give us a better insight into how changes in the Cu or Ni concentration can influence the structures of the nanoalloys. Fig. 7 shows the similarity of NinCum clusters of sizes N ¼ 15, 17, and 23 as a function of Ni atoms n, when comparing to the pure Ni (left panel) and pure Cu (right panel). As a typical behavior, for most values of N the structure is very similar to that of the both pure clusters (i.e., the similarity function is close to 1), as it is seen for N ¼ 23. The exceptions are clusters with N ¼ 15 and N ¼ 17. For these cases, the calculated similarity functions show a higher similarity of the bimetallic clusters to the structure of the pure NiN cluster than to that of the pure CuN cluster. An additional discontinuity in the similarity functions of clusters with N ¼ 17 happens at n ¼ 5 and indicates that new structures, different from those of the pure Ni and Cu clusters, are formed. The similarity function for N ¼ 38 indicates large structural changes in the composition range n ¼ 525 (Fig. 8). The lowest-energy structure for the bimetallic clusters with n  4 and n  26 is found to be the truncated octahedron (the same as for the pure clusters). But for n ¼ 525, the structures have pentagonal symmetry C5V, which is very different from the pure Ni and Cu clusters. The fact that in many bimetallic nanoalloys the core part of the cluster is occupied by atoms of one type and the surface is formed by the other type can be explained as follows. First, the strong internal strain in an icosahedron can be relieved by replacing the inner atoms with smaller atoms (e.g., in NiCu clusters, the inner Cu atoms with smaller Ni ones). Second, the atomic bonds that are stronger are more favored as they stabilize the structure significantly. In NiCu clusters, this leads to structures with a preference for large Ni coordinations. Third, the element with the smaller surface energy prefers to occupy

FIG. 8 The similarity function versus the number of Ni atoms n. The bimetallic clusters of the size N ¼ 38 are compared to those of the pure Cu38 clusters.

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the positions on the surface of the structure, for example, Cu has smaller surface energy in comparison to Ni [Cu: s(111) 69.5 kJ/mol vs. Ni s(111) 80 kJ/mol], once again suggesting that Ni atoms prefer to occupy positions with the highest coordination numbers (e.g., the center of an icosahedron). Finally, positive or negative heat of solution of the alloy is an important factor for segregation or mixing of the atomic species in the cluster. For example, the positive heats of solution of Ni–Cu alloys lead to a segregation of copper to the surface. However, in a Cu–Au icosahedron, the atoms of one type prefer to be surrounded by the atoms of the other type, that is, a mixing of the two types of atoms is preferred. This can be explained by the negative heats of solution for solid Cu–Au alloys. Thus, despite the relative similar behavior in size and surface energies of these two distinct systems, the question whether segregation or mixing is observed can be determined by several different factors including the heat of solution and the relative cohesive energies. Nickel-silver. The global optimization of all possible compositions, that is, combinations of m and n, of NimAgn clusters were done using BH and EAM for the size range N ¼ m + n ¼ 260 [17]. The results were then analyzed thoroughly for their structural properties with the help of different descriptors. The first descriptor is the excess energy, which compares the alloy clusters to their pure counterparts: Eexc ¼ Eðm, nÞ  m

EðNiN Þ EðAgN Þ n , N N

(6)

where for instance E(NiN) is the energy of a Ni cluster with the same geometry as the alloy cluster, that is, the one with the energy E(m, n). The excess energy is, per construction, zero for pure clusters. Negative values imply that mixing is favored, and the smallest value for a given N corresponds to the most stable cluster for this size when comparing with all possible compositions. Fig. 9 shows the excess energy per atom, Eexc/N, for all considered NiAg clusters. For almost all sizes and stoichiometries, the excess energy is negative, implying that some mixing is (almost) always favored for Ni–Ag clusters. Interestingly, there is a certain size range, that is, m ’ 10 and n ’ 22, for which Eexc/N is particularly negative, suggesting that these clusters are the most stable ones. More detailed information can be obtained by looking at Eexc as a function of the number of one atomic species for some specific cluster sizes. In Fig. 10, Eexc is shown as a function of the number of Cu atoms m for four different sizes. In general, the excess energy decreases monotonically from zero and after arriving at a minimum increases again roughly monotonically to zero. On top of these general trends, small N-specific deviations are seen, especially for N ¼ 55 and 60. These are caused by sudden changes in the structure compared with their neighboring stoichiometries. The structures of all 34-atomic clusters with significantly low excess energies were found to be fivefold pancakes. For N ¼ 38, all low values of Eexc are found for polyicosahedra with a broken-symmetry fivefold pancake structure in which some of the Ag atoms are placed outside Ni atoms near the surface. As Fig. 10 shows, the excess energy of clusters with N ¼ 34 and 38 have a plateau for m ¼ 714 and 813, respectively. This can be attributed to the structural similarity of FIG. 9 The excess energy per atom for NimAgn clusters as a function of stoichiometry (m, n) for N ¼ m + n from 2 to 60.

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 10 The excess energy as a function of (m, n) for four selected sizes of NimAgn clusters, that is, N ¼ 34, 38, 55, and 60 atoms.

these clusters. As mentioned earlier, for N ¼ 34, the structures are all fivefold pancakes, whereas for N ¼ 38, they are all polyicosahedra, in which a part of the fivefold pancake is still formed although containing some deformations and having extra atoms attached to it. The excess energy for N ¼ 55 has very low values for polyicosahedral structures and some deformed 55-atomic icosahedra for compositions with m ¼ 1525. Examples are (m, n) ¼ (16, 39), (22, 33), and (24, 31) with polyicosahedral structures, and (23, 32) which is a slightly deformed 55-atomic icosahedron. For N ¼ 60, Eexc has a minimum for the (20, 40) cluster with a truncated Ih-55 structure with five additional atoms on the sides. The mixing versus segregation of the nanoalloys can be better identified by using the bond-order parameter, s. For an AmBn nanoalloy, it is defined as [17] s¼

N AA + N BB  N AB : N AA + N BB + N AB

(7)

Here, Nij (i, j ¼ A, B) is the number of nearest-neighbor bonds between atoms of type i and j. s becomes positive for segregated, almost zero for disorderly mixed, and negative for mixed and onion-like phases of nanoalloys. In Fig. 11, we depict the bond-order parameter versus number of Ni atoms for all compositions of four selected sizes, N ¼ 34, 38, 55, and 60. In addition, we also show the corresponding numbers of Ni–Ni, Ag–Ag, and Ni–Ag bonds in each case. As mentioned, positive values for s imply some degree of segregation. In the present case, this segregation is mainly due to the formation of core-shell-like structures. For the clusters with comparable numbers of Ni and Ag atoms, there is a relatively large number of Ni–Ag bonds so that s obtains lower values. For example, the lowest value of s for N ¼ 34 is found for the (16, 18) and (17, 17) clusters. However, in case of N ¼ 38, 55, and 60, the lowest values of the bond-order parameter are found for more asymmetric clusters, that is, (m, n) ¼ (16, 22), (23, 32), and (34, 26), respectively. The absolute values of the numbers of different types of bonds (insert in Fig. 11) suggest that by starting from pure Ag clusters the number of Ag–Ag bonds decreases monotonically to a value close to 0 that is obtained not only for the pure Ni clusters. This implies that in Ni-rich clusters, the Ag atoms are well separated. However, the number of Ni–Ni bonds is nonzero even for clusters with just a few Ni atoms. We consider this difference as a consequence of a spatial separation of the Ag and Ni atoms: the former are mainly found in the outer parts of the clusters, and the latter are mainly found in the inner parts. The spatial separation of the Ni and Ag atoms is easily seen in their average radial distances (the radial distance for a given atom in a cluster is defined as the distance of the atom to the average position of all atoms in the cluster). Fig. 12

Bimetallic clusters Chapter

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FIG. 11 Bond-order parameter as a function of composition (number of Ni atoms, m) for the global minima of four sizes of interest (N ¼ 34, 38, 55, and 60). The inserts show the number of the three possible types of bonds versus m. Solid triangles and squares refer to the numbers of Ni–Ni and Ag–Ag bonds, respectively, whereas open circles are for Ni–Ag bonds.

FIG. 12 Ratio of the average radial distance of the Ni atoms to that of the Ag atoms in NimAgn clusters as a function of composition (m, n) for N ¼ m + n  60.

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Atomic clusters with unusual structure, bonding and reactivity

presents the ratio of the average radial distances of Ni and Ag atoms for all stoichiometries of NiAg clusters with sizes N  60 [17]. The segregation of the Ag atoms to the sites with larger distances from the center of the clusters is clearly recognized from the ratio of distances, which is mostly C > BH [4,96]. Comparative theoretical study based on EDA-NOCV (natural orbitals for chemical valence) of the bonding situation in carbones, borylenes, and nitreones is summarized in Table 1. The results show that all these species possess a significant donor ! acceptor interaction similar to that of carbones. However, p-back donation is least prominent in nitreones and maximum in borylenes [4,96]. The electronic parameters are partial atomic charge, electron localization function (ELF), lone pair occupancy (LPO) and local nucleophilicity at central nitrogen. Excess electron density at the central N becomes visible with negative partial atomic charge (3.2 electron) and occurrence of two lone pairs during enforced NBO analysis are preliminary parameters to evaluate the nitreone character. AIM analysis is another confirmatory tool to validate the presence of excess electron density at the central N atom. Local nucleophilicity values for nitreones are usually very low (85%) and excellent yield (>80%) (Scheme 20) [105].

SCHEME 19 Reduction of nitreone 46 to generate corresponding radical; radical anion is localized on carborane cluster and positive charge on nitrogen.

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FIG. 16 Chiral pentanidium chloride as phase transfer catalyst (PTC).

SCHEME 20 Stereoselective PTC-catalyzed Michael addition of Schiff’s base.

Mechanistic details on pentanidium catalysis for Michael addition were discussed by Wong et al. using quantum chemical calculations. It was reported that positive charge is accumulated on central core of pentanidium cation (as in nitreones). Therefore, the negative charge transfer from Michael donor (Schiffs base) to acceptor or electrophile takes place in the proximity of central core of pentanidium in order to enhance the electrostatic interaction. The transition state is further stabilized by strong CdH ⋯ O interaction between N-methyl group of pentanidium and oxygen of the substrate (Fig. 17) [113].

FIG. 17 Phase transfer catalysis mechanism of pentanidium-catalyzed Michael addition.

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SCHEME 21 Asymmetric PTC-catalyzed a-hydroxylation and Michael addition of 3-substituted-2-oxindoles.

In 2012, Tan and coworkers developed chiral PTC-catalyzed a-hydroxylation of 3-substituted-2-oxindoles (Scheme 21) using molecular oxygen as oxidant yielding highly enantioselective products. In addition to PTC, the chiral pentanidium salt assists in in situ kinetic resolution. The proposed mechanism is shown in Fig. 18 [114]. Later on in 2015, same research group demonstrated Michael addition of 3-substituted-2-oxindoles to phenyl vinyl sulfone in the presence of similar chiral pentanidium salts (51) as PTC (Scheme 21) [115]. In 2016, pentanidium salt-catalyzed enantioselective alkylation was demonstrated on dihydrocoumarins using silylamide as nonnucleophilic Bronsted probase. Different substitutions such as allylic, propargylic, esters, and benzylic at a-position of dihydrocoumarins were tolerable to yield high enantioselectivity (Scheme 22) [116]. The probase can give rise to a strong base in situ from silylamide using fluoride anion. This strategy leads to the generation of a transient and optimum amount of strong base while reducing side reactions (Fig. 19) [116]. From the literature, it is evident that the NL+2 based PTCs have many merits over conventional NR+4 based PTCs, such as (i) improved yields and enantioselectivity, (ii) simple reaction conditions, (ii) reducing agent no longer needed (triethyl phosphite was not required in the hydroxylation reaction) (iii) use of a pro-base approach discarded the requirement of stoichiometric amount of base, (iv) additives like CsCl and ROH were not required in the Michael addition reaction, (v) reduced reaction time, low catalyst loading, and (vi) scalability of reaction to gram scale, etc. [45]. Nitreone framework in medicinally important molecules: new perspective of protonated biguanide Biguanides are believed to act in their protonated state under physiological conditions. Most of biguanide moiety containing drugs are active in their protonated form [117–121]. The knowledge of the ionic states of biguanide derivatives helps in understanding the pharmacokinetic and pharmacodynamic properties of the medicinally important biguanides. It was shown that these biguanide derivatives in their protonated states adopt a C2 symmetric arrangment in the central core [122]. The acid dissociation constants (pKa) of many biguanides have been measured, and are ranged from 10.3 to 13.4. The high oral absorption of metformin hydrochloride is reported due to its hydrophilic nature (e.g., calculated logD of 6.13 at pH 7) and net positive charge at intestinal pH values (pKa 13.8) [123]. The ionic states of drugs/leads are varied

FIG. 18 Proposed pathway for enantioselective a-hydroxylation of 3-substituted-2-oxindoles.

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Atomic clusters with unusual structure, bonding and reactivity

SCHEME 22 Pentanidium-catalyzed alkylation of dihydrocoumarins.

FIG. 19 Silylamide as probase for the phase transfer catalysis.

upon pH-dependent enzymatic profiles (site of action of drugs/leads) which play a major role in the electrostatic interactions between the ligand and the receptor. Thus, the knowledge of ionic states of drugs/leads helps in understanding the details of drug-receptor interactions which governs the pharmacokinetic and pharmacological properties of drugs/leads. For example, crystallographic information for anti-malarial lead compound WR99210 (prodrug of phenoxypropoxy biguanide derivative) describes its protonated state which interacts with the anionic side chain of Asp54 in the PfDHFR active site [124]. Similar kind of interaction in the protonated state of other antimalarial drug, like pyrimethamine, was confirmed through NMR studies of both the bound and free states [125]. Metformin hydrochloride, which is a blockbuster antidiabetic drug, was inappropriately represented in the literature. In 2005, the structure of biguanides including metformin was reinvestigated and reported the most appropriate structural representation (a tautomer) [108]. In 2009, these motifs were found to possess nitreone character [97]. There are other classes of biologically active molecules and drugs which contain (NL2)+ architecture. Antimalarial drugs such as proguanil hydrochloride, cycloguanil hydrochloride, and WR99210 also possess this character [107,110]. Besides these drugs, there are other class of compounds like antiseptic drug [chlorhexidine2HCl, antiparasitic drug (pyrimethamineHCl; earlier used as antimalarial), proton pump (H2 receptor) inhibitos (famotidineHCl and ebrotidineHCl), and antiinfluenza drug (moroxydineHCl)] that have been found to possess nitreone character (Fig. 20) [107]. Electronic structure analysis of biguanide derivatives confirms that protonation in neutral biguanide occurs at terminal nitrogen rather than at the central nitrogen [98]. Protonation in neutral biguanide derivatives leads to (i) the breakdown of electron conjugation, (ii) breaking of the intramolecular hydrogen bond, (iii) loss of possibility of facile 1,5-H shift, and (iv) reduction in molecular rigidity in acyclic biguanide derivatives. Inspite of destabilizing factors associated with protonation at terminal nitrogen, why this site is preferable? This is explained in terms of the fact that the central core of all medicinally important biguanides in their protonated states adopts L ! N+ L character (which provides additional stability to the molecule), isoelectronic to the central core of carbones. Similar to biguanides, guanyl thiourea, biuret, thiobiuret, dithiobiuret, and their derivatives, which are important pharmacologically active bioisosters of biguanides, also belong to nitreone class after getting protonated [126–128].

4.

Debates on the bond representation

The donor ! acceptor complexes of main group elements with L ! E interaction were reported in the literature in the recent past. The very first report by the Frenking et al., which discusses the C(0) character of carbon, has led the scientific community to rethink the chemical nature of carbon. Both theoretical and experimental studies have proven that carbon has the propensity to accept the electrons from the flanked ligands and can also participate in the reaction where it can act as an electron donating center toward transition metals. Further studies extended the concept for the existence of donor ! acceptor complexes in main group elements of group 13 and 14. Based on their chemical nature and bonding the dative bonding models for the complexes of main group elements have become popular in the scientific literature. The notion (L ! E L) for representing these donor ! acceptor complexes of main group elements with “arrows” has been

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FIG. 20 Marketed drugs in their salt form possessing nitreone framework in their core.

FIG. 21 Possible resonance representation of carbones [132].

used in the structural formulae. This representation has brought a revolutionary change in the concept of ligand and central atom which was introduced by Alfred Werner about 100 years ago [129]. It seems that the bonding in chemical compounds is still a matter of debate among scientific community. Krossing et al. have different perspective on the use of arrows for the representation of donor ! acceptor complexes of main group elements [130,131]. As per their viewpoint, the charge-separated state L+C2  L+ (for carbones) (IV) in place of (L ! C L) sounds more logistic (Fig. 21). They urged chemists to use “fewer arrows.” In response to the above raised viewpoint of Krossing et al., Frenking encouraged the use of “more arrows” (II) (Fig. 21) as this notion (L ! E L) is in agreement with the results of experimental and quantum chemical analysis [133]. He further added that dative bonding model for main group complexes is proven to be very useful for classifying known compounds as well as for predicting the unusual bonds and reactivities of newly explored compounds in this category. Taking several examples, Frenking demonstrated the advantage of the description with dative bonds. The strongly bent nature of C(PPh3)2 is easily rationalized when the molecule is considered to be a complex of a pseudo-quadruply coordinated carbon atom in the electronic 1D state which exhibits two lone pairs of electrons and two PPh3 ligands. Further, this out-of-the-box consideration provided opportunities and challenges in chemical exploration; for example, our group has been considering the opportunity of designing new chemical entities with L ! N+ L formula in drug discovery. Very recently, bonding situation in carbones was re-explained based on valence bond theory [132]. The nature of bonding in carbones was compared with gold-CO dative bond in Au(CO)+2 through valence bond self-consistent field (VBSCF). The Gallup and Norbeck (GN) weight for possible valence bond structures (Fig. 21) showed high weight for covalent structure (I), fractional weight for dative bond structure (II) and some weightage was given for remaining structures. However, this study indicated the inconsistency in the bonding situation when the varieties of ligands (L) are

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FIG. 22 The symmetry unique valence bond SCF orbitals of carbone I. (Reprinted from reference R.W. Havenith, A.V. Cunha, J.E. Klein, F. Perolari, X. Feng, The electronic structure of carbones revealed: insights from valence bond theory, Phys. Chem. Chem. Phys. 23 (5) (2021) 3327–3334.)

considered. The significant charge shift character was not observed for carbone (Fig. 22). The study claimed that bonding in carbones belongs to partially covalent and partially dative bond [132]. It is also true that one can ignore all the new insights and prospects which were obtained from donor ! acceptor bonding model for main group elements and continue to represent these new chemical entities in terms of charge separated species. The numerous experimental evidences available for compounds with unusual bonds prefer the representation with arrows. The progress of all this exciting chemistry has led the transfer of the dative bonding model from transition metal chemistry to chemistry of main group elements.

5.

Summary

Complexes of main group elements always attract the attention of a chemist due to their chemical properties, which are quite new to the world of chemistry. In this article various one and two-center complexes of main group elements were studied with the prime focus for the complexes of nitrogen. The one center complexes are represented with (L ! E L) structural framework. The bonding situation in these compounds suggests the localization of electron density on the central main group element (E and En moieties), which are present in HOMO of these compounds. These HOMOs of high negative energies result in high basicity for these compounds which in turn undergo protonation readily. This is also supported by their high PA values. The synthetic routes which are followed for synthesizing these challenging compounds are also critically reviewed. As nitrogen is the vital element and its presence is found virtually in all the pharmacological drug molecules, exploring the nitrogen cation complexes with medicinal chemistry perspective is highly significant. The mild basicity of nitrogen cation complexes can be further utilized for carrying out various proton exchanges in the biological environment. Moreover, dative bond representation (L ! E) in compounds with main group elements (E), triggered extensive debate in the recent past which deals mainly with the use of “arrows” for representing this set of compounds. The scope and limits of this nonclassical coordination bond warrants comprehensive exploration and thus critical analysis is needed.

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Unusual bonding between second row main group elements Chapter

4

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Chapter 5

Conceptual density functional theory and all metal aromaticity Debolina Paul and Utpal Sarkar Department of Physics, Assam University, Silchar, India

1. Introduction Density functional theory (DFT) [1,2] and DFT-based tools are widely accepted in solving challenging physical, chemical, and biological problems. DFT is a useful method for predicting the electronic structures of atoms, molecules, solids, and other systems successfully. The results are in substantial agreement with the experimental findings [3–6]. The fundamental foundation of DFT was created on the basis of the two theorems given by Hohenberg and Kohn (HK) [1,2], being known as HK theorems.
 ! The first HK theorem of DFT
states that “up to a trivial additive constant, the external potential v r is uniquely deter !

mined by the electron density r r for the nondegenerate ground state of an N-electron system and an inverse mapping
 ! exists between the two.” r r , which is a single-particle density, provides all information about the electronic structure of

the concerned system and hence constitutes
the  backbone of DFT. The second theorem
 is related to the energy variational !

!

principle and states that “a trial density r0 r , which is not the exact density r r , gives an energy (E) greater than the

exact energy, E0, i.e., E[r0 ] E0[r].”

 ! b is determined by N and v ! For a given system of N electrons in an external potential, v r , the Hamiltonian H, r . Thereafter, the physical information regarding the system can be obtained from the many-electron wavefunction, c(r1, r2, …rN). One may get this wavefunction by solving the Schr€odinger like Kohn-Sham equation after having the knowledge b of H.
 ! Mathematically, the electron density r r may be found by integrating over the coordinates of (N  1) electrons, as ð ð
 ! (1) r r ¼ N … c∗ ðr, r 2 …, r N Þcðr, r 2 …, r N Þdr 2 …dr N Finally, the total number of electrons is given by,

ð
 ! r r dr ¼ N

(2)

 ! ! Hence, with the help of N and v r , r r is determined.
 ! HK have proved from their first theorem, that the external potential v r can be uniquely calculated using the electron
 !

density r r , up to a trivial additive constant. This implies that energy and subsequently, other properties together with the
 ! chemical reactivity parameters of a given system is a unique functional of r r . According to the HK variational principle,  ð
  ! ! ¼0 d E½r  m r r d r  N

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00008-5 Copyright © 2023 Elsevier Inc. All rights reserved.

(3)

87

88

Atomic clusters with unusual structure, bonding and reactivity

The above equation is further simplified to, dE½r ¼m dr

(4)

where m is known as the Lagrangian multiplier [7] or electronic chemical potential h
i [7] of a system. ! The functional form of E[r], along with the HK universal functional (F r r Þ, can be given by h
i h
i ð h
i
 ! ! ! ! ! E r r ¼F r r + r r v r dr

(5)

h
i ! where, F r r contains kinetic energy, Coulomb potential and exchange correlational functional. Reorientation of electron density occurs among the constituent atoms which take part in the chemical reactions. This has urged the need to describe the global as well as local reactivity parameters [7–12], and thereafter the requirement of studying the conceptual density functional theory (CDFT) has arisen [13–17]. CDFT, which is a part of DFT, has initiated its roots in the original work by Parr et al. [10]. This leads to the introduction of the Lagrangian multiplier m [7], ultimately finding the expression for the chemical potential (m) or electronegativity (w) [9]. This became useful in bridging the gap between DFT and classical chemical concepts. In 1983, Parr and Pearson identified chemical hardness with the second derivative of the energy with respect to N [7].    2  ∂ E ∂m
¼ ¼ (6) 2 ∂N vð!r Þ ∂N vð!r Þ where the symbols have their usual meaning. With the help of a finite-difference approximation, the Mulliken electronegativity [18] and chemical hardness [7–9] are obtained as: I+A 2

(7)


¼IA

(8)

w¼

where I and A stands for ionization potential and electron affinity, respectively. The ionization potential (I) and the electron affinity (A) can be calculated using the following relations: I ¼ Eð N  1 Þ  Eð N Þ

(9)

A ¼ Eð N Þ  Eð N + 1 Þ

(10)

where E(N  1), E(N), and E(N + 1) are the energies of the (N  1), N, and (N + 1) electron systems, respectively. Reactivity parameters such as hardness and electrophilicity [7–11] help us to understand the thermodynamic stability of the molecular system and are validated with the help of the electronic structure principles such as Pearson’s hard and soft acids and bases (HSAB) principle [12,19–21], Sanderson’s electronegativity equalization principle [22–24], maximum hardness principle (MHP) [25,26], minimum polarizability principle (MPP) [27,28] and minimum electrophilicity principle (MEP) [29–31]. The global chemical reactivity parameters such as chemical hardness (
) [7–9] given in Eq. (8), chemical potential (m) [7] and electrophilicity index (o) [11,32–34] and local reactivity parameters such as Fukui function [35–41] are determined with the help of the following relations: m¼

ð I + AÞ ¼ w 2

o¼ 2
!3
 ∂r r ! 5 f r ¼4 ∂N

(11)

m2 2


(12) 2

vð r Þ !

3 dm ¼ 4
!5 dv r

N

(13)

Conceptual density functional theory Chapter

where,

ð
 ! ! f r dr ¼ 1

5

89

(14)

Fukui functions can be of three types, which describe the possible reactive sites for the nucleophilic, electrophilic and radical attack. These are termed as condensed Fukui function and can be expressed as [42–44]:
  ∂r +


 ! ! ! ! ffi rN+1 r  rN r  rLUMO r f+ r ¼ (15) ∂N vð!r Þ For nucleophilic attack,


  ∂r 


 ! ! ! ffi rN r  rN1 r  rHOMO r f r ¼ ! ∂N vð r Þ

(16)


  ∂r 0

 1


 1
! ! ! ! rN+1 r  rN1 r rHOMO r + rLUMO r ffi f r ¼  ∂N vð!r Þ 2 2

(17)

 !

For electrophilic attack 0 !

For radical attack
 ! Another parameter called local philicity (oa r ) [45,46] helps us to determine the site selectivity.

 ! ! oa r ¼ of a r

(18)

where a ¼ + ,  , 0 represents electrophilic, nucleophilic and radical attacks. Apart from the site selectivity, the local philicity index also able to explain the other chemical reactivity parameters. Delocalization index d(A, B) [47,48] derived from the atoms in molecules theory [49–51] has been used to measure the number of electrons delocalized between two atoms (in para atoms of an aromatic ring) as described by Sola` and coworkers. Their investigation suggests that para-delocalization index (PDI) is a potential index of aromaticity [52–54]. In general, the delocalization index between two atoms A and B is defined as: ð ð dðA, BÞ ¼ 2 GXC ðr, r 0 Þdr dr 0 (19) A B

where GXC represents the exchange-correlation density and the double integration is performed over the atomic basins of atoms A and B. Sola` and his team have come to an affirmative conclusion that d(1, 4) may be considered as an indicator of aromaticity. They did an extensive survey on polycyclic aromatic rings representing a broad range of aromatic character, from which they came to the above-stated conclusion on aromaticity.

2. History and descriptors of aromaticity Aromaticity is a fundamental concept deeply rooted in chemical literature and plays a vital role in determining the structure, stability, and reactivity of many molecules. It has been known that even before a clear establishment of structural and bonding principles of any system, aromaticity made its debut and was widely accepted. Kekule was the first person who introduced the concept of aromaticity to understand the stability of benzene [55] and later, the quantum mechanical treatment of this concept has been explained by Pauling [56,57]. Aromaticity is one of the useful but at the same time is not a well-defined concept, which follows contemporary concepts’ of chemistry. Apart from their “aroma,” it is established fully that “aromatic” molecules are often found to be stable and possess regular geometries. In addition, they are hardly reactive, despite the presence of several unsaturated bonds in them. These unsaturated bonds are delocalized throughout the molecule but are confined within it. In the year 1931, H€ uckel formulated the (4n + 2) rule [58–60] for studying the ground singlet states of ring-like molecules with delocalized II-type molecular orbitals (MOs; n is the number of delocalized II-type molecular valence electrons). This formula formed the link between aromaticity and molecular electronic structure. Later Baird completed this by finding that modification of H€ uckel’s electron counting rule should be done for spin states of multiplicity which are higher than singlet [61].

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Thereafter, Baird’s rule has also been revised [62–65]. To explain the strange magnetic behavior of cyclopropane, the concept of s-aromaticity has been introduced by Dewar [66,67]. He did it by extending H€uckel’s aromaticity rule to the skeletal s-type electrons. The two types of aromaticities, namely, II-type and s-type are found to occur in many systems simultaneously [68]. In some cases, they cooperate to enhance the aromaticity, while in some other cases, they lower the aromaticity by acting negatively. Breslow finally came forward with the concept of anti-aromaticity in chemistry [69]. Antiaromatic systems are found to be highly unstable as well as highly reactive, unlike aromatic ones. These compounds follow the H€ uckel 4n rule (s or II rule). Initially, there was no way for direct experimental measurement of aromaticity in a system. For the prototypical example, benzene has also been experimentally verified indirectly using deshielded nuclear magnetic resonance (NMR) chemical shift and heat of hydrogenation. There have been several arguments on the origin and definition of aromaticity, yet it went beyond the study of benzenoid hydrocarbons and included other hetero systems, like carbon-free or carbon-related systems. The aromaticity concept was extended in the study of all-metal clusters [70–73]. For instance, photoelectron spectra of CuAl4, LiAl4, and NaAl4 clusters were recorded by Lai-Sheng Wang and coworkers [74]. From the molecular electronic structure point of view, H€uckel’s rules for aromaticity/antiaromaticity give only simple probes of aromatic or antiaromatic nature of systems but provide no information regarding its extent. Hence, to get a quantitative approach, it is required to analyze other probes [75] like the geometric criterion, energetic criterion, reactivity parameters, electronic criterion, and magnetic criterion of any system. Based on the energetic, geometric, electronic, and magnetic criterion, metal clusters have been identified as aromatic, antiaromatic, and nonaromatic over the past few decades [52,76–84]. The reactivity parameters that are used to describe the concept of aromaticity in different types of compounds are based on CDFT [85–87]. Energetic and electronic criteria which are used to describe the aromaticity indices in several systems include energy gap (difference between highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gap), aromatic stabilization energy (ASE) [88], enhanced resonance energy [76], electron localization function [77], etc. Another essential model used for identifying geometry-based indices of aromaticity is the harmonic oscillator model [78]. Based on magnetic criterion, the aromaticity of a molecule has been measured by investigating NMR chemical shift [79], magnetic field induced current density [80] and nucleus-independent chemical shift (NICS) values [81]. PDI [52] and the fluctuation index [82] are put forward based on electron delocalization indices between pairs of atoms. Multicenter bond index (MCI) [84], calculated from generalized population analysis is another crucial parameter for characterizing aromaticity [89–92]. However, in realizing the aromaticity of any system, NICS [81] study stands out to be the most effective and widely used technique till date among the above discussed criteria. Although it has some limitations of its own. The NICS value for a system may be defined as the negative value of the magnetic shielding tensor for the selected point in a molecule. NICS value can be determined usually with the help of two methods such as individual gauge localized orbitals [93,94] and gauge-independent atomic orbitals [95]. In recent times, to evaluate the aromaticity of some cyclic organic compounds, a NICS-based aromaticity index known as NICS-rate [96] is also proposed. A critical value of NICS-rate of 0.5 acts as a boundary between aromatic and nonaromatic systems. CDFT-based reactivity descriptors such as absolute hardness [97,98] and relative hardness [97,99], relative aromaticity indices (△ X) [85–87], relative electrophilicity [100], Shannon entropy [101], etc. are popularly used approaches for interpreting aromaticity. The Shannon entropy or aromaticity of a system may be defined as the probability of electronic charge distribution found between atoms of molecular rings present in the given system. Noorizadeh and Shakerzadeh have suggested limits for defining aromaticity and antiaromaticity of any system as 0.003 < Shannon aromaticity < 0.005. They did it by matching the corresponding index with various other aromaticity indices such as ASE, NICS, etc. Interestingly, Chattaraj and coworkers, reported that, among the CDFT approaches, relative aromaticity indices (△ X) is in good agreement with the other aromaticity indices such as NICS and MCI techniques for estimating aromaticity [85–87]. Apart from this, △ X also gives valuable information about chemical reactivity, stability, and electronic properties of the molecules and metal clusters. The following section presents some important discussion on CDFT-based aromaticity indices.

2.1 Relative aromaticity indices (X) The relative aromaticity indices (D X; X ¼ E, o,
, and a) can be expressed as follows: DX ¼ XCyclic  XOpen=Localized

(23)

Conceptual density functional theory Chapter

5

91

where XCyclic is the energies, electrophilicity, hardness, and polarizability of the cyclic delocalized isomer of the system and XOpen/Localized represents the same for the open/localized isomer. Generally, an aromatic system should have negative DE, D o, D
, and D a values. One may note here that, molecules which are aromatic in nature generally have lower energy, electrophilicity, polarizability and higher hardness values as validated by several electronic structure principles. The antiaromatic molecules on the other hand possess higher values of energy, electrophilicity, polarizability, and lower values of hardness. Zhou and Parr proposed another CDFT-based reactive descriptors namely absolute and relative hardness [97], other than NICS, MCI, and D X indices to judge the aromaticity of cyclic compounds [85–87,97–101]. In their work, they have considered 96 cyclic molecules and based on the absolute and relative hardness values, inferred about the aromatic, antiaromatic and nonaromatic nature of the molecules. In a follow-up work, aromaticity of 14 benzenoid hydrocarbons has been investigated by Zhou and Navangul [98]. Here, they have discussed elaborately aromaticity and hardness concepts in a parallel manner for benzenoid molecules and they have concluded that both these concepts, gives similar trend about aromaticity and it is quite obvious as both the concept originated from the electronic structure principles.

3. Aromaticity in the context of metallic systems Here in this section, we have provided a detailed description about the aromaticity of various earlier studied metal clusters. First, we present the aromaticity of alkali and alkaline earth metals, after that, we have given a presentation on the aromaticity study of transition metal clusters. We have found from the literature that, sometimes purely alkali metal and alkaline metal clusters are involved, whereas in some cases, their mixtures are also studied. In some other instances, alkali and alkaline earth metals are clustered with other organic or inorganic fragments. Next, the transition metal clusters and their blended mixture with alkali or alkaline earth metal clusters are reported. In the year 1979, Bursten and Fenske introduced the term “metalloaromaticity” in order to describe metal complexes of cyclobutadiene containing a metal atom coordinated to C4H4, which is the simplest H€uckel antiaromatic (4n-II-electrons) molecule [102]. Interestingly, a coordinated C4H4 often acts as if it were aromatic rather than antiaromatic. The first organometallic compound was synthesized by Robinson et al., [103] which consists of an aromatic cycle of only metal atoms. This particular aromatic compound, Na2[(Mes2C6H3)Ga]3 (Mes ¼ 2,4,6-Me3C6H2), contains a triangular aromatic Ga2 3 ring embedded in a large organometallic molecule. Twamley and Power [104] prepared the first II-aromatic organometallic compound. It is composed of four gallium atoms, in which an almost perfect square-planar gallium cluster Ga4 is embedded in an organometallic environment in the K2[Ga4(C6H3-2,6-Trip2)2] (Trip ¼ C6H2-2,4,6-Pr3) compound. Among few solid compounds containing aromatic metalloid and metal clusters, experimental characterization has been 2+ 2 2 reported on aromatic square clusters, such as Se2+ 4 , Te4 , Sb4 , and Bi4 [105–110], which are valence isoelectronic to 2 6 the prototypical aromatic hydrocarbon C4H4 , as well as planar pentagonal aromatic clusters, such as As 5 , Sn5 , and 6  Pb5 , which are valence isoelectronic to the prototypical aromatic hydrocarbons C5H5 [111–113].

3.1 Alkali and alkaline earth metal In the year 2001, the concept of aromaticity was extended to all-metal aromatic systems [74]. It was shown that the prototypical all-metal aromatic cluster Al2 4 is an example of a doubly aromatic system. Stability, reactivity and aromaticity of two aluminum clusters, namely, Al7C and Al7O have been done. Further, the stability of these aluminum clusters is clearly understood in the context of both addition and removal of one electron/Al atom and has been discussed based on the chemical reactivity parameters. NICS, which is chosen to be an indicator of aromaticity, is also studied to know the aromatic behavior of these clusters. Based on MHP and MEP, the aluminum clusters are found to be stable and their NICS value suggests that they possess strong aromatic nature [114]. Another aluminum cluster, namely Al 4 and its different isomeric forms have been discussed. It is observed that this cluster is energetically more stable in its linear form as compared to its cyclic isomer. A comparison (in terms of energy, polarizability, and hardness) between a linear system and its cyclic counterpart showed that negative changes are found for energy and polarizability, while positive changes are found for hardness in the case of an aromatic molecule and thus validating the minimum energy, minimum polarizability and MHPs [87]. Aromaticity of Al 3 anion has been examined through atomic probes with the help of information-theoretic approach. The probe atoms are found to switch in between aromaticity and antiaromaticity. Their distinct spin-states are responsible for their largely different aromatic characteristics [115]. 4 2 Chattaraj et al. analyzed the aromaticity of aluminum clusters, Al2 4 and Al4 . The aromatic behavior of Al4 is clearly understood from the three principles, namely, minimum energy, minimum polarizability and maximum hardness. But for the other cluster, i.e., Al4 4 , transparency is not observed. However, NICS and magnetic field-induced current values show

92

Atomic clusters with unusual structure, bonding and reactivity

that the II-antiaromaticity of this cluster is dominated by s-aromaticity [86]. The same group also studies the aromaticity of some other aluminum clusters [116]. Different reactivity descriptors and induced magnetic field values have been used to understand the stability and aro2 2 maticity of Be2 3 , Mg3 , and Ca3 systems. Their mono and di sodium complexes are also analyzed. All these clusters along with their sodium complexes are found to be aromatic as suggested by the induced magnetic field values [117]. Density functional study on one-dimensional hexagonal sodium cluster and two-dimensional potassium cluster show that the Na6 and K6 rings present in the chain are highly aromatic in nature, as confirmed from their NICS values [118]. The counter-ion effect on the stability, aromaticity and bonding of trigonal dianionic metal cluster, Na2Mg3, has been studied. Optimized geometry of this cluster along with its MO is presented in Fig. 1. Electron delocalization above and below the trigonal Mg3 plane reveals the II-bonding nature of the HOMO. In contrast, the LUMO plot indicates the sdelocalization over the entire trigonal Mg3 plane. The separation between the Na atoms and the ring increased to get a comparative idea. One may find that at closer Na-Mg3 distances, the Mg3 ring records highly negative NICS(0) values (singlet geometries) but further rises steadily, causing a substantial loss in aromaticity criterion [119].  Structures of two mixed metal clusters of aluminum, MAl 4 and MAl6 (M ¼ Li, Na, K, and Cu) have been explained [74,122–124]. The presence of multifold aromaticity in these metal clusters predicts their structures as well as their extra stability. Appearance of cyclic isomers in some molecules, viz., XAl 3 (X ¼ Si, Ge, Sn, and Pb) [125], MAl3 (M ¼ P and As) [126,127], and MGa3 (M ¼ P and As) [128] is another example of the usefulness of aromaticity. An interpretation of chemical bonding in metal clusters based on the concept of s-aromaticity is made. The importance of aromaticity in metal clusters is again proved when cyclic s-aromatic structures of tetratomic 6s-electron Li2Mg2 system are found to be more stable than its linear counterpart (i.e., Li-Mg-Mg-Li) [129]. Evidence of aromaticity is presented for purely metallic systems, MAl 4 (M ¼ Li, Na, and Cu) by Li and co-workers [74]. Theoretical and experimental characterization of antiaromaticity in the all-metal system (Li3Al 4 ) has been reported by Boldyrev and Wang [130]. Next, clusters of alkali metal, P 5 , P5M (M ¼ Li, Na, and K) are theoretically investigated and aromaticity of planar P 5 anion is explored. MO analysis and NICS provided insight into the aromaticity of this anion. Results showed that inorganic P 5 anion shows aromaticity with six delocalized II-electrons involving structural as well as magnetic criteria [131]. Again another group studied a series of alkali metal MAs 4 (M ¼ Li, Na, K, Rb, and Cs) clusters and alkaline earth metal MAs4 (M ¼ Be, Mg, Ca, Sr, and Ba) clusters. In two different kinds of MAs 4 and MAs4 species, the aromaticity of square As2 4 dianion is explored. MO analysis, the natural bond orbital analysis and the NICS are explored into the aromaticity of square As2 4 dianion. All the necessary analysis showed that maintaining structural and electronic criteria, inorganic square planar As2 4 dianion exhibited characteristics of aromaticity with six delocalized II-electrons [132]. Aromaticity of polyhexagonal Li-, Na-, and K-clusters has been quantified with the help of NICS study. Aromaticity of hexagonal Li- and Na-clusters has a similar NICS value as the aromaticity of linear polyacenes. But the K6 rings are considerably found to be less aromatic than the corresponding benzenoid systems [117,133,134]. Hybridization can impact the shapes of small all-alkali metal clusters and thus oppose s-aromaticity for defining cluster shapes. One example for this study is made by considering valence-isoelectronic anionic Li-doped alkali metal clusters, namely, LiNa4 and LiK4. Then a direct comparison is done with the negatively charged clusters of Li, and Li5, which are studied extensively, both experimentally and theoretically. It is indeed found that hybridization of atomic orbitals exists in these clusters and opposes s-aromaticity in maintaining cluster structure. In addition, aromaticity overpowers covalency upon substituting Na to K, thus changing the cluster shape accordingly [135,136].

FIG. 1 (A) Optimized geometry, (B) highest occupied molecular orbital (HOMO), and (C) lowest unoccupied molecular orbital (LUMO) of the Na2Mg3 system in its singlet state (produced using B3LYP/aug-cc-pVDZ level of theory as given in Ref. [119], with Gaussian 09 version D [120] and Gauss View 05 [121] softwares).

Conceptual density functional theory Chapter

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3.2 Transition metal All-transition-metal aromatic system Hg6 4 as a part of the Na3Hg2 amalgam was reported by Kuznetsov et al. for the first time in the year 2001 [137]. From then onwards, the study of all-transition-metal aromatic/antiaromatic systems has grown enor+ + 2  mously. Some of the examples are Cu+3 ; X 3 (X ¼ Sc, Y, La); X3 (X ¼ Zn, Cd, Hg); Hf3; Ta3 ; Au5Zn ; Cu5Sc, Cu6Sc ,  Cu7 and Cu7Sc; M4Li2 (M ¼ Cu, Ag, Au); M4L2 and M4L (M ¼ Cu, Ag, Au; L ¼ Li, Na); Al2(CO)2; cyclo-CunHn (n ¼ 3–6); cyclo-MnHn (M ¼ Ag, Au; n ¼ 3–6); cyclo-Au3LnH3 n (L ¼ CH3,NH2, OH, and Cl; n ¼ 1–3); and many more [72]. Chain-like clusters of Zn atoms have been studied considering Zn2 3 as the base structure. Thereafter, alkali and alkaline earth metal counter cations are taken to generate mixed metal clusters with the zinc moiety. Optimized geometries of Zn4 cluster and its alkali and alkaline earth metal doped structures are provided in Fig. 2. Some possible reactions are also   studied by substituting Zn2 3 by C5H5 and Be5 rings. The stability, aromatic characteristics, as well as reactivity trends of these clusters have been determined using the global and local chemical reactivity parameters along with NICS study. In  terms of stability, it is found that the counter cations stabilize the Zn2 3 anion. However, the substitutional analogues (C5H5  and Be5 rings) do not produce more stability than all zinc clusters. Also to be noted here is that the HOMOs of the studied clusters show both II and s symmetries [138]. Wannere et al. have initially studied the aromaticity in M2 4 (M ¼ Cu, Ag, Au) dianions considering M4Li2 (M ¼ Cu, Ag, Au) as neutral species. They have reported NICS values in centers of Cu4Li2, Ag4Li2, and Au4Li2 clusters, which showed that M4 dianions are aromatic in nature. It is noted that the d-orbital aromaticity of Cu4Li2 is observed because of the presence of high atomization energy [139]. On the other hand, using the gauge-including magnetically induced current (GIMIC) method, Lin et al. [140] found that HOMO of Cu 4s atomic orbitals mainly sustain strong ring currents. Hence from GIMIC calculation, one may say that due to 4s atomic orbital, Cu4 ring is s-aromatic, where d-orbitals are not significantly involved for electrondelocalization effects. But their study did not support the work by Wannere et al. [139], who found that first example of aromatic molecules involving d-orbital is the square-planar Cu4. Cu4 and Ag4 rings can be realized as systems with six valence s-electrons and regarded as s-aromatic systems (according to 4n + 2 rule), provided that bonding in these rings is mainly due to s-orbitals. Therefore these systems may be compared to main-group clusters having six bonding s-electrons (Li4, Mg2+ 4 , and Li2Mg2) as found from the study of Alexandrova and Boldyrev [129]. The presence of six delocalized electrons and appropriate nodal pattern in Au5Zn+ (bearing some resemblance with the prototype C6H6 and C5H5 aromatic organic species in terms of MO pattern) satisfy the 4n + 2 rule for s-aromaticity [141]. NICS calculations for these three structures have been done by Tanaka et al. [142], who concluded that negative NICS values of Au5Zn+ are greater than C6H6 and C5H5 molecules, which confirms the presence of aromaticity in Au5Zn+. FIG. 2 Optimized geometries of (A) Zn4, (B) [Zn3Li], (C) [Zn3Na], (D) [Zn3K], (E) Zn3Be, (F) Zn3Mg, and (G) Zn3Ca clusters (produced using B3LYP/6-311 + G(d) level of theory as given in Ref. [138], with Gaussian 09 version D [120] and Gauss View 05 [121] softwares).

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Atomic clusters with unusual structure, bonding and reactivity

All in all, Au5Zn+ cluster can be considered as s-aromatic bimetallic cluster bearing six delocalized s-electrons. In addition, one may also note that the enhanced stability of Au5Zn+ can be ascribed to aromaticity. Theoretical studies on cyclic organocopper(I) compounds, CunHn (n ¼ 3–6) cyclic species, such as four-membered square-planar ring, Cu4R4 (R ¼ CH2SiMe3) with short CudCu distances of 2.42 A has been performed by Tsipis and Tsipis [143]. According to them, equivalence of CudCu and CudH bonds in these systems indicates the aromatic nature of cyclic hydrocopper(I) compounds. In addition, they reported NICS values which again support the aromatic character of these species. Chandrasekhar et al. [144] introduced the concept of double aromaticity (presence of s- and II-aromaticity simultaneously) for explaining the properties of 3,5-dihydrophenyl cation. Again to understand the chemical bonding in small carbon rings, aromaticity and antiaromaticity was first given by Martin-Santamaria and Rzepa [145]. Small carborane molecules having 3- and 4-membered rings also show s- and II-aromaticity both [145,146]. And to be noted that, Hg6 4 cluster was considered first transition-metal system in which double (s-II) aromaticity was discovered due to p-atomic orbitals [145]. Transition-metal systems can diversely provide combination of aromaticity-antiaromaticity due to the presence of a more complicated nodal structure of d-atomic orbitals, which is used to form d-bond, besides s- and p-bonds.

4.

Conclusion

This review deals with the study of aromaticity in different systems involving various metals. Several approaches by which aromaticity can be understood in systems have been studied. We started our discussion by covering a small introduction on DFT, followed by its part on conceptuality, i.e., conceptual DFT and the associated reactivity parameters. Next, we have put a section on the history and origin of aromaticity. Here, the different ways to determine any system’s aromaticity are also discussed, which involves the four criteria, namely, energetic, structural, magnetic, and reactivity-based measures. This section also contains discussion on different types of aromaticity or antiaromaticity of the concerned systems. Subsequently, the discussion is moved to present the aromaticity found in the metallic systems, which is one of the main parts of this particular chapter. In this section, aromaticity found in the alkali metals, alkaline earth metals, and transition metal complexes has been considered from a series of earlier publications. Hence, a short summary of the conceptual DFT and its approach to understanding the aromaticity in metallic systems have been presented here.

Acknowledgments DP thanks Council of Scientific & Industrial Research (CSIR), India for her CSIR SRF. US would like to thank Prof. Pratim K. Chattaraj, Prof. Gabriel Merino, and Dr. Sudip Pan for kindly inviting him to write this book chapter. US would also like to thank DST, New Delhi, India for the SERB project (File No. EMR/2016/006764) for financial assistance.

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Chapter 6

Structural evolution, stability, and spectra of small silver and gold clusters: A view from the electron shell model Pham Vu Nhata, Nguyen Thanh Sia, and Minh Tho Nguyenb a

Department of Chemistry, Can Tho University, Can Tho, Vietnam, b Institute for Computational Science and Technology (ICST), Quang Trung Software

City, Ho Chi Minh City, Vietnam

1. Introduction Over the past decades, numerous studies, experimental and theoretical alike, have been devoted to the noble metal clusters owing to their special roles in photography, catalysis of chemical reactions, electronic materials, and medical treatments [1–5]. With a broad spectrum of antibacterial and antifungal activities, these systems are well suited for clinical and therapeutic applications [6,7]. More importantly, they are able to conjugate to a variety of bimolecular systems and much less toxic to human bodies in comparison to many other metallic compounds [8]. In addition, noble metal derivatives exhibit superior optical properties as compared with other transition metals and have therefore attracted a great deal of interest in the field of sensors, biosensors, and biomedical diagnostics [9–11]. It can be argued that both silver and gold clusters are among the most characterized atomic aggregates to date by both experimental techniques [12–15] and quantum mechanical calculations [1,16–20]. Identification of the most stable structures of elemental clusters has long been, and still is, a challenging but vital task in the field of cluster science, as it would provide us with deeper insights into many functional properties of this class of compounds [2]. Several research groups conducted extensive investigations to elucidate geometrical structures of silver/ gold clusters [21–23]. The atomic arrangements of the small pure Mn (M ¼ Ag, Au) clusters in the range of n ¼ 2–20 and their growth pattern are now well characterized [24–29]. As a result of relativistic effects [30–32], the preference for planar shape of neutral gold clusters likely continues up to 11 atoms [31]. Larger systems typically exist in hollow cages, and a structural transition from an oblate form to a pyramid-like shape is observed at Au17 [26]. Au20 has also been the focus of numerous studies, and a consensus, which was long reached, is that it prefers a Td symmetry macro-tetrahedron characterized by a single peak infrared spectrum [25,26]. Although coinage metals share several analogies in their bulk states, their small clusters exhibit several differences in the gas phase, again as a result of relativistic effects [30,32]. For example, while gold clusters favor planar forms up to surprisingly large sizes, Cu clusters tend to adopt three-dimensional (3D) structures already at much smaller sizes [33]. Silver clusters are expected to behave in between, i.e., their 3D structures come out a little earlier than gold clusters, but later than copper counterparts. Similar to alkali clusters, electron shell effects in silver and gold clusters are also operative since their nd states are completely occupied [34,35]. It has well been established that both Ag+n and Au+n cationic clusters exhibiting a high thermodynamic stability are found at n ¼ 3, 9, 21, 35, 41, 59, …, whereas the negatively charged counterparts are exceptionally abundant at n ¼ 7, 19, 33, 39, 57, …[36]. A widely accepted explanation for such a phenomenon is based on a strong delocalization of the external s electrons [37,38]. Accordingly, they can be treated as particles moving around a spherical pseudo-potential composed of the inner electrons along with the nuclei. This highly delocalized behavior of valence electrons brings about the main characteristics of simple metal clusters, including formation of electron shells and occurrence of shell closing effect that are somewhat similar to those in free atoms. Consequently, a cluster in which the number of valence electrons matches the shell closure, namely 1S2/1P6/1D10/2S2/1F14/2P6/…, is produced more abundantly as compared with the immediately preceding or following ones and is called a magic cluster. This approach, currently known as the phenomenological shell model (PSM), has been proven to be a simple but effective model to interpret the stability pattern and electronic structure of small-size metal clusters [39]. Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00020-6 Copyright © 2023 Elsevier Inc. All rights reserved.

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100 Atomic clusters with unusual structure, bonding and reactivity

In this chapter, we would review not only the structural evolution but also the stability trend of a series of small-sized silver and gold clusters Mn with n ¼ 2  20. Geometric shapes of the lowest-energy structures and their basic thermodynamic parameters including the binding energy per atom, the second-order difference of energy, and the one-step fragmentation energy are determined and confirmed using density functional theory (DFT) computations. The electronic structure, which is at the origin of astonishing properties of clusters, is further elucidated within the perspective of the PSM. As such electronic distributions are well reflected in the excited states, their optical spectra are also analyzed to figure out the similarities and differences between two noble metal elements.

2. Equilibrium structures and growth mechanism In the following sections, the possible equilibrium structures of silver and gold clusters, along with their growth mechanism, will be determined and confirmed. Conventionally, the structures presented hereafter are denoted as Mn-X in which M ¼ Ag and Au, n ¼ 2–20, and X ¼ I, II… being isomers with increasing relative energy. Thus, Mn-I consistently stands for the most stable isomer of size n. For the sake of simplicity, relative energies are given hereafter in eV in single decimal figure. The present review mostly discusses the computational results that were recently obtained making use of DFT approaches [22,26–28]. In our DFT computations, local energy minima are fully optimized, without any symmetry or geometry restrictions, employing a density functional with long-range-corrected exchange effects, namely the LC-BLYP functional [44], in conjunction with the effective core potential (ECP) cc-pVDZ-PP basis set [45]. This level of calculation has been tested earlier for quantitative examination of systems containing noble metals [28,46]. Harmonic vibrational frequency computations are also carried out at the same level to confirm the detected structures as local minima and to estimate their zero-point energy (ZPE) corrections. With the aim to calibrate the performance of DFT methods, we first conduct some benchmark calculations for the dimers Au2 and Ag2 using various types of functionals along with the cc-pVDZ-PP basis set. Computed results are present in Table 1, which also includes available experimental data for the purpose of comparison. Generally, all functionals tested tend to overestimate the bond lengths of both Au2 and Ag2, except for the LC-BLYP. Deviations from experiment vary from ˚ for Au2 and from 0.03 to 0.13 A ˚ for Ag2 using traditional GGA and hybrid functionals. The LC-BLYP 0.05 to 0.11 A ˚ (Table 1). This funcexhibits smallest deviations from experimental bond lengths with an error margin smaller than 0.03 A tional is also reliable in predicting vibrational frequencies. The LC-BLYP values oe(Au2) and oe(Ag2) amount to 190 and 203 cm1, respectively, which agree well with experimental values of 191 and 192 cm1 [40,42]. Though the geometric and spectroscopic information can sufficiently be established by the LC-BLYP functional, it is likely to yield considerably inconsistent results for the bond dissociation energy (De). In fact, De values predicted by LC-BLYP/cc-pVDZ-PP computations for Au2 and Ag2 amount to 1.9 and 1.5 eV, respectively, which significantly underestimate the experimental values of 2.3 eV for Au2 and 1.7 eV for Ag2 [40,43]. GGA and meta-GGA functionals such as BP86, PW91, PBE, and TPSS produce better results for this quantity (Table 1). On the basis of a good agreement with the experiment for both Au2 and Ag2, the PBE functional appears to be more suitable for energetic quantities. The above discussion clearly shows that it is rather difficult to judge the overall accuracy of a specific functional as each naturally has its own advantages and drawbacks, good performance, and shortcomings when treating a certain property. As discussed above, benchmark studies testing the accuracy of functionals were frequently performed on relatively small systems whose accurate experimental information or theoretical data from high-level MO calculations are available. Because most targets of actual interest have larger size and more complicated structure, the benchmarks do not always reflect the special structural or electronic features of medium and large-size compounds. Therefore, it is crucial for each case considered to carefully examine the applicability of different computational options for every new type of compounds. Although it is rather not wise to make a clear statement about the overall performance of a functional for both gold and silver clusters in all computations, the LC-BLYP appears reliable in predicting molecular geometries and vibrational signatures, but it is likely to be deficient in computing the bond energy for which the PBE seems to describe a bit better. For the TPSSh and M06-L functionals, results in Table 1 show that they are not more reliable than the LC-BLYP in predicting atomic geometries and vibrational signatures and also not better than the PBE in determining the bond energies. Therefore, we select both LC-BLYP and PBE functionals for most calculations carried out in this report. While the former is employed for prediction of the lowest-energy structures, the latter is used for determination of energetic parameters. Both silver and gold clusters prefer planar structure up to n ¼ 6 in their ground state (Fig. 1). The M2 dimers have a low ˚ (Ag2) at the LC-BLYP/cc-pVDZ-PP level. These are spin (singlet) ground state with bond lengths of 2.50 (Au2) and 2.53 A ˚ ˚ (Ag2) [41]. The M3 trimers adopt a bent closely comparable to the experimental values of 2.47 A (Au2) [40] and 2.53 A structure, which is distorted from a Jahn-Teller effect on a D3h form. A single occupancy of the doubly degenerate e orbital

Structural evolution, stability, and spectra Chapter

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101

TABLE 1 Theoretical and experimental bond lengths (R, A˚), vibrational frequencies (ve, cm21), and dissociation energies (De, eV) of dimeric Au2 and Ag2 species. Ag2

Au2 DFT functional

Re

ve

De

Re

ve

De

B3LYP

2.56

167

1.94

2.59

179

1.56

B3P86

2.52

176

2.08

2.56

190

1.63

PW91

2.53

173

2.28

2.56

188

1.81

BB95

2.53

171

2.25

2.57

187

1.81

B3PW91

2.53

174

1.97

2.57

186

1.51

BP86

2.53

173

2.23

2.56

188

1.78

BPW91

2.54

171

2.13

2.57

186

1.66

M06

2.58

162

2.14

2.59

192

1.83

M06-2X

2.56

159

1.49

2.68

154

1.40

M06-L

2.56

160

2.26

2.57

189

1.90

PBE

2.53

172

2.27

2.57

186

1.78

PBE0

2.53

175

2.00

2.57

185

1.55

TPSS

2.52

176

2.25

2.56

191

1.76

TPSSh

2.52

177

2.15

2.56

190

1.67

LC-BLYP

2.50

190

1.88

2.53

203

1.46

Experiment

a

2.47

a

191

a

2.29

b

2.53

c

192

1.66d

a

Taken from Ref. [40]. Taken from Ref. [41]. c Taken from Ref. [42]. d Taken from Ref. [43]. b

of the equidistant triangle (D3h) leads to a geometric distortion giving rise to an orbital splitting to a pair of (a1 + b2) orbitals (Fig. 2) in the C2v form. The unpaired electron in both Au3 and Ag3 tends to occupy the b2 SOMO, resulting in a 2 B2 ground state. A planar rhombus having a D2h point group is predicted to be most stable form of the tetramers M4. Starting from the square D4h cycle with a singlet 1A1g state, a Jahn-Teller vibrational deformation following a normal mode, being a simultaneous ring deformation mode B1g, leads to a substantially stabilized rhombic D2h structure. The M4 species with D4h structure would have had two electrons in a degenerate eu orbital (Fig. 3). Such a degenerate state is not stable upon Jahn-Teller distortions, and the system undergoes a geometry relaxation to remove that degeneracy. The resulting stable structure of M4 is thus distorted from D4h to D2h with a low spin 1Ag state. Let us note that a Y-shaped isomer of Au4, not shown in Fig. 1, is calculated to have a similar energy content to the rhombic one. In agreement with previous predictions, the global minima of Au5 and Au6 are a planar W-type shape (C2v) and a planar triangle (D3h), respectively. Generally, the equilibrium geometry of a specific system Mn in the size range of n ¼ 2–6 is generated by simply adding an extra M atom to the lowest-lying isomers of the smaller-size Mn1. The lowest-energy structures and isomers for Mn from n ¼ 7 to 11 clusters are shown in Fig. 4. The most stable form Au7-I is reached from Au6-I by adding an extra Au atom, followed by a 3D configuration Au7-II. The 3D structure has a higher symmetry (C2v) but is less stable than Au7-I by 0.2 eV. As in Au7, the planar shape Au8-I (D2h) is predicted as the global minimum of Au8, whereas the second most stable isomer Au8-II has a 3D conformation (Td), being 0.7 eV higher. The planar D2h structure was also reported as the lowest-energy isomer when CCSD(T) computations with the correlation consistent basis sets were employed [47]. The most stable form Au9-I of Au9 has again a planar arrangement and is generated by attaching one extra gold atom to the most stable isomer of Au8-I. The next isomer Au9-II is computed to be about 0.1 eV higher. Previous DFT calculations

102 Atomic clusters with unusual structure, bonding and reactivity

Ag3-I (0.00, C2v, 2B2)

Au3-I (0.00, C2v, 2B2)

Ag4-I (0.00, D2h, 1Ag)

Au4-I (0.00, D2h, 1Ag)

Ag5-I (0.00, C2v, 2A1)

Au5-I (0.00, C2v, 2A1)

Ag6-I (0.00, D3h, 1A1’)

Au6-I (0.00, D3h, 1A1’)

FIG. 1 The lowest-energy isomers for Mn clusters, M ¼ Ag, Au, n ¼ 3–6.

a1

b2

FIG. 2 Plots of the a1 and b2 orbitals in the C2v M3 clusters.

[48,49] also reported Au8-I and Au9-I as the best candidates for equilibrium geometries. In the case of Au10, the 3D structure Au10-I (Fig. 4) is predicted to be preferred over the planar elongated hexagon Au10-II, with an energy of 0.2 eV (LC-BLYP value). As mentioned above, the 2D ! 3D transition in neutral Aun is still a matter of debate as it remains sensitive to the computational method employed. Some previous investigations recorded a crossover already at n ¼ 7 [33], while other studies predicted it to occur somewhere between n ¼ 12 and 15 [24]. Our present geometrical search points out that such a transition from planarity to nonplanarity is likely to be initiated at n ¼ 10, and Au10 represents the first size whose lowerlying structures do not have a 2D shape. This is consistent with a recent investigation [48] in which this transition is predicted to start from Au10. The 3D Au11-I is confirmed in agreement with previous reports [48,49]. This global minimum contains a trigonal prism as the main structural feature and lies 0.5 eV lower in energy than the planar Au11-II. In contrast to the propensity of small gold clusters to favor a planar conformation, densely packed 3D structures tend to dominate the lower-lying population of silver Agn clusters with n > 6. Indeed, two 3D configurations, namely a D5h pentagonal bipyramid Ag7-I and a C3v tricapped tetrahedron Ag7-II, are competing for the lowest-lying equilibrium geometry of Ag7. Similar to earlier results [18,22,50], the former is the most stable isomer. For Ag8, the Td Ag8-I arising from a regular tetrahedron capped with four Ag atoms is predicted to lie 0.3 eV higher in energy than a D2d geometrical shape

Structural evolution, stability, and spectra Chapter

Square M4 (D4h)

HOMO (eu)

HOMO (eu)

HOMO (b3u)

LUMO (b2u)

Rhombic M4 (D2h)

FIG. 3 Degenerate HOMOs in a square M4, and their counterparts in a rhombic M4.

Au7-I (0.00, Cs, 2A’)

Au7-II (0.20, C2v, 2A1)

Ag7-I (0.00, D5h, 2A1g)

Ag7-II (0.07, C3v, 2A1)

Au8-I (0.00, D2h, 1A1g)

Au8-II (0.68, Td, 1A1)

Ag8-I (0.00, Td, 1A1)

Ag8-II (0.30, D2d, 1A1)

Au9-I (0.00, C2v, 2A1)

Au9-II (0.10, C4v, 2A1)

Ag9-I (0.00, C2v, 2A1)

Ag9-II (0.23, Cs, 2A’)

Au10-I (0.00, C3v, 1A1)

Au10-II (0.24, D2h, 1Ag)

Ag10-I (0.00, D2d, 1A1)

Ag10-II (0.05, C1, 1A)

Au11-I (0.00, C3h, 2A”)

Au11-II (0.47, Cs, 2A’)

Ag11-I (0.00, C2, 2A)

Ag11-II (0.02, C2v, 2A1)

FIG. 4 Lower-energy isomers of Mn from n ¼ 7 to 11 clusters.

6

103

104 Atomic clusters with unusual structure, bonding and reactivity

Ag8-II (LC-BLYP/cc-pVDZ-PP). Our current prediction for the most stable isomers of both Ag7 and Ag8 is again in line with most previous calculations [2,51–53]. Consistent with earlier DFT calculations [52,54,55], the most stable isomer of Ag9, i.e., Ag9-I with C2v symmetry, is formed by adding one Ag atom to the Td Ag8-I. However, this form was considered as a local minimum in some other studies [18,56]. The Ag9-I is computed to be 0.2 eV more stable than Ag9-II (LC-BLYP value). In going further, Ag10 is created by adding one extra silver atom to the most stable D2d isomer of Ag9. Such a twinned pentagonal bipyramid has also been considered to be the ground state by Fournier [16] and Yang et al [55]. The next most stable isomer Ag10-II is higher than Ag10-I by only 0.05 eV in energy (LC-BLYP value). This asymmetric form has also been suggested to be the lowest-energy structure of Ag10 in some other previous reports [50,56]. For Ag11, two nearly degenerate isomers within 0.02 eV energy range are competing for the ground state. The C2 Ag11I was reported as a local minimum in earlier studies [50,56], but the C2v Ag11-II has recently been considered to be the best candidate for the ground state of Ag11 [22]. Overall, starting from the size n ¼ 6, both silver and gold clusters grow in completely different patterns. Some lower-lying isomers for Mn systems from n ¼ 12 to 16 are displayed in Fig. 5. Accordingly, both Au12 and Au13 exhibit 3D ground state. Flat structures are also found for these sizes but only as high-energy isomers. At the level of theory used, 2D conformations Au12-II and Au13-II are 0.2 eV higher than their 3D counterpart. The results obtained here thus disagree with BLYP calculations reported in ref. [57] in which the equilibrium geometries of both Au12 and Au13 are planar forms, namely Au12-II and Au13-II, respectively. The two most stable isomers detected for Au14, i.e., Au14-I and Au14-II, are all 3D (Fig. 5). Similarly, the nonplanar-shape Au15-I is reached by capping an extra Au atom upon one side of Au14-I. The second most stable Au15-II is only 0.2 eV higher. The structures Au14-I and Au15-II located here were also previously suggested as the global minima for Au14 and Au15, respectively [49,57–59]. In accordance with earlier reports [26,50], the Td Au16-I is predicted to be 0.1 eV lower than the Cs Au16-II.

Au12-I (0.00, C2v, 1A1)

Au12-II (0.14, C2v, 1A1)

Ag12-I (0.00, Cs, 1A’)

Ag12-II (0.18, Cs, 1A’)

Au13-I (0.00, C2v, 2B2)

Au13-II (0.21, C2v, 2A1)

Ag13-I (0.00, C2, 2A)

Ag13-II (0.03, Cs, 2A’)

Au14-I (0.00, C3v, 1A1)

Au14-II (0.42, C2v, 1A1)

Ag14-I (0.00, C2, 1A)

Ag14-II (0.07, C2v, 1A1)

Au15-I (0.00, Cs, 2A’)

Au15-II (0.15, Cs, 2A’)

Ag15-I (0.00, C1, 2A)

Ag15-II (0.24, C2v, 2B1)

Au16-I (0.00, Td, 1A1)

Au16-II (0.14, Cs, 1A’)

Ag16-I (0.00, C1, 1A)

Ag16-II (0.18, Cs, 1A’)

FIG. 5 Lower-energy isomers of Mn clusters from n ¼ 12 to 16.

Structural evolution, stability, and spectra Chapter

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For silver counterparts, the lowest-lying Ag12-I has a Cs symmetry and can be derived from Ag11-I. The next most stable Ag12-II also has Cs symmetry and lies 0.2 eV higher. In agreement with a recent report [18], the most stable Ag13-I is obtained by capping one Ag atom on an edge of Ag12-I. The next isomer Ag13-II, which was suggested to be the most stable structure by Yang et al. [55], is computed to be only 0.03 eV less stable than Ag13-I in our current work. Both forms are thus quasidegenerate isomers. The C2 Ag14-I, reached by capping one Ag atom to Ag13-I, is recently computed to be the lowest-energy Ag14 [22]. The next most stable structure, i.e., Ag14-II, is somewhat more symmetric (C2v) and lies only 0.1 eV higher (LC-BLYP value). From attachment of one silver atom Ag14-I and Ag14-II, two lowest-lying isomers Ag15-I (C1) and Ag15-II (C2v) are built upon. They possess a substantial energy gap, in which the former is 0.2 eV below the latter. This result is inconsistent with recent calculations by Rodrı´guez-Kessler et al. [60] in which Ag15-II was predicted as the global minimum, whereas Ag15I has not been seen. Accordingly, the low symmetry Ag15-I emerges as the new global minimum not detected before [61]. In agreement with this report [60], two oblate configurations Ag16-I and Ag16-II with an energy difference of 0.2 eV (LC-BLYP value) are the most stable forms of Ag16. Again, the growth patterns of the two series of noble metal clusters differ much from each other and do not induce any correlation. We now examine the lower-lying structures of systems having 17–20 metal atoms that are shown in Fig. 6. Previously, the asymmetric Au17-II, which can be considered as a result of removal of three atoms on one edge of the tetrahedral Au20 cluster, was reported to be the lowest-energy Au17 [49]. However, our current LC-BLYP results point out that this size prefers to exist in the highly symmetric Td Au17-I form. Following attachment of one gold atom to Au17-II, the lowestenergy Au18-I of the octadecamer Au18 is built upon. The next low-lying structure Au18-II is computed to stay 0.1 eV higher. Continuously, the most stable forms of Au19 and Au20, i.e., Au19-I and Au20-I in Fig. 4, are obtained from Au18-I upon addition of one and two Au atoms. The regularly tetrahedral pyramid Au20-I and the truncated pyramid Au19I have also unambiguously been assigned as the global energy minima for both Au20 and Au19 systems, respectively [25,62]. Competitive candidates with tiny energy gaps have been found for systems containing from 17 to 20 silver atoms [22,50]. Two foremost lower-lying isomers of Ag17 are the close-packed structure Ag17-I and the hollow cage Ag17-II (Fig. 6). Ag17-I turns out to be a new structural motif, which was not found in previous studies. At the LC-BLYP/ccpVDZ-PP level, the former is computed to be 0.2 eV higher in energy than the latter. Similarly, several configurations

Au17-I (0.00, D2d, 2A1)

Au17-II (0.35, C1, 2A)

Ag17-I (0.00, C3, 2A)

Ag17-II (0.16, C2v, 2A1)

Au18-I (0.00, C2v, 1A1)

Au18-II (0.10, Cs, 1A’)

Ag18-I (0.00, Cs, 1A’)

Ag18-II (0.12, C2v, 1A1)

Au19-I (0.00, C3v, 2A1)

Au19-II (0.92, C2v, 2A1)

Ag19-I (0.00, Cs, 2A’)

Ag19-II (0.05, C3v, 2A1)

Au20-I (0.00, Td, 1A1)

Au20-II (1.22, C3, 1A)

Ag20-I (0.00, C3, 1A)

Ag20-II (0.03, Td, 1A1)

FIG. 6 Lower-energy isomers for Mn clusters with n from 17 to 20.

106 Atomic clusters with unusual structure, bonding and reactivity

also exist in competing for the ground state of Ag18. Consistent with a recent study [18], a derivative of the Ag13 icosahedron with Cs symmetry, i.e., Ag18-I in Fig. 6, is detected as the ground-state geometry of Ag18. The C2v hollow cage Ag18II, which was considered as the best candidate for the global minimum in ref. [50], is now computed to lie 0.1 eV above. For Ag19, while Tsuneda [28] suggested a pyramid motif, i.e., the truncated trigonal pyramid Ag19-II (C3v), to be the lowest-energy structure, we identify this form to be marginally less stable by 0.05 eV than the Cs isomer Ag19-I. Similarly, the pyramidal motif Ag20-II was located as the most preferred structure of Ag20 in several reports [18,50,54]. However, the newly detected eicosamer Ag20-I, having a C3 symmetry, is computed to be only 0.03 eV more stable than Ag20-II (LC-BLYP value) [22,39]. Based on computational results reported in recent literature for small Mn clusters [22,60] and combined with our present calculations, structural evolutions in going from M3 to M20 are summarized in Figs. 7 and 8. Overall, the preference of a successive growth pattern in gold clusters is not as much recognizable as in silver counterparts. Particularly, in the size range of Ag12–Ag16, the most stable form of a specific size n can regularly be obtained upon addition of an extra Ag atom to the lowest-lying isomer of the smaller n  1 size. Such an evolution, which forms the basis for the successive growth algorithm for structural search, can be employed to predict the favored structures of silver clusters. Nevertheless, such a pattern is broken at a certain size where a completely different structural feature emerges. It is, for example, the case of Ag17 whose corresponding global minimum is not derived from the most stable Ag16. It could arise from a higher energy isomer of a smaller size but the reasons for such a strong stabilization are not clear. In summary, while Agn clusters prefer a planar structure up to n ¼ 6, a structural transition from 2D to 3D in gold systems is likely to take place at Au10. From n ¼ 7, Agn species and their Aun counterparts exhibit much different ground-state geometries. The Agn clusters going from n ¼ 7 to 11 adopt three-dimensional structures that can be constructed following a successive growth algorithm, that is, by adding extra Ag atoms either on the D5h bipyramid Ag7-I or the Td tetrahedral Ag8-I. For systems from n ¼ 13 to 16, hollow flat cages become the most favored configurations. As the cluster size increases, a common trend emerges for formation of close-packed structures from a 13-atom icosahedral core by adding extra atoms on the triangular faces. As compared with the silver Agn, the gold Aun clusters generally exhibit more highly symmetric ground-state structures. The local minima of a specific Aun system are usually generated from the lowest-lying isomer of the smaller-size Aun–1 by adding an extra gold atom around. Remarkably, in contrast to a preference of amorphous structures obtained for silver clusters, the pyramid-like motif is the most energetically favorable structure in Aun systems with n ¼ 18–20. This can be understood by the fact that the contribution of d electrons in the small silver clusters is more significant than in gold counterparts [63]. On the one hand, the effects of d electrons and s–d hybridization strongly FIG. 7 Structural evolution from Au3 to Au20 clusters. The added Au atom is labeled as ◯ in red (dark gray in print version). (Results are taken from P.V. Nhat, N.T. Si, J. Leszczynski, M.T. Nguyen, Another look at structure of gold clusters Aun from perspective of phenomenological shell model, Chem. Phys. 493 (2017) 140–148 and present work.)

Structural evolution, stability, and spectra Chapter

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FIG. 8 Structural evolution from Ag3 to Ag20 clusters. The added Ag atoms is labeled as ◯ in red (dark gray in print version). (Results are taken from P.L. Rodrı´guez-Kessler, A.R. Rodrı´guez-Domı´nguez, D.M. Carey, A. Mun˜oz-Castro, Structural characterization, reactivity, and vibrational properties of silver clusters: a new global minimum for Ag16, Phys. Chem. Chem. Phys. 22 (46) (2020) 27255–27262, M.L. McKee, A. Samokhvalov, Density functional study of neutral and charged silver clusters Agn with n ¼ 2– 22. Evolution of properties and structure, J. Phys. Chem. A 121 (26) (2017) 5018–5028 and present work.)

influence the ground-state configuration of silver clusters. On the other hand, the relativistic effects that are very important in heavier atoms such as gold [33] play a vital role in stabilizing the pyramidal structures.

3. Thermodynamic stabilities To gain some quantitative insights into the thermodynamic stabilities of these neutral noble metal clusters, we now evaluate the change of binding energy per atom (BE), the second-order difference of energy (D2E), and the one-step fragmentation energy (Ef) with respect to the cluster sizes. For these clusters, such parameters of a specific system Mn (M ¼ Ag, Au) are defined by following equations: BE ¼ ½nEðMÞ  EðMn Þ=n D2 E ¼ ½EðMn+1 Þ + EðMn1 Þ  2EðMn Þ Ef ¼ EðMn1 Þ + EðMÞ  EðMn Þ where E(Mn) is the total energy of the lowest-lying Mn cluster shown above. The graph illustrating the variation of BE as a function of gold cluster size is plotted in Fig. 9. This parameter can be regarded as the energy gained in assembling a definite cluster from isolated gold constituents. Previously, a theoretical study [64] employed a local density approximation (LDA) to calculate the Eb of gold clusters up to Au20, but experimental data are not systematically reported yet. Our present results are quantitatively different from earlier values [61]. As shown in Fig. 9, the BE plot of Aun clusters roughly exhibits a gradual growth and reaches the maximal value of 2.3 eV/atom for Au20. This is still much smaller than the cohesive energy of 3.8 eV obtained for the bulk gold [65]. In addition, computed results show local maxima at n ¼ 6, 8, and 20, suggesting that these clusters are expected to be more stable than their immediate neighboring ones. The value Ef ¼ 2.3 eV obtained for Au2 (at PBE/cc-pVDZ-PP level) is in line with the experimental value of 2.29  0.02 eV [40]. Generally, calculated Ef values as a function of cluster size obey an odd-even oscillation. Accordingly, clusters with an even number of atoms appear to be more stable than the neighboring odd-numbered ones. Of these clusters, Au3 and Au7 are characterized by the lowest Ef values, implying their low thermodynamic stability. On the contrary, Au6 and Au20 are expected to be the most stable ones as they exhibit exceptionally high Ef values. Other remarkably high peaks are found at n ¼ 2, 8, 12, and 14, indicating that these systems are also particularly stabilized.

108 Atomic clusters with unusual structure, bonding and reactivity

FIG. 9 Binding energies per atom (Eb), one-step fragmentation energies (Ef), and second-order difference of energy (D2E) for neutral Aun clusters as a function of cluster size. Results are obtained at the PBE/cc-pVDZ-PP + ZPE level.

As discussed above, the optimal structure of Aun at a certain size is normally generated from that of the smaller one by adding an extra gold atom. Accordingly, the energy gain in incorporating an Au to the smaller size can be considered as the embedding energy (EE). Such a parameter can also be characterized by the one-step fragmentation energy (Ef), i.e., the energy needed to detach one gold atom from Aun, giving rise to Aun1. Evolution of the Ef values for the Aun clusters considered, calculated at the PBE/cc-pVDZ-PP + ZPE level, is also plotted in Fig. 9. The second-order difference of energy (D2E) is an important indicator that measures the relative stability of clusters. In particular, peaks in the graph of D2E as a function of cluster sizes were found to be well correlated with peaks in the experimental mass spectra [66]. As illustrated in Fig. 9, an extreme odd-even oscillation appears indicating that a cluster having an even number of atoms is more stable than the odd-numbered ones. Consistent with the above analyses based on fragmentation energies, the Au6 and Au20 species are found to have the largest D2E values, indicating their peculiarly high thermodynamic stability. On the contrary, the Au3 and Au7 systems are again found to be the least stable sizes. Concerning the relative stability of silver clusters, different density functionals have been employed to predict the total atomization energies of Agn clusters up to n ¼ 99 [18]. Accordingly, the closed-shell systems having an even number of silver atoms were generally found to be more stable than the open-shell counterparts exhibiting an odd number of electrons. In addition, total atomization energies of Agn start converging slowly to the bulk at the size n ¼ 55 and reach the maximal value of 2.2 eV/atom for Ag99. Our computed atomization energy or the binding energy per atom as function of Agn size up to n ¼ 20 is displayed in Fig. 10. As in gold systems, the Eb of Agn species also increases slowly with the cluster size and reaches the maximum of 1.9 eV/atom at n ¼ 20. This is again much smaller than the cohesive energy of 2.9 eV of bulk silver [18]. The energetics for detachment of an Ag atom from a cluster Agn to form a smaller species Agn1 up to n ¼ 20 are also illustrated in Fig. 10. At the PBE/cc-pVDZ-PP level, the fragmentation energy (Ef) of Ag2 amounts to 1.8 eV, which is comparable to the experimental value of 1.7 eV [43]. Such a result in addition suggests that the chemical bond in a silver cluster is more breakable than that of the gold counterpart. The largest fragmentation energy for Ag20 (2.7 eV) is 90% of 2.9 eV for bulk silver [18]. Similar to gold clusters, an even-odd oscillation appears in the calculated Ef values as a function of Agn size (Fig. 10), in which even-numbered systems are thermodynamically more stable than odd-numbered ones. Of the

Structural evolution, stability, and spectra Chapter

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FIG. 10 Binding energies per atom (Eb), one-step fragmentation energies (Ef), and second-order difference of energy (△2E) for neutral Agn clusters as a function of cluster size. Results are obtained at the PBE/cc-pVDZ-PP + ZPE level.

clusters considered, Ag3 and Ag9 are predicted to be the least stable species, being characterized by the lowest Ef values. In contrast, Ag6, Ag8 Ag14, and Ag18 are expected to be particularly stable as their Ef values are exceptionally high. The even-odd fluctuation becomes more obvious when we look at the second-order difference of energy (D2E). This thermodynamic parameter can also be considered as a disproportionation energy, being the energy change of such reaction 2Agn ! Agn+1 + Agn1

(1)

In agreement with the analysis based on dissociation energies given above, the Ag6, Ag8, and Ag14 species are found to be particularly stable with sharply high D2E values. Fig. 10 also reveals that while the disproportion of even Agn clusters is endothermic, that of odd counterparts is generally exothermic. Furthermore, the energy difference tends to decrease as the cluster size increases. For instance, the energies of reactions (1) are calculated to be 0.8 and 0.5 eV for n ¼ 6 and 12, respectively. Two factors that determine the metal cluster stability include the electron shell effect and atomic arrangement [67]. Similarly to alkali clusters, the former can also be observed in clusters of coinage metals, because the nd levels of these elements are (almost) completely occupied [34,35]. However, a particular attention should be paid to the behavior of their d electrons. The participation in bonding of these fermions can be seen experimentally via cohesive energy of the bulk materials. For example, the experimental cohesive energy of Sc amounts to 3.9 eV/atom, then it increases to 4.9 eV/atom for Ti and attains a maximum value of 5.3 eV/atom for V. After this maximum, it decreases, reaching a bottom for Mn (2.9 eV/atom), then increases, having a broad maximum (4.3–4.4 eV/atom) for Fe, Co, and Ni, and finally, it decreases again, approaching another valley (3.5 eV/atom) at Cu [68]. Morse [40] pointed out earlier that the 4 s orbitals of the first transition metal series have more significant contribution to the chemical bonding than the 3d orbitals because the latter are more contracted. Moving down in the Periodic Table, while the d orbitals tend to expand, the s orbitals are getting contracted [69]. Hence, the nd and (n + 1)s orbitals of the second and third transition metal series become nearly comparable in size, thereby facilitating the d bonding. This may result in a greater stabilization of compounds containing heavier transition metals, as compared with those of the first-row metal elements, even though such a correlation remains not linear. In fact, the bond dissociation energies of V2 and Nb2 are determined to be 2.75  0.01 [70] and 6.20  0.05 eV [71], respectively, while that of Ta2 is predicted to be in the range from 5.0 to 5.4 eV [72].

110 Atomic clusters with unusual structure, bonding and reactivity

For gold-containing compounds, it should be noted that another crucial factor making their properties unique is the relativistic effect [49], rather than the “lanthanoid contraction” as mentioned above. Actually, the relativistic maximum of gold, which was initially pointed out by Pyykk€o [30], results in many unusual molecular shapes and properties of gold compounds. In fact, while the ground state of Ag7 has a 3D structure, that of Au7 is planar, separated from the optimal 3D isomer by 0.2 eV (LC-BLYP value). The propensity of neutral gold clusters to favor planar structures continues up to surprisingly large sizes, being likely up to 10 atoms (see above). The strong relativistic effects in addition enhance the d–d interaction, leading to a stabilization of AudAu bond [73,74]. As a result, the gold-gold distance in metal is getting even ˚ , respectively [75]. A similar phenomenon also shorter than the corresponding silver-silver distance, being 2.88 and 2.89 A appears in dimers in which the bond dissociation energy of Au2 (2.29  0.02 eV) is significantly larger than that of Ag2 (1.65  0.03 eV) [40]. Without relativistic effects, interactions between fulfilled d orbitals become negligible; the bond strength between gold atoms could thereby be weaker than that between silver counterparts, because the s orbital shrinks down in the group. Also owing to strong relativistic effects, Au(I) compounds tend to aggregate (polymerize) via formation ˚ and a strength of 7–12 kcal mol1, a phenomenon known by the terms of of weak gold-gold bonds with a length of 3.0 A aurophilicity or aurophilic attraction [76].

4. Phenomenological shell model (PSM) As demonstrated above, not only the geometrical and electronic structure but also the (physical and chemical) properties of atomic clusters strongly depend on their size and differ considerably from both individual atoms and bulk materials [66,77]. Experimental observations become more and more available owing to the use of modern spectrometric techniques, and theoretical steps can be taken for understanding and elucidating the observed findings. However, similar to the situation in other fields, it is not always straightforward to understand experimental or calculated results, which often need to be rationalized with the help of some simpler approaches. In this context, the PSM [78] provides us with a simple but effective model to interpret the electronic structure and stability pattern of simple metal clusters [39]. In the original formulation of PSM, the shape of stable clusters is presumed to be spherical. The highly delocalized valence electrons that play a central role in the formation of chemical bonding in clusters are treated as itinerant particles moving around a spherical pseudo-potential composed of the inner electrons along with the nuclei, giving rise to the spherical shell closures, namely 1S/1P/1D/2S/1F/2P. Subsequently, the model was extended to treat elliptical shapes, known as the Clemenger-Nilsson model (CNM) [79], which was introduced to account for both spherical and oblate/prolate clusters, and from which the energy ordering of different levels with respect to the distortion of the cluster shape is qualitatively shown in Fig. 11. As well established in the earlier literature [78,80,81], alkali clusters bearing 8, 18, 20, 40, 58, and 92 valence electrons exhibit an outstanding thermodynamic stability. A widely accepted explanation for such a phenomenon is based on a strong delocalization of the external s electrons bound in a spherically symmetric potential well [37,38]. Noble metal systems related to closed valence shell with 1S2/1P6/1D10/2S2 electronic configurations have also been found to show particularly low polarizabilities, high ionization energies, and large dissociation energies [22]. These observations indicate the dominant contribution to formation of chemical bonding of 5s (Ag) and 6s (Au) electrons, reproducing the comparable closed electron shell numbers of the alkali clusters. Hence, to some extent, the electron distribution of silver clusters can be considered as similar to those of the alkalis. The Mn species with n ¼ 6, 8, 20 considered in the present work provide us with some interesting samples for applications of the PSM. We now examine the electronic structure of some spherical systems based on a perspective of the PSM. A cluster would exhibit a nearly spherical shape when all 3 P or all 5 D shell orbitals are filled with 6 or 10 electrons, respectively, in order to form a closed electronic structure (Fig. 11). As presented in Fig. 1, the lowest-energy structure of both Au6 and Ag6 is an equivalent triangle with a D3h point group. This form was also reported as the ground-state geometry of Cu6 [82]. The number of itinerant electrons amounts to six (one electron from each coinage metal atom) that formally satisfy the H€ uckel (4n + 2) counting rule for planar aromatic compounds. The clusters are characterized by planar shape; hence, an 1S2{1P2x 1P2y } electronic configuration is expected (the curly brackets represent a double degeneracy). The energetic ordering of the valence molecular orbitals shown in Fig. 12 clearly approves the fact that the cluster with six itinerant electrons exhibits a 1S2 {1P2x 1P2y } configuration. The two higher-lying orbitals, i.e., the degenerate HOMO, show an p-character, whereas the HOMO-1 is actually an s-orbital. Earlier, the electronic mechanism of how the Au6 ring encapsulates a transition metal dopant atom was also successfully interpreted using such a simple model [83]. As mentioned above, irrespective of some similarities in bulk states, coinage metal clusters exhibit several significant differences in terms of geometrical shapes. For Ag8, the 3D Td structure (Fig. 4) is located as the ground-state geometry, and so is for Cu8 [82]. On the contrary, Au8 prefers the D2h planar shape, being more stable than the Td conformation by 0.7 eV

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FIG. 11 Schematic energy levels of the shell orbitals with respect to the distortion toward prolate (right) and oblate (left) cluster shapes. (Taken from P.V. Nhat, N.T. Si, J. Leszczynski, M.T. Nguyen, Another look at structure of gold clusters Aun from perspective of phenomenological shell model, Chem. Phys. 493 (2017) 140–148.)

1Py

1Px

1S FIG. 12 Shell orbitals of the oblate M6 clusters. (Taken from P.V. Nhat, N.T. Si, J. Leszczynski, M.T. Nguyen, Another look at structure of gold clusters Aun from perspective of phenomenological shell model, Chem. Phys. 493 (2017) 140–148.)

(LC-BLYP value). A closer look at the electronic structure of these systems allows the differences in going from Cu via Ag to Au to be understood and provides us with an interesting sample for the validity of PSM. The most stable form of either Cu8 or Ag8 is predicted to be a tetrahedron in such a way that it can be treated as a spherical-shaped system. Indeed, Fig. 13 confirms that the electron configuration of this cluster can be described by the initial shell of 1S2/1P6. With eight itinerant electrons and a sphere-like tetrahedron, these species constitute a closed-shell configuration giving them the status of magic clusters and thereby particular stability [22]. In contrast to the preference of sphere-like Ag8 and Cu8 structures, the most stable D2h form of Au8 can be regarded as bearing an oblate shape. Therefore, a different landscape emerges as partially occupied quasidegenerate orbitals are involved. From the PSM view for oblate clusters n(Fig. 11), Auo8 possesses a 1S2/1P4/1D2 electron configuration with two electrons distributed on twofold degeneracy

1D1xy 1D1x2 y2

orbitals. Such an electron shell tends to result in a

Jahn-Teller distortion, and its frontier orbitals are likely to split into two different energy levels, namely 1Dxy and 1Dx2 y2 . As shown in Fig. 13, Au8 with eight itinerant electrons and an oblate shape is characterized by an electron shell configuration of 1S2/{1P2x 1P2y }/1D2xy.

112 Atomic clusters with unusual structure, bonding and reactivity

FIG. 13 Shell orbitals of the oblate Au8 (left) and the spherical-like Ag8 (right). (Taken from P.V. Nhat, N.T. Si, M.T. Nguyen, Elucidation of the molecular and electronic structures of some magic silver clusters Agn (n¼ 8, 18, 20), J. Mol. Model. 24 (8) (2018) 1–14.)

1Dxy

1Px

1Py

1P

1S

1S

Au8

Ag8

We now examine the electronic structure of M20 with M ¼ Cu, Ag, Au species. The ground-state geometry of Ag20 and Cu20 is almost the same [27], which can be generated by adding three Ag/Cu atoms to a (12,3) Frank-Kasper polyhedron, in such a way that it can also be treated as a spherical shape system. For a nearly spherical cluster with 20 itinerant electrons, it again corresponds to a closed electron shell structure. Fig. 14 shows the shell orbitals for the lowest-energy neutral Ag20 cluster denoted as Ag20-I in Fig. 6. Both the lowest-energy and highest-energy shells of this neutral cluster are actually s orbitals, whereas the three higher-lying orbitals show a p-character. Other orbitals lying lower in energy than the HOMO (the 2S shell) clearly have a d-character, and they contribute predominantly to the 1D shell orbitals. With 20 itinerant electrons and a closed electron shell, the neutral Ag20 thus also prefers a nearly spherical shape over an elliptical form. The Au20 cluster also exhibits an unusually high stability as compared with its neighboring clusters. The most stable form of Au20 is a tetrahedron (Td symmetry) as presented in Fig. 6; it can thus also be treated as a spherical-shape system.

2S

1D

1P

1S FIG. 14 Electronic structure of the spherical-like cluster Ag20 with a closed electron shell configuration {1S21P61D102S2}. (Taken from P.V. Nhat, N.T. Si, M.T. Nguyen, Elucidation of the molecular and electronic structures of some magic silver clusters Agn (n¼ 8, 18, 20), J. Mol. Model. 24 (8) (2018) 1–14.)

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2S

1D

1P

1S FIG. 15 Electronic structure of the spherical-like cluster Au20 with a closed electron shell configuration {1S21P61D102S2}. (Taken from P.V. Nhat, N.T. Si, M.T. Nguyen, Elucidation of the molecular and electronic structures of some magic silver clusters Agn (n¼ 8, 18, 20), J. Mol. Model. 24 (8) (2018) 1–14.)

For a nearly spherical cluster enclosing 20 itinerant electrons, it clearly corresponds to a closed electron shell structure. Indeed, Fig. 15 confirms that the electron configuration of this cluster can be described by the shell 1S2/1P6/1D10/ 2S2 configuration. Strictly speaking, notably, five orbitals on the 1D shell split into two different sublevels, namely e and t2, as a result of the Td-symmetric crystal field. Relativistic effects are strong in heavy atoms such as gold [30,32] and apparently play a vital role in stabilizing the high symmetry tetrahedral Td structure [84]. On the contrary, for Ag counterpart, due to the significant contributions of its d states [85], the effect of geometric packing, i.e., atomic order, is likely competing with an electronic order and leads to a more compact structure for Ag20.

5. Electronic absorption spectra Electron transitions giving rise to absorption and emission spectra are a direct consequence of orbital configurations. Let us therefore consider in some detail the electronic spectra of these clusters. In addition, optical properties of noble metal clusters also attract a great deal of interest owing to their importance in both basic and applied research studies [4,86]. Study of electronic properties and optical response of these systems is an interesting but also challenging task due to the closely lying s, d-electrons and strong relativistic effects. One of the most intriguing phenomena is the surface plasmon resonance, i.e., a strong UV absorption due to collective oscillations of conduction electrons [58]. In small clusters, the absorption becomes molecule-like reflecting a well-defined electronic structure [14,87], and the determination of the size at which molecular-type electronic transitions evolve into a plasmon-like absorption is also a fascinating task [88]. In this context, optical properties of noble clusters have been a subject of several studies using both experimental and computational approaches alike [14,58,89]. Small noble metal clusters up to 30 atoms show molecular-like excitations with several typical peaks in their electronic absorption spectra. In addition, substantial differences in optical properties of both Agn and Aun clusters emerge because the d states in gold are more directly involved in electronic excitations as compared with those of silver [4]. Such alterations furthermore result from a stronger s-p-d hybridization in molecular orbitals of Aun and maximum relativistic effects of gold. In what follows, we shall examine size by size the predicted absorption spectra of silver and gold species Mn in the range of n ¼ 1–20. Theoretical spectra are simulated from time-dependent density functional theory (TD-DFT) calculations using the long-range corrected XC functional LC-BLYP and the cc-pVDZ-PP basis set. A number of results have previously been reported and are confirmed by the present study. The experimental spectrum of the gold atom contains two well-defined peaks at 4.8 and 5.4 eV, corresponding to the spin-orbit split 2S1/2 ! 2P1/2 and 2S1/2 ! 2P3/2 transitions [90]. The simulated spectrum of Au atom shown in Fig. 16,

114 Atomic clusters with unusual structure, bonding and reactivity

FIG. 16 TD-DFT absorption spectra for the lowest-lying structures of gold Aun clusters.

however, exhibits only one intense transition near 5.3 eV, which can be attributed to the S ! P transition. This is presumably due to the fact that TD-DFT calculations do not accurately account for the spin-orbit coupling effects. The absorption spectrum of the gold dimer has also been experimentally studied, and several absorption bands are found to be situated in the energy region of 2.5–5.5 [86,91,92]. With a reasonable certainty, we would assign the so-called X ! B transition measured at 3.18 eV to the predicted value near 3.0 eV, whereas the X ! D measured transition at 5.3 eV corresponds to the computed one at 5.2 eV. However, both X ! A and X ! C transitions observed experimentally at 2.44 and 4.80 eV are not reproduced in the TD-DFT absorption spectrum of Au2. The LC-BLYP absorption spectrum for Au3 consists of several significant peaks in the range between 3.0 and 4.5 eV, for which experimental data have also been available [86]. Accordingly, the low-intensity peak at 3.1 eV is likely associated with the 3.05–3.50 eV transitions found in previous experiments [86,92]. In addition, the higher-intensity absorption line at 3.6 eV can be accounted for by the measured transition at 3.64 eV [86,92]. The absorption spectrum of gold tetramer exhibits a highly intense peak near 3.2 eV and another weaker transition at 4.1 eV. The experimental spectrum [86] also exhibits a prominent transition at 3.3 eV, whereas the peak at 4.05 eV splits into two components at 4.0 and 4.1 eV. For Au5, the predicted transitions with particularly large oscillator strengths are located at 2.7, 3.3, and 3.5 eV. Previous measurement [86] also detected such transitions at a somewhat lower-energy region between 2.95 and 3.25 eV. In addition, one can detect some low-intensity peaks above 4.0 eV. To date, the experimental UV spectrum for Au6 is not available, even though previous analyses indicate this is a “magic” cluster with a rather large excitation energy [93]. Thus, it can serve as a proper model system for assessing the reliability of excited states calculations for gold clusters. The lowest excited state of Au6 is located at 3.4 eV (LC-BLYP). A comparable gap is obtained with the oB97X functional (3.3 eV) and EOM-CCSD (3.3 eV) approaches, whereas both TPSS and B3LYP functionals yield significantly reduced transition energies to 2.4 and 2.8 eV, respectively [89]. The measured and computed absorption spectra of Au7 have been reported in the literature [58,86]. Accordingly, locations of strong peaks are observed in the range of 2.6–3.6 eV that are also well reproduced in the simulated spectrum (Fig. 16). As compared with the odd numbered clusters such as Au3, Au5, and Au7, the absorption of the gold octamer is much simpler. The calculated spectrum for the ground-state Au8-I covers a range below 4.1 eV with three characteristic peaks centered at 3.4, 3.6, and 4.1 eV that were also experimentally seen with close values [86]. For Au9, the experimental spectrum shows a large amount of peaks situated at 2.9, 3.2, 3.7, 3.9, and 5.1 eV [86]. Previously, the observed

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photo-dissociation spectrum emphasized two broad absorption bands at 2.2 and 3.0 eV [94]. The spectra of two lowestenergy structures Au9-I and Au9-II (Fig. 4) simulated at the LC-BLYP/cc-pVDZ-PP level and shown in Fig. 12 suggest that the observed spectrum likely arises from a superposition of both isomers rather than from a sole carrier. Accordingly, the peaks located at 2.2, 2.7 and 3.2 eV can be assigned for intense transitions at 2.2, 2.8, and 3.2 eV in the predicted spectrum of Au9-I. On the contrary, the observed bands at 2.9 eV, which are absent in the spectrum of Au9-I, are consistent with the transition at 3.0 eV coming from the second isomer Au9-II. In the case of Au10, TD-DFT calculations give a main peak at 3.2 eV and followed by a gap of  0.6 eV to a characteristic trio of lower-intensity features separated by about 0.05–0.1 eV. For Au11, previous experimental study observed a local absorption maximum at 2.9 eV [12]. This peak is quite well reproduced in the calculated spectrum of Au11-I as it is located around 2.8 eV. In addition, a weaker signal at 3.0 eV and a stronger one at 3.3 eV could also be detected. Calculations performed for both Au12-I (3D) and Au12-II (2D) isomers (Fig. 16) show that the spectrum of Au12-I contains a strong doublet between 3.5 and 3.6 eV, along with lower-intensity peaks at 3.1 and 3.3 eV. The 2D isomer Au12-II exhibits a simpler spectrum with two prominent bands centered at 3.2 and 3.7 eV. Calculated absorption spectra for larger Aun clusters including n ¼ 13–20 in the energy range of 2.5–5.0 eV are also presented in Fig. 16. In general, the absorption lines of odd-numbered systems, when compared with those of the evennumbered counterparts, are typically more complicated and appear in the lower-energy region. For example, the optical absorption spectra of odd clusters with n ¼ 13, 15, 17, and 19 show particularly high peaks between 2.4 and 2.7 eV that are significantly shifted to 3.7 eV for Au14 and Au16. Concerning Au18 and Au20, while the former gives several lines with a maximum at 3.7 eV and two lower-intensity transitions below 3.6 eV, the latter exhibits a strong absorption located at 3.7 eV. Our present LC-BLYP calculations are thus comparable to those obtained with the CAM-B3LYP and LC-M06L functionals. Previously, the plasmon-like band of the tetrahedron Au20 was located at 3.5 and 3.8 eV by the CAM-B3LYP and LC-M06L, respectively [89,95]. However, LDA and GGA predictions produced peaks at much lower energies [96]. For example, while the BP86 calculation identified the most intense transition in Au20 at 2.9 eV [97], LDA predicted such a band at 2.8 eV [58]. Several studies on both experimental and theoretical aspects have been devoted to the optical properties of neutral silver clusters [14,90,95]. Generally, these nanostructures display the stepwise, multiple-band spectra that bear a common feature of noble metal clusters [90,98,99] as shown in Fig. 17. The simulated spectrum of the Ag atom produces one main peak at 3.8 eV (331 nm). This band can be assigned to the measured band at 326 nm (3.80 eV) or 328 nm (3.78 eV) in Ar and Kr matrices, respectively [100]. For the dimer Ag2, the lowest excited state is observed at 3.0 eV, which is marginally closer to the given experimental value of 2.96 eV [90] than the previous PBE prediction of 3.1 eV [101]. In addition, we detect two other intense transitions in the higher-energy region at 4.3 and 4.8 eV. The theoretical UV-visible optical absorption of the trimer Ag3 becomes more complicated with two intense signals at 3.6 and 3.8 eV. Besides, some lower-intensity peaks can be detected below 3.0 eV. To date, no experimental spectrum of the trimer is available. Let us now examine the UV-vis spectra of Agn in the sizes n ¼ 4–14 in comparison with existing experimental results previously obtained by Harb et al. [14]. The simulated absorption spectra the D2h isomer Ag4-I exhibit the highest transition at 3.1 eV, which has also been observed by all TD-DFT calculations [14]. Other lesser intense signals at 4.0, 4.2, 4.8, and 5.0 eV are further detectable. As compared with the experiment, the main transition is well reproduced at 3.1 eV, while other lower-intensity peaks are measured at 4.2, 4.5, and 4.8 eV. In the case of Ag5, two extensive bands were experimentally measured at 3.3 and 3.8 eV. TD-DFT calculations for Ag5-I also reproduce two main transitions at 3.2 and 3.8 eV. Accordingly, the agreement between theory and experiment is quite good for both Ag4 and Ag5 systems. The computed spectrum of the D3h Ag6-I contains a distinct transition at 3.6 eV along with some smaller signals at 3.1 and 5.1 eV. Such a prediction is not properly consistent with the experimental one, which shows two distinct peaks at 3.6 and 4.2 eV, as well as smaller bands at 4.9 and 5.1 eV. Harb et al. [14] also pointed out in taking the presence of the less stable C5v isomer into account, the agreement between theoretical and experimental spectra becomes better. Regarding Ag7, the experimental spectrum contains primarily one broad band with a doublet at 3.6 and 3.8 eV. As discussed above, two quasidegenerate isomers D5h Ag7-I and C3v Ag7-II (Fig. 4) with a tiny energy gap of 0.07 eV are strongly competing for the ground state of Ag7. However, the calculated spectrum of Ag7-I does not match the experiment well. Instead, that of Ag7-II reproduces better the experimental spectrum and the agreement between both computed and measured spectra turns out to be more convincing. The doublet between 3.6 and 3.8 eV is properly generated and can be assigned to the broad signal at 3.7 eV in the C3v spectrum. The reason for such a presence of only one isomer of a degenerate couple in the molecular beam is not clear to us. For Ag8, LC-LYP/cc-pVDZ-PP calculations predict that important absorption bands of Ag8-I occur at 4.1 and 3.1 eV that are comparable to previous predictions at 4.0 and 3.0 eV (BP86/LANL2DZ), 4.0 and 3.1 eV (B3LYP/LANL2DZ), 4.2 and 3.3 eV (EOM-CCSD), 4.0 and 3.1 eV (CASPT2) [102]. Such transitions can also be assigned for intense experimental

116 Atomic clusters with unusual structure, bonding and reactivity

FIG. 17 TD-DFT absorption spectra for the lowest-lying structures of silver Agn clusters.

peaks observed at 4.0 and 3.0 eV, respectively [14]. However, the spectrum of this form does not contain a narrow peak at 3.6 eV detected in the experimental spectrum. Instead, such transition can be derived from the D2d Ag8-II whose absorption spectrum exhibits noticeable peaks centered at 3.1, 3.8, and 4.1 eV. Thus, both the lowest-energy isomers are likely present under experimental conditions, and the observed spectrum arises from a superposition of both spectra, rather than from a sole carrier. The experimental spectrum of Ag9 contains a low intensity peak at 2.8 eV and several narrow ones between 3.3 and 4.3 eV. We actually reproduce this absorption spectrum in considering the lowest-energy Ag9-I located in this study. The latter can account for the measured bands in the region of 3.3–4.3 eV, but not for the experimental band at 2.8 eV. The experimental spectrum of Ag10 shows three distinct lines at 3.8, 4.0, and 4.2 eV, which are very similar to that of Ag9. Since all three peaks are present in the Ag9 spectrum at the same positions, a plausible explanation is that in reality the Ag9 size was observed. As shown in Fig. 17, the calculated spectrum of Ag10-I featuring three important peaks centered at 3.5, 4.1, and 4.2 eV does not reproduce well the reported spectrum in ref. [14], which does not show any transition below 3.5 eV. The absorption spectra of two lowest-lying isomers Ag11-I and Ag11-II, which are basically degenerate, are presented in Fig. 12. Overall, the former reproduces the experimental data better than the latter. Accordingly, sharp lines at 3.6 and 4.2 eV in the computed spectrum of Ag11-I can be assigned to those at 3.7 and 4.3 eV in the measured one [103,104]. The Ag11 cluster was also studied in detail by Idrobo et al. [105], and although the experimental spectrum could be due to a superposition from different isomers, the lowest-energy one brings in the most significant contribution. For Ag12, the predicted spectrum of the lowest-energy Ag12-I contains one intense peak near 3.5 eV and two lower ones at 4.0 and 4.4 eV, which are in good agreement with the experiment [14]. Accordingly, the spectrum obtained for Ag12 in Ar exhibits a very similar shape with three distinct peaks in the region of 3.4–4.4 eV. In the case of Ag13, LC-BLYP calculations for both lowest-energy structures find an intense transition at 3.6 eV and some lower ones between 4.0 and 4.5 eV. Overall, the agreement with the experiment is convincing as the measurement obtained for Ag13 consists of a well-defined peak at 3.4 eV and some lower-intensity lines between 3.9 and 4.4 eV. Positions of the distinct transitions are rather well reproduced if we take a small blue shift into account. For silver cluster containing

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14 atoms, it was experimentally found to have a very similar absorption spectrum to those of Ag12 and Ag13 with a broad band near 3.5 eV, along with two shoulders at 4.0 and 5.0 eV [14]. The spectra computed for both lowest-energy Ag14-I and Ag14-II exhibit very comparable shapes with a main transition near 3.6 eV and some less intense bands at higher energy. Next we consider the absorption spectra for the larger series of Agn (n ¼ 15–20) systems in which our LC-BLYP results can be compared with experimental data previously reported by Fedrigo and coworkers [106]. Overall, the measured spectra of these clusters are characterized by the dominance of a broad band lying between 3.2 and 3.8 eV, accompanied by some shoulders at higher energies with much lower intensities. This common feature is coupled to a slight shift to the higher energy region of the highest transition when the cluster size increases. While our predicted spectra for even systems are quite consistent with the experiment, the agreement of those of odd counterparts is less convincing. Let us look at Ag15 as for an example. While present calculations give a main transition centered at 3.8 eV for Ag15-I (Fig. 17), the experimental spectrum shows a broad band at 3.5 eV. Similarly, the predicted transitions between 2.5 and 3.3 eV in the absorption spectra of Ag17 and Ag19 are also not observed experimentally. However, the mostly intense peaks between 3.5 and 3.7 eV in the measured spectra of these clusters are well detected in the TD-DFT spectra. As reported in ref. [106], the experimental absorption spectrum of Ag18 is characterized by a dominant peak at 3.6 eV and accompanied by a shoulder at 4.1 eV. Our LC-BLYP calculations for the lowest-energy isomer Ag18-I give a main peak at 3.7 eV along with a rather lower-intensity peak at 4.1 eV, which are in very good agreement with experiment. However, the predicted transition at 3.9 eV is, nevertheless, not observed experimentally. Previous CAM-B3LYP calculations similarly detected several peaks scattered on the 3.5–4.5 eV range with two maxima at 3.6 and 4.1 eV [107]. We also regenerate the absorption spectrum of the lower-lying isomer Ag18-II and find that it differs much from that of Ag18-I and does not match the experiment. The experimental absorption spectrum of Ag20 was also recorded and reported in ref. [106]. Accordingly, the recorded spectrum is composed of a broad band near 3.7 eV and a much less intense one 4.0 eV. Our LC-BLYP computations for the Ag20-I yield a strong transition centered at 4.1 eV, which is not in line with the experiment. The computed spectrum for the tetrahedron Ag20-II with some dominant peaks at 3.7 and 4.0 eV, for its part, agrees better with the experiment. Overall, following an increase of cluster size, the absorption spectrum becomes simpler with only one main strong band, in correlation with the fact that the silver structure becomes more compact and spherical. Overall, the magic clusters represent somewhat different absorption behaviors. The UV-Vis spectra of Ag8, Ag20, and Au20 systems are in fact much simpler, and each is characterized by the dominance of a broad, highly intense transition. There is no direct correlation between the electron shell model and absorption spectra. It seems that the model is too simple to account for the complex electronic excitations of clusters.

6. Concluding remarks We report in this chapter a comprehensive review on the structural evolution, stability trend, and electronic properties of a series of small silver and gold clusters in the range size from 2 to 20 atoms. The lowest-energy structures and some basic thermodynamic parameters are either confirmed or determined using the LC-BLYP and PBE functionals in conjunction with the pseudo-potential cc-pVDZ-PP basis set. The structural evolution of both series is analyzed in terms of the electron shell model. Their optical spectra are also simulated and compared with available experimental data. A number of interesting results emerge as follows: (i) While the silver Agn early adopts three-dimensional (3D) structures at n ¼ 6, a structural transition from 2D to 3D in gold systems appears to take place at around Au10. From n ¼ 7, both Agn and Aun species exhibit very different geometries in their ground state and follow distinct growth patterns. The Agn with n ¼ 7–16 prefers 3D structures that can be constructed by adding extra Ag atoms either on the D5h bipyramid Ag7-I or the Td isomer Ag8-I. As the cluster size increases, a common trend of forming close-packed structures emerges from a 13-atom icosahedral core by adding extra atoms on triangular faces. As compared with the silver Agn, the gold Aun generally exhibits more symmetric ground state structures. A local minimum of a specific Aun system is usually generated from the lowest-lying isomer of the smaller size Aun1 by adding an extra gold atom at different spatial position. In particular, the global energy minima obtained for species with n ¼ 18–20 are formed by capping extra gold atoms on the 16-vertex Frank-Kasper Au16 core. (ii) Some basic energetic properties including binding energies per atom, the one-step fragmentation energy, and stepwise dissociation energies are also presented. Compared with experimental data, the PBE functional turns out to be more reliable in predicting the binding energies per atom and dissociation energies, whereas the LC-BLYP functional is

118 Atomic clusters with unusual structure, bonding and reactivity

likely to be more suitable for structural predictions. Analysis of basic energetics demonstrates that the sizes Au6, Ag8, Ag20, and Au20 attain remarkable thermodynamic stability with more symmetric structures than their neighbors. (iii) The stability pattern and electronic structure of these magic clusters are also rationalized and enlightened in terms of the phenomenological shell model (PSM). Accordingly, the Ag8 with eight itinerant electrons features a sphere-like form ground state (Td symmetry) and a closed 1S2/1P6 electronic configuration. Similarly, the nearly spherical Au20, which favors a tetrahedron shape (Td symmetry), also has a closed 1S2/1P6/1D10/2S2 electron shell. Consequently, such species exhibit high symmetry structures and are particularly stable members of the series examined. (iv) The computed electronic spectra of the clusters present several interesting observations. Both systems exhibit a strong optical response in the UV-visible range, but some major inherent differences appear in their spectra. While the spectra of gold clusters are normally characterized by sharply well separated peaks, those of silver clusters are dominated by a relatively broad peak, accompanied with some lower intensity absorption bands. The main absorption peaks in each series seem to appear in a similar energy region, with some small shifts. However, the spectral pattern becomes much simpler with the emergence of a dominant peak when the clusters size becomes larger and more spherical. Such differences in behavior of the two series of clusters allow a suitable choice for the cluster to be used, for example, as sensors for detecting some organic compounds and pollutants.

Funding information This work was funded by VinGroup (Vietnam) and supported by VinGroup Innovation Foundation (VinIF) under project code VINIF.2020.DA21.

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Chapter 7

Optical response properties of some metal cluster supported host-guest systems Arpita Poddara and Debdutta Chakrabortyb a

Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India, b Department of Chemistry, Birla Institute

of Technology, Mesra, Ranchi, Jharkhand, India

1. Introduction Nonlinear optical (NLO) materials could be classified as entities that upon interacting with light produce a nonlinear response. The crystal structures of NLO materials are anisotropic in the presence of electromagnetic radiation. NLO materials have applications in photonic system including high-speed optical modulators, ultrafast optical switches, and highdensity optical storage media [1]. When the intensity of the electric field is very high, it creates a very large displacement of the electrons in the material from their equilibrium position. As a result of this, anharmonic behavior comes into the picture of electronic oscillation. So the general linear relationship becomes nonlinear. The polarization (P) of the medium is a nonlinear function of the electric field (E) and it could be expressed as follows: ð1Þ

ð2Þ

ð3Þ

P ¼ wij Ej + wijk Ej Ek + wijkl Ej Ek El + ⋯

(1)

Herein, w is the electrical susceptibility. w(n) is the tensor quantity and n is the order of the process. (w(1)¼ linear polarizability, w(2), w(3)…¼ the first, second … hyperpolarizability coefficient, etc.). The nonlinearity is observed only at very high light intensities such as those provided by lasers. When P(2) ¼ w(2) ijk EjEk, it represents second-order optical nonlinearity which consists of second harmonic generation (SHG), sum/difference frequency generation (SFG/DFG), and parametric conversion. In SHG process, frequency gets doubled with respect to incident radiation. There is also third harmonic generation (THG) and higher order harmonic generation (HHG). A synchronization is required between the phase velocities of fundamental and second harmonic (SH) wave for obtaining maximum SHG. There are many inorganic and organic materials which show significant amount of NLO properties. Inorganic NLO materials such as LiNbO3 or KH2PO4 are very known to exhibit SHG effect significantly [2]. Atomically precise aggregates of metal atoms are called metal cluster. Metal clusters consist of the particles which have countable number of atoms, starting with the diatomic molecule and reaching, with an undefined upper bound of several hundred thousand atoms into the definite size range [3]. Theoretical description of clusters strongly depends on the geometry of the single cluster and the topology of the cluster sample. Size plays an important role for the formation of metal clusters. Two different types of cluster size effect can be considered [4]. One is intrinsic effect which concerns the specific changes in volume and surface material properties, whereas other is known as extrinsic effects which is mainly size-dependent responses to external fields or forces irrespective of the intrinsic effects. Intrinsic effect of metal cluster mainly focus on the changes of electronic and structural properties such as ionization potentials (IPs), polarizabilities, binding energies, chemical reactivity, hyperpolarizabilities, crystallographic structure, or optical properties as a function of particle size and geometry. Electronic shell model was developed to verify in these electronic response properties. The optical responses of clusters, in particular metal clusters are unique since pronounced changes occur across the largest size range of all known cluster effects. For large clusters, electrodynamic theory can be applied using bulk optical constants, whereas for small size clusters, the optical functions are size dependent. Due to quantum confinement effects, the properties of metal clusters are quite different from that of the molecules and bulk solids [5]. Since little can be determined about the structure and properties of the clusters a priori, accurate characterization relies on synergetic efforts between experiments and corresponding theoretical computations. In the past few decades, several progress has been made in the synthesis, characterization, and fundamental understanding of materials by using atoms as a building block of matter.

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00015-2 Copyright © 2023 Elsevier Inc. All rights reserved.

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124 Atomic clusters with unusual structure, bonding and reactivity

Clusters made of atoms with appropriate size and composition could be designed to mimic the chemistry of atoms in periodic table. Those clusters are described as superatoms. The most studied subsets of superatoms are superhalogens and superalkalies [6–8]. Superhalogens with high electron affinities (EAs) are of a great importance in chemistry since they can be used for the oxidation of counterpart (such as O2, Xe) systems with relatively high IPs. On the other hand, superalkalies are characterized by low IPs. The superalkali-superhalogen compounds with proper orientations may show large NLO responses due to the interaction between two subsets compared to the traditional alkalihalide. Since the electrons are delocalized in superatoms (e.g., the molecular orbitals (MOs) of BLi6-BF4 compound show significant resemblances to those of the NaCl molecule [6]), the compounds that consist of superatoms are easily polarizable and can be expected to show large polarizability (a) value. For a fixed superalkali, it can be shown that hyperpolarizability (b0) value decreased with increasing EAs of superhalogen/halogen atoms [6]. Computational studies exemplify the NLO response of inorganic and organic compounds doped with alkali metal atoms [9–11]. The investigations revealed that diffuse excess electrons play an important role in increasing the hyperpolarizability values. The diffusion of excess electrons can be generated by doping of a system with alkali metal atom [12,13]. The increment of b0 value is due to the generation of diffuse excess electrons which is pushed out by the valence electrons of alkali metal atom. This approach of enhancing NLO properties of a material has successfully been used to design new high performance NLO materials. Electronic and optical property analysis of doped Al12N12 nanocages approved that alkali doping can significantly increase the hyperpolarizability value [14]. Exohedral doping of alkali metal atoms (e.g., B12N12 nanocages) increases the hyperpolarizability [15]. Alkali metal doped Al12N12, B12N12, Al12P12, and B12P12 [16,17] can reduce the band gap (Eg has been calculated by Gaussian 09 program) of these complexes whereas hyperpolarizability (b0) are increased many times. Alkali metal doped phosphide and nitride nanocages reveal that the nitride nanocages are superior for NLO responses [18]. Carbon allotropes such as graphene, fullerene, nanotube, etc. have played key role in developing tunable nanostructure material [19] which have numerous applications [20,21]. Graphene has limited applications in the domain of nanoelectronics as it has a zero band gap. To overcome this limitation, chemical modification is required by introducing dopants and/or functional groups into graphene via covalent or noncovalent routes [22–24]. Monolayer boron nitride flake (BNF) is an inorganic counterpart of graphene [25–29] which has similar geometric structure with graphene although its electronic properties are quite differed from that of graphene [30]. Finite band gap is observed in BNF unlike graphene. It has less delocalized p electron distribution compared to graphene. The interaction between a surface (such as graphene) and a guest (such as a metal cluster) has become an important area of research [31–34]. Graphdiyne (GDY; which is an allotrope of carbon) contains a large delocalized p conjugated framework. The optical response properties of Li3NM @ GDY (M ¼ Li, Na, and K) systems have been studied recently [20]. Herein, we will discuss on the effect of functionalization of BNF, GR, BGR, and BNGR by some selected main group metal clusters on their optoelectronic properties. We will also discuss the impact of host-guest interactions between octa acid and some selected metal clusters on the electronic properties of OA [35].

2.

Computational details

To model the BNF surface, a finite chunk with molecular formula B58N58H28 has been considered where all the end atoms have been considered to saturate with H in order to curtail unphysical boundary effects. The metal clusters, pristine BNF as well as metal cluster supported BNF moieties have been optimized at the wb97xd/6-311G(d,p) level of theory. Harmonic vibrational frequency calculation results that all the reported geometries resides on the minima of their respective potential energy surfaces. On the other case, to model the graphene nanoflakes, a finite chunk of graphene surface has been considered with respective molecular formulae C48H18 (GR), C46H18B2 (BGR), and C42H18B3N3 (BNGR). All the end carbon atoms have been passivated with the help of H atoms. The cationic super alkali systems, graphene nanoflakes have been optimized at the wb97xd/6-31G(d,p) level of theory in concurrence with their harmonic vibrational frequency calculations. Similarly here also all the reported structures resides on the minima of their corresponding PES. In order to compute the linear and nonlinear responses of the selected systems for both the cases, average polarizability (a) and first static hyperpolarizability [36–39] (b) calculation have been done according to the standard finite field [39] approach at the wb97xd [40]/6-311G(d,p) [41] and for other case wb97xd/6-31G(d, p) level of theory. The following expressions have been used to evaluate a [42] and b [6,22]:  1
a¼ axx + ayy + azz (2) 3

Optical response properties of metal clusters Chapter

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2x + b2y + b2z

(3)

bx ¼ bxxx + bxyy + bxzz

(4)

by ¼ bxxy + byyy + byzz

(5)

bz ¼ bxxz + byyz + bzzz

(6)

b¼

3 5

where

Diagonal components of the polarizability tensor have been represented as axx/ayy/azz and the components of the hyperpolarizability tensors have been expressed as biii/bjjj/bkkk (i, j, k ¼ x, y, z). It is to be mentioned that evaluation of b is sensitive to the level of theory employed. For TDDFT calculation wb97xd/6-311 + G(d, p) level of theory has been selected for all the metal cluster supported BNF moieties, where 15 electronic excited states and only spin allowed transition have been considered. M3O+@GR/ BGR/BNGR systems wb97xd/6-31 + G (d, p) [43] level of theory has been selected. Here also spin allowed singlet to singlet transitions have considered in all 20 electronic excited states (within Franck-Condon approximation). All the computations have been performed with the help of Gaussian 09 [44] program package. To study the nature of binding between the host and guest, atoms-in-a-molecule (AIM) [45,46] analysis has also been done through Multiwfn software [47]. By using ADF software [48], EDA has been performed at the rev-PBE-D3/TZ2P level of theory by considering the selected metal clusters as one fragment and the BNF as the other one for studying the interaction between host and guest moieties. On the other case, EDA has been done for all the M3O+@GR/BGR/BNGR species by considering M3O+ as one fragment and GR/BGR/BNGR moieties as other fragment by using the same level of theory.

3. Results and discussion 3.1. Geometrical structures and thermodynamic feasibility of obtaining the corresponding host-guest moieties Firstly, the geometrical structures of the OLi4/NLi5/CLi6/ BLi7/Al12Be@BNF will be discussed and for this purpose, the minimum energy structures of those are given in Fig. 1. sp2 hybridized B and N atoms form the monolayer structure of BNF moieties. The corresponding structure is the collection of honeycomb network of hexagonal rings. Due to the adsorption of the metal clusters on the BNF surface the planarity of the host remains intact. OLi4/NLi5/CLi6/BLi7 moieties adjust itself in FIG. 1 Minimum energy structures of (A) BNF, (B) OLi4@BNF, (C) NLi5@BNF, (D) CLi6@BNF, (E) BLi7@BNF, and (F) Al12Be@BNF, respectively. (This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Effect of functionalization of boron nitride flakes by main group metal clusters on their optoelectronic properties, J. Phys. Condens. Matter 29 (42) (2017) 425201, © 2017, IOP Publishing.)

126 Atomic clusters with unusual structure, bonding and reactivity

a slightly tilted fashion with respect to the host surface in the minimum energy structure except Al12Be cluster, which interacts almost perpendicularly to the host surface. The cluster geometries are slightly deviate from their respective minimum energy structure during the cluster-BNF interactions. The formation of the cluster-BNF moieties is thermodynamically favorable as shown the data of the Gibbs free energy change and enthalpy change from Table 1. The metal clusters-BNF interaction is also energetically favorable as the zero point energy corrected dissociation energy (D0) value suggests. The optimized geometries of GR/BGR/BNGR and their respective contour line diagrams of Laplacian electron density on the molecular plane (XY) have been depicted in Fig. 2. The electron density distribution for GR suggests that it consist of polyaromatic hydrocarbons, whereas for BGR, a few localized pockets of electron rich centers have been generated instead of extended p network. From the contour line diagram of Laplacian of electron density, it can be visualized that B centers remain electron deficient for BGR and BNGR, whereas N centers act as electron rich sites in case of BNGR. Incorporating M3O+ ions (Fig. 3) on GR/BGR/BNGR, a negligible amount of distortion takes place on the host surface in case of GR. For M3O+ @ BGR/BNGR, a slight distortion has noticed on the host surface because of the presence of guest cations. The superalkali cations have lost their respective symmetry when they interact with the graphene nanoflakes. The distances between the M3O+ ions and GR/BGR/BNGR surfaces as obtained from their respective optimized geometries have been represented in Table 2. The distance between the O center of the super alkali cation and the GR surface increases as going from Li3O+ to K3O+ in case of M3O+@GR. It should be mentioned that for all cases, metal centers remain more closer than the O centers to the GR surface. This is due to the slightly greater electrostatic interaction between the electron rich GR surface and electropositive metal centers than the corresponding O center and the GR surface. In case of M3O+@BGR/BNGR, the distance between the O centers and BGR/BNGR surfaces decreases on going from Li3O+ to K3O+ which is exactly opposite with respect to the previous case. The geometrical feature suggests that the superalkali cations have oriented in a slightly tilted fashion toward the GR surface, whereas for M3O+@BGR/BNGR, the cations are situated in an outward fashion with respect to the BGR/BNGR surface. From the thermodynamic point of view, all the M3O+ functionalization processes of GR/BGR/BNGR are favorable as proven by the negative value of Gibbs free energy (DG) change. Also negative enthalpy suggests that all the concerned processes are exergonic in nature. All the M3O+ @ GR/BGR/BNGR species are forced to stay in a particular geometrical alignment because of the complexation process which indicates a significant loss of entropy. Despite of these apparent constrains in complexation processes, the above process is thermodynamically favorable and the interaction between the guest and host species is very strong. The positive zero point energy corrected dissociation energies (D0) suggests that all the M3O+ species interact with the host in a favorable way. For all super alkali cations, it has been noted that interaction between M3O+ and GR surface decreases as moving from Li3O+ to K3O+,whereas for BGR/BNGR a reverse trend is observed. TABLE 1 Free energy change (DG, kcal/mol) and enthalpy change (DH, kcal/mol) at 298.15 K for the process: Guest + BNF ! Guest@BNF; ZPE corrected dissociation energy (D0, kcal/mol) for the dissociation process: Guest@BNF ! Guest + BNF; HOMO-LUMO gap (Gap) (in eV), polarizability (a) (in a.u. [3]), and static first hyperpolarizability (b) (in a.u.); dipole moment (m) (in Debye); closest distances in between the metal cluster moieties from the host surface (RHost-Guest) (in A˚); most important stabilizing donor-acceptor interaction (E(2)) as given by secondorder perturbation theory analysis of Fock matrix in the NBO basis for the Guest@BNF moieties (in kcal/mol). Systems

DG

DH

D0

Gap

a

b

m

RHost-Guest

E(2)

OLi4@BNF

3.78

12.47

13.00

3.73

1510.34

15,030.38

11.04

N-Li ¼ 2.39

BDB-N to LP*Li ¼ 4.40

NLi5@BNF

4.65

14.12

14.62

3.64

1550.89

10,817.33

10.88

N-Li ¼ 2.33

BDN-B to LP*Li ¼ 4.19

CLi6@BNF

2.51

11.49

12.04

2.49

1589.41

9557.64

6.72

N-Li ¼ 2.41

BDN-B to LP*Li ¼ 5.54

BLi7@BNF

2.72

13.86

14.20

3.62

1714.68

23,118.74

9.34

N-Li ¼ 2.44

BDB-N to LP*Li ¼ 6.87

Al12Be@BNF

23.72

38.02

37.73

4.21

1469.49

2243.92

7.26

N-Al ¼ 2.06

LP*Al to BD*AlN ¼ 68.52

This table is reproduced from D. Chakraborty. P.K. Chattaraj, Effect of functionalization of boron nitride flakes by main group metal clusters on their optoelectronic properties, J. Phys. Condens. Matter 29 (42) (2017) 425201, © 2017, IOP Publishing.

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FIG. 2 Minimum energy structures of GR/BGR/BNGR and Laplacian of the electron density for BNGR/BGR/GR, respectively, at the XY plane (in clockwise direction). (This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Optical response and gas sequestration properties of metal cluster supported graphene nanoflakes, Phys. Chem. Chem. Phys. 18 (28) (2016) 18811–18827, © 2016, Royal Society of Chemistry.) FIG. 3 Minimum energy structures of (A) Li3O+@GR, (B) Na3O+@GR, (C) K3O+@GR, (D) Li3O+@BGR, (E) Na3O+@BGR, (F) K3O+@BGR, (G) Li3O+@BNGR, (H) Na3O+@BNGR, and (I) K3O+@BNGR, respectively. (This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Optical response and gas sequestration properties of metal cluster supported graphene nanoflakes, Phys. Chem. Chem. Phys. 18 (28) (2016) 18811–18827, © 2016, Royal Society of Chemistry.)

3.2. Optical and electronic properties of the selected metal cluster-host complexes The changes in the electronic properties during the metal cluster-host interactions are discussed below. The computed HOMO-LUMO gap, mean polarizability, and static(first) hyperpolarizability values for the functionalized BNF moieties are presented in Table 1. Due to the functionalization of the BNF, the HOMO-LUMO gap decreases and consequently mean polarizability value increases although the static hyperpolarizability values exhibit a dramatic increase. The functionalization of BNF by the main group metal clusters results a very high NLO response. Electron density distribution of host BNF

TABLE 2 Free energy change (DG, kcal/mol) and reaction enthalpy change (DH, kcal/mol) at 298 K for the process: Guest + GR/BGR/BNGR ! Guest@GR/BGR/ BNGR; ZPE corrected dissociation energy (D0, kcal/mol) for the dissociation process: Guest@GR/BGR/BNGR ! Guest + GR/BGR/BNGR; HOMO-LUMO gap (Gap) (in eV), polarizability (a) (in a.u. [3]), and static first hyperpolarizability (b) (in a.u.); NBO charges on M3O+ moieties (QK(M3O+)); distance in between the O center of the M3O+ moieties from the host surface (RO-GR/BGR/BNGR) (in A˚); distance in between the metal centers of the M3O+ moieties from the host surface (RLi/Na/K-GR/BGR/BNGR) (in A˚); most important stabilizing donor-acceptor interaction (E(2)) as given by second order perturbation theory analysis of Fock matrix in the NBO basis for the Guest@ GR/BGR/BNGR moieties (in kcal/mol). RO-GR/BGR/

RLi/Na/K-GR/BGR/

BNGR

BNGR

QLi ¼ 0.87, 0.82, 0.82 QO ¼ -1.58

O-C ¼ 2.99

Li-Cavg ¼ 2.69

BDC-C to LP*Li ¼ 1.59

953.91

QNa ¼ 0.84, 0.85, 0.87 QO ¼ 1.58

O-C ¼ 3.17

Na-Cavg ¼ 2.90

LPO to RY*C ¼ 0.77

735.91

3115.38

QK ¼ 0.85, 0.83, 0.85 QO ¼ 1.53

O-C ¼ 3.32

K-Cavg ¼ 3.23

BDC-C to LP*K ¼ 0.73

2.72

678.63

18,838.26

QLi ¼ 0.89, 0.84, 0.89 QO ¼ 1.24

O-B ¼ 1.59

Li-Cavg ¼ 2.37

LPO to LP*B ¼ 431.61

75.66

3.02

834.52

30,957.28

QNa ¼ 0.93, 0.92, 0.90 QO ¼ 1.24

O-B ¼ 1.55

Na-Cavg ¼ 2.49

LPO to LP*B ¼ 464.12

87.93

88.22

3.06

829.27

32,546.06

QK ¼ 0.93, 0.94, 0.92 QO ¼ 1.19

O-B ¼ 1.52

K-Cavg ¼ 2.83

LPO to LP*B ¼ 500.34

37.99

51.14

50.84

3.57

737.99

6239.62

QLi ¼ 0.89, 0.89, 0.89 QO ¼ 1.32

O-B ¼ 1.55

Li-C ¼ 2.15, Li-Navg ¼ 2.11

LPO to LP*B ¼ 300.05

Na3O+@BNGR

52.49

66.19

66.28

3.52

751.43

6670.73

QNa ¼ 0.92, 0.93, 0.92 QO ¼ 1.29

O-B ¼ 1.49

Na-C ¼ 2.47, Na-Navg ¼ 2.46

LPO to LP*B ¼ 342.93

K3O+@BNGR

66.47

79.66

79.81

3.46

769.10

7847.28

QK ¼ 0.94, 0.94, 0.93 QO ¼ 1.24

O-B ¼ 1.45

K-C ¼ 2.81, K-Navg ¼ 2.81

LPO to LP*B ¼ 433.18

Systems

DG

DH

D0

Gap

a

b

QK(M3O+)

Li3O @GR

30.65

42.20

42.50

3.94

700.06

841.66

Na3O+@GR

26.28

37.81

38.41

3.95

707.76

K3O+@GR

22.42

33.94

34.56

3.38

Li3O+@BGR

49.75

63.43

63.15

Na3O+@BGR

61.87

75.50

K3O+@BGR

74.94

Li3O+@BNGR

+

E(2)

This table is reproduced from D. Chakraborty, P.K. Chattaraj, Optical response and gas sequestration properties of metal cluster supported graphene nanoflakes, Phys. Chem. Chem. Phys. 18 (28) (2016) 18811–18827, © 2016, Royal Society of Chemistry.

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has been distorted due to the functionalization of guest species which results a significant increase in linear as well as nonlinear response properties in presence of external static electric field as compared to that in pristine BNFs. TDDFT analysis (Table 3) has been performed to justify the observed increase in static hyperpolarizability values for the functionalized BNF moieties. From the qualitative two state model studies, it has been observed that b is proportional to the oscillator strength ( f ), where it is inversely proportional to the electronic transition energy (DE) associated with the particular transition. TABLE 3 TDDFT results for BNF and Guest@BNF moieties where the various important excited states, oscillator strengths (f) associated with that particular electronic transition, electronic transition energy (DE) associated (in eV) with that particular electronic transition and the percentage contribution of the dominant electronic transitions have been presented. Systems

Excited state number

f

DE

Dominant transition

BNF (HOMO ¼ 362, LUMO ¼ 363)

(1) 2

0.7766

6.57

359–>365 (7.67%) 359–>363 (7.60%) 354–>364 (7.15%)

(2) 13

1.2299

7.07

361–>366 (5.19%) 359–>366 (5.27%) 359–>371 (4.81%)

(1) 2

0.3921

1.37

372–>374 (99.95%)

(2) 3

0.5171

1.69

372–>375 (97.97%)

(1) 1

0.3776

1.25

373–>374 (99.90%)

(2) 3

0.3753

1.51

373–>376 (86.40%)

(1) 1

0.3453

1.27

374–>375 (98.22%)

(2) 2

0.3731

1.40

374–>376 (98.45%)

(1) 1

0.2168

1.06

375–>376 (74.40%)

(2) 5

0.2241

1.41

375–>380 (63.18%)

(1) 11

0.0006

2.16

441–>444 (20.20%) 441–>446 (21.90%) 442–>445 (21.61%)

(2) 15

0.0048

2.27

440–>444 (35.72%) 441–>446 (19.05%) 442–>447 (9.98%)

OLi4@BNF (HOMO ¼ 372, LUMO ¼ 373)

NLi5@BNF (HOMO ¼ 373, LUMO ¼ 374)

CLi6@BNF (HOMO ¼ 374, LUMO ¼ 375)

BLi7@BNF (HOMO ¼ 375, LUMO ¼ 376) Al12Be@BNF (HOMO ¼ 442, LUMO ¼ 443)

This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Effect of functionalization of boron nitride flakes by main group metal clusters on their optoelectronic properties, J. Phys. Condens. Matter 29 (42) (2017) 425201, © 2017, IOP Publishing.

130 Atomic clusters with unusual structure, bonding and reactivity

As a result of functionalization, DE associated with the crucial electronic transition get reduced by a significant amount, particularly for BLi7 cluster, it has reduced DE value at a very high extent. Therefore, BLi_7 @BNF shows highest b value. On the other hand, Al12Be@BNF shows highest DE value associated with that particular transition which results least NLO response among all the functionalized BNF moieties. Now the optical properties of M3O+ @ GR/BGR/BNGR complexes will be discussed and for this purpose HOMO-LUMO gap, a, and b values are presented in Table 2. It has been noticed that due to the presence of M3O+ species, the HOMO-LUMO gap decreases from their respective pristine GR/BGR/BNGR. For all the cases of M3O+ @ GR, both a and b values increases to a significant extent as compared to the pristine GR. But for M3O+ @ BGR/BNGR moieties, the variation is observed as the a and b values are concerned. In case of M3O+ @ BGR, a value decreases as compared to pristine BGR where b increases sharply. On the other hand, a reverse trend is observed for a and b values in case of M3O+ @ BNGR. To justify the observed b values, TDDFT analysis has been done (Table 4). From this result, it can be observed that for BGR and BNGR, it exhibit decrease in the DE associated with the crucial electronic transition in comparison with the electronic transition in case of GR. Therefore, BGR/BNGR exhibit significant NLO response than GR. Also, the reasonable oscillator strength (f) values associated with the electronic transition demonstrate the higher b values of BGR/BNGR moieties than GR. Among the BGR and BNGR moieties, since BGR exhibits smaller DE value associated with the crucial electronic transition and also smaller f values in comparison with BNGR. That is why BGR shows a smaller b value than BNGR. Now by incorporating M3O+ cations on GR, a significant increase of b values has been noted. b values increases sharply as going from Li3O+ to K3O+. This is due to reduce the DE value associated with the crucial electronic transitions for M3O+ @ GR as compared to pristine GR. Among the M3O+ species, K3O+ moieties act as the most efficient guest and that is why it exhibit the largest b value within this particular series. So, it can be concluded that K3O+ cation acts as a most powerful guest to increase the NLO response of the respective host moieties. Absorption spectrum analysis for pristine BNFs revealed that BNFs can only absorb radiation in the UV-vis domain. But the functionalized BNFs can absorb the radiation in UV-vis as well as IR domains. The intensity of two bands is different in case of OLi4/NLi5/CLi6/BLi7 @ BNF moieties. The intensity of the band in IR domain is more intense than the other one. But for Al12Be@BNF the intensity of the absorption band is more which is in UV-vis domain. On the other case, the M3O+ @ BGR/BNGR moieties absorb radiation in the UV-vis as well as IR domain, whereas M3O+ @ GR moieties absorb radiation mostly in UV-vis domain. So, M3O+ @ BGR/BNGR shows good NLO response than M3O+ @ GR. It should be mentioned that the materials which absorb radiation in IR domain, can be used in photovoltaic cells. Therefore, the chemical moieties that can absorb radiation in the IR domain have great importance. For all functionalized BNF moieties, overlap between the orbitals of guest and host takes place where the host species contribute through the P orbitals of the constituent atoms. TDOS plots in case of M3O+ @ GR moieties reveals that, due to the presence of M3O+ ions, it lowers the Fermi levels (EF) of the concerned moieties as compared to the EF value of pristine GR. Li3O+ reduces maximum EF value followed by Na3O+ and K3O+ for M3O+ @ GR species. In case of M3O+ @ GR due to the functionalization, the DOS values increase slightly near the Fermi levels as compared to pristine GR. Similar trend is observed for M3O+ @ BGR species where in additional guest ions reduce EF values of the complex as compared to pristine BGR. Here it should be noted that lighter Li3O+ species can more efficiently reduce EF values as compared to its heavier analogues. OPDOS values suggest that, within 19 to 12 eV energy range, the binding between the host and guest is most favorable. In this energy range, Li3O+ and Na3O+ both bind strongly with the host BGR, whereas for K3O+, the binding energy domain increases to 21.77 to 11 eV. For M3O+ @ BNGR the trend is almost similar with the previous one. Here one point should be noted that, lighter metal clusters interact with BNGR in a slightly smaller energy domain than the corresponding heavier metal clusters.

3.3. AIM analysis AIM theory analyzes the nature of the bond critical points (BCPs) with the help of local electron density based descriptors viz. the magnitude of electron density (r(rc)), Laplacian of electron density (r2r(rc)), local electron energy density (H(rc)) as well as ratio of local kinetic energy density (G(rc)) and local potential energy density (V(rc)). Generally r2r(rc) > 0 and H(rc) < 0 suggest the partial covalent nature of bonding. On the other hand, r2r(rc) < 0 indicates the nature of covalent bonding. If ( G(rc)/V(rc)) < 1, then also the bonding is classified as partially covalent type. The extent of localization of electron density within a specified region of space can be studied by analyzing the electron localized function (ELF). Following these information, it has been concluded that OLi4/NLi5/CLi6/BLi7 clusters interact with the BNF moieties in a noncovalent manner and Al12Be cluster interacts with the host in a partly covalent manner. AIM result of these compounds is depicted in Table 5.

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TABLE 4 TDDFT results computed at the wb97xd/6-31 + G(d,p) level for GR/BGR/BNGR and M3O+@GR/BGR/BNGR moieties where the various important excited states, oscillator strengths (f) associated with that particular electronic transition, electronic transition energy (DE) associated (in eV) with that particular electronic transition, the percentage contribution of the dominant electronic transitions, and the maximum wavelength (lmaxin nm) of the p and b bands in the absorption spectra have been presented. Excited state number

f

DE

GR (HOMO ¼ 153, LUMO ¼ 154)

(a) 1 (b) 9

0.5406 0.3356

1.72 3.72

153–>154 (97.59%) 153–>160 (40.99%) 148–>154 (14.81%) 153–>157 (14.03%) 150–>154 (13.23%)

pmax ¼ 722.07 bmax ¼ 294.24

BGR (HOMO ¼ 152, LUMO ¼ 153)

(a) 1 (b) 7

0.0784 0.3539

0.54 2.76

pmax ¼ 2296.11

(c) 18

0.2951

3.75

152–>153 (94.43%) 148–>153 (45.83%) 151–>154 (28.73%) 152–>157 (45.61%) 149–>154 (11.91%)

(a) 1 (b) 8

0.4054 0.3622

1.40 3.47

pmax ¼ 886.92 bmax ¼ 298.84

(c) 12

0.4555

3.95

(d) 13

0.6780

4.15

153–>154 (96.04%) 151–>154 (34.28%) 149–>154 (19.76%) 153–>157 (16.35%) 153–>159 (22.93%) 150–>154 (17.50%) 152–>155 (10.91%) 153–>162 (30.77%) 152–>155 (21.91%) 153–>159 (7.41%)

Li3O+@GR (HOMO ¼ 161, LUMO ¼ 162)

(a) 1 (b) 18

0.4984 0.5508

1.68 4.17

161–>162 (97.05%) 152–>162 (29.65%) 160–>165 (7.70%) 160–>164 (6.71%)

pmax ¼ 738.68 bmax ¼ 297.11

Na3O+@GR (HOMO ¼ 173, LUMO ¼ 174)

(a) 1 (b) 2

0.0066 0.4872

1.50 1.70

172–>174 (94.93%) 173–>174 (94.75%)

pmax ¼ 731.24 bmax ¼ 338.37

K3O+@GR (HOMO ¼ 185, LUMO ¼ 186)

(a) 1 (b) 4

0.0034 0.4919

0.82 1.70

185–>186 (97.11%) 182–>186 (96.88%)

pmax ¼ 730.42 bmax ¼ 454.86

Li3O+@BGR (HOMO ¼ 160, LUMO ¼ 161)

(a) 1 (b) 2

0.0012 0.1461

0.35 1.59

160–>161 (91.99%) 160–>162 (65.60%) 159–>161 (23.51%)

pmax ¼ 779.68 bmax ¼ 376.27

Na3O+@BGR (HOMO ¼ 172, LUMO ¼ 173)

(a) 1 (b) 2 (c) 15

0.0939 0.2087 0.0491

0.61 1.69 3.34

172–>173 (87.56%) 172–>174 (79.60%) 167–>173 (22.59%) 168–>173 (21.94%) 172–>180 (16.31%)

pmax ¼ 2017.48 bmax ¼ 376.97

K3O+@BGR (HOMO ¼ 184, LUMO ¼ 185)

(a) 1 (b) 2 (c) 13

0.1139 0.1863 0.1317

0.70 1.69 3.25

184–>185 (87.19%) 184–>186 (77.98%) 184–>192 (53.46%) 184–>189 (4.90%)

pmax ¼ 1774.04 bmax ¼ 380.95

Li3O+@BNGR (HOMO ¼ 161, LUMO ¼ 162)

(a) 1 (b) 9

0.3445 0.2955

1.23 3.35

pmax ¼ 1005.30

(c) 15

0.4530

3.85

161–>162 (95.21%) 159–>162 (52.55%) 157–>162 (17.37%) 158–>162 (21.39%) 161–>170 (18.39%) 157–>162 (16.49%)

(a) 1 (b) 11

0.3313 0.2996

1.19 3.39

173–>174 (94.97%) 171–>174 (50.77%)

pmax ¼ 1038.52 bmax ¼ 309.33

Systems

BNGR (HOMO ¼ 153, LUMO ¼ 154)

Na3O+@BNGR (HOMO ¼ 173, LUMO ¼ 174)

Dominant transition

lmax

bmax ¼ 330.95

bmax ¼ 322.29

Continued

132 Atomic clusters with unusual structure, bonding and reactivity

TABLE 4 TDDFT results computed at the wb97xd/6-31 + G(d,p) level for GR/BGR/BNGR and M3O+@GR/BGR/BNGR moieties where the various important excited states, oscillator strengths (f) associated with that particular electronic transition, electronic transition energy (DE) associated (in eV) with that particular electronic transition, the percentage contribution of the dominant electronic transitions, and the maximum wavelength (lmaxin nm) of the p and b bands in the absorption spectra have been presented—cont’d Systems

K3O @BNGR (HOMO ¼ 185, LUMO ¼ 186) +

Excited state number

f

DE

(c) 16

0.2975

3.83

(d) 19

0.4121

4.01

(a) 1 (b) 12

0.3101 0.3157

1.14 3.39

(c) 17

0.2503

3.82

Dominant transition

lmax

173–>179 (12.21%) 169–>174 (25.69%) 173–>183 (13.61%) 172–>180 (14.30%) 173–>183 (10.50%) 185–>186 (94.56%) 183–>186 (48.03%) 185–>191 (14.11%) 181–>186 (20.51%) 185–>205 (12.73%)

pmax ¼ 1085.21 bmax ¼ 312.83

This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Optical response and gas sequestration properties of metal cluster supported graphene nanoflakes, Phys. Chem. Chem. Phys. 18 (28) (2016) 18811–18827, © 2016, Royal Society of Chemistry.

TABLE 5 Electron density descriptors (in a.u.) at the bond critical points (BCP) for the Guest@BNF moieties. Systems

BCP

r(rc)

—2r(rc)

H(rc)

2G(rc)/V(rc)

ELF

OLi4@BNF

LidN

0.011

0.062

0.003

1.309

0.016

NLi5@BNF

LidN

0.013

0.071

0.003

0.987

0.019

CLi6@BNF

LidN

0.011

0.058

0.003

1.305

0.016

BLi7@BNF

LidN

0.009

0.052

0.003

1.306

0.014

Al12Be@BNF

AldN

0.051

0.231

0.005

0.924

0.097

This table is reproduced from D. Chakraborty, P.K. Chattaraj, Effect of functionalization of boron nitride flakes by main group metal clusters on their optoelectronic properties, J. Phys. Condens. Matter 29 (42) (2017) 425201, © 2017, IOP Publishing.

On the other hand for all M3O+ @ GR cases, the interaction between host and guest is noncovalent manner. All the concerned descriptors (Table 6) suggest that the only stabilizing interaction in the M3O+ @ GR species is van der Waals type. In case of M3O+ @ BGR/BNGR systems, the O atom of the cations interacts with the B centers of the host system in a partially covalent fashion. The interaction strength increases as going from Li3O+ to K3O+ for both GR, BGR, and BNGR cases as specified by the increasing value of r(rc) as well as ELF around the OdB BCPs. For all cases, the metal interacts with the host in a noncovalent manner. From the thermodynamic point of view, M3O+ @ BGR/BNGR moieties are more stable compared to M3O+ @ GR moieties because of the covalent nature of OdB BCPs.

3.4. EDA study Electron decomposition analysis results the nature of interaction between the host and guest moieties. The total interaction energy (DEtotal) can be divided into different components viz. Pauli repulsion energy (DEpauli), electrostatic interaction energy (DEel), orbital energy (DEorb), and dispersion energy (DEdisp). The polarization effect as well as charge transfer effect coming from the orbital interaction term whereas the electrostatic interaction term tells about the coulomb type of interaction between the various atom centers. When the two electrons come closer, the Pauli repulsion term plays an important role. On the other hand, dispersion energy term represents the noncovalent forces which operate on the system. In our first case study, by performing EDA (Table 7), it can be noted that OLi4/NLi5/CLi6/BLi7@BNF complexes can be stabilized through the electrostatic interaction. For all cases, the relative contribution of the magnitude of the orbital

Optical response properties of metal clusters Chapter

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133

TABLE 6 Electron density descriptors (in a.u.) at the bond critical points (BCP) for the Guest@GR/BGR/BNGR moieties. Systems

BCP

r(rc)

—2r(rc)

H(rc)

2G(rc)/V(rc)

ELF

Li3O @GR

OdC LidC

0.01 0.01

0.04 0.03

0.00 0.00

1.03 1.31

0.06 0.01

Na3O+@GR

OdC NadC

0.01 0.01

0.03 0.03

0.00 0.00

1.16 1.27

0.04 0.01

K3O+@GR

OdC KdC

0.01 0.01

0.02 0.03

0.00 0.00

1.25 1.36

0.03 0.02

Li3O+@BGR

OdB LidC

0.12 0.02

0.39 0.08

0.08 0.00

0.69 1.25

0.18 0.04

Na3O+@BGR

OdB NadC

0.13 0.02

0.44 0.09

0.09 0.00

0.70 1.27

0.19 0.03

K3O+@BGR

OdB KdC

0.14 0.02

0.49 0.01

0.09 0.00

0.70 1.24

0.20 0.04

Li3O+@BNGR

OdB LidN LidC

0.13 0.02 0.02

0.43 0.14 0.12

0.09 0.01 0.01

0.69 1.28 1.24

0.20 0.03 0.03

Na3O+@BNGR

OdB NadN NadC

0.15 0.02 0.02

0.55 0.09 0.09

0.10 0.00 0.00

0.70 1.25 1.23

0.21 0.03 0.03

K3O+@BNGR

OdB KdN KdC

0.17 0.02 0.02

0.67 0.07 0.09

0.11 0.00 0.00

0.71 1.20 1.19

0.21 0.04 0.04

+

This table is reproduced from D. Chakraborty, P.K. Chattaraj, Optical response and gas sequestration properties of metal cluster supported graphene nanoflakes, Phys. Chem. Chem. Phys. 18 (28) (2016) 18811–18827, © 2016, Royal Society of Chemistry.

TABLE 7 EDA results of Guest@BNF studied at the revPBE-D3/TZ2P level (kcal/mol). Systems

Fragments

DEel

DEPauli

DEorb

DEdisp

DEtot

OLi4@BNF

[OLi4] + [BNF]

63.44

78.57

16.02

16.22

17.11

NLi5@BNF

[NLi5] + [BNF]

67.89

84.11

18.83

16.38

19.00

CLi6@BNF

[CLi6] + [BNF]

65.98

83.15

16.67

17.90

17.40

BLi7@BNF

[BLi7] + [BNF]

89.69

116.32

23.87

22.35

19.59

Al12Be@BNF

[Al12Be] + [BNF]

156.96

193.52

73.64

26.63

63.71

This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Effect of functionalization of boron nitride flakes by main group metal clusters on their optoelectronic properties, J. Phys. Condens. Matter 29 (42) (2017) 425201, © 2017, IOP Publishing.

interaction and dispersion interaction are quite comparable. For all cases, the relative contribution of orbital interaction and dispersion interaction are quite comparable in terms of its magnitude. For Al12Be@BNF moieties, the contribution of orbital interaction is greater than the dispersion interaction. This is due to the partial covalent type of interaction between Al center of the guest and N center of the host species. The DEtotal value suggests that all the host guest complexes are energetically stable. On the other case, since the interaction between the electron deficient guest moiety and the electron rich host moiety takes place thus it is obvious that, electrostatic interaction plays the important role to stabilize the system. Also DEorb plays a significant role because, the bonds which are formed contains some covalent type character particularly for M3O+ @ BGR/BNGR cases. All the EDA results are depicted in Table 8. In case of M3O+ @ GR, the main stabilizing parameters are DEel and DEorb. Moving from Li3O+ to K3O+, contribution of DEdisp increases as the distance between

134 Atomic clusters with unusual structure, bonding and reactivity

TABLE 8 EDA results of guest@GR/BGR/BNGR studied at the revPBE-D3/TZ2P//wb97xd/6-31G(d,p) level (kcal/mol). The values in bracket are the contribution in percentage toward the total attraction (DEel + DEorb + DEdisp). Systems

Fragments

DEelstat

DEPauli

DEorb

DEdisp

DEtot

Li3O @GR

[Li3O ] + [GR]

23.64 (35.32%)

32.58

32.66 (48.79%)

10.64 (15.89%)

34.36

Na3O+@GR

[Na3O+] + [GR]

25.23 (39.94%)

32.64

20.48 (32.42%)

17.46 (27.64%)

30.52

K3O+@GR

[K3O+] + [GR]

18.78 (34.36%)

27.49

18.62 (34.07%)

17.25 (31.56%)

27.16

Li3O+@BGR

[Li3O+] + [BGR]

179.95 (51.15%)

281.88

161.59 (45.93%)

10.24 (2.91%)

69.90

Na3O+@BGR

[Na3O+] + [BGR]

185.89 (49.51%)

289.82

175.15 (46.65%)

14.44 (3.85%)

85.65

K3O+@BGR

[K3O+] + [BGR]

188.45 (47.01%)

303.68

197.98 (49.38%)

14.47 (3.61%)

97.21

Li3O+@BNGR

[Li3O+] + [BNGR]

191.42 (50.61%)

302.65

172.28 (45.55%)

14.50 (3.83%)

75.56

Na3O+@BNGR

[Na3O+] + [BNGR]

210.60 (50.67%)

321.68

188.25 (45.29%)

16.78 (4.04%)

93.94

K3O+@BNGR

[K3O+] + [BNGR]

223.86 (48.76%)

346.63

218.20 (47.53%)

17.04 (3.71%)

112.47

+

+

This figure is reproduced from D. Chakraborty, P.K. Chattaraj, Optical response and gas sequestration properties of metal cluster supported graphene nanoflakes, Phys. Chem. Chem. Phys. 18 (28) (2016) 18811–18827, © 2016, Royal Society of Chemistry.

the guest and host moieties decreases. DEel shows an inverse relationship with respect to the distance between the concerned moieties. For M3O+ @ BGR/BNGR moieties, the principle stabilizing parameters are DEel and DEorb, whereas the contribution from DEdisp decreases significantly. This is due to the partial covalent type of binding between the host and guest species. In this case, as moving from Li3O+ to K3O+, DEtotal increases, whereas a reverse trend is observed in case of M3O+ @ GR. For all the cases, DEpauli term is the main destabilizing parameter, the magnitude of which decreases as moving from Li3O+ to K3O+ in case of GR and a reverse trend is observed in case of BGR/BNGR. This is mainly due to the distance factor between the host and guest moieties.

3.5. TDDFT analysis of the guest@OA complexes TDDFT calculations have been carried out at the wb97xd/6-311 + G(d,p) level of theory to study the effect of guest encapsulation on the various occupied as well as low lying excited states of OA. The cations Li3O+, Na3O+, and K3O+ have been considered as guest moieties in this case. For TDDFT calculation, 12 electronic excited states have been considered. The TDDFT result for guest@OA complexes is presented in Table 9. In case of Li3O+@OA, the changes observed for DE, f as well as the absorption spectrum are small compared to pristine OA. For Na3O+/K3O+ metal clusters, DE associated with the particular transitions decreases at a significant amount. As a result, the absorption maxima shift toward the infrared domain although a less intense band is observed near the UV-visible domain. Therefore, Na3O+/ K3O+@OA could be gauged by studying their corresponding absorption spectra. Na3O+ and K3O+ decrease the HOMO-LUMO gap of OA to the maximum extent. These moieties also lower the DE value to the maximum extent and, therefore, the absorption spectrum of OA is mostly influenced by these cationic species. The occupied as well as unoccupied MOs are primarily located on the benzene rings of the outer cavity in case of pristine OA which takes part in the electronic transitions. In case of Li3O+@OA, the occupied MOs are mainly composed of contribution from OA moiety and they are located at the outer cavity. In the unoccupied MOs, the orbital overlap between OA and metal cluster is observed. In case of Na3O+/ K3O+@OA, occupied MOs control the crucial electronic transition of the metal cluster and the overlap between Na3O+/ K3O+ and OA orbitals is noted in the unoccupied MOs only. From this above discussion, it can be concluded that as a result

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135

TABLE 9 TDDFT results computed at the wb97xd/6-311 + G(d,p) level for OA and Guest@OA moieties where the various important excited states, oscillator strengths (f) associated with that particular electronic transition, electronic transition energy (DE) associated (in eV) with that particular electronic transition, the percentage contribution of the dominant electronic transitions have been presented. Systems

Excited state number

f

DE

Dominant transition

OA (HOMO ¼ 448, LUMO ¼ 449)

(1) 2

0.2063

4.96

433–>450 (9.77%) 432–>449 (6.05%)

(2) 7

0.1528

5.07

444–>456 (8.77%) 445–>449 (7.63%)

(1) 7

0.1945

4.96

456–>471 (7.92%) 456–>463 (5.90%)

(2) 10

0.1753

4.99

452–>468 (8.14%) 452–>470 (7.63%)

(1) 1

0.0541

2.99

468–>474 (51.54%) 468–>469 (21.68%)

(2) 3

0.0548

3.45

466–>474 (50.73%) 466–>469 (22.39%)

(1) 1

0.0413

2.90

480–>485 (36.13%) 480–>488 (18.77%)

(2) 2

0.0300

3.01

479–>485 (35.47%) 479–>488 (18.69%)

Li3O+@OA (HOMO ¼ 456, LUMO ¼ 457)

Na3O+@OA (HOMO ¼ 468, LUMO ¼ 469)

K3O+@OA (HOMO ¼ 480, LUMO ¼ 481)

This table is reproduced from D. Chakraborty, P.K. Chattaraj, Host–guest interactions between octa acid and cations/nucleobases, J. Comput. Chem. 39 (3) (2018) 161–175, © 2018, John Wiley and Sons.

of guest encapsulation, the appropriate orbitals that involve in the crucial electronic transitions of OA are modulated to a magnificent extent and as a result of which the intramolecular charge-transfer within OA changes. For Na3O+, K3O+ encapsulation, the intermolecular charge transfer is greater in magnitude compared to other one.

4. Conclusion Based on the discussions stated in this article, we can say that metal clusters could be successfully utilized in order to tune the optical properties of some host molecules. The reason for that could be attributed to the fact that the electronic transition energy as well as oscillator strengths of crucial electronic transitions get suitably tuned to facilitate optical response properties, as corroborated by the results obtained from TDDFT calculations. Metal clusters could help the host molecules

136 Atomic clusters with unusual structure, bonding and reactivity

to absorb light covering both the UV-vis as well as the IR domains. Therefore, these host-guest systems could be utilized to harness solar radiation which in turn could be useful in several applications. Further analysis in this direction should be carried out so that the full potential of the metal clusters in tuning the optical response properties of host systems could be realized.

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[30] D. Chakraborty, P.K. Chattaraj, Effect of functionalization of boron nitride flakes by main group metal clusters on their optoelectronic properties, J. Phys. Condens. Matter 29 (42) (2017), 425201. [31] J.N. Tiwari, V. Vij, K.C. Kemp, K.S. Kim, Engineered carbon-nanomaterial-based electrochemical sensors for biomolecules, ACS Nano 10 (1) (2016) 46–80. [32] S. Sahoo, M.E. Gruner, S.N. Khanna, P. Entel, First-principles studies on graphene-supported transition metal clusters, J. Chem. Phys. 141 (7) (2014), 074707. [33] Y. Zhang, K. Zhou, K. Xie, H. Zhang, Y. Peng, C. Wang, Tuning the magnetic and transport properties of metal adsorbed graphene by co-adsorption with 1, 2-dichlorobenzene, Phys. Chem. Chem. Phys. 14 (33) (2012) 11626–11632. [34] S. Tang, W. Wu, J. Yu, Interfacial interaction of Ag nanoparticles with graphene oxide supports for improving NH3 and no adsorption: a firstprinciples study, Phys. Chem. Chem. Phys. 18 (11) (2016) 7797–7807. [35] D. Chakraborty, P.K. Chattaraj, Host–guest interactions between octa acid and cations/nucleobases, J. Comput. Chem. 39 (3) (2018) 161–175. [36] P. Karamanis, N. Otero, C. Pouchan, Unleashing the quadratic nonlinear optical responses of graphene by confining white-graphene (h-BN) sections in its framework, J. Am. Chem. Soc. 136 (20) (2014) 7464–7473. [37] Z.J. Zhou, X.P. Li, F. Ma, et al., Exceptionally large second-order nonlinear optical response in donor–graphene nanoribbon–acceptor systems, Chem. Eur. J. 17 (8) (2011) 2414–2419. [38] P. Karamanis, C. Pouchan, Second-hyperpolarizability (g) enhancement in metal-decorated zigzag graphene flakes and ribbons: the size effect, J. Phys. Chem. C 117 (6) (2013) 3134–3140. [39] D.R. Kanis, M.A. Ratner, T.J. Marks, Design and construction of molecular assemblies with large second-order optical nonlinearities. Quantum chemical aspects, Chem. Rev. 94 (1) (1994) 195–242. [40] J.D. Chai, M.H. Gordon, Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections, Phys. Chem. Chem. Phys. 10 (2008) 6615–6620. [41] R. Krishnan, J.S. Binkley, R. Seeger, J.A. Pople, Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions, J. Chem. Phys. 72 (1980) 650. [42] P.K. Chattaraj, U. Sarkar, D.R. Roy, Electronic structure principles and aromaticity, Chem. Rev. 84 (2) (2007) 354. [43] M.J. Frisch, et al., Self-consistent molecular orbital methods 25. Supplementary functions for Gaussian basis sets, J. Chem. Phys. 80 (1984) 3265. [44] M.J. Frisch, et al., Gaussian09, Revision C.01, 2010. [45] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Oxford, Clarendon, 1990. [46] D. Cremer, E. Kraka, Chemical bonds without bonding electron density—does the difference electron-density analysis suffice for a description of the chemical bond? Angew. Chem. Int. Ed. 23 (8) (1984) 627–628. [47] T. Lu, F.W. Chen, Multiwfn: a multifunctional wavefunction analyzer, J. Comput. Chem. 33 (5) (2012) 580–592. [48] G. te Velde, et al., Chemistry with ADF, J. Comput. Chem. 22 (9) (2001) 931–967.

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Chapter 8

Group III–V hexagonal pnictide clusters and their promise for graphene-like materials Esha V. Shah and Debesh R. Roy Materials and Biophysics Group, Department of Physics, Sardar Vallabhbhai National Institute of Technology, Surat, India

1. Introduction Atomic clusters are aggregation of atoms ranging from two to few hundreds whose properties exclusively depends on size, shape, charged state, etc. An atomic cluster can be homoatomic, i.e., all the atoms of the cluster are made of same element, and it can also be heteroatomic, i.e., atoms of different elements combine to form a cluster. Atomic clusters are subnanoscale quantum systems synthesized in gas phase. Although all of the ground state and meta-stable state structure of the atomic clusters are not geometrically well symmetric, but many of the atomic clusters have reportedly shown good symmetry [1]. Atomic clusters are confined in all the three directions and do not interact with the basis atoms of the subsequent clusters. The different atomic clusters present in gas phase are well isolated from each other and only bind with atoms within the cluster. A cluster assembled material is a bottom-up approach of designing novel nanomaterials to obtain distinct properties. In this case, a nanomaterial is built up from subnanoscale to nanoscale, i.e., from atomic clusters to nanowires, nanosheets, and even bulk. First, an atomic cluster study is executed for a targeted compound, which then aids in identifying the degree of stability of several possible geometrical configuration of the compound under study. It is also reported in past that stability of atomic clusters often preferred in aromatic clusters [2–5]. The most stable configuration of the atomic cluster can then be used as a building block for the modeling of nanomaterials in one, two and three dimensions. Hence the name ‘cluster assembled materials’, where the assembly of an atomic cluster fashions the materials and its properties. Nevertheless, the suitable atomic cluster and its cluster assembled materials will always be expected to exhibit unique and dimensionality specific properties. Some of the well-referred studies on cluster assembled materials include work by Y. Song et al. where they report synthesis and properties of Au60 cluster assembled material [6]. The building block atomic cluster of the novel material consists of five icosahedral Au13 units linked by Se atoms [2]. Another gold nanocluster assembled material was reported by Shichibu et al. [3] They reported the study of electronic levels of gold nanocluster complex [Au25(PPh3)10(SCnH2n+1)5Cl2]2+ (n ¼ 2–18) when building from nanocluster to cluster assembled material [7]. The work by J-Y. Wang et al. reports interesting tailoring of two silver nanocluster complex assembled material; Ag12bpa and Ag12bpe [bpa ¼ 1,2-bis(4-pyridyl) ethane; bpe ¼ 1,2-bis(4-pyridyl)ethylene] which exhibit the unique property of cluster controlled luminescence [8]. J. Zhao and R-H Xie report Na6Pb clusters assembled materials exhibiting inert nature [9]. The sodium-lead atomic clusters were found to be stable and plentiful in an experimental mass spectra study by Yeretzian et al. [10] The M12N12 (M ¼ Al, Ga) atomic cluster assembled two-dimensional nanowires exhibit gas sensing properties, whereas three-dimensional materials show wide band semiconducting properties as investigated by Yong et al. [11,12]. In the present chapter, the group IIIA pnictide atomic clusters are discussed for their stability in hexagonal symmetry and compared with graphene. Primarily, growth at the clusters level is scrutinized till five units and further the most stable cluster compounds are focused on as they form building motifs for respective cluster assembled nanomaterials. The atomic clusters particularly (MX)2n+1H2n+4 [M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As; n ¼ 1–5] and cluster assembled layered nanomaterials InN and TlN are discussed under the frame work of density functional theory. With the goal to design a novel inorganic graphene-like material through bottom-up approach, the structure, stability, and essential electronic properties of

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00009-7 Copyright © 2023 Elsevier Inc. All rights reserved.

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Atomic clusters with unusual structure, bonding and reactivity

polyacenes, Cn [n ¼ 1–5] is compared with its III–V pnictide cluster analogues. The structure, electronic, and thermoelectric properties of the cluster assembled hexagonal 2D pnictide nanomaterials are also conferred.

2. Computational details All the theoretical calculations presented in this chapter are performed under the framework of Density functional theory (DFT) [13–15]. A well-accepted exchange correlation functional PBE as proposed by Perdew, Burke, and Ernzerhof [16], with generalized gradient approximation (GGA) is considered for all the cases, i.e., atomic clusters and 2D nanomaterials. The basis sets for atomic clusters to perform electronic structure calculations is adopted according to molecular orbital approach which incorporates linear combination of atomic orbitals [17]. For the considered atomic clusters, viz. C6H6, B3X3H6, Al3X3H6, Ga3X3H6, (C6)5, (B3X3)5, (Al3X3)5, and (Ga3X3)5, where X ¼ N, P, and As, an all-electron basis set, viz. 6-31G[d] is used [16,18,19]. And for Tl3X3H6 and (Tl3X3)5 clusters, where X ¼ N, P, and As, a popular basis set which includes scalar relativistic correction, viz. LANL2DZ (Los Alamos ECP plus DZ) is used [16,19,20]. The actual calculations for all the properties of atomic clusters are carried out by utilizing GAMESS [17] and GAUSSIAN 09 [21] codes. These codes allow for minimization of structural parameters without symmetry constrains and hence full variational freedom is attained [17,21]. The DFT implication of 2D layered h-InN and h-TlN nanomaterials exercises the projected augmented wave (PAW) method for core and valence electrons interactions and employs plane wave basis set [13,22]. The real scale calculations are carried out utilizing QUANTUM ESPRESSO code [23] for h-InN monolayer [24] and Vienna ab initio simulation package [25,26] for h-TlN mono and multilayer [22]. The computational modeling, visualization, and analysis of the nanostructures are performed in CHEMCRAFT software [27].

3. Benzene and its group III–V pnictide cluster analogues 3.1 Structural properties Fig. 1 displays the ground state energy geometries of benzene and its analogues single unit hexagonal cluster of the group III boron, aluminum, gallium, and thallium with group V pnictides nitrogen, phosphorus, and arsenic. The lowest energy bond lengths and bond angles of all the structures are also represented. Both the top view and the side view are reported in the above figure for C6H6 and M3X3H6 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) clusters [18,20]. The bond lengths between group III and group V hexagonal cluster units increases as we proceed from boron to thallium, and from nitrogen to arsenic ˚ and for as represented above. The bond lengths in B3X3H6 clusters, i.e., BdN, BdP, and BdAs ranges from 1.44 to 1.94 A ˚ Al3X3H6 clusters, i.e., AldN, AldP, and AldAs ranges from 1.82 to 2.36 A. The bond length ranges for Ga3X3H6 clusters ˚ of GadN, GadP, and GadAs. The bond show numerically similar values to that of Al3X3H6 clusters as 1.85 to 2.35 A ˚ [20]. lengths for Tl3X3H6 show significant rise in values for TldN, TldP, and TldAs clusters ranging from 2.16 to 2.81 A The bond length between group V atoms and saturated hydrogen atoms also increases as we proceed from N to As showing ˚ . It can be observed that all the M3X3H6 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) single units very a range of 1.02–1.52 A well retain the hexagonal symmetry in their lowest energy configurations. It is highly intriguing to notice that out of the total of 12 considered group III–V pnictide hexagonal cluster units, 10 exhibit planar and well symmetric structure in their ground state. The favorable planar structures of the pnictide single cluster units (except for TlP and TlAs) can be confirmed from the side view of the clusters, as represented in Fig. 1. The hexagonal and planar conformations of the group III–V pnictide clusters is analogues to their prototypical structure of organic benzene (C6H6). It may be noted that transferring a homoatomic (CdC) ring with higher symmetry (D6h) into a heteroatomic (III–V) analogues system reduces the symmetry (D3h) due to difference in size and relevant properties of heteroatoms. Further, when interaction between elements in such a hetero-ring enhances, possibility of symmetry lowering or distortion also increases, as observed for the phosphorous and arsenic compounds of thallium (TlP and TlAs), although nitride unit (TlN) retains planarity. Such distortion may also be understood as an effect of quantum confinement for zero-dimensional systems [28a]. TlP and TlAs show a distinct hexagonal buckled symmetry instead of planar which is analogues to the hexagonal single unit of silicene [28b]. The resemblance between the obtained bond lengths which are theoretically calculated by our group and the experimentally reported in the literature is illustrated in Table 1. The ground state parameters of all the hexagonal single units were calculated employing the exchange correlation functional Perdew, Burke, and Ernzerhof (PBE), a GGA [16]. For M3X3H6 (M ¼ B, Al, and Ga; X ¼ N, P, and As) an all-electron basis sets 6-31G[d] [18,19] was used, whereas for TlX3H6 (X ¼ N, P, and As), LANL2DZ (Los Alamos ECP plus DZ) with scalar relativistic correction was used [19,20]. The small difference (10%) as observed between experimental and theoretical values for TldP and TldAs, may be for nonconsideration of complete core-electrons effect for thallium, due to limitation in available basis set in

Group III–V hexagonal pnictide clusters Chapter

8

FIG. 1 Ground state structures of benzene (C6H6) and its analogues hexagonal M3X3H6 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) clusters.

141

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Atomic clusters with unusual structure, bonding and reactivity

TABLE 1 Comparison of theoretically calculated and experimentally reported bond lengths (in A˚) of the considered hexagonal cluster units M3X3H6 (M 5 B, Al, Ga, and Tl; X 5 N, P, and As). M-X

Group III (M)

Group V (X)

Bond lengths

B

Al

Ga

Tl

N

Calculated

1.4

1.8

1.8

2.2

Experimental

1.4

1.8

–

–

Calculated

1.8

2.3

2.3

2.7

Experimental

–

–

–

3.0

Calculated

1.9

2.4

2.4

2.8

–

–

–

3.2

a

P

a

As

a

Experimental a

Refs. [29–34].

the utilized program package [21], which deserves a future scrutiny. Based on the accuracy level of the theory involved in the calculations, the theoretical values of the bond lengths confirm a good accord with the experimental values reported in the literature [32–34].

3.2 Electronic properties Table 2 reports various electronic properties of benzene (C6H6) and its group III–V pnictide M3X3H6 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) atomic cluster analogues. The HOMO–LUMO energy gap (HLG) is the energy difference between highest occupied molecular orbital and lowest unoccupied molecular orbital by electrons of the atomic cluster, which is analogous to the band gap in solid-state materials. It can be observed that as we proceed from hexagonal boron pnictide clusters toward thallium pnictides, the overall HLG decreases. This indicates easy transition of electrons from occupied orbitals to unoccupied orbitals, suggesting good conductivity among the hexagonal cluster units of higher atomic sizes TABLE 2 HOMO–LUMO energy gap (HLG), ionization potential (IP), electron affinity (EA), chemical hardness (h), and electrophilicity index (v) in electron volts (eV) of C6H6 and M3X3H6 (M 5 B, Al, Ga, and Tl; X 5 N, P, and As) clusters. Hexagonal cluster units

HLG

IP

EA

Η

v

C6H6

5.12

6.11

0.99

2.56

2.46

B3N3H6

6.15

6.81

0.66

3.07

2.27

B3P3H6

3.58

5.89

2.31

1.79

4.69

B3As3H6

3.29

5.66

2.37

1.64

4.90

Al3N3H6

4.84

6.03

1.19

2.42

2.70

Al3P3H6

3.28

5.46

2.18

1.64

4.46

Al3As3H6

2.97

5.28

2.30

1.49

4.83

Ga3N3H6

4.63

5.78

1.14

2.32

2.59

Ga3P3H6

3.20

5.38

2.18

1.60

4.46

Ga3As3H6

2.87

5.19

2.32

1.44

4.90

Tl3N3H6

2.30

4.74

2.45

1.15

5.62

Tl3P3H6

2.54

5.41

2.87

1.27

6.74

Tl3As3H6

2.48

5.40

2.91

1.24

6.95

Group III–V hexagonal pnictide clusters Chapter

8

143

in group III period (Table 2). Also, a decrease in HLG can be observed in each considered group III pnictide clusters, going from nitrogen toward arsenic, toward higher atomic sizes of group V period, except for thallium pnictides which may be due to structural distortion of Tl3P3H6 and Tl3As3H6 clusters. We can note that many of the inorganic single hexagonal units show better conductivity compared to their organic counterpart benzene. Ionization potential (IP) is an ability of an atomic cluster to release an electron/s from its occupied orbitals and become single positively charged. Similar to HLG, IP also reduces from boron pnictides toward thallium pnictide clusters, and IP values also lower from nitrogen toward arsenic in case of each group III pnictide clusters except Tl3X3H6. Electron affinity (EA) is tendency of an atomic cluster to attract and add electron/s to its unoccupied orbitals and become negatively charged. Contrary to IP, EA for all group III pnictide clusters increases from nitrogen toward arsenic. All M3X3H6 clusters exhibit higher EA values compared to C6H6 except for B3N3H6. The chemical hardness (
) is an electronic property which indicates how harder a molecule/cluster chemically is and the stability of the atomic clusters as well. The higher stability of clusters like C6H6, B3N3H6, Al3N3H6, and Ga3N3H6 indicates these clusters tendency to engage in lower electron exchange, hence may be less preferable in the cluster growth/assemble process. On the other hand, the lower chemical hardness (
) values of remaining hexagonal units favor electronic interactions and reveal as promising motifs for cluster assembled materials. The measure of reactivity in terms of maximal electron transfer for an atomic cluster may be understood as electrophilicity index (o) [35,36]. It is the ability of an atomic cluster to attract a pair of electrons to form bonds. It follows similar trend as EA, the lower values of o for implies their inability to form covalent bonds with surrounding atoms or atomic clusters. The remaining inorganic hexagonal cluster units are appearing to be unstable as single unit, and by bonding with neighboring clusters results in augmentation which in controlled fashion produces cluster assembled materials. The hexagonal cluster units C6H6, B3N3H6, Al3N3H6, and Ga3N3H6 are found to be highly stable as single unit and may be explored for their unique properties as quantum dots. Tl3P3H6 and Tl3As3H6 do not follow the expected trend of electronic properties as other M3X3H6cluster does, which may be due to their nonplanar geometry and different interatomic interactions.

4. Polymeric growth of benzene and its III–V analogues 4.1 Structural properties The ground state geometries of the growth of benzene (C6)5 and its analogues (M3X3)5 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) till their fifth units is presented in Fig. 2 [18,20]. Both the top and side views of all the optimized structures along with their respective bond lengths and bond angles are presented in the figure. Similar to single hexagonal cluster units, the bond lengths of the five assembled hexagonal cluster units also exhibit increasing order as we go from boron to thallium in group III and from nitrogen to arsenic in group V. The bond length for (B3X3)5 clusters depending on the boron atom ˚ . Likewise, binding with hydrogen, nitrogen, phosphorus, or arsenic atoms, the values are ranging from 1.02 to 1.95 A ˚ , 1.02 to 2.37 A ˚, for (Al3X3)5, (Ga3X3)5, and (Tl3X3)5 clusters bond lengths ranges are found to be from 1.03 to 2.36 A ˚ , respectively. It is fascinating to note that all the grown M3X3 units till fifth unit retain their planar and 1.04 to 2.82 A (2D) architecture alike their carbon counterpart (polyacene). The growth resulting in planar two dimensionality recognizes the considered M3X3 pnictide clusters as promising aspirants for the designing and fabrication of their cluster assembled (M3X3)H6 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) hexagonal 2D nanomaterials like g-MX nanosheets, nanotubes, etc. The (TlP)5 and (TlAs)5 show a nonplanar arrangement as a reflection of its nonplanar Tl3P3H6 and Tl3As3H6 single units as shown in Fig. 1. The elevated inter atomic binding between higher elements of group III, i.e., Tl and group V, i.e., P and As may be the possible reason for the periodic distortions in (TlP)5 and (TlAs)5 cluster symmetry. However, Tl3P3H6 and Tl3As3H6 units may be recognized as prospective candidates for the cluster assembled nonplanar 2D nanomaterials having buckled or puckered arrangements.

4.2 Electronic properties Fig. 3 shows the profile of HLG with respect to the growth in the hexagonal clusters (MX)n (M ¼ B, As, Ga, and Tl; X ¼ N, P, and As) from units one to five, i.e. n ¼ 1–5. For their comparison with carbon analogue, profile of polyacenes (Cn) is also presented [18,20]. A uniform trend can be observed where almost all the (MX)n shows steep reduction of HLG when the single hexagonal unit grows to two units. Further, the HLG values nearly remain constant for growth from two to five units. In case of polyacene (Cn), the HLG values constantly decreases with increase in the number of benzene units, which may possibly be understood a direction toward their counterpart of zero band gap 2D graphene material [37]. The dropping value of HLG indicates diminishing stability of a compound. Hence it can be deduced that the benzene’s (C6H6) polymeric growth to polyacenes (Cn) is less endured compared to its inorganic group III–V hexagonal counterpart’s (M3X3)

144

Atomic clusters with unusual structure, bonding and reactivity

FIG. 2 Ground state structures of polyacene (C6)5 and its analogues hexagonal (M3X3)5 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) clusters, both in top and side views.

Group III–V hexagonal pnictide clusters Chapter

(a)

5

5

4

4

3 2 1

0 2

3 X (no. of ring)

4

5

1

(c)

5

5

4

4

3

1

1

(GaP)x (GaAs)x

0 1

2

3 X (no. of ring)

4

5

3 X (no. of ring)

4

5

(d)

3 2

Cx (GaN)x

2

6

HLG (eV)

HLG (eV)

6

2

(AIP)x (AIAs)x

(BAs)x 1

Cx (AIN)x

1

(BP)x

0

3 2

Cx (BN)x

145

(b)

6

HLG (eV)

HLG (eV)

6

8

0

Cx (TIN)x (TIP)x (TIAs)x 1

2

3 X (no. of ring)

4

5

FIG. 3 Profile of the HOMO-LUMO energy gap (HLG) with respect to the growth of polyacenes Cn (n ¼ 1–5) in comparison with its III–V analogues (A) (BX)n, (B) (AlX)n, (C) (GaX)n, and (D) (TlX)n hexagonal pnictide atomic clusters where X ¼ N, P, and As; n ¼ 1–5. (Reprinted from J. Mol. Struct., 1007, D. R. Roy, A DFT study on group III and V combined hexagonal clusters as potential building motifs for inorganic nanomaterials, 203–207, Copyright (2012), with permission from Elsevier.)

H6 growth to the (MX)n (M ¼ B, As, Ga, and Tl; X ¼ N, P, and As) assembled compounds. The common HLG nature noted in all the (MX)n pnictides growth is not observed in case of TlP and TlAs, as expected. The approximate constant HLG values when growing from one to five units shows their ability for polymeric growth. Fig. 4 presents the EA profile with respect to the growth in the hexagonal clusters (MX)n (M ¼ B, As, Ga, and Tl; X ¼ N, P, and As) from units one to five, i.e. n ¼ 1–5 [18,20]. The EA of polyacenes (Cn) is also included for the comparison with (MX)n. It can be observed that the EA of all the (MX)n clusters increases with their growth from one to five units. This evidently indicates the group III–V hexagonal clusters tendency of attracting and accumulating more similar clan clusters from surroundings to form higher dimensional materials. Based on these appealing results selected subnanoscaled (MX)n clusters are investigated and reported toward their growth and potential application as two-dimensional planar nanosheets. Table 3 presents various electronic properties of polyacene (C6)5 and its group III and group V pnictide (M3X3)5 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) hexagonal atomic cluster analogues. It can be observed that going from (M3N3)5 toward (M3As3)5 down the group V elements, the values of HLG decreases and the electrophilicity index (o) increases which indicates improved reactivity of the (M3X3)5 hexagonal clusters. The higher atomic number pnictide atomic clusters show reduced IP and enhanced EA depicting better reactivity in the larger scale cluster assembled nanomaterials. The chemical hardness (
) of polyacene (C6)5 is lowest than all the hexagonal clusters in the considered series. Hence, the inorganic group III–V atomic clusters (M3X3)5 are more stable and appropriately reactive compared to their carbon analogue, viz. the polyacene (C6)5 to substantiate the process of cluster assembly for creation of novel nanomaterials.

5. Group III–V graphene-like materials from potential cluster units The brilliant stability as well as good binding nature of the hexagonal single their grown till fifth units assembled atomic clusters of the group III–V elements establishes them to be promising candidates for custom-built cluster assembled

146

Atomic clusters with unusual structure, bonding and reactivity

4.0 3.5

(a)

(BP)x (BAs)x

3.0 2.5

EA (eV )

EA (eV )

4.0

Cx (BN)x

2.0

3.0

(AIAs)x

(b)

(AIP)x

2.5 2.0

1.5

1.5

1.0

1.0 0.5

0.5 1

4.0

2

3.5

Cx (GaN)x

3.0

(GaAs)x

3 X (no. of ring)

4

1

5

2

4.0

(c)

3 X (no. of ring)

4

5

(d)

3.5

(GaP)x 3.0

2.5

EA (eV)

EA (eV)

3.5

Cx (AIN)x

2.0 1.5

2.5 2.0 Cx

1.5

(TIN)x

1.0

1.0

0.5

0.5

1

2

3 X (no. of ring)

4

5

(TIP)x (TIAs)x 1

2

3 X (no. of ring)

4

5

FIG. 4 Profile of the electron affinity (EA) with respect to the growth of polyacenes Cn (n ¼ 1–5) in comparison with its analogues (A) (BX)n, (B) (AlX)n, (C) (GaX)n, and (D) (TlX)n hexagonal pnictide clusters where X ¼ N, P, and As; n ¼ 1–5. (Reprinted from J. Mol. Struct., 1007, D. R. Roy, A DFT study on group III and V combined hexagonal clusters as potential building motifs for inorganic nanomaterials, 203–207, Copyright (2012), with permission from Elsevier.)

nanomaterials. The remarkable planar symmetry of the majority of the group III–V hexagonal pnictide clusters signify that the further assemblages of these clusters to higher scale would result in a two-dimensional graphene-like nanosheets materials. Based on the ground state parameters of the hexagonal single units of the considered series of pnictide atomic clusters, the unit cell for the respective pnictide nanomaterial is designed. Couple of the hexagonal pnictide cluster assembled nanosheets based on the atomic cluster parameters reported here in is elaborately discussed in this section. Further the standard structure, electronic and thermoelectric properties of the same are also conferred.

5.1 Monolayer indium nitride for thermoelectrics This section will address hexagonal unit of indium nitride based 2D InN nanosheet for its possible promises toward thermoelectric applications. The initial lattice parameters for conducting monolayer h-InN study are assumed to be close to that ˚ which is in good of the h-InN atomic cluster [38], where the optimized lattice parameter for h-InN is found to be 3.58 A agreement to the reported value in similar monolayer works [39]. Fig. 5 presents the structure and electronic properties of the 2D monolayer hexagonal indium nitride. Fig. 5A shows the crystal structure of the h-InN. It can be observed that a hexagonal super cell was modeled which is similar to the h-graphene [37]. The unit cell consists of one atom of indium (In) and one atom of nitrogen (N). The minimum energy lattice constant ˚ through PBE-PAW calculations under DFT [24]. It may be noted that hexagonal symmetry remains turned out to be 3.54 A intact similar to graphene in the converged structure of h-InN nanosheet. The indirect band gap value of 0.43 eV (G-Κ point) and direct band gap value of 0.88 eV (G point) can be observed in the electronic band structure of h-InN as shown in Fig. 5B. The h-InN monolayer is found to be an indirect bandgap material possessing semiconducting properties, and may have potential semiconductor applications. Total density of states (TDOS) and partial density of states (PDOS) of the h-InN

Group III–V hexagonal pnictide clusters Chapter

8

147

TABLE 3 HOMO–LUMO energy gap (HLG), ionization potential (IP), electron affinity (EA), chemical hardness (h), and electrophilicity index (v) in electron volts (eV) of (C6)5 and (M3X3)5 (M 5 B, Al, Ga, and Tl; X 5 N, P, and As) clusters. (M3X3)5

HLG

IP

EA

h

v

(C6)5

1.18

4.30

3.12

0.59

11.66

(B3N3)5

4.60

5.95

1.35

2.30

2.89

(B3P3)5

1.65

5.08

3.43

0.83

10.96

(B3As3)5

1.40

4.86

3.47

0.70

12.41

(Al3N3)5

3.75

5.73

1.99

1.87

3.98

(Al3P3)5

2.63

5.23

2.60

1.31

5.82

(Al3As3)5

2.15

5.04

2.90

1.07

7.35

(Ga3N3)5

3.35

5.44

2.08

1.68

4.21

(Ga3P3)5

2.39

5.09

2.70

1.20

6.34

(Ga3As3)5

1.64

4.87

3.23

0.82

10.03

(Tl3N3)5

1.07

4.60

3.53

0.53

15.50

(Tl3P3)5

1.98

5.43

3.45

0.99

9.97

(Tl3As3)5

1.96

5.42

3.46

0.98

10.06

FIG. 5 The (A) crystal structure, (B) electronic band structure, and (C) total DOS and partial DOS (PDOS) of monolayer hexagonal indium nitride nanosheet. (Reprinted from J. Mater. Sci., 53, V. Kumar, D. R. Roy, Structure, bonding, stability, electronic, thermodynamic and thermoelectric properties of six different phases of indium nitride, 8302–8313, Copyright (2018), with permission from Springer.)

8 7 6 5 4

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3

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Energy (eV)

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3 2 1 0 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 Energy (eV)

(C)

K

M

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 6 The profile of thermoelectric properties of hexagonal indium nitride monolayer in comparison to its various bulk phases: (A) Electrical conductivity and (B) thermal conductivity. (Reprinted from J. Mater. Sci., 53,V. Kumar, D. R. Roy, Structure, bonding, stability, electronic, thermodynamic and thermoelectric properties of six different phases of indium nitride, 8302–8313, Copyright (2018), with permission from Springer.)

monolayer can be observed in Fig. 5C. Energy bands of both the conduction and valence state of a compound are the cumulative involvement of the atomic orbitals of the composed elements. The PDOS indicates the contribution from each atom and its specific orbitals and the TDOS shows the vacancy at each energy level for accommodating the quantity of states in a particular range of energy. It can be observed that conduction band receives utmost contribution from p-orbital of Indium atom and valence band receives its maximum contribution from p-orbital of nitrogen atom [24] (Fig. 5C). The different phases of bulk Indium nitride (InN) are reported to show band gap ranges from 0.6 to 1.89 eV depending on the level of theories applied [24,40–42]. It can be observed from Fig. 6 that h-InN monolayers show very less change in electrical conductivity as well as thermal conductivity with rise in temperature when compared to bulk wurtzite (w-InN), zinc blende (zs-InN), and rock-salt (rs-InN) phases. The 2Dh-InN can be viewed as thermoelectrically intact candidate whose semiconducting nature is not affected due to the high temperatures. Zhuang et al. [39] theoretically reported 2D hexagonal indium phosphide (h-InP) and indium arsenide (h-InAs) com˚ , respectively. The lattice constant pounds to be with low buckled geometry with buckling displacement of 0.52 and 0.67 A ˚ correspondingly. The band gap of h-InP (1.80 eV) is employing HSE06 hybrid functional is reported to be 4.25 and 4.38 A reported to be larger than that of h-InAs (1.41 eV), and similar to h-InN both h-InP and h-InAs displays an indirect band gap [39]. As both the h-InP and h-InAs monolayers exhibit higher band gap values, they may be used as potential insulating materials in nanocapacitors for 2D nanoelectronics applications [43]. Compared to various bulk phases of InP and InAs, zinc blende atomic arrangement is reported to be the most stable [44]. ˚ and that of InAs unit cell is 6.06 A ˚ . Both InP and InAs exhibit The lattice constant value of InP unit cell is reported as 5.87 A direct band gap of 1.34 and 0.35 eV, respectively, at room temperature [45]. The bulk InP and InAs have primary application as substrate material in many electronics and optoelectronics devices such as photodiodes, detectors, sensors, LEDs, switches, etc. [46–51]. The narrow gap h-InN (0.43 eV) monolayer is expected to have potential applications in infrared detectors and thermoelectrics [52].

5.2 Mono- and multilayer thallium nitride for thermoelectrics Fig. 7A displays the top view of hexagonal thallium nitride nanosheet. It can be observed that the lowest energy bond length ˚ [22]. And the minimized energy lattice constant turns out to be between thallium and nitrogen atoms is found to be 2.19 A ˚ 3.79 A which is higher than InN [24]. A 2D hexagonal unit cell consisting of one thallium (Tl) and one nitrogen (N) atoms is optimized and the resulting ground state maintains the hexagonal assembly as in case of TlN hexagonal atomic clusters. A single layer TlN nanosheet can be viewed in Fig. 1B. The minimized energy of the TlN monolayer system turns out to be ˚ and the stabilizing distance between both the 7.67 eV. The TldN bond length in case of TlN bilayer increases to 2.21 A ˚ layer is 2.62 A as shown in Fig. 1C. The lowest energy of this system is found to be 15.89 eV and the unit cell comprises of two Tl and two N atoms. Similarly, in case of trilayer TlN nanosheets, the hexagonal unit cell consists of three atoms of Tl and three atoms of N. The N atoms arranged alternatively to Tl atoms along the z-direction. The TldN bond length for the ˚ and that of top and bottom layer as 2.20 A ˚ . The inter layer distances is found to be 2.84 A ˚ middle layer is calculated as 2.19 A as in Fig. 1D. The ground state total energy for the trilayer TlN nanosheet is found to be 23.90 eV. It is interesting to note that all the TlN nanosheet variants not only maintains the hexagonal arrangements and planarity but also are highly stable.

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FIG. 7 Relaxed structures of the mono- and multilayer of hexagonal thallium nitride nanosheets: (A) top view of the monolayer and side views of (B) monolayer, (C) bilayer, and (D) trilayer. (Reprinted from Appl. Nanosci. 9, E. V. Shah, D. R. Roy, Density functional investigation on hexagonal nanosheets and bulk thallium nitrides for possible thermoelectric applications, 33–42, Copyright (2019), with permission from Springer.)

This could be due to the s-bonds between Tl and N atoms of each ring and the distribution of the p-bonds density cloud over the surface between neighboring atoms of the rings [22]. Fig. 8 displays the band structures of the mono-, bi-, trilayer h-TlN, and the bulk (wurtzite) counterpart. All TlN nanosheets, i.e., monolayer, bilayer, and trilayer show zero band gap at G–point. For TlN monolayer band, lines meet 0.20 eV below the Fermi level. Similarly, for TlN bilayer and trilayer band, lines meet 0.05 and 0.02 eV below the Fermi level. Therefore, as the number of layers increases the zero-gap band meet shifts toward the Fermi level. These h-TlN nanosheets show good analogy with the h-graphene [53]. The band structures obtained for the mono- and multilayer

FIG. 8 Electronic band structures of hexagonal thallium nitride nanosheets: (A) Monolayer, (B) Bilayer, (C) Trilayer and (D) Bulk. (Reprinted from Appl. Nanosci. 9, E. V. Shah, D. R. Roy, Density functional investigation on hexagonal nanosheets and bulk thallium nitrides for possible thermoelectric applications, 33–42, Copyright (2019), with permission from Springer.)

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h-TlN indicate semimetallic nature of these 2D nanomaterials. For a comparison, band structure of bulk (wurtzite) TlN is also presented in Fig. 2D. The band lines meet 0.06 eV below the Fermi level at G–point itself which is similar to the results obtained by Saidi-Houat et al. [54]. Fig. 9 illustrates the PDOS and TDOS of h-TlN monolayer, bilayer, trilayer, and bulk. The density of states at Fermi level and adjoining region shows nonzero values in all the h-TlN variants confirming the presence of zero band gap or band lines meet [55]. It can be observed in Fig. 3A that the p-orbital of nitrogen atom forms valence band maximum (VBM) and p-orbital of the thallium atom forms the conduction band maximum (CBM) of the h-TlN monolayer. Fig. 3B–D reveals that contrary to monolayer, the VBM and the CBM both of multilayer and bulk h-TlN variants comprises of the contributions from the p-orbital of their nitrogen atoms [56]. A comparative plot between the h-TlN monolayer, bilayer, trilayer, and bulk w-TlN on their thermoelectric properties, namely, thermal conductivity (Κ), electrical conductivity (s), Seebeck coefficient (S), and figure of merit (ZT) is represented in Fig. 10. All the h-TlN variants show rise in thermal conductivity (Κ) with increase in temperature which can be noted in Fig. 10A. The increase in thermal conductivity for h-TlN monolayer is almost linear and maximum with rise in temperature. The monolayer h-TlN can effectively circulate heat at higher temperatures as its K values range till 12  1014 W/ms K at 800 K which is highest compared to the reported multilayer and bulk h-TlN. The electrical conductivity (s)of h-TlN monolayer shows a linear response with respect to the rise in temperature, which can be observed from Fig. 10B. It can maintain good electric current especially at high temperature, i.e., 3.0  1019 at 800 K. The electrical conductivity (s) of bulk w-TlN also shows linear response though lower rise in values with increasing temperature. The s of bilayer and trilayer h-TlN does not show significant response to the rise of temperature. Seebeck coefficient (S) indicates a

8

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7 6

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FIG. 9 Total DOS and partial DOS(PDOS) plots of hexagonal thallium nitride nanosheets: (A) monolayer, (B) bilayer, (C) trilayer, and (D) bulk. (Reprinted from Appl. Nanosci. 9, E. V. Shah, D. R. Roy, Density functional investigation on hexagonal nanosheets and bulk thallium nitrides for possible thermoelectric applications, 33–42, Copyright (2019), with permission from Springer.)

Group III–V hexagonal pnictide clusters Chapter

Mono Bi Tri Bulk

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Electrical Conductivity s/t (S/ms) x 1019

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FIG. 10 Profile of various thermoelectric properties of hexagonal thallium nitride nanosheets and bulk: (A) thermal conductivity, (B) electrical conductivity, (C) Seebeck coefficient, and (D) figure of merit. (Reprinted from Appl. Nanosci. 9, E. V. Shah, D. R. Roy, Density functional investigation on hexagonal nanosheets and bulk thallium nitrides for possible thermoelectric applications, 33–42, Copyright (2019), with permission from Springer.)

materials ability to sustain electrical potential at a given temperature. The h-TlN monolayer and bulk exhibit good S values only at lower temperatures, i.e., near 250 K which is illustrated in Fig. 10C. The bilayer h-TlN can produce thermoelectric power sufficiently well compared to its other h-TlN variants as its figure of merit (ZT) values increase linearly with rise in temperature as reported in Fig. 10D. The h-TlN monolayer and bulk show good ZT values at lower temperatures and decrease with rise in temperature [56]. As the two-dimensional hexagonal thallium phosphide (h-TlP) and thallium arsenide (h-TlAs) does not form a symmetric and planar architecture, not much notable study is reported about the nanosheets of these thallium pnictides. Although the bulk thallium phosphide (zb-TlP) and bulk thallium arsenide (zb-TlAs) are observed to show the ground state ˚ , respectively [54,57]. Both zb-TlP and zb-TlAs exhibit in zinc blende phase with lattice parameters 6.12 and 6.37 A zero-gap band structures like graphene at Fermi level and G point. A slight CBM crossover is noted between Brillouin zone points L and G in case of TlAs [54,57]. These zero-gap materials can find potential applications in fields of electronics, optics, spintronics, topological insulators, etc. [58].

5.3 Other two-dimensional group III–V materials The 2D monolayer phase of boron nitride (h-BN) is observed to form the hexagonal and planar geometry similar to gra˚ and an indirect band gap around phene [59]. At LDA-GW0 level of theory Sahin et al. reported lattice constant as 2.51 A 6.0 eV [59] for h-BN. Due to its insulating property, it finds possibility to be used as dielectric material along with graphene in electronic devices [60–62]. The 2D layers of h-BN can form potential chemical sensors for sensing highly toxic warfare chemicals [63]. The hexagonal boron phosphide (h-BP) and boron arsenide (h-BAs) show a perfect planar honey comb monolayer with ˚ , respectively. Both are accounted to exhibit a direct band gap in semiconductor domain as lattice constants 3.18 and 3.35 A 1.81 eV for h-BP and 1.24 eV for h-BAs at LDA-GW0 level of theoretical calculations [59]. Due to the small band gaps, both

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of h-BP monolayer and h-BAs monolayer are investigated to be ideal for their application as anode materials in alkalimetal-based batteries [64,65]. Both are reported to exhibit similar properties that are of good chemical stability, easy charge transfers with good carrier mobility and strong optical absorptions in UV-vis range proposing their versatile uses in nanoelectronics and optoelectronics [59,66–69]. The 2D h-BAs is also accounted to show good thermodynamic as well as thermoelectric properties making it appropriate candidate for applications in thermal management systems [69]. Similar to two-dimensional h-BX (X ¼ N, P, and As), the 2D hexagonal aluminum nitride (h-AlN) and hexagonal aluminum phosphide (h-AlP) are investigated and display planar structure [39]. At HSE06 level of theory, h-AlP possess ˚ compared to h-AlN of 3.12 A ˚ . Conversely, the 2D hexagonal aluminum a slightly higher lattice constant of 3.94 A ˚ . It retains lattice arsenide (h-AlAs) monolayer display a low buckled structure with buckling displacement of 0.45 A ˚ constant of 4.06 A. The h-AlN, h-AlP, and h-AlAs exhibit an indirect band gap of 4.85, 3.24, and 2.49 eV, respectively. The 2D h-AlN may find applications as an insulating material, whereas 2D h-AlP and 2D h-AlAs as wide-band gap semiconducting nanomaterials [26]. The 2D h-AlN also observed to possess higher mechanical stability compared to h-BN [70]. Among two-dimensional monolayer hexagonal GaX (X ¼ N, P, and As), only gallium nitride (h-GaN) acquires a perfect honey comb geometry, gallium phosphide h-GaP and gallium arsenide h-GaAs attains buckled structure with buckling ˚ , respectively [39]. The lattice constant of GaX increases in the order of distance of 0.43 and 0.59 A ˚ , respectively. The entire GaX exhibits indirect h-GaN < h-GaP< h-GaAs with numerical values as 3.25, 3.91, and 4.06 A band gap with decreasing values in order as h-GaN > h-GaP > h-GaAs. The h-GaAs monolayer exhibits a semiconducting band gap value of 1.83 eV. The h-GaP and h-GaN show a wide band gap semiconducting values of 2.51 and 3.23 eV correspondingly as accounted in work by Zhuang et al. [39]. The 2D h-GaN show activity in visible as well as deep ultra-violet region of the absorption spectrum and hence is studied widely for its potential applications in single-photon emitters, widespectrum LEDs, solar cells, and flexible nano-optoelectronic devices [22,71–73].

6. Conclusions The structural and electronic properties of benzene (C6H6) molecule and its analogues group III–V hexagonal pnictides, viz. M3X3H6 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) are discussed in details. Almost all the M3X3H6 optimizes into a hexagonally well symmetric and planar atomic clusters. The nonplanar hexagonal geometries are found in case of zerodimensional confined Tl3P3H6 and Tl3As3H6 cluster systems, whereas their possible structures in periodic counterparts in other dimensions deserves a careful scrutiny. The electronic properties reveal good stability and reactivity of the considered III–V hexagonal pnictide clusters. For further confirmation on the individual atomic clusters for cluster assemblage process, the investigation on polymeric growth till five units of benzene and its group III–V analogues is conferred. The grown polyacenes (C6)5 and its analogues hexagonal (M3X3)5 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) clusters discloses the structural and electronic properties similar to their single unit counterparts. Thus, our collective study on M3X3H6 hexagonal atomic clusters provide a wide pool of candidates from group III–V which are apposite for the exploration and application as cluster assembled hexagonal nanomaterials like graphene. In the same direction, we have provided here in depth study of the cluster assembled 2D nanosheets: h-InN and h-TlN which are array construct from their hexagonal atomic clusters, viz. In3N3H6 and Tl3N3H6, respectively. It is inspiring to note that the Tl3N3H6 cluster assembled h-TlN nanosheets possess a zero-band gap similar to graphene making it an ideal inorganic graphene counterpart. The zeroband gap structure is not observed in many other 2D hexagonal group III–V pnictide nanomaterials. The h-InN holds good semiconducting properties and h-TlN illustrates good thermoelectric properties, as reported in the present chapter. The HOMO-LUMO/band gap of TlN cluster/2D material decreases with its growth from cluster to materials as follows: 2.30 eV (Tl3N3H6) > 1.07 eV {(TlN)5} >0.0 eV (2D h-TlN nanosheets). The group III–V pnictide clusters, i.e., M3X3H6 and (M3X3)5 (M ¼ B, Al, Ga, and Tl; X ¼ N, P, and As) sustain much better stability and reactivity properties compared to geometrically similar benzene (C6H6) and polyacene (C6)5. Likewise, the hexagonal group III–V cluster assembled 2D nanomaterials also hold the best potential to superior properties compared for, to their conventional counterpart graphene given to their tailor-made course.

Acknowledgments We thank Professor Pratim K. Chattaraj for kindly inviting us to write this chapter. DRR is thankful to the SERB, New Delhi, Government of India for financial support (Grant No. CRG/2020/002634).

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[31] G. Giesecke, H. Pfister, Precision determination of the lattice constants of the A III B V-compounds, Acta Crystallogr. 11 (1958) 369. [32] O. Olofsson, J. Gullman, The crystal structure of TlP5, Acta Chem. Scand. 25 (1971) 1327–1337. [33] J.R. Wilson, P.K. Sengupta, P.D. Robinson, A.J. Criddle, Fangite, Tl3AsS4, a new thallium arsenic sulfosalt from the Mercur Au deposit, Utah, and revised optical data for gillulyite, Am. Mineral. 78 (1993) 1096–1103. [34] L. Pauling, L.O. Brockway, Carbon—carbon bond distances. The electron diffraction investigation of ethane, propane, isobutane, neopentane, cyclopropane, cyclopentane, cyclohexane, allene, ethylene, isobutene, tetramethylethylene, mesitylene, and hexamethylbenzene. Revised values of covalent radii, J. Am. Chem. Soc. 59 (1937) 1223–1236. [35] P.K. Chattaraj, D.R. Roy, Update 1 of: electrophilicity index, Chem. Rev. 107 (9) (2007) PR46–PR74. [36] P.K. Chattaraj, U. Sarkar, D.R. Roy, M. Elango, R. Parthasarathi, V. Subramanian, Is electrophilicity a kinetic or a thermodynamic concept? Ind. J. Chem. A 45A (5) (2006) 1099–1112. [37] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669. [38] B.R. Mehta, E.V. Shah and D.R. Roy, unpublished work. [39] H.L. Zhuang, A.K. Singh, R.G. Hennig, Computational discovery of single-layer III-V materials, Phys. Rev. B: Condens. Matter Mater. Phys. 87 (2013) 165415. [40] O.K. Semchinova, J. Aderhold, J. Graul, A. Filimonov, H. Neff, Photoluminescence, depth profile, and lattice instability of hexagonal InN films, Appl. Phys. Lett. 83 (2003) 5440. [41] H. Jin, G.L. Zhao, D. Bagayoko, Calculated optical properties of wurtzite InN, J. Appl. Phys. 101 (2007), 033123. [42] T.L. Tansley, C.P. Foley, Optical band gap of indium nitride, J. Appl. Phys. 59 (1986) 3241. [43] Y.Y. Illarionov, T. Knobloch, M. Jech, M. Lanza, D. Akinwande, M.I. 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Chapter 9

M(L)8 complexes (M 5 Ca, Sr, Ba; L 5 PH3, PF3, N2, CO): Act of an alkaline-earth metal as a conventional transition metal Hai-Xia Lia, Zhong-Hua Cuia, Dandan Jiangb, Lili Zhaob, and Sudip Panc a

Institute of Atomic and Molecular Physics, Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), Jilin University,

Changchun, China, b Institute of Advanced Synthesis, School of Chemistry and Molecular Engineering, Jiangsu National Synergetic Innovation Center for Advanced Materials, Nanjing Tech University, Nanjing, China, c Institute of Atomic and Molecular Physics, Jilin University, Changchun, China

1. Introduction Langmuir postulated the so-called 8, 18, and 32 electron counting rules for sp block (main group elements), spd block (transition metals), and spdf block (lanthanides and actinides), respectively, in 1921, prior to the advent of quantum chemistry [1,2]. These rules, which are originated from the extraordinary stability gained in the cases of fully occupied atomic valence shell, have been widely used, even in this modern quantum era, as an important directive to dictate the maximum coordination number in the first sphere and the related structure and stability of the complexes. Alkaline-earth elements, Be-Ba, in general, have been treated as the sp block elements, and, therefore, the octet rule is prevalent for them. However, for the heaviest member, Ba, the participation of its 5d orbital to a varying extent was reported towards chemical bonding, which brings the suggestion of naming it as an “honorary transition metal” [3–6]. The recent report about barium carbonyl ions Ba(CO)q (q ¼ +1 and 1) is another set of examples in favor of this demand [7,8]. In these complexes, Ba+/Ba is in 2D (5d1) electronic state and the strongest orbital interaction is originated from the Ba+/Ba (5d1p) ! CO(p* LUMO) p backdonation, indicating the behavior of Ba as a classical transition metal. This observation led us to examine the larger carbonyl complexes which fulfill the 18-electron rule like transition metal. In this regard, our experimental collaborator isolated not only the octacarbonyl complex, Ba(CO)8, in low-temperature matrix, but also, counter-intuitively, its nearing lighter homologs, M(CO)8 (M ¼ Sr, Ca) [9,10]. These complexes possess cubic (Oh) symmetry with 3A1g electronic ground state where M has an excited (n)s0(n  1)d2 electron configuration and M(dp) ! (CO)8 p backdonation yields 70%–86% of the total orbital interaction. Interestingly, they all satisfy the 18-electron rule, as shown in Scheme 1, where the eg components are singly occupied. The electron correlation diagram shows that a2u molecular orbital (MO) is purely ligand-based. There is not any perfectly symmetric atomic orbital on M to mix with that. This makes M(CO)8 an effective 16-electron complex, which adopts a triplet spin state to symmetrically occupy the highest lying eg orbitals. This is very surprising that even Ca can mimic a transition metal. Given that the excitation energy for ns2 ! ns0(n  1) 2 d transition is very high, being 159.5 (Ca) and 150.9 (Sr) kcal/mol at the M06-2X/TZ2P-ZORA level, and, therefore, very efficient M(dp) ! (L)8 p backdonation is required to overcompensate this as in the cases of M(CO)8. This can be satisfied by the low-lying p* unoccupied orbital of CO, which makes it an excellent p acceptor ligand. Now, the question is: is such transition metal-like behavior of alkaline-earth metals exclusive to CO or even to other higher p-accepting ligands? This is even more astonishing that dinitrogen, which is a weaker s-donor as well as a weaker p-acceptor than CO, can also form octa-coordinated complexes, M(N2)8 (M ¼ Ca, Sr, and Ba) [11]. These complexes were isolated at a lowtemperature matrix and are stable with respect to single dinitrogen dissociation. These complexes are effective 16-electron complexes. Therefore, adding two extra electrons could result in a single closed-shell complex. That aspect is understood with the report of the M(Bz)3 complex which is an effective 18-electron complex. We herein considered two new ligands, L ¼ PH3 and PF3, which have somewhat lower p accepting ability than CO but still they possess ample p-accepting capability. For PH3 and PF3, the steric repulsion would also play an important role to decide the overall structure and stability of the M(L)8 complexes. Our density functional theory (DFT) calculations Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00011-5 Copyright © 2023 Elsevier Inc. All rights reserved.

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Atomic clusters with unusual structure, bonding and reactivity

SCHEME 1 Splitting of the spd valence orbitals of an atom M with the configuration (n  1)d2ns0np0 in the octacoordinate cubic (Oh) field of eight CO ligands and the related occupations.

M

M(CO)8

8CO

indicated that not only the title M(L)8 complexes are stable with respect to either single L dissociation or eight L dissociation altogether, but also they behave like classical transition metals akin to M(CO)8 complexes. Therefore, we expect this behavior of Ca-Ba elements to be quite general for a larger number of ligands than previously thought!

2. Computational details The geometry optimizations followed by the harmonic frequency calculations for all the systems presented here were carried out at the M06-2X-D3/def2-TZVPP level [12–14] using the Gaussian 16 suit of program [15]. Scalar-relativistic effective core potentials were used for the 28 and 46 core electrons of Sr and Ba, respectively. A superfine integration grid is considered for all cases. The energy decomposition analysis (EDA) [16] in combination with the natural orbital for chemical valence (NOCV) method [17,18] was performed at the M06-2X/TZ2P-ZORA//M06-2X-D3/def2-TZVPP level using the ADF (2017.101) program package [19,20]. The zeroth-order regular approximation (ZORA) [21] was used to include scalar relativistic effects for the metals. All electrons were considered in the computations. In the EDA method, the interaction energy (DΕ int) between two prepared fragments is divided into three energy terms, viz., the electrostatic interaction energy (DEelstat), which represents the quasiclassical electrostatic interaction between the unperturbed charge distributions of the prepared atoms, the Pauli repulsion (DEPauli), which is the energy change associated with the transformation from the superposition of the unperturbed electron densities of the isolated fragments to the wavefunction that properly obeys the Pauli principle through explicit antisymmetrization and renormalization of the product wavefunction, and the orbital interaction energy (DEorb), which is originated from the mixing of orbitals, charge transfer and polarization between the isolated fragments. Therefore, the interaction energy (DΕint) between two fragments can be defined as: DΕint ¼ DEelstat + DEPauli + DEorb

(1)

Since metahybrid M06-2X functional is considered in the present study, the so-called transition state procedure uses an approximate Fock operator in the computations, and, therefore, it adds additional metahybrid correction, DΕ hybrid, towards the DEorb term. The EDA-NOCV calculation combines charge and energy decomposition schemes to divide the deformation density, Dr(r), associated with the bond formation into different components (s, p, and d) of a chemical bond. From the mathematical point of view, each NOCV, ci is defined as an eigenvector of the deformation density matrix in the basis of fragment orbitals. DPci ¼ ni cI

(2)

In EDA-NOCV, DEorb is given by the following equation: DEorb ¼

N=2 X X   TS DEorb ¼ nk FTS k k + Fk k

k¼1

(3)

M(L)8 complexes (M=Ca, Sr, Ba; L= PH3, PF3, N2, CO) Chapter

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TS where  FTS  k and Fk are diagonal Kohn-Sham matrix elements corresponding to NOCVs with the eigenvalues –nk and nk, respectively. The DEorb k terms are assigned to a particular type of bond by visual inspection of the shape of the deformation density, D rk. The EDA-NOCV scheme thus provides both qualitative (D rorb) and quantitative (D Eorb) information about the strength of orbital interactions in chemical bonds. More details about EDA-NOCV and its application can be found in recent reviews [22–27]. EDA-NOCV is a very powerful tool to get deep insight into the bonding in various systems [28–45].

3. Structure and stability of M(L)8 complex Fig. 1 displays the M06-2X-D3/def2-TZVPP equilibrium geometries for M(L)8 complexes and the corresponding energies needed for the dissociation of single L and all eight L. All the complexes belong to the triplet electronic ground states, which {1.415} [1.414]

1.416 3.054

Ca(PH3)8 (3Ag, Th) De = 6.0 (47.1) kcal/mol

{3.177} [3.349]

M(PH3)8 (3A1g, Oh) De = 12.4 (51.7) kcal/mol; M = {Sr} De = 12.8 (62.1) kcal/mol; M = [Ba]

{1.562} [1.563] 1.561

2.981

Ca(PF3)8 (3A1, O) De = 11.8 (61.1) kcal/mol 1.127 {1.126} [1.125]

{3.109} [3.338]

M(PF3)8 (3A1, Td) De = 14.4 (58.5) kcal/mol; M = {Sr} De = 12.7 (65.9) kcal/mol; M = [Ba] 1.095 {1.095} [1.093]

2.602 {2.751} [2.960] 2.518 {2.670} [2.892]

M(CO)8 (3A1g, Oh) De = 10.6 (71.7) kcal/mol; M = Ca De = 12.6 (65.9) kcal/mol; M = {Sr} De = 11.2 (69.9) kcal/mol; M = [Ba]

M(N2)8 (3A1g, Oh) De = 9.0 (23.0) kcal/mol; M = Ca De = 10.6 (21.3) kcal/mol; M = {Sr} De = 9.4 (30.1) kcal/mol; M = [Ba]

FIG. 1 The optimized structures of M(L)8 (M ¼ Ca, Sr, Ba; L ¼ PH3, PF3, N2) complexes at the M062X-D3/def2-TZVPP level. The dissociation energy, De values without parentheses are for single L dissociation, M(L)8 (T)! M(L)7 (T)+L (S) and the italic values in parentheses are for M(L)8 (T)! M (S)+8L (S).

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Atomic clusters with unusual structure, bonding and reactivity

are 3.2 (Ca(PF3)8)–11.8 (Ba(PH3)8) kcal/mol lower in energy than the corresponding singlet isomers. As expected, the steric crowding effects indeed play an important role in deciding the geometries. While for L ¼ CO and N2, all M(L)8 complexes belong to the Oh point group, for L ¼ PH3, only Sr and Ba analogs possess Oh symmetry, and Ca(PH3)8, because of the lower size of Ca, the Oh geometry is not even a minimum (having an imaginary 31.7i frequency), rather it corresponds to a chiral polyhedral group, pyritohedral symmetry, Th. In the cases of L ¼ PF3, even for the Sr and Ba complexes, the Oh symmetric complexes are not minima. While they belong to the tetrahedral (Td) point group, the lightest analog, Ca(PF3)8, has a chiral octahedral (O) symmetry. Similar to M(L)8, M(L)7 is also in the triplet ground state, and the corresponding singlets are 2.5–5.9 kcal/mol higher in energy than the triplet ones. There needs sizable energy to dissociate a single L from M(L)8 complexes which ranges from 6.0 (Ca(PH3)8) to 14.4 (Sr(PF3)8) kcal/mol, indicating the increased stability of octacoordinate complex in comparison to the heptacoordinate one. Except for Ca(PH3)8, PH3 and PF3 bound complexes have comparable stability with L ¼ CO, whereas M(N2)8 complex has slightly decreased stability compared to the other ligands. For the dissociation process, M(L)8 ! M + 8 L, the corresponding dissociation energy, De, for L ¼ PH3 and PF3, lies within 47.1 (Ca(PH3)8)–65.9 (Ba (PF3)8) kcal/mol, which is somewhat smaller than those values (65.9–71.7 kcal/mol) for L ¼ CO, whereas for L ¼ N2 it is significantly smaller (23.0–30.1 kcal/mol) than the former ones. Therefore, the ligand binding ability towards a given M can be arranged as CO > PF3 > PH3 >> N2. ˚ ), N2 (by 0.007– The complexation causes an elongation in the CdO, NdN and PdH bonds in CO (by 0.005–0.007 A ˚ ˚ 0.009 A), and PH3 (by 0.002–0.004 A), respectively, with respect to the free ones. However, in the case of PF3, the PdF ˚ ). The same is reflected in their corresponding IR (infrared) bonds get shortened upon the binding (by 0.005–0.007 A stretching frequencies given in Table 1. All these highly symmetric M(L)8 complexes show only one very intense peak in their respective IR spectra which corresponds to the asymmetric PdH or PdF or NdN or CdO stretching mode.

TABLE 1 The IR active PdH stretching frequencies (n, cm21) in M(PH3)8 complexes, the IR active PdF stretching frequencies in M(PF3)8 complexes, the IR active NdN stretching frequencies in M(N2)8 complexes and the IR active CdO stretching frequencies in M(CO)8 complexes at the M06-2X/def2-TZVPP level. Complex

n(L)

Dnd

Ca(PH3)8 (Th)

2294 (tu)

51

Sr(PH3)8 (Oh)

2302 (t1u)

43

Ba(PH3)8 (Oh)

2314 (t1u)

31

Ca(PF3)8 (O)

893 (t1)

33

Sr(PF3)8 (Td)

891 (t2)

31

Ba(PF3)8 (Td)

889 (t2)

29

Ca(N2)8 (Oh)

2156 (t1u)

174

Sr(N2)8 (Oh)

2166 (t1u)

164

Ba(N2)8 (Oh)

2198 (t1u)

132

Ca(CO)8 (Oh)

2018 (t1u)

125

Sr(CO)8 (Oh)

2027 (t1u)

116

Ba(CO)8 (Oh)

99

2044 (t1u) a

b

c

c

The scaling factors are 0.952 for PH3, 0.973 for PF3, 0.921 for N2, and 0.941 for CO. a Computed from the ratio of experimental frequency for e mode of PH3 (2345 cm1) and computed 2462 cm1. b Computed from the ratio of experimental frequency for e mode of PF3 (860 cm1) and computed 884 cm1. c Computed from the ratio of experimental and computed stretching frequencies. d Shift with respect to the experimental NdN stretching frequency of N2, experimental CdO stretching frequency of CO, and experimental e mode of PdH and PdF frequencies of PH3 and PF3, respectively.

M(L)8 complexes (M=Ca, Sr, Ba; L= PH3, PF3, N2, CO) Chapter

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While M(N2)8 complexes exhibit a large red-shift (ranging from 132 to 174 cm1) in NdN stretching frequency (n(N2)) in comparison to the free N2 [46], M(CO)8 complexes show large red-shift (ranging from 99 to 125 cm1) in CdO stretching frequency (n(CO)) in comparison to the free CO. On the other hand, M(pH3)8 complexes have a rather small but still redshift (by 31–51 cm1) in their PdH stretching frequency (n(PH)) with respect to the asymmetric e mode of frequency in free PH3 [47]. In contrast, the PdF stretching frequency shows a tiny blue-shift (by 29–33 cm1), compared to the asymmetric e frequency of bare PF3 [48]. This observation corroborates with the shortening of PdF bond distance upon the binding. In other words, PdF bonds get strengthened upon the complexation which will be explained from the EDA-NOCV analysis (vide infra). For a given L, the degree of red-shift or blue-shift is the largest for M ¼ Ca, and the smallest for M ¼ Ba.

4. MOs and correlation diagram The shape of the related occupied MOs, which are relevant to the M-L interaction, is displayed in Figs. 2 and 3 for Oh symmetric M(N2)8 and O symmetric Ca(PF3)8 complex. Irrelevant to the point group, they possess five types of MOs. The two degenerate SOMOs mainly possess d AOs of M with a significant coefficient on the (L)8 moiety, and they represent M(d) ! (L)8 p-backdonation. The remaining four types of MO are mainly ligand-based orbitals with small coefficients on M, except for HOMO. The HOMO is purely ligand-based orbital as there is no proper symmetric valence AO in the spd shell of M that could act as an acceptor. On the other hand, while the HOMO-1 and HOMO-2 correspond to the M(d) (L)8 and M(p) (L)8 s-donation, respectively, HOMO-3 stands for the M(s) (L)8 s-donation. This can be directly correlated with the correlation diagram given in Scheme 1. Note that the title M(L)8 complexes have a total of 18 valence electrons and, therefore, as such it can be argued that they satisfy the 18-electron rule. However, under the polyhedral field, there is always a pure ligand-based MO (like a2u in Oh symmetry, a2 in O symmetry, au in Th symmetry, and a1 in Td symmetry) which cannot be donated to M, because of the lack of appropriate acceptor AO. FIG. 2 The shape of the occupied molecular orbitals of triplet Oh symmetric M(N2)8 (M ¼ Ca, Sr, and Ba) complex that are responsible for M-N2 interaction.

162

Atomic clusters with unusual structure, bonding and reactivity

FIG. 3 The shape of the occupied molecular orbitals of triplet O symmetric Ca(PF3)8 complex that are responsible for Ca-PF3 interaction. The orbital energy values are in eV.

They only contribute little stabilization because of the presence of f polarization functions at M which have correct symmetry to interact with the ligand orbitals. Therefore, for the present cases, the effective electron counting makes them a 16 valence electron complex. In order to produce a somewhat fulfilled valence shell scenario, the title complexes prefer to adopt a triplet state where two degenerate highest-lying MOs are occupied by two electrons of the same spin. In this sense, only a 20 valence electron complex would give a completely fulfilled valence shell of M, as found in the cases of recently detected 20 valence electronic [TM(CO)8] (TM ¼ Sc, Y, and La) complexes [49].

5. Energy decomposition analysis The quantitative stabilization, originated from each of the above orbital interactions, can be evaluated with the EDA-NOCV analysis. The inspection of the SOMOs for the M(L)8 complexes clearly indicates that the M atom in these complexes corresponds to an excited state with ns0(n  1)d2 valence electron configuration. The numerical values for the EDA-NOCV analysis of M(L)8 complexes considering M in triplet excited reference state with ns0(n  1)d2 electronic configuration and singlet (L)8 fragment at the frozen geometry of M(L)8 are provided in Tables 2–5. In all the cases, the intrinsic interaction energy, DEint, between the fragments is quite high, implying the interplay of strong attractive energy therein, and the same follows the order as Ca > Sr >> Ba. Note that the De values do not show the same order as DEint since the major part of the increased latter term for Ca and Sr is used to compensate for the larger preparation energy (DEprep) for the excitation, M(ns2) ! M(ns0(n  1)d2) which is computed as 159.5 (Ca), 150.9 (Sr), and 68.2 (Ba) kcal/mol. The major

M(L)8 complexes (M=Ca, Sr, Ba; L= PH3, PF3, N2, CO) Chapter

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TABLE 2 EDA-NOCV results for triplet M(PH3)8 (M 5 Ca, Sr, and Ba) complexes at the M06-2X/TZ2P-ZORA//M06-2X-D3/ def2-TZVPP level taking (PH3)8 in singlet ground state and M in triplet excited state with a (n)s0(n 2 1)d2 valence electronic configuration as interacting fragments. Energies

Interaction

Ca (T) + (PH3)8 (S)

Sr (T) + (PH3)8 (S)

Ba (T) + (PH3)8 (S)

D Eint

219.3

209.3

135.7

D Ehybrid

11.4

15.4

13.8

D EPauli

8.4

20.3

32.4

D Eelstata

57.2 (23.9%)

58.8 (24.0%)

91.1 (50.1%)

D Eorba

181.9 (76.1%)

186.3 (76.0%)

90.7 (49.9%)

162.2 (89.2%)

164.8 (88.5%)

59.6 (65.7%)

D Eorb1b,c

[M(d)] ! (PH3)8 p backdonation

D Eorb2b,c

[M(d)]

(PH3)8 s donation

14.1 (7.8%)

15.3 (8.2%)

21.3 (23.5%)

D Eorb3b

[M(s)]

(PH3)8 s donation

2.0 (1.1%)

2.1 (1.1%)

2.1 (2.3%)

D Eorb4b,c

[M(p)]

(PH3)8 s donation

0.1 (0.1%)

1.2 (0.6%)

1.5 (1.7%)

D Eorb5b

(PH3)8 polarization

0.5 (0.3%)

0.2 (0.1%)

1.2 (1.3%)

4.2 (2.3%)

5.5 (2.9%)

5.0 (5.5%)

D Eorb(rest)

Energy values are given in kcal/mol. a The values in parentheses give the percentage contribution to the total attractive interactions DEelstat + DEorb. b The values in parentheses give the percentage contribution to the total orbital interactions DEorb. c The sum of the two or three components is given.

TABLE 3 EDA-NOCV results for triplet M(PF3)8 (M 5 Ca, Sr, and Ba) complexes at the M06-2X/TZ2P-ZORA//M06-2X-D3/ def2-TZVPP level taking (PF3)8 in singlet ground state and M in triplet excited state with a (n)s0(n 2 1)d2 valence electronic configuration as interacting fragments. Energies

Ca (T) + (PF3)8 (S)

Sr (T) + (PF3)8 (S)

Ba (T) + (PF3)8 (S)

D Eint

209.6

200.5

128.4

D Ehybrid

14.5

17.0

18.9

D EPauli

17.3

27.0

37.3

D Eelstata

65.0 (26.9%)

64.3(26.3%)

91.5(49.5%)

D Eorba

176.4 (73.1%)

180.2 (73.7%)

93.2 (50.5%)

151.2 (85.7%)

154.5 (85.7%)

57.9 (62.1%)

D Eorb1b,c

Interaction

(e)

[M(d)] ! (PF3)8 p backdonation

D Eorb2b,c

[M(d)]

(PF3)8 s donation

18.9 (10.7%)

19.2 (10.7%)

24.0 (25.8%)

D Eorb3b

[M(s)]

(PF3)8 s donation

2.0 (1.1%)

2.2 (1.2%)

2.2 (2.4%)

D Eorb4b,c

[M(p)]

(PF3)8 s donation

0.6 (0.3%)

0.6 (0.3%)

2.4 (2.6%)

D Eorb5b

(PF3)8 polarization

0.2 (0.1%)

0.1 (0.1%)

1.5 (1.6%)

3.9 (2.3%)

4.8 (2.6%)

5.2 (5.5%)

D Eorb(rest)

Energy values are given in kcal/mol. a The values in parentheses give the percentage contribution to the total attractive interactions DEelstat + DEorb. b The values in parentheses give the percentage contribution to the total orbital interactions DEorb. c The sum of the two or three components is given.

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TABLE 4 EDA-NOCV results for triplet M(N2)8 (M 5 Ca, Sr, and Ba) complexes at the M06-2X/TZ2P-ZORA//M06-2X-D3/ def2-TZVPP level taking (N2)8 in singlet ground state and M in triplet excited state with a (n)s0(n 2 1)d2 valence electronic configuration as interacting fragments. Energies

Ca (T) + (N2)8 (S)

Sr (T) + (N2)8 (S)

Ba (T) + (N2)8 (S)

DEint

190.7

175.2

104.0

DEhybrid

48.9

51.4

35.1

DEPauli

25.8

31.3

37.0

49.2 (18.5%)

45.2 (17.5%)

54.7 (31.1%)

216.2 (81.5%)

212.5 (82.5%)

121.4 (68.9%)

184.3 (85.2%)

180.6 (85.0%)

85.0 (70.0%)

DEelstat

Interaction

a

DEorba DEorb1

b,c

[M(d)] ! (N2)8 p backdonation

DEorb2

b,c

[M(d)]

(N2)8 s donation

18.0 (8.3%)

17.4 (8.2%)

18.0 (14.8%)

DEorb3

b

[M(s)]

(N2)8 s donation

3.5 (1.6%)

3.9 (1.8%)

3.7 (3.0%)

DEorb4

b,c

[M(p)]

(N2)8 s donation

2.4 (1.1%)

2.7 (1.3%)

4.2 (3.5%)

DEorb5

b

(N2)8 polarization

0.8 (0.4%)

1.1 (0.5%)

1.9 (1.6%)

7.2 (3.4%)

6.8 (3.2%)

8.6 (7.1%)

DEorb(rest)

This table is taken from reference Q. Wang, S. Pan, S. Lei, J. Jin, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Nat. Commun. 10 (2019) 3375. Energy values are given in kcal/mol. a The values in parentheses give the percentage contribution to the total attractive interactions DEelstat + DEorb. b The values in parentheses give the percentage contribution to the total orbital interactions DEorb. c The sum of the two (eg) or three (t2g, t1u) components is given.

TABLE 5 EDA-NOCV results for triplet M(CO)8 (M 5 Ca, Sr, and Ba) complexes at the M06-2X/TZ2P-ZORA//M06-2X-D3/ def2-TZVPP level taking (CO)8 in singlet ground state and M in triplet excited state with a (n)s0(n 2 1)d2 valence electronic configuration as interacting fragments. Energy terms

Ca (T) + (CO)8 (S)

Sr (T) + (CO)8 (S)

Ba (T) + (CO)8 (S)

DEint

243.1

224.1

145.2

DEhybrid

41.8

46.9

37.4

DEPauli

19.5

30.5

30.8

65.3 (21.5%)

61.7 (20.5%)

78.5 (36.8%)

239.1 (78.5%)

239.7 (79.5%)

135.0 (63.2%)

206.2 (86.2%)

206.4 (86.2%)

95.0 (79.8%)

DEelstat

Interaction

a

DEorba DEorb(1)

b,c

[M(d)] ! (CO)8 p backdonation

DEorb(2)

b,c

[M(d)]

(CO)8 s donation

21.3 (9.0%)

20.7 (8.7%)

22.8 (16.8%)

DEorb(3)

b

[M(s)]

(CO)8 s donation

2.4 (1.0%)

2.9 (1.2%)

3.2 (2.4%)

DEorb(4)

b,c

[M(p)]

(CO)8 s donation

0.9 (0.3%)

0.9 (0.3%)

2.7 (2.1%)

DEorb(5)

b

(CO)8 polarization

0.6 (0.3%)

1.0 (0.4%)

2.3 (1.7%)

7.7 (3.2%)

7.8 (3.3%)

9.0 (6.7%)

DEorb(rest)

b

This table is taken from reference X. Wu, L. Zhao, J. Jin, S. Pan, W. Li, X. Jin, G. Wang, M. Zhou, G. Frenking, Science 361 (2018) 912–916. Energy values are given in kcal/mol. a The values within the parentheses show the contribution to the total attractive interactions DEelstat + D Eorb. b The values within the parentheses show the contribution to the total orbital interaction, DEorb. c The sum of the two (eg) or three (t2g, t1u) components is given.

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contribution (c. 69%–83%) towards the DEint comes from the orbital term, D Eorb for all cases, except for Ba(PH3)8 and Ba(PF3)8. For the latter two cases, both the orbital and electrostatic (D Eelstat) terms contribute almost equally. For a given L, while the size of the D Eorb term varies as Ca  Sr >> Ba, D Eelstat shows the reverse trend, Ca  Sr < Ba for L ¼ PH3 and PF3. For L ¼ CO and N2, D Eelstat follows the order as Sr < Ca < Ba. The most meaningful information from the EDA-NOCV analysis is obtained from the breakdown of D Eorb term into the pairwise orbital interactions and the corresponding plots of deformation densities (Dr) (see Figs. 4 and 5 for the shapes of Dr in Ca(PF3)8 and Ca(N2)8 complexes, respectively). In the *deformation densities plot, electron density shifts from red to blue regions. The size of electron transfer is indicated by the charge eigenvalues, n. The major part of the M-L orbital

FIG. 4 The plot of deformation densities of triplet O symmetric Ca(PF3)8 complex at the M06-2X/ TZ2P-ZORA level. Associated orbital energies are in kcal/mol. Electron density shifts from red to blue region. The charge eigenvalues are given by jn j.

'Eorb(1a) = -75.6 »Q1aα°= 0.68

'Eorb(2a) = -6.3 »Q2aα/2aβ°= 0.13/0.12

'Eorb(1b) = -75.6 »Q1bα°= 0.67

'Eorb(2b) = -6.3 »Q2bα/2bβ°= 0.13/0.12

'Eorb(2c) = -6.3 »Q2cα/2cβ°= 0.13/0.12

'Eorb(3) = -2.0 »Q3α/3β°= 0.08/0.06

'Eorb(4a) = -0.2 »Q4aα/4aβ°= 0.07/0.05

'Eorb(4b) = -0.2 »Q4bα/4bβ°= 0.07/0.05

'Eorb(5) = 0.2 »Q5α/5β°= 0.04

'Eorb(4c) = -0.2 »Q4cα/4cβ°= 0.07/0.05

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FIG. 5 The plot of deformation densities of triplet Oh symmetric Ca(N2)8 complex using neutral partitioning scheme at the M06-2X/TZ2P-ZORA level. Energies are in kcal/mol. Electron density shifts from red to blue region. The charge eigenvalues are given by jnj. (This figure is taken from reference Q. Wang, S. Pan, S. Lei, J. Jin, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Nat. Commun. 10 (2019) 3375.)

'Eorb(1a) = -92.1 »Q1aα°= 0.73

'Eorb(1b) = -92.2 »Q1bα°= 0.73

'Eorb(2a) = -6.0 »Q2aα/2aβ°= 0.09

'Eorb(2b) = -6.0 »Q2bα/2bβ°= 0.09

'Eorb(2c) = -6.0 »Q2cα/2cβ°= 0.09

'Eorb(3) = -3.5 »Q3α/3β°= 0.07/0.06

'Eorb(4a) = -0.8 »Q4aα/4aβ°= 0.05

'Eorb(4b) = -0.8 »Q4bα/4bβ°= 0.05

'Eorb(4c) = -0.8 »Q4cα/4cβ°= 0.05

'Eorb(5) = -0.8 »Q5α/5β°= 0.04

interaction (c. 85%–89% for Ca and Sr and 62%–80% for Ba) comes from the two sets of M(dp) ! (L)8 p backdonation. For M(CO)8 and M(N2)8, the (n  1)d2 electrons are transferred into unoccupied p* MO of (L)8 moiety causing a very large redshift inn(CO) and n(N2), whereas in the cases of L ¼ PH3 and PF3, two degenerate LUMOs of (L)8 fragment, which consist of mainly vacant 3p orbitals of P centers, act as acceptors for this backdonation [21]. Specifically, the LUMOs for L ¼ PH3 have small coefficient from s orbitals of H centers which represent the PdH antibonding s* orbitals, and, thus, the backdonation weakens the PdH bonds which interpret the small red-shift in n(pH). On the other hand, for L ¼ PF3, the LUMOs in (PF3)8 fragment possess very small (smaller than in PH3) coefficient from p orbitals of F center, and therefore, the effect of the backdonation on PdF bond is very minute. The next highest contribution comes from the M(d) (L)8 s donation (c. 7%–11% for Ca and Sr and an improved 14%– 26% for Ba). The s donation ability gets enhanced from PH3 to PF3 as an electron donation from lone pairs of P center to M

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induces an electron transfer from electron-rich F centers towards P centers. This is the reason for the obtained small blueshift in n(PF). The combination of acceptor and donor d orbitals of M is responsible for 85%–97% of total orbital interaction, proving the act of M as a conventional transition metal. The next term is the M(s) (L)8 s donation which results in only 1%–3% of the D Eorb. The contribution from the M(p) (L)8 s donation and (L)8 polarization is even smaller than the former ones. The polarization term corresponds to the HOMO which brings tiny stabilization through polarization because of the field effect in the complex. Note that in some cases, small positive orbital contributions are obtained which may be because of the numerical errors of the employed method. In M(L)8 complexes, the M(dp) ! (L)8 p backdonation ability is noted to follow the order as CO > N2 > PH3 > PF3.

6. M(Bz)3: 20-electron complex The above M(L)8 complexes are 18-electron complexes which make them triplet electronic ground state. The addition of two more electrons should lead to a closed-shell complex. We studied M(Bz)3 complex (M ¼ Ca, Sr, Ba; Bz ¼ benzene) where benzene would act as a six electron donor making it a 20-electron complex [50]. As expected, M(Bz)3 complex has a D3 symmetric singlet (1A1) electronic ground state as energy minima which is 0.6–3.8 kcal/mol lower in energy than the corresponding triplet state which has C1 symmetry for M ¼ Ca and C2 symmetry for M ¼ Sr, Ba (see Fig. 6). In the M(Bz)3 complexes, the geometry of benzene rings get distorted from D6h symmetry in the free state to C2 symmetry where ˚ ) than that in free benzene (1.388 A ˚ ), whereas the other CdC two CdC distances are considerably longer (1.420–1.423 A bond lengths are only marginally changed. Similar to M(Bz)3 complexes, M(Bz)2 complexes also have a singlet electronic ground state with a C2v point group. Table 6 provides the bond dissociation energy (BDE) values at 0 and at 298 K for the dissociation of one and all three benzene ligands from the M(Bz)3 complex. The results show that the BDE values for one benzene dissociation at 0 K lie in the range of 19.4 kcal/mol for M ¼ Ca and 26.1 kcal/mol for M ¼ Sr. The inclusion of thermal correction and entropy factor

2.820 {2.928} [3.095]

2.768 {2.895} [3.082]

2.822 {2.920} [3.070] 1.387 {1.389} [1.396]

FIG. 6 Calculated geometries of M(Bz)3 and M(Bz)2 complexes (M ¼ Ca, Sr, and Ba) at M06-2X-D3/ ˚ , Relative energies are def2-TZVPP. Bond distances are in A in kcal/mol. (This figure is taken from reference Q. Wang, S. Pan, Y. Wu, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Angew. Chem. Int. Ed. 58 (2019) 17365–17374.)

1.396 {1.396} [1.389]

2.677-3.075 {2.815-3073} [2.977-3.236]

1.422 {1.423} [1.420]

M(Bz)3 (C1, 3A for Ca; C2, 3A for Sr, Ba) Erel = 3.8 {2.3} [0.6]

M(Bz)3 (D3, 1A1) M = Ca, {Sr}, [Ba] E rel = 0.0 {0.0} [0.0]

2.659 {2.838} [3.046]

2.632 {2.797} [2.988]

2.597 {2.788} [3.009] 1.427 {1.424} [1.419]

1.378 {1.378} [1.379]

2.510 {2.669} [2.855]

1.436 {1.434} [1.429]

M(Bz)2 (C2v, 1A1) M = Ca, {Sr}, [Ba]

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TABLE 6 Calculated bond dissociation energies of M(Bz)3 complexes for the loss of one Bz and three Bz molecules at the M06-2X-D3/def2-TZVPP level. Reaction

M

De

D0

M(Bz)3 (D3, 1A1) ! M(Bz)2 (C2v, 1A1) + Bz

Ca

19.4

18.3

6.5

Sr

26.1

25.3

14.2

Ba

24.0

23.6

13.4

Ca

38.6

38.3

8.7

Sr

39.2

39.7

11.0

Ba

46.0

47.1

19.5

M(Bz)3 (D3, A1) ! M ( S) + 3 Bz 1

1

DG298K

This table is reproduced from reference Q. Wang, S. Pan, Y. Wu, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Angew. Chem. Int. Ed. 58 (2019) 17365–17374. The De values give the electronic energies, the D0 data include the corrections by zero-point vibrational frequencies and the DG298K values consider the thermodynamic and vibrational corrections at room temperature.

reduces the BDE values at 298 K. The corresponding dissociation free energy value ranges between DG298 ¼ 6.5 kcal/mol (Ca) and DG298 ¼ 14.2 kcal/mol (Sr). All the three benzene ligands dissociation needs significant energies ranging between 38.6 kcal/mol (Ca) and 46.0 kcal/mol (Ba) whereas corresponding free dissociation energies at room temperature are considerably lower having values between DG298 ¼ 8.7 kcal/mol (Ca) and DG298 ¼ 19.5 kcal/mol (Ba). The orbital correlation diagram for M(Bz)3 at the equilibrium geometries in the D3 field of the benzene ligands and the splitting of the (n)s(n)p(n  1)d AOs of M atoms and the shape of the occupied MOs of M(Bz)3 are depicted in Fig. 7. In D3 field, the five d orbitals of M are split into a1 + e + e irreducible representations. The a1 orbital that represents the d2z AO of the metal involves in the backdonation to the antibonding p* orbital of the benzene ligands. This interaction makes the bonding MO (2a1). The remaining eight metal AOs act as acceptor orbitals to accommodate the electron donation from the p orbitals of the benzene ligands. There are nine symmetry-adapted linear combination orbitals from the bonding p orbitals of three benzene ligands. The 2a2 MO is purely ligand-based orbitals as M lacks any valence orbitals having such symmetry to interact. 1a2 and 1e MOs present (Bz)3 ! M(p) electron donation, whereas 2e and 3e MOs represent FIG. 7 MO correlation diagram of the valence (n) s(n)p(n  1)d AOs of M and (C6H6)3 in D3 symmetry. Shapes of the MOs of Ba(Bz)3 are also shown. (This figure is reproduced from reference Q. Wang, S. Pan, Y. Wu, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Angew. Chem. Int. Ed. 58 (2019) 17365–17374.)

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TABLE 7 Results of EDA-NOCV calculations at BP86-D3(BJ)/TZ2P for M(Bz)3 (D3, 1A1) complexes using neutral atoms M in the singlet state with ns0np0(n 2 1)d2 electron configuration and (Bz)3 (singlet) as interacting fragments. Ca (S, 4s04p03d2) + (Bz)3 (S)

Sr (S, 5s05p04d2) + (Bz)3 (S)

Ba (S, 5s05p04d2) + (Bz)3 (S)

D Eint

197.4

191.5

135.5

D EPauli

124.5

140.5

156.8

D Edispa

7.1 (2.2%)

8.4 (2.5%)

16.8 (5.7%)

D Eelstata

82.7 (25.7%)

88.8 (26.8%)

116.4 (39.8%)

D Eorba

232.1 (72.1%)

234.7 (70.7%)

159.1 (54.4%)

(Bz)3 M(d) backdonation

191.8 (82.6%)

189.9 (80.9%)

119.9 (75.4%)

D Eorb(2)b (3e)

(Bz)3 ! M(d) donation

14.0 (6.0%)

16.8 (7.2%)

16.8 (10.6%)

D Eorb(3)b (2e)

(Bz)3 ! M(d) donation

13.6 (5.9%)

15.0 (6.4%)

11.8 (7.4%)

D Eorb(4)b (1a1)

(Bz)3 ! M(s) donation

2.7 (1.2%)

2.9 (1.2%)

2.1 (1.3%)

D Eorb(5)b (1a2)

(Bz)3 ! M(p) donation

1.4 (0.6%)

1.3 (0.6%)

0.8 (0.5%)

D Eorb(6)b (1e)

(Bz)3 ! M(p) donation

3.2 (1.4%)

2.4 (1.0%)

0.8 (0.5%)

D Eorb(7)b (2a2)

Polarization

1.3 (0.6%)

1.8 (0.8%)

2.5 (1.6%)

4.1 (1.8%)

4.6 (2.0%)

4.4 (2.8%)

Term

D Eorb(1)b (2a1)

D Eorb(rest)b

Interaction

This table is reproduced from reference Q. Wang, S. Pan, Y. Wu, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Angew. Chem. Int. Ed. 58 (2019) 17365–17374. a The values in parentheses give the percentage contribution to the total attractive interactions DEelstat + DEorb + DEdisp b The values in parentheses give the percentage contribution to the total orbital interactions DEorb.

(Bz)3 ! M(d) electron donation. Lastly, (Bz)3 ! M(s) electron donation is reflected from the 1a1 MO. Therefore, this makes 20-electron complex an effective 18-electron complex. From the above diagram, it is understandable that the M-Bz interaction in M(Bz)3 complex can be expressed as the interaction between M in its singlet excited state with an (n  1)d2ns0np0 electron configuration and three benzene ligands in the singlet state. The corresponding EDA-NOCV results considering the above partitioning scheme are provided in Table 7. Here, it should be mentioned that the calculations at M06-2X/TZ2P level with the ADF program for M in the singlet excited state with d2 configuration were not converged, and, therefore, we switched to BP86-D3(BJ)/TZ2P level. The results in Table 7 imply that M-(Bz)3 interactions are predominantly covalent in nature where DEorb value is responsible for 54%–72% of the total attraction whereas DEelstat term is accountable for 25%–40% of the total attraction. Further division of total orbital term into pair-wise orbital interaction shows that the strongest contribution, DEorb(1) (75%–83%) is originated from the (Bz)3 M(d) backdonation. The next two sets of degenerate orbital contributions, DEorb(2) and DEorb(3) come from the (Bz)3 ! M(d) donation. Therefore, the combined contribution of d AO of M in the bonding is very substantial. The contribution of s (DEorb(4)) and p AOs (DEorb(5) and DEorb(6)) of M towards stabilization is quite small. Note that the orbital stabilization (DEorb(7)) originated from the 2a2 MO is also negligible. The shape of the deformation densities, Dr(1)  Dr(7) corresponding to DEorb(1)  DEorb(7) for Ba(Bz)3 is shown in Fig. 8. In the figure, the electron density is shifted from red to blue region. These plots nicely confirm the donation and backdonation assignment in Table 7. The shape of Dr(7) clearly indicates that the small stabilization energy DEorb(7) is merely due to the polarization of the ligand orbitals. The deformation densities Dr(1)  Dr(7) for the lighter homologs, Ca(Bz)3 and Sr(Bz)3 are very similar to those of Ba(Bz)3.

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Atomic clusters with unusual structure, bonding and reactivity

Eorb(1) = -119.9,

1

= 1.58

Eorb(2a) = -8.4,

2a

= 0.30

Eorb(3b) = -5.9,

3b

= 0.27

Eorb(4) = -2.1,

4

= 0.13

Eorb(6b) = -0.4,

6b

= 0.06

Eorb(7) = -2.5,

7

= 0.12

Eorb(2b) = -8.4,

Eorb(5) = -0.8,

2b

5

= 0.30

= 0.09

Eorb(3a) = -5.9,

3a

= 0.27

Eorb(6a) = -0.4,

6a

= 0.06

FIG. 8 Shape of the deformation densities Dr(1)–(7), which are associated with the orbital interactions DEorb(1)–(7) between neutral fragments Ba and (Bz)3 in Ba(Bz)3 (D3, 1A1) complex, using Ba (S, 6s06p05d2) + (Bz)3 (S) as interacting complexes, and eigenvalues jnn jof the charge flow. The color code of the charge flow is red (light gray in print version) ! blue (light gray in print version). The isosurface values are 0.002 for Dr(1), 0.0005 for Dr(2)–(4) and 0.0004 for Dr(5)–(7). (This figure is reproduced from reference Q. Wang, S. Pan, Y. Wu, G. Deng, G. Wang, L. Zhao, M. Zhou, G. Frenking, Angew. Chem. Int. Ed. 58 (2019) 17365–17374.)

7. Conclusions The present study shows that a wide range of ligands can induce the transition metal-like behavior in alkaline-earth metals, M ¼ Ca, Sr, and Ba. The title M(L)8 complexes are energetically stable with respect to both single ligand dissociation and all eight ligands dissociation. The bonding analysis shows that the M center is located in an excited triplet state with ns0(n  1) d2 valence electron configuration and the M-L interaction is mainly comprised of M(dp) ! (L)8 p backdonation and of M(d) (L)8 s donation to a minor extent, mimicking like a conventional transition metal. The present results also show that the mere sum of valence electrons of ligand and metal can be sometimes wrong, rather one should only count those electrons which interact with the valence AO of metal. M(Bz)3, which is a 20-electron complex, satisfies the 18-electron rule as one MO is a purely ligand-based orbital, and M does not have a suitable symmetric AO to interact with that ligand orbital. Ba was already regarded as an “honorary transition metal.” Our results with four different ligands showed that even Ca and Sr may also be eligible to get the membership.

Acknowledgments LZ acknowledges the financial support from National Natural Science Foundation of China (Grant No. 21973044), Nanjing Tech University (Grant Nos. 39837123 and 39837132), the State Key Laboratory of Materials-oriented Chemical Engineering (project No. KL19-11), and the high performance center of Nanjing Tech University for supporting the computational resources. DJ thanks the Postgraduate Research &

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Practice Innovation Program of Government of Jiangsu Province (No. KYCX21_1184). ZC acknowledges the financial support from National Natural Science Foundation of China (Nos. 11922405, 11874178, and 91961204).

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Chapter 10

Structures, reactivity, and properties of low ionization energy species doped fullerenes and their complexes with superhalogen Abhishek Kumara, Ambrish Kumar Srivastavab,#, Gargi Tiwaric, and Neeraj Misraa a

Department of Physics, University of Lucknow, Lucknow, Uttar Pradesh, India, b Department of Physics, Deen Dayal Upadhyaya Gorakhpur University,

Gorakhpur, Uttar Pradesh, India, c Department of Physics, Patna University, Patna, Bihar, India

1. Introduction Among all the elements that have been discovered to date, carbon is surely the most intriguing element. As the core element of thousands of organic compounds, carbon is associated closely with the origin of life [1]. Carbon-based nanostructures, such as graphene [2], carbon nanotube [3], and fullerene [4], have been known for a broad range of remarkable properties. The discovery of fullerenes, a collection of spherical molecules or clusters consisting of pure carbon atoms, has not only added a new form of elemental carbon but also opened an avenue for research in molecular architecture. The most popular fullerene (C60) consists of 12 pentagonal rings and 20 hexagonal rings that form a closed-cage geometric structure of C60 molecule with 32 faces, which was discovered by Kroto et al. [4] in 1985 as displayed in Fig. 1. Its macroscopic preparation [5] has opened multiple research areas, now commonly embraced under the label of “fullerene chemistry.” It is well known that the C60 molecule may have unusual magnetic properties and also possesses an ambiguous aromatic character. The hollow space enables fullerene to trap atoms or small molecules/clusters, and these were referred to as endofullerenes also depicted in Fig. 1. Endofullerenes consist of fullerene cages encapsulating small species, which are free to rotate and translate inside the cage [6]. The dihydrogen endofullerenes (H2@C60), water endofullerenes (H2O@C60), and their isotopologues have been synthesized by the procedure known as “molecular surgery.” In this process, synthetic operations are used to open a hole in the fullerene allowing encapsulation of the guest, followed by a suturing technique to reform the pristine fullerene shell [7–9]. Endofullerenes [10] have been widely known for their unique structures, novel electronic properties, and potential applications in a variety of fields, such as superconductors [11], optical switches [12], solar cells [13], nanotechnology and biomedical applications [14]. The discovery of the first endofullerene La@C60 named as endohedral metallofullerene complex, by Heath et al. [15], has opened the route to the study of endofullerenes ranging from metal-doped C60 [16– 19], nonmetal doped C60 [20–23], noble atom doped C60 [24–26] as well as various molecules doped C60 [8,27–29]. The properties of encapsulated metal atoms or species contribute largely to the electronic and chemical properties of the endofullerenes. Endohedral fullerene, formed by encapsulation of atoms or molecules within the fullerene cages [30,31] makes them more reactive. In the case of endohedral metallofullerenes, the C60 sphere works as a protective cover such that the properties of the dopant remain unchanged. There have been many attempts on the doping of C60 using alkali [17], transition, and rare-earth metals [19] during the past decades, because of their valence level structures exhibiting different spin configurations. The doping of C60 with alkali metals and resulting materials exhibit promising superconducting properties with high critical magnetic fields [32], which have been assessed via experimental techniques [33]. Further, the doping of fullerene with a transition element could result in very interesting properties, which can be utilized in the design, synthesis, and growth of future nanomaterials. Nevertheless, such complex structures are important in many disciplines, such as surface science, nanoscience, catalysis, inorganic chemistry, and materials science. These structures are also interesting because they serve as precursors to potential new materials for nanotechnology and semiconductors. Therefore, many theoretical and experimental studies have aimed to discover as well as synthesize these novel nanomaterials [33–39]. # This author has equal contribution as the first author. Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00002-4 Copyright © 2023 Elsevier Inc. All rights reserved.

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174 Atomic clusters with unusual structure, bonding and reactivity

Atom or Small molecule/cluster

Pentagon ring

Hexagon ring

Fullerene

Endofullerene

FIG. 1 The structures of fullerene (C60) and endofullerene.

Some other endohedral fullerenes such as H2O@C60 have also been experimentally determined [8] and well studied [40– 44]. The properties of endohedral fullerenes with encapsulated noble gas atoms such as Ne@C60 have also been theoretically reported [26,27,45,46]. There are also some theoretical [47–52] as well as experimental [53,54] reports on superheavy elements such as lanthanides and actinides doped fullerene. This chapter is intended to provide a concise account of some endofullerenes doped with the species having low ionization energy (IE). Alkali atoms are known to possess the lowest IE among all the elements in the periodic table, 5.39 eV (Li)–3.89 eV (Cs) [55]. However, there are certain clusters, which have lower IE than alkali atoms. These are known as “superalkalis,” which also act as powerful reducing agents and building blocks of compounds with several interesting applications. These superalkali species were suggested and designed by Gutsev and Boldyrev in 1982 [56]. Superalkalis are small hypervalent clusters consisting of excess electropositive atoms, typical examples are FLi2, OLi3, and NLi4 [57–59]. These compounds can be employed to design several unusual compounds such as supersalts [60–65] with aromaticity and nonlinear optical (NLO) behavior, superbases [66,67] with the basicity larger than strong bases, alkalides [68–70] with the negatively charged alkali atoms. Considering the ever-increasing applications of superalkalis in a variety of fields, superalkalis are attracting continuous attention to date. The counterparts of superalkalis are known as “superhalogens”. The superhalogens were also introduced by Gutsev and Boldyrev [71]. They devised a systematic approach for the design of the species that possess higher electron affinity than halogen atoms and classified them as superhalogens [71–73]. Superhalogens are hypervalent clusters, which need an extra electron to complete their octet. These species play an important role in the design of unusual compounds with high oxidizing capabilities. In the early 1960s, Bartlett has synthesized the XePtF6 complex demonstrating the capability of PtF6 to oxidize Xe (a noble) atom, which was attributed to the high EA of PtF6, about 7 eV [74]. Superhalogen can also be used in the design of superacids [75,76], materials for hydrogen storage [77], and lithium-ion batteries [78,79]. Superalkalis interact with superhalogens to form supersalts with pronounced NLO characteristics [66,68,80] and other interesting properties [60,61,70]. In this chapter, we will also focus on the superhalogen complexes of such low IE species doped endofullerenes.

2.

Computational techniques

The computational technique used in this chapter incorporates density functional theory (DFT)-based B3LYP/6-31G(d) scheme as implemented in GAUSSIAN 09 program [81]. This B3LYP method is a hybrid form of exchange-correlation functional which combines parameterized exchange term of A.D. Becke [82] and correlation term devised by Y. Lee, W. Yang, and R. G. Parr [83]. For Lr atom, we have used SDD pseudo-potentials in which 60 core electrons of Lr are replaced by electronic core potentials. The present scheme has already been used to provide accurate results at an affordable cost in various C60 derivatives and related systems [84–87]. The geometry optimization has been carried out without any symmetry constraint in the potential energy surface. Below, we provide some formulae used to obtain the parameters discussed in this chapter. The dipole moment (m) and mean polarizability (ao) are obtained as,  1=2 m ¼ m2 x + m2 y + m2 z ao ¼

 1 axx + ayy + azz 3

Low ionization energy species doped fullerenes Chapter

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The symbols mi, and aij represent the components of dipole moment vector and polarizability tensors, respectively, along the directions specified by subscripts, i, j, and k ¼ x, y, and z. The components of polarizability are computed by numerical differentiation using static electric field of magnitude 0.001 a.u. The depression of polarizability (Da) is calculated as, Da ¼ ao  (aC60 + aM)

3. Low IE species doped endofullerenes Lithium (Li) is representative element of the alkali group having the lower IE across the periodic table. The charge transfer takes place to C60 when trapped by Li atom. Consequently, C60 behaves as an electron-acceptor in the case of Li@C60. As mentioned in the preceding section that superalkalis (SAs) possess lower IE than alkali atoms. This invited us to introduce SA doped C60 fullerene. We will compare the properties of SA@C60 with those of well-studied Li@C60 by choosing typical SA clusters, i.e., FLi2, OLi3, and NLi4. Lawrencium (Lr) belongs to the actinide series with atomic number 103, discovered by Ghiorso et al. [88] during the 1960s. Although the position of Lr in the periodic table has been debated since long ago [89], a recent study by Sato et al. [55] reported the first IE of Lr 4.96 eV by experimental measurements as well as theoretical calculations. This value is not only smaller than those of other elements in the actinide series but also lower than those of Li and Na. Therefore, the finding of Sato et al. [55] further continues the debate whether Lr resembles the properties of actinides or alkali metals. This motivates to encapsulate Lr atom into C60 fullerene and compare its properties with those of Li@C60.

3.1 Li@C60 vs SA@C60 endofullerene (SA 5 FLi2, OLi3, and NLi4) The equilibrium structures of SA@C60 (SA ¼ FLi2, OLi3, and NLi4) are shown in Fig. 2 and corresponding parameters are listed in Table 1. As compared to Li@C60, the dcenter significantly decreases with the introduction of superalkali inside C60 ˚ for FLi2, 0.21 A ˚ for OLi3, and merely 0.02 A ˚ for NLi4. It has been reported [92,93] that the such that it becomes 0.23 A displacement of alkali metal from the center decreases as one moves down in the periodic table, i.e., Li ! Na ! K such ˚ away. Note that the atomic size of alkali metal increases that K is situated near the center of the C60 cage, only 0.3 A and IE decreases as move down in the group. This fact can be successfully correlated with the size and IE of encapsulated ˚ in Li@C60 as compared to empty C60 superalkalis. As mentioned earlier, the dH-H and dH-P bond lengths increase by 0.01 A ˚. but remain unaffected in FLi2@C60 and OLi3@C60. In NLi4@C60, however, these bond lengths are increased up to 0.04 A This causes to slightly destabilize NLi4@C60 relative to Li@C60 and other SA@C60, as discussed later. It is also interesting to analyze the effect of confinement on the structure of superalkalis. Note that the bond lengths of encapsulated FLi2, OLi3, ˚ , respectively. To discuss the charge transfer from SA to C60 in and NLi4 superalkalis are reduced by 0.03, 0.06, and 0.07 A SA@C60 and the back-donation from C60, the net NBO charge on encapsulated superalkalis (Q) are also listed in Table 1. The Q is slightly larger in Li@C60 due to the fact that the natural bond orbital (NBO) scheme assigns the maximum possible occupancy to each atomic orbital [94]. For FLi2@C60 and OLi3@C60, QSA is significantly smaller suggesting the back donation from C60 to encapsulated SA, which increases from FLi2 to OLi3. The encapsulated Li atom transfers valence electrons to the fullerene and stabilizes in an off-center equilibrium position, as noted above, resulting in a net electric dipole moment for the molecule. The dipole moment (m) of Li@C60 and SA@C60 are also listed in Table 1. The dipole moment of Li@C60 results due to polarization of cage, i.e., transfer of negative charge in the vicinity of Li atom. On the contrary, the vicinity of encapsulated SA seems to be positively charged, which clearly explains the back-donation from C60. In SA@C60, therefore, the significantly reduced off-center distance and backdonation of charge tend to reduce the dipole moment, which is compensated by increased charge transfer from SA. Therefore, the m value of SA@C60 is comparable to or smaller than that of Li@C60. The direction of m in SA@C60 is along the cage center to encapsulated SA, like Li@C60. Note that all these endofullerenes are neutral (doublet), their frontier orbital energy gap (Egap) was calculated by the energy difference of the singly occupied molecular orbital and the lowest unoccupied molecular orbital. The Egap values of SA@C60 are also listed in Table 1. In Li@C60, our computed Egap value reads 0.88 eV, which is comparable to those of SA@C60, namely, 0.87 eV. Since SA@C60 endofullerenes are stabilized due to polarization of the C60 cage by charge transfer from SA, it might be interesting to analyze their polarizability. For encapsulated C60, the ao increases monotonically from Li@C60 (501.0 a.u.) to NLi4@C60 (538.5 a.u.). As mentioned in the preceding section, the encapsulated metal atom or species, inside larger fullerene such as C60, is compressed such that the total polarizability decreases. For Li@C60, Da < 0, i.e., the depression of polarizability takes place (see Table 1). This depression increases for the encapsulation of SA systems due to an increase in the negative Da values, as expected. Like the encapsulated metal atom, encapsulated SA species are also compressed, which is reflected in their reduced bond lengths.

176 Atomic clusters with unusual structure, bonding and reactivity

FIG. 2 Optimized geometry of endofullerenes, M@C60 discussed here. (Recreated from reference A.K. Srivastava, S.K. Pandey, A.K. Pandey, N. Misra, Chem. Phy. Lett. 655–656 (2016) 71–75.)

3.2 Li@C60 vs Lr@C60 endofullerene The optimized structure of Lr@C60 is displayed in Fig. 3. Unlike Li@C60 in which the Li atom resides on the pentagon ˚ . In the previous section, it has already been meninside C60, the Lr atom prefers hexagon site, whose dcenter is only 0.22 A tioned that the displacement of an encapsulated metal atom inside the C60 cage can be successfully correlated with the IEs of metal atoms. Please note that the IE of Lr is very close to that of K, therefore, it resembles the position of K doped into C60. It has also been urged [95] that the equilibrium position of the alkali atom is at the center, or very near the center, of the C60

TABLE 1 The distance of dopant from the center of C60 (dcenter), the NBO charge on dopant (Q), frontier orbitals’ energy gap (Egap), dipole moment (m) and mean polarizability (ao), and depression of polarizability (Da) of endofullerene systems at B3LYP/6-31G(d) level of theory.a Parameters

Li@C60

Lr@C60

FLi2@C60

OLi3@C60

NLi4@C60

dcenter (A˚)

1.52

0.22

0.23

0.21

0.02

Q (e)

0.44

0.68

0.38

0.10

0.78

Egap (eV)

0.88

0.84

0.87

0.87

0.87

m (Debye)

0.50

0.22

0.32

0.50

0.19

ao (a.u.)

501.0

578.1

509.0

517.0

538.5

Da (a.u.)

98.7

172.4

217.6

460.2

471.0

a

These data are taken from Refs. [86,90,91].

Low ionization energy species doped fullerenes Chapter

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FIG. 3 Optimized geometry of endofullerenes, M@C60 discussed here. (Recreated from reference A.K. Srivastava, S.K. Pandey, N. Misra, Mater. Chem. Phys. 177 (2016) 437–441.)

cage for the electronically excited neutral molecules. The calculated dH-H and dH-P bond lengths of Lr@C60 are slightly larger and smaller than those of Li@C60. In Table 1, we have also listed atomic charges (Q) calculated on Li and Lr atoms. The Q value of Li, 0.44e indicates the charge transfer of 44% to the C60 cage. It has been suggested by the HF calculations of Aree et al. [96] that the Li atom ˚ from the central position such that the interaction between Li completely ionizes to form Li+ at a distance of less than 1 A and C60 has ionic character. In the case of Lr@C60, the charge transfer from Lr to C60 increases to 68%, which can be expected due to the lower IE of Lr than that of the Li atom. Like Li@C60, the Q value of Lr@C60 clearly indicates that there is a considerable back-donation from C60 to encapsulated atom. The dipole moment (m) of Li@C60 (0.70 D) is found to be larger than that of Lr@C60 (0.22 D). This value is consistent with the distances of Li and Lr atoms from the center of the C60 cage. The Egap of Lr@C60 is listed in Table 1, and the value of Lr@C60 (0.84 eV) is comparable to that of Li@C60. The smaller Egap of Lr@C60 further suggests the behavior of Lr@C60 similar to that of Li@C60, whose reactivity is comparable to or slightly greater than Li@C60. The mean polarizability of Lr@C60 is also listed in Table 1. Like Li@C60, the encapsulation of Lr makes the system more polarizable due to charge transfer. For both Li@C60 and Lr@C60, Da < 0, i.e., the depression of polarizability takes place. Thus, most of the properties of Lr atoms resemble those of Li inside C60.

4. Endofullerene-superhalogen complexes The reactivity of Li@C60 endofullerene leads to the formation of the dimer (Li@C60)2 [97] and complexes, Li@C60SbCl6 [19], Li@C60-PF6 [98], etc. The theoretical investigation by Wang et al. [99] revealed the NLO properties of Li@C60dBX4 (X ¼ F, Cl, and Br). The large NLO response of these complexes is expected due to charge transfer from Li@C60 to highly electronegative BX4 moieties, which are actually superhalogens. In this section, we first discuss some properties of Li@C60dPF6 studied theoretically [100]. Subsequently, we will focus on SA@C60dBF4 (SA ¼ FLi2, OLi3, and NLi4) [101] and compare their properties with Li@C60dBF4. The binding energy of endofullerene complex (M@C60dX) is given by, DE ¼ E½X + E½M@C60   E½M@C60  X Here E[..] represents the electronic energy of the respective neutral species including zero-point correction.

4.1 Li@C60 2PF6 endofullerene complex The equilibrium geometry of Li@C60dPF6 is shown in Fig. 4 and corresponding parameters are listed in Table 2. The ˚ , by the interaction of Li@C60 with PF6, due to the large electronegativity distance from center (dcenter) reduces to 1.26 A difference between P and F. The electrons are delocalized over F atoms such that P becomes positively charged (2.73e). This causes to repel electropositive Li and decrease dcenter value. PF6 is a superhalogen species, which needs an extra electron for stability. Net NBO charges on Li and PF6 are obtained to be 0.65e and 0.87e, respectively, which suggests that the Li@C60-PF6 complex is stabilized by the electron transfer from Li atom to PF6 superhalogen moiety. The binding

178 Atomic clusters with unusual structure, bonding and reactivity

FIG. 4 Optimized geometry of endofullerene complexes, M@C60dX. (Recreated from references A.K. Srivastava, A. Kumar, N. Misra, Phys. E 84 (2016) 524–529 and A.K. Srivastava, A. Kumar, A.K. Pandey, N. Misra, Chem. Phys. Lett. 682 (2017) 20–25.)

TABLE 2 The distance from center (dcentre), NBO charges (Q) on dopant, frontier orbital energy gap (Egap), dipole moment (m) and mean polarizability (ao), and binding energy (DE) of endofullerene complex systems at B3LYP/6-31G(d) level of theory.a System

dcentre (A˚)

Q (e)

Egap (eV)

m (Debye)

ao (a.u.)

DE (eV)

Li@C60dPF6

1.26

0.65

2.69

20.2

494

4.45

Li@C60dBF4

0.79

0.65

2.61

6.64

489

3.20

FLi2@C60dBF4

0.63

0.34

2.56

6.97

509

3.13

OLi3@C60dBF4

0.65

0.07

2.36

6.97

517

3.18

NLi4@C60dBF4

0.68

0.10

1.00

7.18

538

2.79

a

These data are taken from Ref. [100,101].

energy (DE) value calculated for the Li@C60-PF6 complex is found to be 4.45 eV, which is large enough to establish the stability of the complex. The frontier orbitals of Li@C60dPF6 are shown in Fig. 5. One can see that the HOMO is delocalized over the whole complex including PF6, whereas LUMO is localized on Li@C60, excluding PF6. The Egap of Li@C60dPF6 is 2.69 eV, which can explain the charge transfer interactions within the complex. This HOMO-LUMO gap can be used as a stability index of the system as well. Note that the calculated Egap of Li@C60 is 0.88 eV (see Table 1). Therefore, Li@C60 is significantly stabilized by interacting with PF6 such that the Egap of the resulting complex is increased. It is also due to this charge transfer that Li@C60dPF6 possesses a remarkably high dipole moment. As listed in Table 2, the dipole moment (m) of the complex is calculated to be 20 Debye. Note that the m of Li@C60 is only 0.50 D (see Table 1) due to polarization of

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FIG. 5 HOMO-LUMO surfaces of endofullerene complexes. (Recreated from reference A.K. Srivastava, A. Kumar, N. Misra, Phys. E 84 (2016) 524–529 and A.K. Srivastava, A. Kumar, A.K. Pandey, N. Misra, Chem. Phys. Lett. 682 (2017) 20–25.)

C60 cage by encapsulation of Li atom. This large dipole moment of the complex undoubtedly suggests that the complex is highly polar, which can be treated as [Li@C60]+ PF 6 salt. In order to explore the electric properties of the Li@C60dPF6 endofullerene complex, mean polarizability (ao) is also listed in Table 2. The ao value of the Li@C60dPF6 is slightly smaller than that of Li@C60 (Table 1) but larger than that of Li@C60dBF4 complex, which is reported to be 489 a.u. (see Table 2). This can be expected due to the larger size of PF6 as compared to the BF4 superhalogen.

4.2 SA@C60dBF4 endofullerene complex The optimized structures of SA@C60dBF4 endofullerene complexes are also displayed in Fig. 4 for SA ¼ FLi2, OLi3, NLi4. ˚ from 1.52 A ˚ in Li@C60. In SA@C60dBF4, on the contrary, the In Li@C60dBF4, the dcenter of Li is decreased to 0.79 A ˚ ˚ ˚ for NLi4 as compared to SA@C60. This suggests dcentre of SA is increased to 0.63 A for FLi2, 0.65 A for OLi3, and 0.68 A that the interaction of SA@C60 with BF4 is somewhat different from that of Li@C60. The calculated △ E values are listed in Table 2, which suggests that all complexes are stabilized by interaction with BF4 superhalogen, like Li@C60dBF4 complex. Their stability tends to decrease with the increase in the size of superalkalis, i.e., FLi2 < OLi3 < NLi4. To study the nature of the interaction of the C60 cage with SA and BF4 superhalogen, the NBO charges on SA in SA@C60dBF4 complexes are also calculated. Being a superhalogen, BF4 needs an extra electron for stability, net NBO charges on BF4 are expected to be close to unity. This suggests that complexes are stabilized by the electron transfer from SA@C60 endofullerene to BF4 superhalogen moiety, like Li@C60. Consequently, all complexes are essentially ionic, which can be represented as (SA@C60)+ BF 4 . The NBO charge on Li is 0.65e in Li@C60dBF4. On the contrary, this electron is effectively transferred by the C60 cage in SA@C60dBF4 as the NBO charges of SA are 0.50e, 0.78e, and 0.75e for SA ¼ FLi2, OLi3, and NLi4, respectively. Thus, BF4 superhalogen ionizes C60 fullerene when interacts with SA@C60, unlike Li@C60 in which Li is ionized leaving C60 almost neutral. This fact is consistent with the previous study [102] in which superhalogens are found to be able to ionize C60 cage. The net charge transfer in all SA@C60dBF4 complexes is approximately the same as that for Li@C60dBF4 (Table 2). This leads to almost equal dipole moments of SA@C60dBF4 complexes. As listed in Table 2, the dipole moments (m) of these complexes are calculated to be 6.64–7.18 Debye. The direction of m in all SA@C60dBF4 complexes, is from C60 center to B atom, including Li@C60dBF4. The mean polarizability (ao) of SA@C60dBF4 complexes are also calculated and listed in Table 2. The ao of Li@C60dBF4 complex is calculated to be 489 a.u., which agrees with the earlier reported value [99]. The ao values of SA@C60dBF4 complexes lie in the range 509–533 a.u., which increases linearly with the size of superalkali, i.e., SA ¼ FLi2 to SA ¼ NLi4. In Fig. 5, the frontier molecular orbitals of SA@C60dBF4 are also displayed. For SA ¼ FLi2, the HOMO of SA@C60dBF4 is delocalized over C60 and BF4 moieties whereas it is localized mainly on SA moiety in the case of SA ¼ OLi3 and NLi4. The Egap decreases with the increase in the size of superalkalis. From Table 2, the trend of ao values is consistent with their Egap values.

180 Atomic clusters with unusual structure, bonding and reactivity

FIG. 6 The plots of parameters for SA@C60 endofullerenes and SA@C60dBF4 complexes.

In order to explicitly analyze the effect of BF4 interaction on SA@C60 endofullerene, we have plotted some parameters of SA@C60 and SA@C60dBF4 in Fig. 6. As mentioned earlier, the distance of SA from the center of C60 (dcenter) decreases with the increase in the size of SA in SA@C60. On the contrary, the dcenter in SA@C60dBF4 increases with the increase in the size of SA (see Fig. 6A). This can be expected due to the fact that the charge transferred from SA is modified due to the interaction with BF4 superhalogen. As shown in Fig. 6B, the NBO charge (Q) on SA is smaller for FLi2 and higher for NLi4 in SA@C60 but it is higher for FLi2 and smaller for NLi4 in SA@C60dBF4. Similarly, the dipole moment (m) of SA@C60 lies in the range 0.2–0.5 Debye (Fig. 6C), which is enhanced to 7.0 Debye by interaction with BF4 superhalogen. This is in accordance with the enhanced charge transfer and separation in the complex. However, this interaction does not affect the mean polarizability (ao) appreciably such that both SA@C60 endofullerene and SA@C60dBF4 complex possess the almost same ao values (see Fig. 6D).

5.

Conclusions and perspectives

This chapter discussed various aspects of low IE species doped endofullerenes Li@C60 and Lr@C60 including superalkali doped C60, SA@C60. Using typical endofullerenes such as Li@C60, we established the effect of the doping of Li, the representative element of the group with the lowest IE in the periodic table. The results of the B3LYP/6-31G(d) calculations on endofullerenes suggested that the properties of all low IE species doped C60 resemble each other, irrespective of the nature of dopants. Further, their structural parameters such as the distance from the center appear as a function of their IE. The NBO charge on doped atoms/species, the dipole moment of endofullerenes, and their polarizability seem to be also related to the IE of dopants. The NBO charge on SA in SA@C60 indicated not only the charge transfer from SA to C60 but also a significant back donation from C60, unlike Li@C60. Therefore, superalkalis can be exploited to design a new series of endofullerenes having some unique properties. In this quest, we have considered the superhalogen complexes of some

Low ionization energy species doped fullerenes Chapter

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endofullerenes such as Li@C60dPF6 and SA@C60dBF4 (SA ¼ FLi2, OLi3, and NLi4). Their dipole moments and mean polarizability have been found to be quite large, which suggest that these complexes are highly polar and polarizable. We also analyzed the effect of BF4 interaction on SA@C60 by comparing the properties of SA@C60 endofullerenes and SA@C60dBF4 complexes. This chapter should provide a concise account of some special endofullerenes and their complexes, which need further attention and exploration for their possible technological applications.

Acknowledgments AKS is thankful to CSIR, SERB, and UGC for providing funding at various stages of the research.

Conflict of interests The authors declare no conflict of interest.

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Chapter 11

Generation of global minimum energy structures of small molecular clusters using machine learning technique Gourhari Janaa and Ranita Palb a

Department of Chemistry, Indian Institute of Technology Bombay, Mumbai, India, b Advanced Technology Development Centre, Indian Institute of

Technology Kharagpur, Kharagpur, India

1. Introduction The term “global optimization” (GO) refers to finding the overall best solution for a given problem in a search space. GO techniques have now become essential in the fields of theoretical and computational chemistry to avoid several difficulties associated with experiments, such as low-quality data obtained from X-ray diffraction study, small sample quantity and instrumental limitations or other difficulties regarding phase-related issues arising in structure determination through X-ray diffraction. In recent years, computational chemists have paid special emphasis to solve such difficulties by predicting the structure of molecular clusters with the use of some optimization tools. Several traditional methods (e. g., conjugate gradient, gradient descent, steepest descent, Newton method, etc.) and deterministic algorithm-based methods (e.g., Newton-Raphson [1], line search [2], Simplex Method [3], Hill Climbing [4], and so on) have failed to detect global minimum energy structure due to the existence of a large number of local minima (inherent property of molecular configuration), and the inability of the system to climb up high potential energy barriers within the large multidimensional search space. Deterministic algorithms demand some tuning parameters and constraints to obtain a global minimum energy structure or the lowest-lying isomer. In this connection, stochastic search algorithms are put forward to deal with such problems by adopting randomness in their simulation strategies. Some of them, namely, Ant Colony Optimization (ACO) [5], Genetic Algorithm (GA) [6–12], random sampling method [13–15], tabu search [16,17], basin hopping [18,19], simulated annealing [20], meta-dynamics [21,22], data mining [23], Artificial Bee Colony (ABC) [24,25], Cuckoo Search (CS) [26], Firefly Algorithm (FA) [27–38], and Particle Swarm Optimization (PSO) [39–44] are widely used. Swarm intelligence (SI)-based techniques, PSO and FA, show quick convergence because of their combined exploitation and/or exploration mechanism. Initially, the algorithm starts with a population of random particles (individual cluster unit) which fly with randomized velocities through the search space. The particles use the concept of individual intelligence and neighborhood intelligence for swarm purposes. The population and self-organization-based PSO, first developed by Kennedy and Eberhart [45,46] in 1995, is based on some natural phenomena. Further improvements on the original PSO algorithm are introduced depending on its applicability in various emerging fields. Chen, Jiao, and Yan have developed a novel cooperative co-evolutionary PSO (NCPSO) [47] and showed that it provides better solutions than the original PSO by working through a niche sharing mechanism. In 1998, Angeline has introduced a hybrid PSO algorithm (HPSO) [48] by adding a selection operation taken from the evolutionary computation. In 2000, Løvbjerg, Rasmussen, and Krink [49] have presented two hybrid PSOs by combining the standard PSO parameter with an arithmetic crossover operator. 2002 brought in Kennedy and Mendes’ improvement on the performance of PSO based on different population topologies for cluster analysis [50]. In 2007, Boldyrev and coworkers have introduced a novel PSO technique, which detects a global minimum structure for a given chemical composition [51]. Jiao et al. in 2008 have developed an improved PSO technique [52] by using dynamic inertia weight. Hamta et al. have addressed a Hybrid PSO algorithm for multiobjective (MO) optimization of a single-model assembly line balancing problem (ALBP) by knowing the upper and lower bound operation time for each task [53]. In 2013, Chou designed a

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00001-2 Copyright © 2023 Elsevier Inc. All rights reserved.

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cocktail decoding method to minimize the makespan of the hybrid flow shop (HFS) problems with multiprocessor tasks [54]. Apart from its different applicability of standard PSO in many different research fields, several modifications, adaptation [55], and developments [56–64] like constraint optimization with PSO [65–69], single solution PSO [70–79], MO optimization [80–88], niching with PSO [89–93], discrete PSO [94–97], and dynamic environment of PSO [98–107] along with the parameterization [108–114] are carried out in many stages to compose an efficient algorithm. Recently, democratic and discrete PSO [115–117] is designed for structural optimization [118–121] and both their advantages and disadvantages [39] are also discussed. Corresponding constriction coefficients are also estimated [122,123]. Both synchronous and asynchronous PSO updates are analyzed in this connection [124–131]. Due to the flexibility and robustness, the use of SI-based approaches has proved to be highly efficient in finding out the optimal or near-optimal solutions in a search space. Out of several modern metaheuristic algorithms used in different domains, the applicability of PSO and FA for the search of global best configuration or determination of lowest-lying isomer are introduced in this chapter. To develop an efficient “global optimizer” tool and fast cluster optimization technique, we have implemented a python code-based Gaussian software [132] integrated PSO and FA approach (density functional theory-PSO or DFT-PSO and DFT-FA) in the present chapter. Advantages of these techniques are their inconsideration of any symmetry constraint, bond characterization of matrix in the implementation and their single way of information sharing strategies, make these two algorithms unique. DFT-PSO iteratively finds the global optimum structure by simply adjusting the trajectory of each particle of the entire swarm at each time step and strictly analyzes the convergence criteria. Here, we have also established the superiority of DFT-FA in performance over DFT-PSO. Further, one of the most critical challenges that researchers face is to design an algorithm that promotes first-principle calculations without incurring high computational costs. Deep learning and artificial neural networks (ANNs) can handle several levels of complexity in a diverse range of problems, making them highly efficient in extracting chemical insight from data. To overcome one of the challenges like global best structure search and energy prediction of a randomly generated unknown atomic cluster by learning the features of constructing elements, atom to atom distances, and corresponding energies of training set clusters, we have proposed a PSO coalesced convolutional neural network (CNN)-based technique. CNN training through fully connected or dense layers (supervised learning) is performed on an initial set of clusters generated by Atom Centered Density Matrix Propagation molecular dynamics simulation (ADMP) [133–137]. Such an approach is forecast for widespread potential application in the large-scale high-throughput computation of electronic energy of homogenous clusters due to its robustness and computational efficiency. The proposed CNN-DFT-PSO model become advantageous because it can handle a huge number of data set of cluster units while searching for the putative global minimum structure, instead of using time-consuming quantum mechanical calculations in an iterative process. 4
Small sized nonmetallic clusters like Boron (B5 and B6) [40], Carbon (C5) [44], and polynitrogen clusters (N2
4 and N6 ) 2
[138], and metallic clusters like Al4 [38], Aun (n ¼ 2–8) and AunAgm (2  n + m  8) [138] clusters are considered for determination of their respective global minima (GM) energy structures since doing the same using standard quantum chemistry methods is not easy. An increase in the importance of quantum size effects makes the quantum chemical description of small-sized clusters (or subnanometer particles) essential. Owing to the intrinsic electronic deficiency in boron, its clusters manifest some notable structural complexity, fluxional behavior [139,140], and novel bonding interactions like multicenter-2e bonds. Moreover, it shows a wide range of important technological and biological applications [141–147], antiviral, antifungal, antiseptic and antitumor agents being some of the examples of the latter. On the other hand, the investigation on pure carbon molecules is of interest as it exists in various structural forms like cyclic/ring and linear/ branched chains [148,149]. Due to the high reactivity and transient behavior of carbon-riched molecules, their production in 4
the laboratory is notoriously difficult. N2
4 and N6 clusters [150], being thermochemically stable, are also included in the present study since they are known to be good source of clean energy and can be used as good high energy density materials [151–153]. Also, their formation is highly endothermic in nature, making their synthesis and general use difficult. Aluminum clusters with size specific geometries show properties of superatoms [154]. Moreover, due to the all-metal aromatic and antiaromatic characteristics, aluminum clusters have drawn the attention of the researchers [155–163]. It is, however, to be noted that the minimum structure determination of aluminum clusters containing more than a few atoms is virtually impossible since they have floppy potential hypersurfaces with numerous local minima. Small clusters are usually not very stable and their synthesis can only successfully take place in near-vacuum conditions. The small-sized metal clusters have potential applications in developing Nano functional materials such as quantum dots, nanoclusters, and nanowires. Mixed metallic or multi-metallic clusters are more attractive in the field of nanomaterials having wider range of properties. Exhibiting enhanced catalytic performance, dominating quantum effects of bimetallic clusters containing noble metals compared to those of mono-metallic clusters, and their distinctive properties including electronic, mechanical, and optical, have been extensively researched [164–168]. Here we have considered bimetallic gold-silver clusters AunAgm (2  (n + m)  8) together with pure gold clusters Aun (n ¼ 2–8) to attain their GM.

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Considering the abovementioned difficulties concerning these metallic and nonmetallic clusters, along with their important applications in various fields and distinct properties, we have developed more rigorous quantum chemical techniques to achieve effective GO of their structures.

2. Our proposed methodology and algorithm (parallel implementation) A cluster comprising of n atoms is represented in three-dimensional (3D) vector space by (x0, x1, x2, …, x3n
1), where (x0, x1, x2) is the position of the first atom, followed by the second atom (x3, x4, x5) and so on. This 3n dimensional vector is regarded as one particle in both PSO and FA. In Fig. 1, a cluster unit or a particle is considered as a vector in the 3D search space.

2.1 Particle swarm optimization Initially, a set of random configurations (three-dimensional cartesian coordinates) is generated with the aid of Gaussian 09 (abbreviated as G09) program call. Each randomly generated cluster configuration is considered as swarm particle which begins its search for new position or generation of new configuration (called local optimum solution, pbest configuration) through PSO algorithm. Each atom in those particles take three-coordinates (x, y, z) within a certain range [
3.3] in “.gjf” file format which is executable in G09 program. Velocity vector drives the particles to reach a new position and the corresponding single point energy (SPE) of the cluster is calculated using the G09 program. The energy so obtained serves as objective function for the corresponding swarm particle. A lower energy (negative value) corresponds to a better objective function. Upgradation of configuration generations or improvement of energy (towards lower energy) of the swarm particle is carried out by analyzing the lowest energy configuration obtained locally so far and their corresponding energy value to find out the global best (gbest) configuration. To avoid the trapping of pbest configuration into local minimum energy well, some sort of randomness is imposed (called exploration technique) to escape from the well. t+1 Upgradation of new positions (xt+1 i ) and velocities (vi ) of configuration in ith generation follows the equations as given by

¼ w∗ vti + c1 ∗ g1 ∗ pbest
xti + c2 ∗ g2 ∗ gbest
xti vt+1 (1) i ¼ xti + vt+1 xt+1 i i

(2)

where xti stands for current position and vti refers to the current velocity.

FIG. 1 A schematic representation of a cluster in multidimensional search space. Adapted from G. Jana, A. Mitra, S. Pan, S. Sural, P.K. Chattaraj, Modified particle swarm optimization algorithms for the generation of stable structures of carbon clusters, Cn (n ¼ 3–6, 10), Front. Chem. 7 (2019) 485, https://doi.org/10.3389/fchem.2019.00485, with permission from Frontiers. © 2019 Jana, Mitra, Pan, Sural and Chattaraj.

188 Atomic clusters with unusual structure, bonding and reactivity

FIG. 2 Overall flowchart of the proposed DFT-PSO approach. Adapted from P. Mitikiri, G. Jana, S. Sural, P.K. Chattaraj, A machine learning technique toward generating minimum energy structures of small boron clusters, Int. J. Quantum Chem. 118 (17) (2018), e25672, https://doi.org/10.1002/qua.25672, with permission from John Wiley and Sons. © 2018 Wiley Periodicals, Inc.

Inertia coefficient (w) refers to the ability of a particle to remain in its current position and takes the value within range [0.4–0.8]. g1; g2 E [0;1] are random coefficients. c1 and c2 (which vary as per requirement), drive the particles to converge in the gbest configuration, known as individual coefficient of acceleration [2] and global coefficient of acceleration [2], respectively. After completion of the final iteration through PSO algorithm, a postprocessing step further proceeds via G09 software, followed by frequency calculation confirming the exact global best configuration. The overall flowchart of the proposed technique is depicted in Fig. 2.

2.2 Firefly algorithm Similar to PSO, as mentioned above, FA also starts with random set of particles having zero velocity as an initial guess, where each particle represents atomic cluster unit. The energy of each atomic structure as generated after SPE calculation from G09 serves as the objective function for the corresponding firefly particle. Firefly algorithm generates better particles (cluster units with lower energy values) in the next generation of initiation, i.e., after generation of random particles at initial step, the energy values of the particles are computed and compared. If the energy of one particle is higher than another particle, it moves towards the former particle. This process is tested for each particle followed by updating the particle coordinates. This process is continued until the global best configuration is achieved. A parameter named patience is used for termination of the process in order to reach the minimum energy configuration of a cluster. A maximum value of patience is represented as max_patience. For a given iteration, if the lower energy value or better energy configuration of a system is not achieved from the previous iteration, the patience value is increased, otherwise it is reset to zero. If the number of iterations and (/or) the value of patience are (/is) reached its (/their) maximum value(s), the algorithm terminates, otherwise, the iterative procedure takes new configuration (new position) as an input to the algorithm. Similar to DFT-PSO, once the termination in FA is achieved, the postprocessing step is carried out using G09 software. A particle having lower energy value, i.e., with a better objective function shines brighter. The nearby fireflies move more towards the bright firefly compared to the ones that are farther away. From real world experience, we know the intensity of any system to decrease with an increase in distance. This relation can be mathematically described using an exponential function exp.(
Ɣr2), r and Ɣ being the Euclidean distance between the particles and the intervening material separating them, respectively. As Ɣ value increases, the atmosphere becomes increasingly foggy which causes the light intensity to reach a smaller distance. Evidently, with increasing distance, the value of the exponential term decreases. Since the particle with lower energy (say j) has a better objective function than that with higher energy (say i), the latter updates its position as follows:
2 xi ¼ xi + b∗ xj
xi ∗e
g∗r + a∗e (3)

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b, Ɣ, a, and e are known as hyper parameters. The rate of convergence changes according to the b. A value of zero indicates that the particles randomly walk in the region of consideration, thus achieving a very small exploitation of the space. As b increases, the said particle reaches other particles faster, resulting in higher exploitation. Hence it is very important to choose a balanced value of b. A random vector e (magnitude smaller than the firefly particles) is initialized in the region of convergence. a is known as the randomization parameter which represents the reach of the generated values from the present values. A lower a prevents any further vector space exploration, whereas a higher a produces particles with similar magnitude to the ones already present, making the system behave like a random walk. a value gradually decreases with the increasing number of iterations, i.e., with the approaching global optima. Here the values of b, Ɣ, and a are set to 0.2, 0.9/12, and 0.1, respectively, whereas e is a random vector. As mentioned before, SPE are calculated at each iteration using B3LYP/6-311 + G(d) level [169–171] as implemented in G09. The generated .log file after the execution of .gjf file in G09 is analyzed for reading the energy value. In order to compare all such energy values and keep tracking of the minimum energy structure and corresponding energy value, all the configurations along with their energy are stored in a text file. The FA is worked out until the following convergence criteria are achieved. (i) Number of iterations attaining its maximum value. (ii) Energy value remaining unaltered for several iteration. We have considered both planner and nonplanner approach for the optimization in FA and checked the performance in both the cases. The overall flowchart of the proposed DFT-FA technique is depicted in Fig. 3.

2.2.1 Performance improvement of FA over the basic approach It is found that the faster convergence is driven by reducing the dimension of the optimization vector. We have applied our FA approaches for Al2
4 cluster for firefly to get its global best configuration and since it is reported that the global best structure for Al2
with optimal energy value is planar, the dimension of the vector is reduced by setting the Z component to 4 zero. Therefore, initial random consideration of Al2
4 configurations are always on the X–Y plane. After execution of both planner and nonplanner FA program, it is found that considering planar structural approach results in a faster convergence. This is due to the reduction of 3n unknowns to 2n in multidimensional vector space for atomic clusters containing n atoms (similar to the initial step), reducing the randomly generated 3D structures to their planar projections on the X–Y plane. So, the program searches the structures or optimal solution in 2n dimensional vector space instead of 3n, resulting in a much faster convergence.

FIG. 3 Overall flowchart of the proposed DFT-FA approach. Adapted from A. Mitra, G. Jana, P. Agrawal, S. Sural, P.K. Chattaraj, Integrating firefly algorithm with density functional theory for global optimization of Al42– clusters, Theor. Chem. Acc. 139 (2020) 32, https://doi.org/10.1007/s00214-0202550-y, with permission from Springer Nature. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

190 Atomic clusters with unusual structure, bonding and reactivity

2.3 ADMP-CNN-PSO approach In this approach, after generation of random structures, ADMP simulation at G09 is performed to create a large number of dataset comprising atomic clusters with corresponding energy values. A majority of the cluster units from the aforementioned dataset is used to train the CNN model and the rest are kept in the test set. ADMP simulation is carried out at B3LYP/ 6-311 + G(d,p) level [169,170,172,173]. At a later stage of this approach is the PSO run to optimize the structures of the minimum energy configuration. Finally, the PSO-executed best configuration is sent for postprocessing purpose to obtain the putative global minimum energy structure. Initially, we have generated 101 random structures from which a set of 101,101 cluster configurations (full dataset) is generated through ADMP simulation. Each of the initial 101 clusters generates 1001 structures, leading to a total of 101,101 structures and their corresponding energies. From the full dataset, 80,880 of these configurations train the CNN model and validation set. The rest (20,221 configurations) are kept for testing purpose. In order to keep the same distribution of the training and test set concerning the energy, the dataset is then shuffled. Each structure in the whole sample space consists of the carbon cluster position consisting five atoms, depicted by a 5  3 vector and its energy. To increase the training set size and reduce overfitting, data-augmentation is implemented. After training with a varying number of samples, the loss of the test set is estimated to find the trade-off between time and accuracy. 10, 50, 100, 500, 1000, 5000, 10,000, and 50,000 sample sizes are selected from the dataset and again division can be done in the ratio of 80:20, 80% for the training set and 20% for the validation set.

2.3.1 CNN architecture The network architecture is depicted in Fig. 4. The CNN architecture consists of two fully connected layers and one convolutional layer. In first convolutional layer the carbon clusters are taken as input and filtered with a stride of 3. It is then sent to the densely connected layer consisting of 64 neurons and its output is flattened. The output of the last layer is then fed to a single neuron and as our network is a regression solving problem, so a linear activation function is employed at the output. Except for the last layer, all the layers of the model have a sigmoid activation function. In order to prevent overfitting, early-stopping is also deployed. The program terminates when the loss of the validation set remains unaltered for 100 epochs. To update the model, the mean square loss is evaluated by making a comparison between the output of the model and the actual energy. Adam (Adaptive Moment Estimation) optimizer is chosen to get the optimal values for the weights to the neural networks since it is known to produce the best results compared to all other optimization algorithms where 9849 trainable parameters exist. PSO is used to search for an optimal solution iteratively after passing the ADMP generated clusters through CNN model. CNN provides the energy for a set of cluster coordinates. Here, each C5 cluster obtained from ADMP simulation is referred as one PSO particle. Such unit, containing five atoms each, appears in search space with vector representation (x0, x1, x2, … x3n
1) (n ¼ 5), where (x0, x1, x2), (x3, x4, x5), (x6, x7, x8), (x9, x10, x11), and (x12, x13, x14) are the position of the first, second, third, fourth, and fifth carbon atoms, respectively. Over a certain number of iterations, if the best value from the objective function does not improve by a particular percentage, the algorithm terminates. It is found that our current computational time is reduced to a few seconds which took 3 h in our previous PSO approach. Thus, we can say that the computational time is reduced by a hundredfold. The overall flowchart of the ADMP-CNN-PSO approach is depicted in Fig. 5.

FIG. 4 Architecture of the convolutional neural network model. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi. org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

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FIG. 5 Overall flowchart of the ADMP-CNN-PSO approach. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi. org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

2.3.2 Postprocessing At the last step, the final structure generated from the PSO or FA run is then processed through G09 package for optimization and frequency analysis of the cluster configuration. After executing the PSO code, the resulting configuration sometimes may not be the exact global minimum energy structure. Therefore, a postprocessing method is required which would optimize “the configuration” having energy very close to the putative global minimum energy to converge into the exact value. Gaussian 09 program uses Berny Algorithm to optimize cluster by taking an input in the form of .gjf or .com file containing the 3D (x, y, z) coordinates of a single carbon cluster (C5). It generates .log/.out output files containing energy values at a given level as implemented in G09.

3. Computational details Python programing language is used for the parallel implementation strategies invoked in our proposed PSO, FA, and ADMP-CNN-PSO approach. Gaussian 09 program package is used to calculate the SPE of particles by a system call and at the postprocessing step. Each of the parallel runs is dependent on the number of processor cores available on the system. We have used B3LYP/6-311 + g(d,p) [169,170,172,173] level of computation for B5 and B6 clusters, B3LYP/6-311 + G 2
4
(d) [166–168] level for Al2
4 , C5, and N4 clusters, B3LYP/6-311G(d) [169,170,172] level for N6 , and B3LYP/LANL2DZ [169,170,174–177] level with effective core potentials (ECPs) for Aun (n ¼ 2–8) and AunAgm (2  (n + m)  8). The center of the molecular plane is considered for the calculation of NICS (0) value of Al2
4 ring at each iteration step using B3LYP/6-311 + G(d) level.

4. Experimental setup Our proposed implementation enables such parallel synchronization mechanisms to use as portable across multiple platforms.

192 Atomic clusters with unusual structure, bonding and reactivity

4.1 PSO, FA, and ADMP-CNN-PSO The whole programming is scripted in Python 3.7 [178]. Gaussian 09 package is interfaced with the python code for SPE calculation and postprocessing optimization. The experiment is carried out on computers that have the following specifications: one server containing two Intel 2.70 GHz Xeon E5-2697 v2 processors and a RAM of 256 GB. Each processor has 12 cores and 30 threads for executing our PSO/FA algorithm saving a few cores for other housekeeping processes and operating system. For each Gaussian input execution, 2 threads and 8 GB of RAM are used. In general, each of the n particles uses └30 n ┘ threads which indicates the equal distribution of the load, ensuring higher efficiency of the overall program as compared to static threading. For each Gaussian call, the number of threads, number of PSO or FA particles, and RAM assignment are set as input hyper parameters. Since sequential updating of both the positions of PSO and FA particle is quite inefficient and all the particles are not updated in any given iteration, we parallelize this task using multithreading on the vCPUs and the threads are allocated dynamically during each iteration to overcome the static assignment of particles to threads of some idle vCPUs. This is the uniqueness of our work, which is yet to be reported in the literature for stable structure prediction. To create the initial dataset, 12 cores are used at a time for ADMP simulation. Keras, an open-source TensorFlow library, is used to train the model interfacing with Python 3.7 for convolution neural networks. Eight cores are used for the postprocessing steps while interfacing with the Gaussian software.

5.

Results and discussion

5.1 PSO: Boron clusters, Bn (n 5 5, 6) In potential energy surface (PES), different stationary points like first order saddle points (such as minima or maxima) and higher order saddle points are connected. Our implemented PSO is employed for finding of global optimized structures of B5 and B6 clusters as shown in Figs. 6 and 7.

FIG. 6 The randomly chosen 14 different molecular frameworks of B5 converge to the global minimum energy structure (bond lengths are given in Angstrom unit). Adapted from P. Mitikiri, G. Jana, S. Sural, P.K. Chattaraj, A machine learning technique toward generating minimum energy structures of small boron clusters, Int. J. Quantum Chem. 118 (17) (2018), e25672, https://doi.org/10.1002/qua.25672, with permission from John Wiley and Sons. © 2018 Wiley Periodicals, Inc.

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FIG. 7 The randomly chosen 15 different molecular frameworks of B6 converge to the global minimum energy structure (corresponding bond lengths are given in Angstrom unit). Adapted from P. Mitikiri, G. Jana, S. Sural, P.K. Chattaraj, A machine learning technique toward generating minimum energy structures of small boron clusters, Int. J. Quantum Chem. 118 (17) (2018), e25672, https://doi.org/10.1002/qua.25672, with permission from John Wiley and Sons. © 2018 Wiley Periodicals, Inc.

5.1.1 B5 cluster Initially, we have generated 14 different random structures for each of the B5 and B6 clusters. Initial position of all these cluster units is set accordingly having zero velocity. From Figs. 6 and 7, we find that some atoms in some of the randomly generated cluster units overlap with each other or are not in their bonding inception yet they are converged iteratively to their most stable structures or global best configurations by minimizing their energy functional. We have chosen 1000 number of iteration steps arbitrarily. The observed bond lengths remain almost unchanged in the global best configurations of both the B5 and B6 clusters obtained after last iteration step of PSO run. Then these best energy structures are optimized in postprocessing step at B3LYP/6-311 + G(d,p) level. Basically, PSO run reduces the number of iteration steps to achieve the global best structure. Postprocessing step helps to conceive the structure with appropriate symmetry and exact energy value. At the end of the PSO run, the obtained structure and the postprocessing structure are energetically too close and their energy difference is 0.0015 eV (0.035 kcal/mol) which is the success of our implemented PSO code as observed. The putative global minimum energy structure corresponds to C2v point group having zero point corrected energy (ZPE ¼
123.9873 a.u.), energy without ZPE correction (
124.0030 a.u.) as well as free energy (
124.0135 a.u.) and enthalpy (
123.9821 a.u.) at 298 K, respectively. Note that time taken at the postprocessing step to optimize the PSO best configuration (E ¼
124.002973) is very less (20 s). Thus, global optimum energy structure determination is very crucial using our easily implemented PSO technique for the structural prediction.

5.1.2 B6 clusters Fifteen different random configurations are considered for PSO run at their triplet spin multiplicity as the triplet state isomer of B6 cluster is reported to be the global minimum energy structure. We have presented randomly generated 15 different configurations (Fig. 7) to converge to the presumable global best configuration. PSO best structures are again optimized at the postprocessing step to get the minimum energy with exactly same energy as that of the global best one. PSO obtained structure have an energy of
148.8219a.u. and corresponds to C1 point group symmetry, which leads to C2h point group after symmetry constrained optimization at postprocessing step. The exact global minimum energy structure having C2h point group attains the following energies: energy with ZPE correction¼
148.8100 a.u., energy without ZPE correction¼
148.8294 a.u., free energy ¼
148.8381 a.u., and enthalpy ¼
148.8038 a.u., at the B3LYP/6-311 + G(d,p) level.

194 Atomic clusters with unusual structure, bonding and reactivity

Single point Energy (a.u.)

-123.75

-123.80

-123.85

-123.90

-123.95

-124.00 0

100

200

300

400

500

600

Iteration Step Number

FIG. 8 Single point energy evolution landscape of B5 cluster during each generation of convergence at the B3LYP/6-311 + G(d,p) level. Adapted from P. Mitikiri, G. Jana, S. Sural, P.K. Chattaraj, A machine learning technique toward generating minimum energy structures of small boron clusters, Int. J. Quantum Chem. 118 (17) (2018), e25672, https://doi.org/10.1002/qua.25672, with permission from John Wiley and Sons. © 2018 Wiley Periodicals, Inc.

Initially, we have considered arbitrarily 1000 steps of iteration steps in our implemented PSO code, though it is found that after 138 iterations the energy values become unaltered for B5 clusters as a case study. We have plotted the single point energies of self-consistence field (SCF, in atomic unit) along y-axis with respect to the number of iterations step up to 600 at 25 step iteration number intervals (1, 25, 50, …, 600) by considering 24 SPE states per iteration. The representative graph is shown in Fig. 8 to depict the convergence result. It is noted that the dotted line in some of the figures of randomly generated configurations of B5 and B6 clusters do not mean the interactions within them. These dotted bonds refer that the atoms are included in the same configurations. Here we have compared the efficiency of our implemented DFT-PSO approach with other evolutionary SI-based techniques like DFT-SA and DFT-BH considering same DFT-based energy functional as object function and presented their results in Table 1. The tabulated results clearly reflect that DFT-PSO method works superior to other such methods based on the search time for finding the putative global minimum energy structure, the energy values after completion of all runs of the studied methods, and the number of iteration steps needed to get the final structure. We have provided the average (maximum) CPU time and the average (maximum) number of iterations up to which we have tested for BH, SA, and PSO considering B5 cluster as a case study.

5.1.3 Carbon clusters, Cn (n ¼ 3–6, 10) The GO of carbon clusters Cn (n ¼ 3–6, 10) with a maximum of 1000 iterations beginning with 10 different random choices of input configurations for each cluster with random initial positions are presented in Table 2. The velocities of these 10

TABLE 1 Comparison of PSO result with other more popular evolutionary GO techniques as applied to the B5 cluster starting from the corresponding local minima structures. Comparison in terms of

Advanced basin hopping (BH)

Simulated annealing (SA)

Modified PSO

Average CPU time to locate the global minimum (GM) (maximum CPU time)

455.43 min (516 min)

369.64 min (639 min)

80.50 min

Energy of the global minimum (average energy after 600 iterations)


124.0 a.u. (
123.9 a.u.)


124.0 a.u. (
124.0 a.u.)

-124.0 a.u. (
124.0 a.u.)

Average number of iterations needed to get a structure close to GM (maximum number of iterations)

600 (600) (did not converge till 600 iterations)

324 (600)

138 (converged)

Adapted from P. Mitikiri, G. Jana, S. Sural, P.K. Chattaraj, A machine learning technique toward generating minimum energy structures of small boron clusters, Int. J. Quantum Chem. 118 (17) (2018), e25672, https://doi.org/10.1002/qua.25672, with permission from John Wiley and Sons. © 2018 Wiley Periodicals, Inc.

TABLE 2 The randomly chosen 10 different molecular frameworks of Cn (n 5 3–6, 10) with singlet and triplet spin multiplicity converge to the global minimum energy structures (bond lengths are given in A˚ unit and the relative energies,
E w.r.t the global minimum energy structures in brackets are given in kcal/mol). Run no. C3 cluster

Initial structure

Final structure using PSO

Final optimized energy (bond lengths) (1.294) 1.291

1.291

Dfh, S (E = –114.0769 a.u.)

C4 cluster

(1.306) 1.309

(1.284) 1.289

1.309

Dfh, T E = –152.1320 a.u. [0.0]

(1.306) 1.310

(1.284) 1.292

1.310

Dfh, S E = –152.1036 a.u. [17.8]

1.446

1.447

1.447

1.446

(1.442) 62.4 (62.8)

D2h, S E = –152.1062 a.u. [16.2] Continued

TABLE 2 The randomly chosen 10 different molecular frameworks of Cn (n 5 3–6, 10) with singlet and triplet spin multiplicity converge to the global minimum energy structures (bond lengths are given in A˚ unit and the relative energies,
E w.r.t the global minimum energy structures in brackets are given in kcal/mol)—cont’d Run no. C5 cluster

Initial structure

Final structure using PSO

Final optimized energy (bond lengths) (1.290) 1.286

1.281

(1.282) 1.281

1.286

Dfh, S E = –190.2546 a.u. [0.0]

1.427 1.310

1.427 1.310

1.487

C2v, S E = –190.1350 a.u. [75.1]

C6 cluster

(1.301) (1.286) (1.274) 1.299

1.286

1.273

1.286

1.299

Dfh, T E = –228.3181 a.u. [0.0] (1.301) (1.286) (1.274) 1.298

1.289

1.274

1.289

1.298

Dfh, S E = –228.2969 a.u. [13.3] 1.323

(1.329) 1.323

1.323

1.323

1.323

1.323

92.3 (93.2)

147.7 (146.8)

D3h, S E = –228.3071 a.u. [6.9]

C10 cluster

1.285 1.285

1.285

1.285

1.285

144.01 144.00 144.00 144.00

144.00

144.00

144.00

144.00 144.00 144.01 1.285

1.285

1.285

1.285

1.285

D10h, S (E = –380.7543 a.u.)

Experimental bond lengths and angles are provided within the parenthesis in the final optimized structure. Adapted from G. Jana, A. Mitra, S. Pan, S. Sural, P.K. Chattaraj, Modified particle swarm optimization algorithms for the generation of stable structures of carbon clusters, Cn (n ¼ 3–6, 10), Front. Chem. 7 (2019) 485, https:// doi.org/10.3389/fchem.2019.00485, with permission from Frontiers. © 2019 Jana, Mitra, Pan, Sural and Chattaraj.

198 Atomic clusters with unusual structure, bonding and reactivity

different random configurations are set to zero at the initial step followed by the Gaussian-interfaced PSO-driven GM search. The GM energy structures for these carbon clusters are obtained by satisfying the termination criteria. For the C3 cluster, the geometry obtained from the PSO run (linear, D ∞h point group) matches exactly with that obtained after the postprocessing step in terms of bond length and energy. GM structure search for C5 cluster also give geometry with linear configuration having Dah point group symmetry and singlet electronic spin state. A cyclic isomer with higher energy is also obtained for C5 cluster. Whereas, the GM of C4 and C6 clusters (containing even number of C atoms) obtained both have linear (D ∞h) as well as cyclic geometries (D2h for C4 and D3h for C6). Table 2 summarizes all bond lengths and energies of the obtained structures. It is seen that the computed final geometries along with the energies are exactly equal to the ones reported in the experimental studies [148,179–183]. The lowest energy isomers of C4 and C6 are in the linear form in their triplet spin state, whereas the linear isomers with singlet spin state are higher in energy by 17.8 (C4) and 13.3 (C6) kcal/mol. Apart from small clusters, we have also been able to successfully obtain the GM for C10 cluster (D10h symmetric ring), which reflects the efficiency of the present PSO code. Here, we have also performed a comparison based on the efficiency of DFT-PSO with other methods like DFT-SA and DFT-BH considering C5 clusters as a case study. The obtained results are presented in Table 3. We get similar findings, i.e., the PSO method is faster than other two methods in terms of the time required to reach the GM, corresponding final energy values, and the number of iterations to get there. From Fig. 9, we can also find the fulfillment of convergence of C5 cluster up to 600 iteration steps a representative plot (as reference) to ensure the same.

TABLE 3 Comparison of PSO results with other more popular evolutionary GO techniques as applied to the C5 cluster starting from the corresponding local minima structures. Comparison in terms of

Advanced basin hopping (BH)

Simulated annealing (SA)

Modified PSO

Execution time to locate the global minimum (GM)

305,140 s

12,959 s

8898 s

Energy of the global minimum (energy after completion of iterations)


190.2546 a.u. (
190.2460 a.u.)


190.2546 a.u. (
189.5141 a.u.)


190.2546 a.u. (
190.2436 a.u.)

Number of iterations needed to get a structure close to GM

1703 (converged)

92 (not converged)

331 (converged)

Adapted from G. Jana, A. Mitra, S. Pan, S. Sural, P.K. Chattaraj, Modified particle swarm optimization algorithms for the generation of stable structures of carbon clusters, Cn (n ¼ 3–6, 10), Front. Chem. 7 (2019) 485, https://doi.org/10.3389/fchem.2019.00485, with permission from Frontiers. © 2019 Jana, Mitra, Pan, Sural and Chattaraj.

–189.7

Energy

–189.8

–189.9

–190.0

–190.1 0

100

200

300 Iteration

400

500

600

FIG. 9 Single point energy evolution landscape of C5 cluster during each generation of convergence at the B3LYP/6-311 + G(d) level. Adapted from G. Jana, A. Mitra, S. Pan, S. Sural, P.K. Chattaraj, Modified particle swarm optimization algorithms for the generation of stable structures of carbon clusters, Cn (n ¼ 3–6, 10), Front. Chem. 7 (2019) 485, https://doi.org/10.3389/fchem.2019.00485, with permission from Frontiers. © 2019 Jana, Mitra, Pan, Sural and Chattaraj.

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42 TABLE 4 PSO results for N22 4 and N6 clusters starting from the random structures.

Kind of information

Results of N22 4

Results of N42 6

PSO execution time to locate the global minimum (GM)

3 h 48 min

5 h 23 min

Energy of the global minimum after postprocessing (energy after completion of iterations)


218.7811 a.u. (
218.6945 a.u.)


327.4117 a.u. (
327.3838 a.u.)

Number of iterations needed to get a structure close to GM

483 (converged)

627 (converged)

Number of Gaussian calls (number of normal terminations occurred)

4217 (4840)

5093 (5967)

Success rate

(4217/4840)  100 87%

(5093/5767)  100 85%

Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

4
5.1.4 N2
4 clusters and N6 clusters

Similar to the Cn clusters (n ¼ 3–6, 10) clusters, we have performed the same technique, i.e., a maximum of 1000 iteration 4
steps are given as input. It is found that 483 and 627 steps are required for N2
4 and N6 clusters, respectively, to converge into their global best configurations. The time required to complete the PSO run, i.e., 483 and 627 iterations are 3 h 48 min 4
and 5 h 23 min for N2
4 and N6 clusters, respectively (see Table 4). The PSO converged structures are very close in energy with the structures that obtained after the end of the postprocessing step, i.e., global minimum energy structure for both the clusters. After the postprocessing optimization step, frequency analysis is performed to check whether the structures are minimum energy on the PES or not. No imaginary frequency (NIMAG ¼ 0) at equilibrium point confirmed that the structures correspond to minimum energy. However, the symmetry constrained optimization at postprocessing step using B3LYP method in conjunction with 6-311 + G(d) level gives a higher order saddle point for N4
6 cluster with D6h point group symmetry, but not for the N2
4 cluster (see Fig. 10). Optimization at B3LYP/6-31G(d) level gives minima for both

FIG. 10 Structures of N2
and N4
clusters after the execution of PSO, postprocessing step, and geometry constrained optimization computed 4 6 at B3LYP/6-311 + G(d) level. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

200 Atomic clusters with unusual structure, bonding and reactivity

the clusters as reported in the previous literature [150]. The structure obtained after PSO execution and postprocessing step with the corresponding bond lengths and energies are provided in Fig. 10. The success rates concerning the Gaussian calls 4
and consequent normal termination for N2
4 and N6 clusters are 87% and 85%, respectively.

5.1.5 Aun (n ¼ 2–8) and AunAgm (2  (n  m)  8) clusters A similar study is carried out on pure gold Aun (n ¼ 2–8) and bimetallic silver-gold neutral AunAgm (2  (n + m)  8) clusters. The PSO executed and postprocessing final structures are depicted in Figs. 11 and 12. Most of the structures obtained after the postprocessing step are identical and exactly same in energy to previously reported data [184,185]. The GM structure of Au3 is linear at singlet state with Cs point group, while its ring or cyclic isomers attain D3h point group symmetry and are lower in energy with higher spin multiplicity, making the latter more stable. It is found that AuAg2 is most stable in its ring form at doublet state, and has Cs point group. The GM of the AunAgm clusters with n + m ¼ 5 are trapezoidal, whereas those with n + m ¼ 6 and 7 at doublet spin multiples are triangular and 3D configuration with C1 point groups symmetry. However, the AunAgm clusters with n + m ¼ 8 exhibit certain anomalies at their SCF convergence owing to their larger size, and are unable to fit within the considered initial random coordinates range [
4, 4]. With the exception of the last case, all the other cases discussed above the structures obtained are identical to the ones reported by Han Myoung Lee et al. [186]. 42 5.2 CNN and PSO: N22 4 , N6 , Aun (n 5 2–8) and AunAgm (2 ≤ n + m ≤ 8) clusters

Instead of using single method-based techniques, we have developed a new and better technique for GO which is a combination of three methods, namely, ADMP simulation (to create huge random data structures), CNN (for learning and

FIG. 11 Global minimum energy structures of pure gold and binary silver-gold clusters after PSO run and postprocessing step. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

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FIG. 12 Global minimum energy structures of pure gold and binary silver-gold clusters after PSO run and postprocessing step. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

predicting the energy of these structures), and PSO (for finding the GM), all combined in one program. Here, we have considered C5 cluster as an example to check the success and robustness of the model.

5.2.1 C5 clusters In this present implementation, initially 101 random structures of C5 cluster are generated. Each randomly generated cluster is taken as ADMP input to create data structures total of 101,101 population (each random structure generates 1001 numbers of random structures). The dataset generated through ADMP simulation have a better backbone in terms of structural diversity (structures belong to saddle points on ADMP trajectories) compared to the randomly generated configurations. The inputs obtained from the ADMP trajectories reduce the widespread raw configurations in the multidimensional search space and discards the configuration having atomic positions within their van der Waal distances. In the next step, the configurations of the cluster and their corresponding SPEs are stored for training a supervised neural network. PSO is employed to find the putative global minimum energy structure along with the corresponding energy value. In order to make the analysis statistically meaningful, the method is made 8-fold and each fold consists of 30 files with the number of iteration steps and the SPE values. The success rate of convergence, loss on the test set, and time taken to train the CNN model-based PSO technique are provided in Table 5. It is a very important aspect that the success rate is found to be significantly high (values ranging within 77%–90%) which clearly suggests that our proposed ADMP-CNN-PSO technique can be satisfactorily used for structure optimization and energy evaluation. It is found that the putative global minimum energy structure of the C5 cluster has a linear configuration, corroborating previous reports [148]. Due to the CNN-based supervised learning model, the convergence rate is higher than the normal optimization using Gaussian 09 program considering any random structure as an input. The putative global minimum energy structure along with two local minimum energy structure also obtained because of premature convergence are presented in Fig. 13. The corresponding equilibrium energy values obtained after postprocessing step is also given in the same figure.

202 Atomic clusters with unusual structure, bonding and reactivity

TABLE 5 Performance of the current optimization technique. Size of the training set

# Percentage of configuration convergence towards global best position

Loss on the test set

Time for model train

Global minimum energy (Hartree)

10

76.7%

0.129397

1 min 8 s


190.255

50

80.0%

0.064768

1 min 36 s

100

83.3%

0.037263

4 min 44 s

500

86.7%

0.012000

8 min 13 s

1000

86.7%

0.007388

20 min 54 s

5000

90.0%

0.002104

4 h 43 min 49 s

10,000

86.7%

0.001368

6 h 22 min 21 s

50,000

90.0%

0.001113

7 h 37 min 11 s

# Percentage of configuration convergence towards global best position ¼ (number of runs in which structure reached global minima/total number of runs)  100. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

Global Best Configuration

E (minimum) = –190.255 Hartree

Local Configurations

E (minimum) = –190.158 Hartree

E (minimum) = –190.135 Hartree

FIG. 13 Global and local minimum energy structures of C5 cluster after optimization in the postprocessing step computed at B3LYP/6-311 + G(d,p) level. Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

In Table 6, we compare and show the results obtained from other more established GO techniques such as SA, BH, and PSO. The current technique offers the best compromise in majority of the factors.

5.3 Firefly algorithm with density functional theory We have proposed an FA technique integrating DFT-based energy calculation for solving the GO problem and established its superiority in performance by reducing the computational time over PSO. The results are shown considering Al2
4 as a prototype example in Table 7. In order to investigate the superiority of FA over PSO, we have made a comparative analysis between FA and PSO augmented by quantum chemical calculations, considering planar and nonplanar approaches. In comparison, it is found that FA performs better than PSO. Furthermore, we have studied restricting the search space only with the planar generation of structures knowing that the putative global minimum energy structure of Al2
4 clusters is planar,

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TABLE 6 Comparison of current PSO results with other more established global optimization techniques as applied to the C5 cluster starting from random structures. Comparison in terms of

Simulated annealing (SA)

Basin hopping (BH)

Current PSO

Execution time to locate the global minimum (GM)

12,959 s

305,140 s

8898 s

Energy of the global minimum (energy after completion of iterations)


190.255 a.u. (
189.514 a.u.)


190.255 a.u. (
190.246 a.u.)


190.255 a.u. (
190.244 a. u.)

Number of iterations needed to get a structure close to GM

92 (not converged) But the final structure and energy are very close to that of the GM

1703 (converged)

331 (converged)

Adapted from A. Mitra, G. Jana, R. Pal, P. Gaikwad, S. Sural, P.K. Chattaraj, Determination of stable structure of a cluster using convolutional neural network and particle swarm optimization, Theor. Chem. Acc. 140 (2021) 30, https://doi.org/10.1007/s00214-021-02726-z, with permission from Springer Nature. © 2021, The Author(s), under exclusive license to Springer-Verlag GmbH, DE part of Springer Nature.

TABLE 7 Comparison between DFT-FA and DFT-PSO considering both planar and nonplanar implementations. Run no.

Convergence time (s)

No. of Gaussian calls

No. of structures

Success rate

Best energy (a.u.)

Firefly algorithm (nonplanar) 1

4838

2781

2762


969.7405

2

3626

2031

1988


969.7405

3

4982

2440

2390


969.7405

4

3898

2207

2171


969.7405

5

4977

2621

2587


969.7405

Mean

4464 (1 h 14 min 24 s)

2416

2380

36(98.5%)


969.7405

Firefly algorithm (planar) 1

3508

2143

2122


969.7405

2

4168

2679

2652


969.7405

3

3612

2348

2273


969.7405

4

4480

2797

2774


969.7405

5

2608

1630

1578


969.7405

Mean

3675 (1 h 1 min 12 s)

2319.4

2280

39(98.3%)


969.7405

Particle swarm optimization algorithm (nonplanar) 1

4906

2800

2436


969.7344

2

4242

2420

2140


969.7344

3

5294

3990

3769


969.74019

4

5582

3710

3371


969.7327

5

5517

3790

3553


969.7404

Mean

5108 (1 h 25 min 8 s)

3342

3054

288(91.4%)


969.7364 Continued

204 Atomic clusters with unusual structure, bonding and reactivity

TABLE 7 Comparison between DFT-FA and DFT-PSO considering both planar and nonplanar implementations—cont’d Run no.

Convergence time (s)

No. of Gaussian calls

No. of structures

Success rate

Best energy (a.u.)

Particle swarm optimization algorithm (planar) 1

5024

4000

3582


969.7405

2

3343

3130

2997


969.7405

3

4177

2830

2498


969.7402

4

3540

3260

3138


969.7405

5

4379

3420

3096


969.7405

Mean

4093 (1 h 8 min 12 s)

3328

3062

266(92%)


969.7404

Adapted from A. Mitra, G. Jana, P. Agrawal, S. Sural, P.K. Chattaraj, Integrating firefly algorithm with density functional theory for global optimization of Al42– clusters, Theor. Chem. Acc. 139 (2020) 32, https://doi.org/10.1007/s00214-020-2550-y, with permission from Springer Nature. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

thereby achieving faster convergence of the algorithm. The current technique is extended by showing a correlation between evolution of cluster stabilization energy with respect to increase of iteration number and aromaticity. In the present FA implementation, an initial arbitrary choice of number of particles, n ¼ 10 for PSO, and n ¼ 4 for FA are considered. To establish the analysis statistically meaningful, we have carried out extensive computational experiments. The results of both FA and PSO considering planar and nonplanar implementations are provided in Table 7 in terms of number of Gaussian 09 calls, convergence time and the success rate of convergence. A noteworthy observation of the results is presented in Table 7 where the average time of convergence for the planar FA implementation is slightly shorter (1 h 1 min 12 s) than the nonplanar one (1 h 14 min 24 s). In case of PSO, the findings of this fact remain same for planar (less time is required) as well as nonplanar (more time consuming). If we notice FA and PSO considering only planar implementations, the former converges at a faster rate, and comparing the energies of four such competitive approaches, it is found that the nonplanar PSO implementation gives the worst mean energy value. Interestingly, the success rate of FA is much higher (98%–99%) than that of the PSO (91%–92%). Thus, it can be stated that for searching of global optima, DFT-FA is more efficient than DFT-PSO.

5.3.1 Effect of planarity From Table 7, we can find that the performance of both FA and PSO is improved when planarity criteria is imposed in the search space. Since the reported global minimum energy structure of Al2
4 cluster is planar, it is logical to consider the planarity criterion over the unrestricted configuration search. Due to the reduced number of variables (like bond angles, torsional angles existing in 3D configurations) in planar structures than in the nonplanar structures, the former shows a faster rate of convergence. Further, it is very important to state that the candidate solution sometimes gets stuck at a local minima or valleys when we consider nonplanar configurations of individual entities due to immature convergence. Moreover, the candidate solution is intended with higher efficacy to gravitate towards the global best position when we impose planarity because of the equivalent decrease in the number of local minimum energy structures.

5.3.2 Energy vs. aromaticity profiles of planar Al2
4 structures H€ uckel first characterized the aromaticity giving the H€uckel’s (4n + 2) p-electron rule [187–189] and applied a quantum mechanical approach to describe the stability of the compounds. The concept of stabilization with respect to aromaticity was extended to inorganic compounds and metal clusters [173,188,190–202] including heterocyclic compounds [203,204]. Since aromaticity is an unobservable quantity, it cannot be measured by any direct experimental procedure or be precisely defined. It can, however, be quantified with some appropriate descriptors, such as, NICS. It has been reported by several research groups [205–208]. The investigation of the aromatic character of Al2
4 summarizes that it can be considered as paromatic and doubly s-aromatic due to the delocalization in its p and s orbitals [157]. The dissected NICS value

Generation of global minimum energy structures Chapter

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205

calculation through CMO-NICS and LMO-NICS suggests whether Al2
4 cluster is aromatic or not [209,210]. NICS values 2
˚ above the Al2
at center and 1 A 4 molecular plane are
39 and
17 ppm, respectively, observed in Al4TiAl4 sandwich 2
complex containing Al4 unit, support its aromatic behavior [156]. From these proven facts, we can say that aromaticity generally describes the energetic stability of a molecule and it is fair to describe it in terms of calculated NICS values. A negative and positive NICS (0) value (NICS value at center of the ring) inferred the aromatic and antiaromatic behavior of the system, respectively. The NICS (0) value of Al2
4 ring is computed per iteration step considering planar implementation using B3LYP/6-311 + G(d) level is presented in Fig. 14. In order to judge the relative stability of Al2
4 ring along with its aromatic stabilization, optimization of energy functional and NICS computations are scanned which are shown in Fig. 14. From this observation,

3 (a) –969.66

Energy

–969.68 –969.70 –969.72 –969.74 –969.76 0

2000

4000 6000 8000 Particle serial numbers arranged in increasing order of energy

10000

3 (b) –22

NICS

–24 –26 –28 –30 –32 –34 0

2000

4000 6000 8000 Particle serial numbers arranged in increasing order of energy

10000

3 (c) –22 –24

NICS

–26 –28 –30 –32 –34 –969.75–969.74–969.73–969.72–969.71–969.70–969.69–969.68–969.67

Energy

FIG. 14 Profiles of the energy (in a.u.) and nucleus-independent chemical shift (NICS) values on Al2
4 (negative NICS values are associated with the aromatic character of the systems), where 3(a), 3(b), and 3(c) represent plots of energy vs. particle serial numbers arranged in increasing order of energy, NICS vs. particle serial numbers arranged in increasing order of energy and NICS vs. energy, respectively. Adapted from A. Mitra, G. Jana, P. Agrawal, S. Sural, P.K. Chattaraj, Integrating firefly algorithm with density functional theory for global optimization of Al42– clusters, Theor. Chem. Acc. 139 (2020) 32, https://doi.org/10.1007/s00214-020-2550-y, with permission from Springer Nature. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

206 Atomic clusters with unusual structure, bonding and reactivity

we can conclude that lower energy values of a particle correspond to higher aromatic nature, which in turn results in a higher negative NICS value and hence higher stability (Fig. 14). This highlights the relationship between the stability and aromaticity for Al2
4 clusters.

6.

Conclusion

Our implemented DFT-PSO, ADMP-CNN-PSO, and DFT-FA techniques for the search of putative global minimum energy structure turns out to be successful for small metallic and nonmetallic clusters. The efficiency of PSO and FA approaches are high, the latter also has much better success rate with lower computational time to minimize the energy functional of molecular systems under investigation. There is no need for any assumptions and/or imposing external factors. The adjustment in global and local best parameters is sufficient in each iteration step, which is the most important advantage of these techniques. Interestingly, the introduction of planarity criterion by reducing number of variables leads to even faster convergence for both FA and PSO implementations. Further, ADMP-CNN-PSO technique also works well with potential efficiency and handles huge data sets of molecular clusters for the prediction of energies and global optimum solution. Moreover, the interpretation of the correlation between NICS and energy values corresponds to the fact that aromaticity increases with increasing molecular stability.

Acknowledgments GJ thanks Professor Pratim Kumar Chattaraj, Professor Gabriel Merino, and Dr. Sudip Pan for inviting him to contribute a chapter to the book, entitled, Atomic Clusters with Unusual Structure, Bonding and Reactivity published by Elsevier. GJ acknowledges Department of Chemistry, Indian Institute of Technology Kharagpur, India where all the works discussed in this chapter was carried out. RP thanks CSIR for her Research Fellowship.

Conflict of interest The authors declare that they have no conflict of interest regarding the publication of this book chapter, financial, and/or otherwise.

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Chapter 12

Studies on hydrogen storage in molecules, cages, clusters, and materials: A DFT study K.R. Maiyelvaganana, M. Janania, K. Gopalsamyc, M.K. Ravvab, M. Prakasha, and V. Subramanianc,d a

Department of Chemistry, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Chengalpattu,

Tamil Nadu, India, b Department of Chemistry, SRM University—AP, Amaravati, Andhra Pradesh, India, c Center for High Computing and Inorganic Physical Chemistry Laboratory, Central Leather Research Institute, Council of Scientific and Industrial Research, Chennai, Tamil Nadu, India, d

Academy of Scientific and Innovative Research (AcSIR), Chennai, Tamil Nadu, India

1. Introduction The inadequate fossil fuel resources and their impacts on the global environment are the reason for the inclination of researchers to improve immaculate energy technologies. Using fuel cells, the chemical energy can directly transform into electrical energy with high efficacy [1]. Hydrogen is the most ensuring alternative candidate for fossil fuel but its storage is prime issue [2]. The prime issues are (i) low T (at 20 K, 70 g/L) condensation (i.e., cryogenic liquid) is not practically feasible and (ii) compressed hydrogen gas only holds 15 g/L at 35 MPa, whereas in high pressure compression holds higher densities [2]. Hydrogen can be stored in two different forms (i) absorption (e.g., metal hydrides) and (ii) adsorption (e.g., carbon-based solid materials) [3]. In the past two decades, plethora of studies have been reported in adsorption techniques due to the viability in higher H-storage capacity [4]. Hydrogen can be artificially produced using water electrolysis [5–7], thermochemical cycles [8–10], and photocatalysis process [11–13].These methods show low efficiency and lead to wide-range commercialization. The US Department of Energy (DOE) has stated that 81 g/L volumetric density and 9.0 wt% gravimetric density as the ultimate requirements for H-storage materials [14]. Over the past decade, numerous molecules [15,16], clusters, and functional materials have been utilized for the H-storage [17]. They are metal hydrides [18], alanates [19], zeolites [20–25], carbon nanomaterials (carbonaceous graphene and carbon nanotubes (CNTs)) [26,27], metal-organic frameworks (MOFs) [28–37], clathrate hydrates [38], hydrates [39], and covalent-organic frameworks (COFs) [40]. The adsorption capacity of hydrogen using CNT shows 2–4 wt% at ambient conditions, when functionalized it increases the H-storage capacity [4,41–45]. Park et al. reported that when the CNT is functionalized with Pt metal, it improves the H-storage capacity due to the hydrogen spillover effect [46]. The hydrogen spillover effect is nothing but the migration of the activated hydrogen atoms from the metal to the catalyst [47]. On investigating various nanotubes such as boron [48], boron nitride (BNNT) [49,50] and silicon [51], it indicates that these materials show the enhanced H-storage capacity compared with the carbon analogs. Exploring CNT to fullerenes, many researchers have appraised the exo/endo-hedral H-storage ability using fullerenes [52–58]. When fullerenes are exohedrally doped with metal/metal ions it remarkably increases the H-storage capacity as well as the adsorption strength [59–62]. Chandrakumar et al. reported that when a fullerene is doped with alkali metal atoms, it enhances the H-storage capacity. The H2 molecule adsorption by exohedrally decorated Na doped fullerene C60 is being
9.5 wt% [59]. Similarly, in boron analogs, the B80Na12 and B80K12 can store up to 72 H2 molecules with 11.2 and 9.8 wt% of gravimetric density, respectively [63,64]. The calculated corresponding binding energies (BEs) per H2 molecules are 1.67 and 1.99 kcal/mol [61]. The boron substituted Li dispersed graphene is observed to show the H-storage capacity around
13.2 wt% [65]. Despite these determinations, none of the materials meet the basic requirement of DOE for H-storage [66]. Thus, the investigation of H-storage at molecules/materials and clusters are receiving widespread attention in terms of adsorption mechanism and site-specific attractions for the enhanced H-storage ability of functional molecules. H-storage in molecules/materials (HSMs) is an important manifestation of the gas molecules adsorption or desorption at room temperature and atmospheric pressure. Molecular level studies on HSMs are very important to understand the Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00019-X Copyright © 2023 Elsevier Inc. All rights reserved.

213

214 Atomic clusters with unusual structure, bonding and reactivity

adsorption or desorption ability of these materials. It is interesting to note that the stability of gas adsorption depends on the interfacial interactions (i.e., the interaction between gas molecules and the surface of functional materials). Hence, numerous experimental and theoretical approaches have been devoted to understanding the molecular level properties of H2 molecules at various isothermal (P-T) conditions [2]. There are several HSMs available to store and utilize depending on their applications. Earlier reports state that the host of small molecules (i.e., H2O, CH4, NH3, and CO2) in larger molecules as crystalline molecular compounds are providing a suitable H-storage capacity [2]. In addition, bulk materials, such as two-dimensional (2D) graphene [65], graphite [67], boron-nitrogen (BN) sheets [68], and three-dimensional (3D) CNT [48], BNNTs, nitrogen doped graphitic carbon along with the various microporous, and nanoporous materials are highly recommended for bulk storage applications. Particularly, MOF [17], zeolitic imidazolate frameworks (ZIFs) [22], and COFs [66] are in the class of porous coordination polymers which are more appropriate for a large amount of gas storage as well as these materials can be extensively utilized for gas separation and conversion application. Molecular cages [66], Medium-sized molecular clusters [64], and clathrate hydrates [38] have also received widespread attention for the potential HSMs according to DOE targets [69]. Subramanian and his coworkers have studied the H-storage capacity in molecular cages using the density functional theory (DFT) approach and reported that molecular cages show effective H2 adsorption than the other materials [66]. The endohedral storage of H2 molecules in the B36N36 cage was studied using the DFT approach [70]. The calculated energy barrier for H2 molecules entering into a cage or escaping from the cage is
1.5 eV. This study also revealed that, as H2 gas concentration inside the cage increases, the cage expands and the bond length of the hydrogen molecule contracts, resulting in significant energy costs. To enhance the adsorption and desorption ability of H2 molecules, the physisorption is more feasible than chemisorption techniques [71]. Further, to enhance the H2 adsorption capacity on molecules, metal ion decorated surfaces are widely used. Thus, identifying of suitable molecules, cages, and molecular clusters for the HSMs is a very important task for developing alternative energy resources. Even though there are numerous examples of H-storage materials, these materials [66] cannot satisfy the requirement of DOE. In this context, the study of small molecules will help to unravel basic interactions and bonding involving these materials with H2. To enhance the adsorption ability of the molecules/materials, the doping of metal ions on the surface is necessary. To get more insights into the structure, stability, and adsorption mechanism of H2 molecules on the surface, we have employed various density functional theory (DFT) methods. Various hydrogen bonded (H-bonded) and charged molecular clusters can interact with H2 molecules through different kinds of noncovalent interactions [e.g., dipole-dipole, charge induced dipole, dispersion, and van der Waals (vdWs) types]. Several DFT methods such as B3LYP, M05-2X, and M06-2X methods were employed in our group. The BE of all clusters was calculated using the supermolecule approach and corrected for basis set superposition error (BSSE) using the counterpoise (CP) procedure suggested by Boys and Bernardi [72]. The BE was calculated using the following energy expression.     (1) jBEj ¼ EComplex  ESurf + N ðEH2 Þ where EComplex, ESurf, and N EH2 are the total energies of complex (with nH2), the energy of selected molecules, clusters or surfaces, and NH2 clusters, respectively. The charges on the metal ions were calculated using the natural population analysis (NPA) method. In addition, the energetics of all the complexes have also been computed with the dispersion correction (DFT + D3). All the electronic structure calculations were performed using Gaussian 16 software [73].

2.

H-storage in various motifs—The road map representation

The binding of atoms gives small molecules. The growth of small molecules makes cages, and the repeating units of small molecules lead to the evolution of nanoclusters. When clusters or cages are combined, it results in the formation of the 2D or 3D material. The Road map representation emphasis the advancement of material from small molecules. This hierarchical representation will help us to understand a concept in a systematic way (shown in Scheme 1). Subramanian and his coworkers systematically studied the H-storage capacity of alkali and alkaline earth metal ions doped carbon-based materials using DFT study [74]. The understanding of interactions with H2 molecules and the impact of the size and charge of the metal ions in small molecules (i.e., cubane, cyclohexane, and adamantane) help to implement the findings for the development of the molecular clusters. The molecular cages pave the way for learning the interactions in confined systems. H-bonded boric acid (BA)-based clusters with pentamer and hexamer motif and BA20 nanoclusters have been explored for H-storage applications [64].

Studies on hydrogen storage Chapter

12

215

SCHEME 1 Road map representation of various motifs transform into functional materials for H-storage applications.

2.1 H-storage in small molecules This section provides interesting information on the design of new linkers for enhancing the H-storage capacity of various systems [74]. We have selected various small molecules such as cubane (Cub), cyclohexane (Cyc), adamantane (Adm), and their alkali and alkaline earth metal cations decorated systems (Fig. 1). All calculations were carried out by using the DFTbased M05-2X/6-31 + G** method. The incorporation of alkali and alkaline earth metal cations significantly improves the H storage.

FIG. 1 Schematic representation of side and top views of metal cation doped carbonaceous materials at M05-2X/6-31 + G** level of theory. (A) Cub-Mm +, (B) Cyc-Mm +, and (C) Adm-Mm + complexes (where M ¼ Li+, Na+, K+, Mg2+, and Ca2+; m ¼ 1 or 2. (Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549– 2559.)

216 Atomic clusters with unusual structure, bonding and reactivity

2.1.1 Cubane (cub) The optimized geometries of Cub with alkali and alkaline earth metal cations and its complexes with hydrogen molecules (nH2@Cub-Mm +, where, Mm + ¼ Li+, Na+, K+, Mg2+, and Ca2+ and n ¼ 4, 5, 9, 5, and 6) were shown in Fig. 2. The distance between the metal cation and Cub molecule is calculated from the metal to the center of four upper ring carbon atoms of Cub (face of Cub). It is the measure of the strength of interaction between the two systems. These values are listed in Table 1. For the IA group metal cation doped complexes, the distance varies in the order of Cub-Li+ < Cub-Na+ < Cub-K+. This is in good agreement with the variation of ionic radius of the corresponding metal cation. The calculated distance between Cub ˚ for Cub-Li+, Cub-Na+, and Cub-K+, respectively. A Similar trend in the and the metal cation is 1.832, 2.322, and 2.757 A distance has also been obtained for the IIA group metal cations doped complexes. The calculated distance is 1.878 and ˚ for Cub-Mg2+ and Cub-Ca2+ complexes, respectively. 2.317 A In the hydrogenated complexes (Fig. 2), the metal cation interacts with both ends of each hydrogen molecule. It results in the formation of a T-shaped complex. A similar observation has also been reported in the previous study on the adsorption of hydrogen molecules on the alkali metal decorated fullerene [75]. After the adsorption of hydrogen molecules on various metal cation doped [17,76] (Cub-Mm +) (hydrogenated complexes), a marginal increase in the distance between Cub and the metal cation is observed. The calculated distances between the metal cations and H2 molecules are given in Table 2. It is visualized from Fig. 2 that a maximum of 4H2 molecules binds with the Cub-Li+ complex. The calculated ˚. distance between the metal cation and the center of the HdH bond distance ranges from 2.243 to 2.332 A + + 2+ 2+ The maximum number of H2 molecules adsorbed on the Na , K , Mg , and Ca doped Cub complexes are five, nine, five, and six, respectively. The calculated BE with BSSE of various metal cation doped Cub-Mm + complexes (bare and after hydrogenation) are listed in Table 3. It is interesting to observe that the BEs for Cub-Mg2+ and Cub-Ca2+ complexes are higher.

4H2@Cub-Li+

5H2@Cub-Na+

5H2@Cub-Mg2+

9H2@Cub-K+

6H2@Cub-Ca2+

FIG. 2 Optimized geometries of nH2@Cub-M (where M ¼ Li , Na , K , Mg2+, and Ca2+; m ¼ 1 or 2) complexes at M05-2X/6-31 + G** level of theory. (Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.) m+

+

+

+

Studies on hydrogen storage Chapter

12

217

TABLE 1 Calculated adsorption distance (A˚) between the host materials and metal cations using M05-2X/6-31 + G** level of theory. H2 adsorption @metal-ion doped cubane

Cub-Li+ +

Cub-Na +

Cub-K

2+

Cub-Mg

2+

Cub-Ca

Beforea

Afterb

1.832

1.925

2.322

2.374

2.757

2.816

1.878

2.005

2.317

2.368

a

Before absorption of nH2 molecules. After absorption of nH2 molecules. Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

b

TABLE 2 Calculated distance (A˚) between the adsorbed H2 molecules and the metal cations at M05-2X/6-31 + G** level of theory. Mm +-nH2 distances

Cubane 4H2@Cub-Li+

2.243–2.332

5H2@Cub-Na

+

2.503–2.596

9H2@Cub-K

+

3.000–3.342

5H2@Cub-Mg

2.168–2.628

6H2@Cub-Ca

2.589–2.667

2+

2+

Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

TABLE 3 Calculated BSSE corrected BEs (kcal/mol) of Cub-Mm + complexes by using M05-2X/6-31 + G** level of theory. Binding energy (kcal/mol) Cub-M

m+

Cub-Li+ +

Cub-Na +

Cub-K

2+

Cub-Mg

2+

Cub-Ca

Beforea

Afterb

28.51

27.88

17.85

17.63

12.08

11.92

106.93

102.93

63.61

62.73

a

Before absorption of nH2 molecules. After absorption of nH2 molecules. Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

b

The calculated BEs, BE/H2 (kcal/mol), and H-storage Capacity (wt%) for hydrogenated Cub-Mm + complexes are listed in Table 4. The calculated BE of 4H2@Cub-Li+ complex is 13.20 kcal/mol and the corresponding BE/H2 molecule is 3.30 kcal/mol. The observed gravimetric density value of 4H2@Cub-Li+ is 6.8 wt%. The vdW radius of the Na+ cation is considerably larger than the Li+ cation. Hence, a greater number of H2 molecules can be accommodated over Na+ cation doped complex. The corresponding BE of Cub-Na+ hydrogenated with 5H2 molecules is 13.78 kcal/mol, which accounts for

218 Atomic clusters with unusual structure, bonding and reactivity

TABLE 4 Calculated BE, BE/H2 (kcal/mol), and H-storage capacity (wt%) for nH2@Cub-Mm + complexes at M05-2X/6-31 + G** level of theory. nH2@Cub-Mm +

4H2@Cub-Li+

5H2@Cub-Na+

9H2@Cub-K+

5H2@Cub-Mg2+

6H2@Cub-Ca2+

BE

13.20

13.78

14.02

46.85

33.69

BE/H2

3.30

2.76

1.56

9.37

5.61

wt%

6.8

7.4

7.3

7.7

11.2

Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

2.76 kcal/mol per hydrogen molecule and the gravimetric density of the complex is 7.4 wt%. The ionic radius of the K+ cation is much larger than that of Li+ and Na+ cations. So it is reasonable to expect that the Cub-K+ complex can absorb a greater number of H2 molecules. As anticipated it is possible to store a maximum of 9H2 molecules on this complex. The calculated BE of the 9H2@Cub-K+ complex is 14.02 kcal/mol, which is equal to 1.56 kcal/mol per hydrogen molecule with a storage capacity of 11.2 wt%. Results reveal that a maximum of five H2 molecules can be attached to the Cub-Mg2+ complex.

2.1.2 Cyclohexane (Cyc) The optimized geometries of hydrogenated (nH2@Cyc-Mm +) complexes along with the geometrical parameters are depicted in Fig. 3. Close scrutiny of BE values of Cyc-Mm + complexes shows that these metal cations interact favorably with the chair conformation of Cyc at the hollow site.

5H2@Cyc-Li+

6H2@Cyc-Na+

4H2@Cyc-Mg2+

9H2@Cyc-K+

7H2@Cyc-Ca2+

FIG. 3 Optimized geometries of nH2@Cyc-M (where M ¼ Li , Na , K , Mg , and Ca2+; m ¼ 1 or 2) complexes at M05-2X/6-31 + G** level of theory. (Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.) m+

+

+

+

2+

Studies on hydrogen storage Chapter

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219

TABLE 5 Calculated BE, BE/H2 (kcal/mol), and H-storage capacity (wt%) for nH2@Cyc-Mm + complexes at M05-2X/6-31 + G** level of theory. nH2@Cyc-Mm +

5H2@Cyc-Li+

6H2@Cyc-Na+

9H2@Cyc-K+

4H2@Cyc-Mg2+

7H2@Cyc-Ca2+

BE

12.78

15.58

15.67

32.19

33.99

BE/H2

2.56

2.59

1.74

8.05

4.86

wt%

9.9

10.1

12.8

6.9

10.2

Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

The calculated important metrics for nH2@Cyc-Mm + Complexes are presented in Table 5 shows the BE per hydrogen molecule (BE/H2) for 5H2@Cyc-Li+, 6H2@Cyc-Na+, 9H2@Cyc-K+, 4H2@Cyc-Mg2+, and 7H2@Cyc-Ca2+ complexes is 2.56, 2.59, 1.74, 8.05, and 4.86 kcal/mol, respectively. The Corresponding H-storage capacity is 9.9, 10.1, 12.8, 6.9, and 10.2 wt%. From the above observation, potassium ion decorated Cyc motifs shows predominant H2 adsorption than the other metal ions.

2.1.3 Adamantane (Adm) The optimized geometries of Adm with alkali and alkaline earth metal cations and corresponding hydrogenated Adm-based complexes [74] are displayed in Fig. 4. It can be found from Table 6 that the BE/H2 molecule for 5H2@Adm-Li+,

5H2@Adm-Li+

6H2@Adm-Na+

4H2@Adm-Mg2+

9H2@Adm-K+

7H2@Adm-Ca2+

FIG. 4 Optimized geometries of nH2@Adm-M (where M ¼ Li , Na , K , Mg , and Ca2+; m ¼ 1 or 2) complexes at M05-2X/6-31 + G** level of theory. (Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.) m+

+

+

+

2+

220 Atomic clusters with unusual structure, bonding and reactivity

TABLE 6 Calculated BE, BE/H2 (kcal/mol), and wt% for nH2@Adm-Mm + complexes at M05-2X/6-31 + G** level of theory. nH2@Adm-Mm +

5H2@Adm-Li+

6H2@Adm-Na+

9H2@Adm-K+

4H2@Adm-Mg2+

7H2@Adm-Ca2+

BE

12.23

15.61

15.86

33.33

33.72

BE/H2

2.45

2.60

1.76

8.33

4.82

wt%

6.6

7.1

9.4

4.8

7.4

Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

6H2@Adm-Na+, 9H2@Adm-K+, 4H2@Adm-Mg2+, and 7H2@Adm-Ca2+ complexes are 2.45, 2.60, 1.76, 8.33, and 4.82 kcal/mol, respectively. The respective gravimetric density is 6.6, 7.1, 9.4, 4.8, and 7.4 wt%. Comparison of BE, BE/H2, and wt% of Cub-Mm +, Cyc-Mm +, and Adm-Mm + (where Mm + ¼ Li+, Na+, K+, Mg2+, and 2+ Ca ) complexes shows that doping of Li+, Mg2+, and Ca2+ cation enhances the H-storage capacity of Cub. In the case of Cyc and Adm, doping of Na+, Mg2+, and Ca2+ cation increases the gravimetric density of the corresponding complexes. Overall doping of Cub, Cyc, and Adm with Mg2+ and Ca2+ augments the H-storage capacity. In addition the H2 storage capacity of polyhydroxy admantane was studied by Subramanian et al. [77] From the analysis, it is well understood that the adsorption property in small molecules depends on the electron density, charge, and size of metal ion bonded with the surface. The small molecules are limited in storing the H2 molecules due to their small size and less atomic interaction with the incoming gas molecules. To store the maximum amount of gases, the development of a cage structure is necessary. In addition to get more insight on the effect of substitution of antipodal atoms can be studied.

2.2 Hydrogen storage in molecular cages For efficient adsorption of molecular hydrogen, we have selected various small molecules such as boranes (BXY, where X and Y ¼ B, C, N, and Si) and alanes (AlXY, where X and Y ¼ C, Si, N, and Al). Gopalsamy et al. have elaborately discussed the H-storage capacity of boranes- and alanes-based cages using density functional studies [66]. It is found from previous reports that the icosahedral (Ih) boranes and alanes were investigated due to their potential applications in various fields [78–80]. The main advantages of Ih closo-boranes and alanes are (i) higher symmetry, (ii) structural stability, (iii) thermal stability, and (iv) chemically inert towards redox reactions [78–80]. The Schematic representation of the closo-borane and alane cages are shown in Scheme 2.

2.2.1 H-storage in boranes- and Alanes-based cages The optimized geometries of nH2@BXY and nH2@AlXY cages are shown in Figs. 5A and B. There are no significant changes in the orientation of H2 units concerning the faces. In the case of H2@BBB, the distance between the center of the delta˚ . Upon loading with the number of H2 molecules, the distance between the deltahedron face and H2 molecule is 3.109 A hedron face and H2 molecule decreases. As reported in Table 7, the range of distances between the deltahedron face of BBB ˚ . The same range for the 10H2@BBB is 2.897–2.940 A ˚ . For 20H2@BBB, this and H2 molecule in 2H2@BBB is 2.898–2.907 A ˚ range is found to be 2.923–2.949 A. It is evident from these findings that the distance between the H2 molecules and the deltahedron face of BBB decreases due to the addition of H2 molecules. Except for BCB, similar findings have been observed for the interaction of hydrogen molecules with BCC, BSiB, BSiSi, BCSi, and BNB. In the case of BCB, there is a noticeable increase in the distances between the deltahedron face and H2 molecule. The calculated HdH bond distance at M05-2X/6-31 + G** level of theory for the free ˚ . The HdH bond distance is sensitive to the complexation with the different cages. In the case hydrogen molecule is 0.741 A ˚ . In the cases of BCB and BCSi, the HdH distance is 0.743 A ˚ , whereas in of BBB, the HdH distance is found to be 0.746 A ˚ BCC, BSiSi, BCSi, and BNB the same distance is 0.742 A. It is evident from these results that the total charge on the cage influences the H2 interaction pattern. The differences observed in the interaction pattern may be attributed due to the changes in the ionic surface and curvature of cage molecules. Similar arguments have also been addressed for hydrogen adsorption in carbon-based model systems. The trend in the interaction pattern of the H2 molecule with the AlXY cages is akin to that of BXY systems.

Studies on hydrogen storage Chapter

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Adsorption sites Adsorption sites

12

221

SCHEME 2 Schematic diagram of the closo-borane and alane cages (X and Y denote antipodal atoms). (I) Upper cap pentagonal ring atoms distance; (II and III) within the upper and lower pentagonal ring atoms distance; (IV)interconnecting pentagon distances; (V) lower cap pentagonal ring atoms distance; and (VI) antipodal atoms distance. (Taken from reference K. Gopalsamy, M. Prakash, R. Mahesh Kumar, V. Subramanian, Density functional studies on the hydrogen storage capacity of boranes and alanes based cages, Int. J. Hydrogen Energy 37 (12) (2012) 9730–9741.)

Icosahedral cage

Boron – B, Carbon – C, Silicon – Si or Nitrogen – N. Boron – B, Carbon – C or Silicon – Si

Aluminium – Al, Carbon – C, Silicon – Si or Nitrogen – N. Aluminium – Al, Carbon – C or Silicon –Si

2.2.2 Energetics of nH2@BXY cages The calculated BE, BE/H2, and gravimetric density in wt% are listed in Table 8 for 20H2@BXY cages. Evidence shows that BE of various 1H2@BXY varies from 0.55 to 2.10 kcal/mol. The BBB cage has the highest BE when compared to the other hetero substituted systems due to the negative (2) charge on the cage. The variation of BE with loaded H2 molecule is shown in Fig. 5A for nH2@BXY cages (where n ¼ 1–20). It can be seen that the enhancement of BE takes place with the increase in the number of H2 molecules. Fig. 6 shows that three different ranges of BEs for di-anion, mono-anion, and neutral cages. The maximum BE is observed for the 20H2@BBB unit which is 43.11 kcal/mol and its corresponding BE/H2 is 2.15 kcal/mol. The respective gravimetric density is 22.2 wt% which is maximum when compared to the other BXY cages. The BE of 20H2@BCB, 20H2@BCC, 20H2@BSiB, 20H2@BSiSi, 20H2@BCSi, and 20H2@BNB is 23.90, 13.31, 23.84, 13.60, 15.58, and 13.69 kcal/mol, respectively. The corresponding BE/H2 (where n ¼ 20) is 1.19, 0.67, 1.19, 0.68, 0.77, and 0.68 kcal/mol. It can be found that BE decreases with the charge on the cages, i.e., di-anion > mono-anion > neutral. When one C or Si atom is substituted in the antipodal position of Ih, one negative charge is replaced by the C or Si valence electron. Thus, the electron density delocalization in the cage decreases, and the decrease in the BE is observed. Hence, the results point out that borane cages exhibit a maximum H-storage capacity of
22.0 wt% when compared to all other cages considered in this study.

2.2.3 Energetics of nH2@AlXY cages The calculated BE, BE/H2 (kcal/mol) and wt% of 20H2@AlXY Cages are listed in Table 9.

222 Atomic clusters with unusual structure, bonding and reactivity

FIG. 5 Optimized geometries of various cages with H2 molecules. (A) BXY cages (where X and Y ¼ B, C, Si, and N) and (B) AlXY cages at M05-2X/6-31 + G** level of theory (where X and Y ¼ Al, C, Si, and N). (Taken from reference K. Gopalsamy, M. Prakash, R. Mahesh Kumar, V. Subramanian, Density functional studies on the hydrogen storage capacity of boranes and alanes based cages, Int. J. Hydrogen Energy 37 (12) (2012) 9730–9741.)

A

5H2@BXY

10H2@BXY

15H2@BXY

20H2@BXY

5H2@AlXY

10H2@ AlXY

15H2@ AlXY

20H2@ AlXY

B

The higher H-storage capacity of AlSiSi can be attributed to the proper fitting of Si atoms in the X and Y positions of alane cages when compared to the BSiSi cage. Among various alanes, AlCC has a maximum H-storage capacity of 11.6 wt%. Based on these findings a new MOF with carborane as a linker has been designed. The overall comparison of BXY and AlXY cages reveals that borane materials have higher H-storage capacity and gravimetric density when compared to that of series of substituted alanes.

2.3 H-storage in molecular clusters Prakash et al. systematically studied the H-storage capacities of alkali metal cations decorated bowl (BA5), sheet (BA6) and fullerene ball (BA20) shaped BA nanoclusters using the DFT method [64]. The molecular cluster is used to measure the H-storage capacity, evaluate the effect of substitution of atoms, it also helps to create large building blocks such as new

Studies on hydrogen storage Chapter

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223

TABLE 7 Calculated distance between H2 and the center of the deltahedron faces of BXY cages at M05-2X/6-31 + G** level of theory. nH2

BBB

BCB

BCC

BSiB

BSiSi

BCSi

BNB

1

3.109

3.077

3.182

3.085

3.166

3.095

3.105

2

2.898–2.907

3.083–3.087

3.118–3.123

3.079–3.091

3.105–3.110

3.095–3.119

3.089–3.112

3

2.908–2.918

3.089–3.092

3.099–3.100

3.074–3.101

3.110–3.166

3.094–3.118

3.093–3.104

4

2.909–2919

3.083–3.097

3.099–3.102

3.070–3.098

3.093–3.107

3.088–3.122

3.089–3.102

5

2.892–2.947

3.079–3.099

3.097–3.102

3.069–3.118

3.103–3.115

3.084–3.119

3.084–3.112

6

2.901–2.921

3.078–3.094

3.093–3.104

3.057–3.097

3.108–3.114

3.088–3.125

3.091–3.114

7

2.899–2.917

3.082–3.097

3.093–3.107

3.059–3.100

3.103–3.116

3.090–3.119

3.079–3.114

8

2.912–2.930

3.082–3.105

3.095–3.102

3.048–3.088

3.100–3.116

3.091–3.117

3.083–3.113

9

2.919–2.939

3.083–3.102

3.098–3.102

3.060–3.110

3.101–3.116

3.092–3.119

3.086–3.109

10

2.897–2.940

3.082–3.115

3.005–3.165

3.065–3.125

3.098–3.108

3.089–3.124

3.085–3.102

15

2.911–2.968

3.021–3.101

3.082–3.165

3.065–3.107

3.086–3.130

3.077–3.189

3.075–3.100

2.923–2.949

3.056–3.151

3.083–3.170

3.061–3.155

3.086–3.330

3.072–3.210

3.084–3.164

0.746

0.743

0.742

0.743

0.742

0.742

0.742

20 a

HdH

The calculated adsorbed H2 molecules H–H bond distances (are in A˚) for the corresponding BXY cages. Taken from reference K. Gopalsamy, M. Prakash, R. Mahesh Kumar, V. Subramanian, Density functional studies on the hydrogen storage capacity of boranes and alanes based cages, Int. J. Hydrogen Energy 37 (12) (2012) 9730–9741. a

TABLE 8 Calculated BE, BE/H2 (kcal/mol), and wt% of 20H2@BXY cages at M05-2X/6-31 + G** level of theory. Boranes

BBB

BCB

BCC

BSiB

BSiSi

BCSi

BNB

BE

43.11

23.90

13.31

23.84

13.60

15.58

13.69

2.15

1.19

0.67

1.19

0.68

0.77

0.68

BE/H2 wt%

22.2

21.9

21.8

20.2

18.6

20.1

21.7

Taken from reference K. Gopalsamy, M. Prakash, R. Mahesh Kumar, V. Subramanian, Density functional studies on the hydrogen storage capacity of boranes and alanes based cages, Int. J. Hydrogen Energy 37 (12) (2012) 9730–9741.

MOF. During the adsorption there are various kinds of interactions that are possible in the molecular cage. For example, Electrostatic, charge-quadrupole, charge-induced dipole, and charge transfer.

2.3.1 BA nanoclusters For enhancing the H-storage capacity of BA-based nanoclusters, it is necessary to understand how various nanostructures are formed from BA (Fig. 7). This study reports a significant addition to the usefulness of BA clusters by way of studying the H-storage capacity of alkali metal ion decorated BA clusters. The H-storage capacity of alkali metal ion decorated BA clusters BAn[M+m(H2)6]N (where n ¼ 5, 6, and 20; M ¼ Li, Na, and K; in bowl type clusters N ¼ 1–3, sheet structure N ¼ 1–4, and ball clusters M ¼ Na; N ¼ 1–4, 6, and 8; m ¼ 1, 2) have studied using DFT-based B3LYP method.

224 Atomic clusters with unusual structure, bonding and reactivity

FIG. 6 Calculated BE (kcal/mol) for BXY cages using M05-2X/631 + G** method. (Taken from reference K. Gopalsamy, M. Prakash, R. Mahesh Kumar, V. Subramanian, Density functional studies on the hydrogen storage capacity of boranes and alanes based cages, Int. J. Hydrogen Energy 37 (12) (2012) 9730–9741.)

TABLE 9 Calculated BE, BE/H2 (kcal/mol), and wt% of 20H2@AlXY cages at M05-2X/6-31 + G** level of theory. Alanes

AlAlAl

AlCAl

AlCC

AlSiAl

AlSiSi

AlCSi

AlNAl

BE

21.34

13.76

10.90

14.08

15.46

15.08

12.86

1.07

0.68

0.55

0.71

0.77

0.75

0.64

BE/H2 wt%

10.7

11.2

11.6

10.7

10.6

11.1

11.1

Taken from reference K. Gopalsamy, M. Prakash, R. Mahesh Kumar, V. Subramanian, Density functional studies on the hydrogen storage capacity of boranes and alanes based cages, Int. J. Hydrogen Energy 37 (12) (2012) 9730–9741.

FIG. 7 Schematic representation of boric acid clusters from BA monomer to BA20 nanoclusters.

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FIG. 8 Optimized geometries for H2 molecules adsorbed bowl shaped (BA5) clusters decorated with single alkali metal ions. Distances are in (where x is indicates the center of H2 molecule same for all figures). (Taken from reference M. Prakash, M. Elango, V. Subramanian, Adsorption of hydrogen molecules on the alkali metal ion decorated boric acid clusters: a density functional theory investigation, Int. J. Hydrogen Energy 36 (6) (2011) 3922–3931.)

2.3.2 H-storage in boric acid pentamer (BA)5 To enhance the H-storage capacity the alkali metal ions are decorated in bowl clusters. The optimized geometries of BA5[M+m(H2)6]1 (where M ¼ Li, Na, and K; m ¼ 1, 2) is shown in Fig. 8. When the metal ions (Li+, Na+, and K+) adsorb six H2 molecules, it exhibits pentagonal pyramid geometry at the metal site. It can be noticed from the optimized geometries of BA5[K+(H2)8]N clusters (where N ¼ 1 and 2) that the geometrical arrangement of H2 molecules is different from that of the clusters decorated with Na ions. BA5[K+(H2)8] structure exhibits hexagonal bipyramid geometry. In BA5[K+(H2)8]2 cluster, five H2 are in one plane and the other three H2 are in the opposite plane. Due to internuclear repulsions between the adjacent metal ions, all the H2 molecules cannot interact from the same plane. It can be seen from Fig. 8 that axial H2 molecule (i.e., the angle between OBANa+H2 angle  180° position) is parallel to the adsorbing surface, whereas other H2 molecules are perpendicular to the surface of the BAn clusters. It is interesting to note that BA5(Na+)2 and BA5(K+)2 systems can store up to 12 and 16 H2 molecules, respectively. The uptake of clusters with three Na+ and K+ ions is the same. Both clusters can adsorb a maximum of 18 H2 molecules. The calculated BEs and BE/NH2 (in kcal/mol) of Bowl Shaped Clusters (BA5[M+m(H2)6]N are summarized in Table 10. It is evident from the results that the calculated BE/H2 molecule varies from one metal ion to other. It is observed that when the number of alkali cations increases the BE/H2 also increases which results in less interaction between the BA and H2 molecules. This study is extended to BA hexamer to investigate the impact of the number of alkali metal ions and the effective adsorption of H2 molecules.

2.3.3 H-storage in boric acid hexamer (BA)6 In the previous study, we have shown that the rosette structure is responsible for the formation of a BA-based nanotubes [64]. Hence, the H-storage capacity of the rosette motif has been studied. The optimized geometries of H2 adsorbed alkali metal decorated sheet structures are depicted in Fig. 9. In BA6M+ (where M ¼ Li and Na), the Li or Na ions interact with one of the oxygen atoms of the BA sheet structure. The BEs of alkali metal ions (in Table 11) to the sheet structure vary as BA6Li+ > BA6Na+ > BA6K+ indicating that the Li+ interacts strongly with the sheet structure. The calculated M-Hx distances of BA6[Li+(H2)3], BA6[Na+(H2)6], and BA6[K+(H2)6] clusters range from 2.096 to 2.111, 2.542 to 2.748, and ˚ , respectively. 3.165 to 3.245 A As observed from bowl-shaped clusters, the adsorption capacity differs for different alkali cations and the number of ions decorated clusters in sheet clusters. The maximum BE/H2 (2.40 kcal/mol) is observed for the BA6[Na+(H2)5]4 clusters. High storage capacity and moderate binding energies are two important requirements for the development of new H-storage materials. As observed in the case of bowl motif, Na+ decorated sheet structure exhibits optimal conditions for H-storage materials. Since the properties of Na+ systems are moderate and suitable for the H-storage, H2 adsorption studies have been carried out only for the Na+ decorated BA20 ball structure.

226 Atomic clusters with unusual structure, bonding and reactivity

TABLE 10 Calculated (BEs) and BE/NH2 (in kcal/mol) of bowl-shaped clusters (BA5[M+m(H2)6]N, where N 5 1–3). BA5M+((H2)6)N BA5Li

+ +

BA5Na BA5K

+ +

BA5Li (H2)6 +

BA5Na (H2)6

BE

BE/NH2

23.91

–

16.15

–

10.94

–

15.44

2.57

11.25

1.88

+

BA5K (H2)6

4.92

0.82

+

6.08

0.76

32.69

3.27

BA5K (H2)8 +

BA5(Li (H2)5)2 +

BA5(Na (H2)6)2

24.15

2.02

+

BA5(K (H2)6)2

11.11

0.93

+

BA5(K (H2)8)2

13.90

0.87

+

53.09

3.59

37.94

2.11

18.12

1.00

BA5(Li (H2)5)3 +

BA5(Na (H2)6)3 +

BA5(K (H2)6)3

Taken from reference M. Prakash, M. Elango, V. Subramanian, Adsorption of hydrogen molecules on the alkali metal ion decorated boric acid clusters: a density functional theory investigation, Int. J. Hydrogen Energy 36 (6) (2011) 3922–3931.

FIG. 9 Optimized geometries of H2 molecules adsorbed sheet shaped (BA6) clusters decorated with ˚ . (Taken from single alkali metal ions. Distances are in A reference M. Prakash, M. Elango, V. Subramanian, Adsorption of hydrogen molecules on the alkali metal ion decorated boric acid clusters: a density functional theory investigation, Int. J. Hydrogen Energy 36 (6) (2011) 3922–3931.)

2.4 H-storage in materials 2.4.1 Small molecules incorporated MOP-9 One of the main objectives of the present work is to develop metal-organic polyhedra (MOPs) with improved H-storage capacity. Many attempts have been devoted in the past decades to design new MOFs by doping metal cations on the organic linkers [17,81–84]. Close analysis of the results obtained from this investigation reveals that H-storage capacity of metal cations (Li+, Na+, K+, Mg2+, and Ca2+) doped Cub, Cyc, and Adm complexes enhance the H2 storage capacity [74,85]. Especially, Li+, Mg2+, and Ca2+ metal cation doped complexes show very good results [86].Among the metal-doped

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TABLE 11 Calculated (BEs) and BE/NH2 (in kcal/mol) of sheet clusters (BA6[M+(H2)6]N, where N 5 1–4). BA6M+((H2)6)N BA6Li

+ +

BA6Na +

BA6K

+

BA6Li (H2)3 +

BA6Na (H2)6

BE

BE/NH2

22.35

–

15.28

–

10.98

–

11.22

3.73

10.06

1.68

+

BA6K (H2)6

4.38

0.73

+

4.96

0.62

31.78

3.18

23.67

1.97

BA6(K (H2)6)2

11.12

0.93

+

50.24

3.35

37.15

2.06

17.46

0.97

47.92

2.40

BA6K (H2)8 +

BA6(Li (H2)5)2 +

BA6(Na (H2)6)2 +

BA6(Li (H2)5)3 +

BA6(Na (H2)6)3 +

BA6(K (H2)6)3 +

BA6(Na (H2)5)4

Taken from reference M. Prakash, M. Elango, V. Subramanian, Adsorption of hydrogen molecules on the alkali metal ion decorated boric acid clusters: a density functional theory investigation, Int. J. Hydrogen Energy 36 (6) (2011) 3922–3931.

complexes, the Cub-Mg2+ complex has been taken as the linker for modifying MOP-9. The modified MOP is denoted as Cub-MOP-9 (M1-MOP-9). The crystal structure of MOP-9 is depicted in Fig. 10. The optimized structures of hydrogenated MOP-9 and M1-MOP-9 are shown in Fig. 11. To calculate the H-storage capacity of the modified MOP-9, the M05-2X method was employed using the LANL2DZ basis set for Cu atom and the 6-311 + G** basis set for remaining atoms. The calculated BE and BE/H2 molecules are listed in Table 12. FIG. 10 Crystal structure of MOP-9. (Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.)

228 Atomic clusters with unusual structure, bonding and reactivity

FIG. 11 Optimized geometries of H2@MOP-9 and linker modified MOP-9 using M052X/6-311 + G** (and LANL2DZ for copper atom) level of theory. (Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.)

TABLE 12 Calculated BE of modified MOP-9 with various linker molecules at M05-2X//(LANL2DZ effective core potential for Cu; 6-311 + G** for remaining atoms) level of theory. nH2 molecules Complexes

Connector

Linker

BE (kcal/mol)

BE/H2 (kcal/mol)

MOP-9

20

2

27.41

1.25

M1-MOP-9

20

5

83.02

3.32

BE: over linker

–

5

49.99

9.99

Taken from reference K. Gopalsamy, V. Subramanian, Hydrogen storage capacity of alkali and alkaline earth metal ions doped carbon based materials: a DFT study, Int. J. Hydrogen Energy 39 (6) (2014) 2549–2559.

Three different sites are available for effective adsorption of H2 molecules in the MOP-9 and M1-MOP-9. They are (i) oxygen atoms (ii) the copper present in the connector and (iii) metal-doped linkers. The calculated BE of MOP-9 with 22H2 molecules (20H2 molecules on the surface of connector and 2H2 on the surface of linker) is 27.41 kcal/mol. The calculated BE/H2 molecule is 1.25 kcal/mol with the gravimetric density of 6.0 wt%. The calculated BE of M1-MOP-9 with 25H2 molecules (20H2 molecules on the connector part and 5H2 on the linker part) is 83.02 kcal/mol which accounts for 3.32 kcal/mol for BE/H2 molecule. In order to understand the importance of the linker, the contribution to the BE from the linker region (hydrogenated CubMg2+) was alone calculated. The contribution to the total BE from the linker is found to be 49.99 kcal/mol which leads to 9.99 kcal/mol for BE/H2 molecule. This result undoubtedly shows that Cub-Mg2+ complex behaves as a very good linker in MOP-9 and increases its H-storage capacity. The calculated gravimetric density value is 6.3 wt%. From the findings, it observes that the modification of the linker enhances the H-storage capacity of MOP-9.

2.4.2 Icosahedral (Ih) cages incorporated MOF-5 It is clear from the results that calculations on closo-boranes and alanes, the carborane has a reasonably strong affinity towards the H2. Hence new MOFs have been designed by using carborane as a linker. Previously, Kumar et al. have illustrated that the enhancement in the H-storage capacity of MOF-5 takes place upon replacement of benzene with benzimidazole [87]. A similar strategy is adopted to enhance the hydrogen uptake of MOF-5 by replacing benzene (i.e., organic linker) with carborane. The modified MOF-5 is designated as MOF-5BCC. To estimate the H-storage capacity of new MOF-5BCC, 42H2 molecules were allowed to interact in all possible adsorption sites. The optimized geometry of

Studies on hydrogen storage Chapter

12

229

FIG. 12 Optimized geometry of MOF-5 BCCH2 complex as obtained from PBE method employing plane wave basis set.

H

B

C

O

Zn

MOF-5BCC is shown in Fig. 12. It can be seen that 16H2 molecules interact with inorganic connectors and 10H2 molecules are adsorbed on the linker site. ˚ . The same distance for The distance between two H2 molecules after adsorption onto the MOF-5BCC is around 3.914 A ˚ MO3 and OM4 sites is 2.939 and 4.375 A, respectively. Comparison of the results obtained from free carborane and carborane in the framework structure shows that there are only marginal changes in the intermolecular distances. The calculated BE at M05-2X/6-311++G** level of theory for MOF-5BCC with 42H2 molecules is 30.11 kcal/mol. The hydrogen adsorption capacity of a full unit cell of the MOF-5BCC was calculated. Based on the number of hydrogens adsorbed at the different sites (8  16H2 for the inorganic connector part and 12  10H2 for the carborane part), the estimated H-storage capacity of MOF-BCC is 6.4 wt%. It can be seen that the H-storage capacity of the new MOF-5BCC is marginally higher than that of MOF-5 (5.5 wt% at 77 K and 50 bar) [88]. It is evident from the results (Table 13) that the modified system exhibits an increase in BE of 0.72 kcal/mol per H2 when compared to MOF-5 and MOF-5 M. In this study, the H-storage capacity of various boranes and alanes has been calculated using the M05-2X method. The effect of substitution of antipodal atoms in cages by C, Si, and N on their H-storage capacity has also been investigated. Results point out that borane cages exhibit a maximum H-storage capacity of
22.0 wt% when compared to all other cages considered in this study. Among various alanes, AlCC has a maximum H-storage capacity of 11.6 wt%. Based on these findings, a new MOF with carborane as a linker has been designed. The DFT calculations on the new MOF-5BCC reveals that it has bulk hydrogen adsorption potential of 6.4 wt%. An increase in the hydrogen uptake can be achieved by introducing carborane as a linker in the MOF materials.

2.4.3 Boric acid cluster (BA20) incorporated fullerene-based material Extending the research from nanotubes to fullerenes, several investigators have evaluated the endohedral and exohedral H-storage ability of fullerenes (C60) using various experimental and theoretical methods [52–55,57]. Previous DFT-based calculations showed an increase in H-storage ability by 8 wt% when fullerenes (Cn (20  n  82)) are either positively or TABLE 13 Calculated BEs of H2 with MOF-5, MOF-5 M, and MOF-5BCC at M052X//(LANL2DZ for Zn; 6-311 ++G** for remaining atoms) level of theory. nH2-adsorbed Complex

BE (kcal/mol)

Connector

Linker

24.81

32

2

MOF-5M

27.03

32

2

MOF-5BCC

30.11

32

10

a

MOF-5

a

a

Values are taken from Ref. [87].

230 Atomic clusters with unusual structure, bonding and reactivity

FIG. 13 Endohedral adsorption of H2 molecules at BA20 fullerene-based nanoclusters were optimized at B3LYP/6-31 + G* method. (White color denotes the h and cyan color indicates the same H2 at center of the BA20.)

negatively charged [59]. The Remarkable increase in H-storage capacity has been observed when fullerenes are doped with metal and metal ions [60–62,89]. Chandrakumar et al. have shown that doping of alkali metal atoms on fullerene remarkably enhances the molecular hydrogen adsorption capacity, the predicted H2 adsorption capacity of exohedral decorated Na-doped fullerene being
9.5 wt% [60]. The H2 adsorption capacity of exohedral approach is verified using BA20 clusters. The optimized geometries of both ˚ . It is found from bare and hydrogen loaded BA20 clusters are shown in Fig. 13 along with the important distances in A ˚ , which is a significantly larger void space than the DFT-B3LYP geometries the diameter of the BA20 cluster is
13 A ˚ ). It means our 3D nanoclusters can accommodate more H2 molecules than the well-known analogs fullerene C60 (
7 A fullerene-based material through the endohedral approach. The stepwise addition of H2 molecules in the pore can accommodate more gases depending on the orientation of the H2 and neighbor molecules. The calculated HdH bond distance ˚ ) suggests that there is no interfacial interaction between gas molecules and the BA20 inner surface. Further, cluster (
0.74 A ˚, analysis reveals that the intermolecular interaction between individual hydrogen is significantly varied from
2.3 to 3.3 A which depends on the formation of H2 cluster size in the center of the BA20. The calculated BSSE corrected BE and BE/nH2 molecules are provided in Table 14. It is interesting to note that BA20 nanocluster can accommodate maximum of 18 H2 molecules with stable form. This is the first report on the H-storage of nH2 molecules inside the fullerene-based BA20 clusters (i.e., vacant site). The cavity of nanoclusters can be utilized for potential H-storage applications, but the adsorption capacity is very low (
2.5 wt%) when compared to other commercial materials such as metal hydride, carbonaceous materials and nanoporous materials. Thus the investigation of the exohedral adsorption process is more viable. When alkali metal ions are decorated at the surface of the nanoclusters or materials the resulting hydrogen density is enhanced. The splendor of our BA20 clusters is highly symmetric (Ih) and spherical in nature. This can be utilized for the decoration of alkali metal ions on the surface of the fullerene-based BA20 materials which is analogs to C60. From the above findings, we proposed that BA20 nanoclusters can accommodate a maximum of 12 metal ions at surface. The optimized geometries of BA20[M+(H2)6]N and partially optimized geometries of BA20[M+(H2)6]8 clusters (where M ¼ Na; N ¼ 1–8) are presented in Fig. 14. The BE and range of OBAM+ distance for all clusters is listed in Table 15. The ˚ . The charge on the Na+ in various clusters ranges from distance between oxygen and metal ions varies from 2.406 to 3.942 A 0.739 to 0.758 a.u. The minimum and maximum H-storage capacities of these clusters are 6 and 48 H2 molecules,

TABLE 14 Calculated BE (in kcal/mol) of nH2 molecules storage at BA20 fullerene clusters (i.e., endohedral) using B3LYP/6–31 + G* level of theory. BEs (in kcal/mol) endohedral storage of H2 molecules@BA20 nH2@BA20

Uncorrected BEs

BSSE corrected BEs

BE/nH2

1

0.89

0.54

0.54

2

2.69

1.89

0.95

3

4.79

3.57

1.19

4

6.84

5.21

1.30

5

9.32

7.26

1.45

6

11.5

8.96

1.49

7

14.17

11.17

1.60

8

16.51

13.09

1.64

9

18.92

15.03

1.67

10

20.17

15.79

1.58

11

22.53

17.71

1.61

12

25.33

20.06

1.67

13

27.81

22.02

1.69

14

28.11

21.74

1.55

15

28.9

21.9

1.46

16

28.74

21.27

1.33

17

28.37

20.37

1.20

18

14.87

6.15

0.34

FIG. 14 Optimized geometries of BA20[M+(H2)6]N (where M ¼ Na; N ¼ 1–8) clusters. Distances are in ˚ . (Taken from reference M. Prakash, M. Elango, A V. Subramanian, Adsorption of hydrogen molecules on the alkali metal ion decorated boric acid clusters: a density functional theory investigation, Int. J. Hydrogen Energy 36 (6) (2011) 3922–3931.)

232 Atomic clusters with unusual structure, bonding and reactivity

TABLE 15 Calculated BE (BE/NH2) and geometrical parameters of boric acid ball clusters (BA20[Na+(H2)6]N, where N 5 1–4, 6, and 8). BA20[Na+(H2)6]N, where N 5 1–8

BE (kcal/ mol)

BE/NH2 (kcal/mol)

Charge on metal ion (in a.u.)

Range of distances in OBANa+

1

11.29

1.88

0.739

2.406

–

2

23.70

1.97

0.745–0.747

2.463

2.483

3

38.61

2.15

0.740–0.747

2.495

2.517

4

53.95

2.25

0.750–0.753

2.592

2.616

6

83.36

2.32

0.743–0.753

2.648

2.598

8

125.86

2.62

0.742–0.758

3.312

3.942

Taken from reference M. Prakash, M. Elango, V. Subramanian, Adsorption of hydrogen molecules on the alkali metal ion decorated boric acid clusters: a density functional theory investigation, Int. J. Hydrogen Energy 36 (6) (2011) 3922–3931.

respectively. It can be seen from the calculations that 6.4 wt% of H2 can be stored in eight Na+ decorated BA20 clusters. Further, Table 15 shows the model-building studies of 72 H2 molecules that can be stored on the BA20 ball surface decorated with 12 Na ions [64]. The H2 adsorption capacity of bowl-shaped cluster [BA5[Na+(H2)6]3] is 8.8 wt% which is equal to that expected for a ball cluster decorated with 12 Na ions (BA20[Na+(H2)6]12). From the observations, it concludes that the H2 storage capacity of Na+ decorated sheet structure is 8.0 wt%, which is relatively higher than the US DOE requirement.

3.

Conclusions

To summarize, we have provided H-storage studies on series of molecules, cages, and clusters and these models can be transformed into 3D bulk materials for the potential H-storage applications. The adsorption and desorption properties of the materials depend on the nature of the binding mode of H2 molecules on the surface (i.e., adsorbent) and the charge density of the complex. It is interesting to note that, alkali metal decorated molecules, clusters and materials exhibit promising hydrogen adsorption capacity. Calculation suggested that this bottom-up approach would give guidance to experimentalists to identify the suitable materials for enhanced H-storage capacity.

Acknowledgments This work was supported by DST India-European Union sponsored project (HYPOMAP 8/233482/2008) and Council of Scientific and Industrial Research (CSIR), New Delhi, India. Also, M.P. thanks the Department of Science and Technology-Science and Engineering Research Board (DST-SERB) of India for the financial support (Grant number: ECR/2017/000891). The authors also thank SRM Supercomputer Centre (HPCC), SRM Institute of Science and Technology for providing the computational facility and financial support.

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Chapter 13

A density functional theory study of H3+ and Li3+ clusters: Similar structures with different bonding, aromaticity, and reactivity properties Dongbo Zhaoa, Xin Heb, Meng Lib, Chunna Guob, Chunying Rongb, Pratim Kumar Chattarajc, and Shubin Liud,e a

Institute of Biomedical Research, Yunnan University, Kunming, Yunnan, PR China, b Key Laboratory of Chemical Biology and Traditional Chinese

Medicine Research (Ministry of Education of China), Hunan Normal University, Changsha, Hunan, PR China, c Department of Chemistry, Indian Institute of Technology, Kharagpur, India, d Research Computing Center, University of North Carolina, Chapel Hill, NC, United States, e Department of Chemistry, University of North Carolina, Chapel Hill, NC, United States

1. Introduction Since its first experimental detection by Thompson [1,2], the triatomic hydrogen molecular ion H3+ has become the target of both intense experimental and theoretical investigations [3]. Gaillard et al. [4] made use of a new technique—the foilinduced dissociation of a fast molecular-ion beam and confirmed that H3+ assumes an equilateral-triangle structure (Fig. 1A). The most probable length of the triangle was determined by three measurements to be 0.97
0.03, ˚ , respectively. Oka later observed the infrared (IR) spectrum of H3+ [5]. This seemingly coun0.95
0.06, and 1.2
0.2 A terintuitively phenomenon of vibrational spectra is ascribed to the fact that though H3+ lacks a permanent dipole moment, the predicted transitions gain intensity due to distortions of the molecule when it rotates; these distortions lead to an instantaneous dipole. In addition, as the electronically simplest stable polyatomic cluster, H3+ is a grand challenge for theory dating back to the old times of quantum mechanics when Bohr first attempted quantum calculations and concluded that H3+ was unstable, while the neutral H3 species was stable. Furthermore, in the literature, aromaticity of H3+ has aroused great interests. Havenith et al. [6] have dissected the ring-current maps and verified that H3+ is s-aromatic. A large negative nucleus-independent chemical shift (NICS) also supports the aromaticity propensity of H3+. Similarly, all-metal triatomic lithium ion cluster, Li3+, has also attracted much attention due to its structural simplicity. Alexandrova et al. [7] found that the equilateral triangular geometry of Li3+ (Fig. 1B) is a global minimum on the potential energy surface. They also concluded that Li3+ should be considered s-aromatic, based on energetics evaluated from a series of homodesmotic reactions, on orbital topology, and on simple electron count. However, this point is negated by Havenith et al. [6] despite its 4n + 2 electron count and negative NICS value. In addition, Li3+ has some potentials for applications, such as trapping noble gases (He-Kr) [8] and molecular nitrogen (N2) [9]. In this contribution, we will revisit bonding, aromaticity, and reactivity properties for these two clusters using analytical tools from density functional theory (DFT) and conceptual density functional theory (CDFT) [10]. They include a few wellestablished tools [such as NICS index for aromaticity, “spike” region for noncovalent interactions (NCIs), etc.] and our newly developed analytical tools in analyzing strong covalent interactions and reactivity patterns in terms of local temperatures, which is intrinsically different to its thermodynamic definition. Taken together, we believe that these results will deepen our understanding about these small clusters.

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00017-6 Copyright © 2023 Elsevier Inc. All rights reserved.

237

238 Atomic clusters with unusual structure, bonding and reactivity

(a)

(b)

FIG. 1 Geometrical structures of H3+ and Li3+ optimized at the B3LYP/aug-cc-pVTZ level of theory.

2.

Methodology

In CDFT [10–13], chemical potential (m) [14] is the first well-established global reactivity descriptor, which can be expressed as the first-order response property of the total energy (E) with respect to the electron density r(r) or alternatively the total number of electrons (N), at a given external potential n(r),

 dE ∂E ¼ ¼ w (1) m¼ drðrÞ u ∂N u In Eq. (1), w, the electronegativity (originating from the definition of Mulliken [15]), generalized by Iczkowski and Margrave [16], is normally obtained by frontier molecular orbital (HOMO and LUMO) energies (eH and eL in short), w ¼ m ¼ 

eH + eL 2

(2)

Another global reactivity descriptor in CDFT is chemical hardness
[17], which is closely related to molecular stability,

2  ∂ E ∂m
¼ ¼ (3) 2 ∂N ∂N u u which can be numerically evaluated through Koopman’s theorem [18],
¼ eL  eH :

(4)

Combining these two quantities as defined in Eqs. (1) and (3), electrophilicity index o [12,19–21] can be formulated as o≡

m2 2


(5)

which gauges the optimal capability of a system to accept maximal number of electrons. For the local descriptor, besides others well known in the literature such as Fukui function [22–24], we very recently evaluated the local temperature T(r) [25] of a molecule using the kinetic energy density (KED) [26–28], T ðrÞ ¼

2τðrÞ 3kB

(6)

where τ(r), r(r), and kB are the KED, electron density, and Boltzmann constant, respectively. Suffice to note that KED is yet not a uniquely defined quantity. Two well-appreciated definitions are the Hamiltonian KED [29] 1X ∗ τðrÞ ¼  f ðrÞr2 fi ðrÞ (7) i i 2 with ’i(r) as the occupied Kohn  Sham orbitals, and the Lagrangian KED [29] 1X ∗ 1X ∗ 1 τ ðrÞ ¼ f ð r Þrf ð r Þ ¼  f ðrÞr2 fi ðrÞ + r2 rðrÞ i i i i i 2 2 4

(8)

A density functional theory study of H3+ and Li3+ Chapter

13

239

where r2r(r) is the Laplacian of the electron density. Additionally, there exist numerous approximate forms of KEDs in the literature [30–35], among which, the Thomas-Fermi formula [36,37] (derived for the homogeneous electron gas) and the Weizs€acker KED [38] (exact for one- and two-electron systems) are the two most famous ones, as defined in Eqs. (9) and (10), respectively:  2=3 τTF ðrÞ ¼ 3=5 6p2 rðrÞ5=3 (9) τ W ðrÞ ¼

1 jrrðrÞj2 8 rðrÞ

(10)

Very recently, based on the early work by Ghosh and Parr [26,27], we proposed to employ the local temperature to appreciate chemical reactivity. In CDFT, it is well known that Fukui function, which is the derivative of the electron density with respect to the number of electrons at constant external potential, can determine the electrophilic or nucleophilic attacking site of a molecular system. Inspired by Fukui function, for nucleophilic attack, using the local temperature, we define
 ∂T ðrÞ + + y ðrÞ ¼ ¼ T N ðrÞ  T N+1 ðrÞ (11) ∂N uðrÞ and for electrophilic attack, we define






y ðrÞ ¼

∂T ðrÞ ∂N

 u ð rÞ

¼ T N1 ðrÞ  T N ðrÞ

(12)

where TN, TN+1, and TN–1 are local temperatures for the system with N, N + 1, and N  1 electrons, respectively, in the N electron (optimized) geometry. Condensed-to-atoms local temperatures were also defined. We have applied local temperatures to a few simple model systems and the results are in good agreement with those of Fukui function. An in-depth discussion of local temperatures can be found in our recent publication [25]. We here just point out that the descriptor of local temperature is unique in several ways, compared to Fukui function. Its connection to the information theory greatly expands its territory of applications. Plus, the local temperature can be used to explore bonding (through a relevant quantum topology analysis), especially for weakly bonded systems as it is based on KED. Of such examples are the electron localization function (ELF) [39,40], which can be used to identify the localization of electron pairs, and our recently proposed SCI (strong covalent interaction) index [41], which is useful in determining multiple bond orders, from double and triple to quadruple and quintuple bond. Suffice to note that CDFT is still an active field with much progress on reactivity parameters [42,43]. Next, we will give a brief introduction of ELF and SCI. Earlier, we assumed that the areas forming multiple covalent bonds should experience strong repulsions due to the existence of the Pauli exclusion principle, and proposed the SCI (strong covalent interaction) index to identify multiple covalent bonds [41]. The Pauli energy (EP) [44] is defined as ð ð EP ½rðrÞ ≡ tP ðrÞdr ¼ TS ½rðrÞ  TW ½rðrÞ ≡ ðtS ðrÞ  tW ðrÞÞdr (13) ð 1 jrrðrÞj2 TW ½rðrÞ ≡ tW ðrÞdr ¼ dr 8 rðrÞ ð

(14)

with the Weizs€acker kinetic energy TW as an integration of Eq. (10), TS as the total noninteracting kinetic energy, and tS(r) and tP(r) the corresponding local energy density. To convert the Pauli energy to a dimensionless quantity, we define a local function z(r) [41], zðrÞ≡

t P ðrÞ t ðrÞ  tW ðrÞ ¼ S tTF ðrÞ tTF ðrÞ

(15)

and then the SCI index is defined as the reciprocal of z(r) [41], 1 zð r Þ

(16)

1 1 + z2 ðrÞ

(17)

SCI ¼ This index is similar to the ELF index [39,40], ELF ¼

240 Atomic clusters with unusual structure, bonding and reactivity

It is worthwhile to mention that the SCI index has its unique features compared with ELF, which was defined to identify regions where electrons are localized while localization orbital locator can do the same thing. However, the SCI index has a clear physical meaning of the Pauli energy contribution. We have revealed that our SCI results are consistent as they yielded the same signature isosurface for the same covalent bond type in various organic/inorganic systems [41]. To be specific, a double covalent bond assumes a dumbbell signature isosurface, whereas a triple covalent bond, the isosurface is like a donut or torus [41]. For new systems with unknown bond types, one can make use of these signature isosurfaces and predict their covalent bond multiplicity. More recent progress along this line of the SCI index can be found elsewhere [45]. To assess aromaticity characteristics of a circular planar structure, there have been tremendous studies in the literature [46]. In this work, we apply NICS [47] and gauge-including magnetically induced currents (GIMIC) [48] for the purpose. Our previous studies with them for other systems have demonstrated their effectiveness and reliability [49–55]. Finally, NCI [56] within H3+ and Li3+ are fully dissected. Details of their definitions and applications are available elsewhere [56].

3.

Results and discussion

We first benchmarked the reliability of a combination of density functional (B3LYP) and basis set (aug-cc-pVTZ) in predicting the geometries and vibrational frequencies of H3+ and Li3+ clusters, with the results of CCSD/aug-cc-pVTZ as the reference. Since the two systems have very few electrons and they are all correlated for advanced electron correlation methods, such as MP2 and CCSD. For CCSD, only numerical frequencies are obtained due to its lack of implementation in Gaussian 16 [45]. All quantum chemical calculations were obtained with the Gaussian 16 package [45] with default parameters, such as tight self-consistent field convergence criteria and ultrafine integration grids for DFT calculations. Collected in Table 1 are the results for bond lengths and harmonic vibrational frequencies of H3+ and Li3+ clusters based on several combinations of methods (B3LYP, MP2, and CCSD) and basis sets (6-311 + G(d), cc-pVTZ, and aug-cc-pVTZ). ˚ ), our predicted value of For H3+, whose experimental bond length is available (0.97
0.03, 0.95
0.06, and 1.2
0.2 A ˚ 0.900 A at the CCSD/aug-cc-pVTZ level is satisfactory considering that many other factors are not considered, such as relativity, temperature, incomplete basis set etc. It is clear from Table 1 that diffuse functions have little effect on the bond lengths of both H3+ and Li3+ compared with the results of CCSD. This is true for both B3LYP and MP2, with a deviation of ˚ for H3+ and  0.01 A ˚ for Li3+. In addition, H3+ has only two vibrational modes, a degenerate bending mode n2 and a 0.02 A symmetric stretching mode n1, which has no dipole associated with it and thus should be IR forbidden. The dipole allowed fundamental frequency for H3+ obtained at the CCSD/aug-cc-pVTZ is 2760.2 cm1 which is in good agreement with the experimental value (2725.9 cm1), with an overestimation of approximately 35 cm1. B3LYP and MP2 underestimate the experimental by about 33 and 120 cm1, respectively. While for the breath mode fundamental frequency, as this mode is Raman active, we find that both B3LYP and MP2 results are very close (3380 cm1), overestimating that of CCSD by 50 cm1. For Li3+, the dipole allowed fundamental frequency is predicted to be 230 cm1 at the CCSD/aug-cc-pVTZ level, in good agreement with that (226 cm1) from CISD (configuration interaction with singles and doubles)

TABLE 1 Benchmark results of equilateral triangle H3+ and Li3+ with different methodologies and basis sets.a IR active (cm21)

Bond length (A˚)

Raman active (cm21)

Methodology/basis set

H3+

Li3+

H3+

Li3+

H3+

Li3+

B3LYP/cc-pVTZ

0.880

2.949

2693.7

242.3

3380.1

308.3

B3LYP/aug-cc-pVTZ

0.880

2.949

2693.2

241.6

3379.0

308.6

MP2/cc-pVTZ

0.871

2.983

2813.0

242.6

3481.3

303.9

MP2/aug-cc-pVTZ

0.871

2.974

2810.3

262.9

3475.2

333.0

0.875

2.972

2763.5

231.0

3432.2

301.7

0.900

2.961

2760.2

230.8

3425.9

301.4

CCSD/cc-pVTZb b

CCSD/aug-cc-pVTZ a

All electrons are correlated for MP2 and CCSD. Numerical frequencies for CCSD are obtained.

b

A density functional theory study of H3+ and Li3+ Chapter

13

241

TABLE 2 Bond order results with Mayer, Fuzzy, Wiberg, Laplacian, and multicenter bond order analyses and intrinsic bond strength index (IBSI) for H3+ and Li3+. Molecule

Bond

Mayer

Fuzzy

Wiberg

Laplacian

IBSI

H3+

H1 ⋯H2

0.444

0.444

0.444

0.512

0.403

H1 ⋯H2 ⋯H3 Li3+

Li1 ⋯Li2 Li1 ⋯Li2 ⋯Li3

Multicenter

0.667 0.422

0.446

0.480

0.197

0.021 0.667

calculations with the frozen core technique. For B3LYP and MP2, the deviations are less than those for H3+. Taken together, B3LYP/aug-cc-pVTZ should be a good choice in balancing the accuracy and efficiency and it is henceforth used through the whole work unless otherwise stated. Next, let us take a look at the bonding of H3+ and Li3+. Shown in Table 2 are bond order results based on popular schemes in the literature, including Mayer, Fuzzy, Wiberg, Laplacian, and multicenter bond order analyses and intrinsic bond strength index (IBSI). To avoid numerical noises, cc-pVTZ is used instead of aug-cc-pVTZ. For H3+, HdH bond order results are consistent with each other for all schemes adopted while for Li3+, the severely underestimated values of LidLi bond order, resulted from the Laplacian (0.197) and IBSI (0.021) schemes, compared with the other three schemes (0.449 on average) are indicative of much more sensitivity to the electron density and its derivatives. Also, it may serve as a reflection of the intrinsic defects of the algorithm. Both for H3+ and Li3+, the multicenter bond analysis gives a value of 0.667, indicative of strong electron delocalization between the three centers. In Fig. 2, we plot both the NCI and SCI isosurfaces. In Fig. 2A, a low-gradient, low-density isosurface is not observed for H3+ while such a surface does exist for Li3+ as shown in Fig. 2B. Also a “spike” region is clearly shown in Fig. 2B and values of sign(l2)r(r) are near zero, indicative of very weak, van der Waals interactions. In addition, our SCI plots for H3+

FIG. 2 Upper panel: NCI plots of the reduced density gradient (RDG) versus the electron density (r) multiplied by the sign of the second Hessian eigenvalue (l2), and lower panel: SCI (strong covalent index) plots for H+3 (left) and Li+3 (right), respectively.

242 Atomic clusters with unusual structure, bonding and reactivity

TABLE 3 Interatomic distances (in A˚), NMR spin-spin coupling constants (in Hz), and NICS(0), NICS [1], and NICS(1)zz values (in ppm) of H3+ and Li3+. Molecule

Distance

H⋯ H

NICS(0)

NICS(1)

NICS(1)zz

H3+

0.882

188.2

33.7

2.2

7.9

Li3+

2.949

96.4

11.1

6.7

7.3

and Li3+ differ significantly from each other. ELF results are very similar to those of SCI and thus not shown. Taken together, both from NCI and SCI analyses, H3+ and Li3+ are different on the origin and nature of multicenter bonding. Next, we will delve into the aromaticity of the H3+ and Li3+ clusters. It is well-documented in the literature that negative NICS value indicates of aromaticity. Our results of NICS(0), NICS [1] and its component at the z-axis NICS(1)zz as collected in Table 3 clearly support this argument. However, from the NCI analysis, LidLi interactions are mainly van der Waals interactions. Accordingly, our results are in line with Havenith et al. [6] supporting that Li3+ is not s-aromatic. To verify this argument, we analyzed the GIMIC distributions of H3+ and Li3+ as shown in Fig. 3. From the top view, the counter-clockwise ring currents of both H3+ and Li3+ are a good indicator of their aromaticity property. However, when we are looking at the GIMIC maps from two side views, the scenario is not the same. Significant differences between the two clusters are observed. Finally, we switch our gear to the CDFT indices, such as electronegativity, hardness, and electrophilicity as collected in Table 4, and local temperatures as shown in Fig. 4. It is clear that from Table 4, both H3+ and Li3+ possess large chemical hardness, indicating of its high molecular stability, and small electrophilicity, suggestive of its lower reactivity. These results are in line with the minimum electrophilicity principle [57]. From Fig. 4, the reactivity scenarios differ for H3+ and Li3+, which are surprising. For Li3+, both nucleophilic and electrophilic attack sites (in dark blue—gray color in print version) are all located near the Li nucleus. This same scenario is observed for the nucleophilic attack of H3+, while for electrophilic attack, the most probable sites (in dark blue—gray color in print version) are within a sphere far from the three H nuclei. It is the first time that chemical reactivities of H3+ and Li3+ are analyzed in terms of local temperature.

FIG. 3 GIMIC distributions of H3+ and Li3+ from top and two side views.

TABLE 4 Conceptual DFT indices (in eV): electronegativity (x), hardness (h), and electrophilicty (v) values of H3+ and Li3+. Molecule

x

h

v

H3+

19.845

27.030

7.285

Li3+

7.327

6.404

4.192

A density functional theory study of H3+ and Li3+ Chapter

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FIG. 4 Local temperature distributions of H3+ and Li3+. Y ¼ T(N  1) T(N) and Y+ ¼ T(N) T(N +1).

4. Conclusions To summarize, we have revisited bonding, aromaticity, and reactivity properties of two small clusters, H3+ and Li3+, with a few well-established and recently developed theoretical tools, among which are the NCI, SCI index, NICS, GIMIC, and local temperature reactivity index. We have shown that though H3+ and Li3+ have a multicenter bonding, NCI analysis suggests that the nature is not the same. Li3+ are bound mainly because of weak stabilizing van der Waals interactions. H3+ are bonded via electron delocalization as evidenced by the SCI index diagrams. Further aromaticity results indicate that Li3+ is not s-aromatic, though its NICS values are negative. Our local temperature isosurfaces indicate that for both nucleophilic and electrophilic attacks, the sites are all located near the Li nucleus, while this is not the same for H3+. Our results reveal that for H3+ and Li3+ clusters, though simple and similar in structure, their physiochemical properties are different. We mention in passing that more work along this line is underway to further demonstrate the robustness and wide applicability of the theoretical tools we have developed and the results will be presented elsewhere.

Acknowledgments DBZ was supported by the startup funding of Yunnan University. SBL would like to thank Prof. Pratim K. Chattaraj, Prof. Gabriel Merino, and Dr. Sudip Pan for kindly inviting him to contribute this book chapter. CR acknowledges support from the National Natural Science Foundation of China (No. 21503076). PKC would like to thank DST, New Delhi, India for the J. C. Bose National Fellowship, grant number SR/S2/ JCB-09/2009.

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Chapter 14

Designing nanoclusters for catalytic activation of small molecules: A theoretical endeavor Anup Pramanika, Sourav Ghoshalb, and Pranab Sarkarb a

Department of Chemistry, Sidho-Kanho-Birsha University, Purulia, India, b Department of Chemistry, Visva-Bharati University, Santiniketan, India

1. Introduction Controlling chemical reactions has always been the central task of chemical community. Homogeneous and heterogeneous catalyses play a significant role there. Both experimental and theoretical chemists contribute to this interdisciplinary area by designing functional materials and proving concrete molecular-level understanding [1–5]. Computational chemistry, in particular, has a significant role in designing materials for catalysis as well as providing insights into the mechanistic pathways, which in turn facilitate developing newer and advanced catalytic systems [6–9]. Modern sophisticated electronic structure theory can provide almost accurate material properties, which are the key factors for controlling a chemical reaction. Nanomaterials, especially, have always been closely associated with catalysis because of their tunable optoelectronic properties correlated to their sizes, shapes, core-shell structures, surface functionalizations, and many other parameters. Although difficult, computing material properties of nanomaterials are now possible with the advent of modern sophisticated techniques which compromise well between accuracy and computing time. Tight-binding-based density functional theory (DFTB) is one of such techniques which can provide reasonable structure-property relationship for larger nanoclusters [10–19]. Now, obtaining minimum energy structure of nanoclusters is also a major concern, relating to their properties. The search for the global minimum structure is in fact very difficult because of the involvement of a large number of degrees of freedom in nanocluster geometry, which in turn provide thousands of local minima. People have developed a number of ways to achieve those structures by using basin hopping and genetic algorithm techniques [20, 21]. Our group has developed genetic algorithm (GA)-based density-functional tight-binding method for searching the lowest-energy structures of nanoclusters of different shapes and sizes [22]. Knight et al. [23] developed homogeneous electron gas model or most popularly known as “jellium model,” which is based on spherical one-electron potential for determining the electronic structure of nanoclusters. Further modifications have been done in “structural jellium model” and “homogeneous jellium model,” which use crystal-field perturbation and average density of valence electrons, respectively, for calculating the electronic structure [24, 25]. These methods are shown to be excellent for determining the characteristic features of small metal clusters including magic sizes [26]. Nevertheless, accurate determination of global minimum structure is still a challenging task [27]. Accurate ab initio calculations provide trustworthy and more realistic results but they become rapidly prohibitive with increasing the cluster size [28]. However, in general, DFT is a good compromise between accuracy and computational cost, while studying the geometry and electronic structure of nanoclusters for catalytic activities [29–31]. During the last few decades, metallic nanoclusters have drawn a considerable attention for activating small molecules, owing to their peculiar optoelectronic properties arising from their finite size- and shape-dependent electronic properties. Such nanoclusters also exhibit high thermodynamic stability along with their different magnetic ground states [32–35]. Now, what makes nanoclusters so different? There are two main reasons behind this. First, surface free energy per atom of the nanoclusters is very high and consequently they can easily accept or donate pair of electrons to the adjacent molecules; second, the electronic bandgap of the nanoclusters can easily be monitored by changing the cluster size, a direct consequence of quantum confinement effect. Theoretical modeling and computation, in this particular field of research, has a profound role, which can interpret the catalytic activity by providing concrete reaction mechanism and

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00004-8 Copyright © 2023 Elsevier Inc. All rights reserved.

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possible intermediates and/or transition states. At the same time, it helps to design further advanced catalytic systems by identifying the bottleneck of the overall reaction scheme. Johnson et al. [36] have nicely demonstrated the importance of theoretical model for investigating nanoscale catalysis. It has been revealed that the reactivity of nanoclusters is very much dependent on the adsorption sites and hence on the structure of the cluster model. Here, the reactivity mainly varies due to an irregular charge distribution on the cluster surface. From practical seance, this is, of course, the matter of surface roughness. Therefore, tuning the cluster size and shape is very important for designing the suitable nanocluster for nanoscale catalysis. Nanoclusters have been widely used for activating C–H, N¼N, O¼O, C¼O, and O–H bond activation. Few review articles have also already appeared [37–40] but concentrating them on in silico design is really scattered. This chapter focuses on the computer-simulated nanoclusters, which play an essential role for developing catalytic materials for small molecule activation. Emphasis is given on the main group elements, especially aluminum and related materials and in some cases some transition and noble metal-based nanoclusters.

2.

N2 activation

Among different nontransitional metal clusters, aluminum and aluminum-based nanoclusters, such as aluminum nitrides, aluminum oxides, etc., have attracted tremendous impact, especially in the field of nanocatalysis [32, 41–46]. Small-sized Aln nanoclusters (n ¼ 16–18) have been investigated to show hydrogen evolution from water [47]. Even, some charged and neutral aluminum nanoclusters are reported to activate molecular hydrogen by means of dissociative chemisorption [48, 49]. Both experimental and theoretical studies have revealed that small-sized aluminum clusters, Al+7 and Al13 are stable as their valence electronic configurations approach closed-shell magic configurations [50–52]. Furthermore, the adsorption of small molecules such as H2, O2, H2O, etc. on quasistable clusters is also possible by redistributing valence electronic configuration in the cluster-molecule composite [53–55]. A number of theoretical and experimental researches have been carried out for understanding the adsorption behavior of such small molecules on different neutral and charged Aln clusters [47, 56–59]. Researches have also been carried out for understanding the adsorption behavior of N2 molecules on different Al nanoclusters. This is fundamentally important since the activation of N2 is really a challenge task because of its very high bond enthalpy and second, aluminum nitride is industrially very important material, especially in the field of electronic industry. So, studying N2 activation over aluminum and identifying optimum catalytic condition is of paramount importance. In order to understand the nature of interaction between Al and N, it requires to study the general properties of AlN small clusters. A partial reduction of the N–N bond is observed in very small clusters like (AlN)2, where the N atom is not in the 3 oxidation state, rather it involves in the formation of anionic species N2 2 [60, 61]. Combined photoelectron spectroscopy and ab initio calculations demonstrate that it forms an octahedral geometry around the N atoms by coordinating Al in different anionic clusters such as Al6N, Al7N, etc. So, N2 is definitely chemisorbed inside the small Al clusters [62, 63]. However, N2 is preferentially situated at the peripheral positions of large Al clusters forming bulk aluminum nitride like four Al–N bond as reported by Bai et al. [64]. On the basis of theoretical study, Romanowski et al. demonstrated that the interaction of N2 with liquid Al metal depends upon the reaction temperature [65]. At low temperature, N2 is just physisorbed onto the Al surface, while at high temperature, it is evidenced that N2 is chemisorbed over the surface [66, 67]. Kulkarni et al. [32] explored the reactivity of Aln nanoclusters, both in the ground and excited states, toward nitrogen molecule on the basis of DFT calculations. The authors demonstrate that the reactivity of aluminum nanoclusters exhibits size sensitivity. The cluster molecule interaction energy is nearly constant irrespective of the N2 adsorption mode at the ground state. However, in the excited state, the interaction energy depends on the shape and orientation of the adsorbate molecule. Optimized geometry and energy profile diagram for activating N2 over different Aln nanoclusters in ground and excited states are shown in Fig. 1. Fixation of N2, that is, reducing it to nitrogenous species is very crucial for the origin of life on Earth [68, 69]. Although molecular nitrogen is the most abandoned species in the atmosphere (>78%), activating it is an extremely difficult task because of its chemical inertness. As already discussed, N–N bond is triple bond in character with a large bond energy (225 kcal mol1) [70] and the species possesses a large HOMO-LUMO gap (10.82 eV) and high ionization energy (15.58 eV) [71]. These make the nonpolar N2 molecule reluctant for donating or accepting electrons, rendering it almost inactive. Thanks to different catalysts which play significant roles for activating dinitrogen molecule [72, 73]. Different transition metals (TMs), in the form of nanocluster or in the complex form, show an excellent catalytic activity for activating and reducing dinitrogen molecules [73–76]. Simultaneous presence of both occupied and vacant d-orbitals in TMs allows them for synergistic electron donation and acceptance with the N2 molecule thereby weakening the N≡N triple bond [73]. Iron and molybdenum sites in naturally occurring nitrogenase enzyme are primarily responsible for biological N2 fixation [77–81]. However, artificial N2 fixation has a great industrial importance for the manufacturing of ammonia

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FIG. 1 (A) Ground-state geometries of Aln (n ¼ 13, 30) nanoclusters and their corresponding N2 complexes. (B) High-energy conformer geometries of Aln (n ¼ 13) nanoclusters and their corresponding N2 complexes. (C) Activation barrier of N2 on various Al13 conformations [32].

(NH3), which is essential for the production of fertilizer, fibers, and energy storage materials [82–84]. Noteworthy, artificial N2 fixation was first made by Haber-Bosch (HB) process, which uses iron-based catalyst [85–88]. Although HB process is the mostly used commercial method for synthesizing NH3, it also has some limitations like using high temperature (350–550°C) and pressure (150–350 atm) [89]. Besides these, it requires H2, obtained from fossil fuels thereby generating huge amount of greenhouse gas, CO2 [90]. Therefore, people are searching for alternative catalytic system based on metals and nonmetals. During the last few decades, enormous efforts are being paid for optimizing thermal [91, 92], photocatalytic [93–97], and electrocatalytic [98–100] reduction of N2 into NH3. Metal clusters in the subnanometer regime offer promise for N2 activation due to their reported higher activity. Various inorganic metal clusters have been explored both from experimental and theoretical researcher group for activating N2 molecule [32, 65, 67, 101, 102]. Alkali metal clusters, Lin (2 < n < 8) are shown to activate N2 by red shifting the N–N bond stretching frequency up to 810 cm1 [102]. It has been revealed that even a small cluster-like Li8 has the potentiality to split the N–N bond completely in an exothermic process. Another light-metallic nanocluster, Aln has also been reported for the activation and fixation of N2 to NH3 [32, 67, 103]. As already mentioned, Pal and coworkers demonstrated that it is also possible to activate N2 on excited Aln nanoclusters of different sizes [32]. As excited-state conformations are metastable in nature and their electronic structures are different, a more general strategy is to develop doped or supported nanoclusters for the activation purpose. Surprisingly, silicon and phosphorus-doped Aln nanoclusters are also shown to activate N2 [104]. The electronically modified nanoclusters, by adsorbing the catalytically active Aln nanoclusters on

250 Atomic clusters with unusual structure, bonding and reactivity

FIG. 2 N2 activation on BN-doped graphene surface supported Al48 nanoclusters. The corresponding charge density differences (CDD) are also shown in right panel (blue and red colors (please see online version) signify charge accumulation and depletion, respectively). The amount of charge transfer from the BN-doped graphene sheet supported aluminum nanocluster to N2 molecule increases with increasing the cluster size [105].

graphene and BN-doped graphene sheets, were also achieved by the same group [105]. The supported surface has been shown to be an excellent way to both stabilizing and enhancing the catalytic activity of the Aln metallic nanoclusters. It has been revealed that the key step for weakening the N–N bond is the charge-transfer interaction between the adsorbate N2 molecule and the supported Aln clusters. Higher the charge transfer from the nanocluster to the N2 molecule, stronger is the adsorption and weakening of N–N bond as demonstrated in Fig. 2. Among the TMs, supported ruthenium is found to be the best alternative to the conventional HB catalyst in industrial NH3 synthesis [106]. Nevertheless, single Ru metal in different forms has been well reported for activating N2 [101, 107, 108]. It has been revealed that N2 dissociates preferentially over some special B5 centers of the Ru metal surface in the ratedetermining step [109]. Doped Ru surfaces also show effectiveness for activating N2 [95, 110–113]. Other than surfaces, activation of N2 has also been performed over neutral Ru clusters in the gas phase by Kerpal et al. [101]. DFT calculations demonstrate that the interfacial electron transfer between Ru cluster and TiO2 significantly increases the catalytic activity of Ru [114]. Ranjit et al. reported that Ru, among the four noble metals (Ru, Rh, Pd, and Pt), exhibits the highest N2 photoreduction capability embedded onto TiO2 surface [112]. In a very recent study, Ghoshal et al. [115] have studied the possible reduction pathways of N2 over bare and TiO2-doped Run cluster. The ruthenium pentamer (Ru5), which is reported as the magic cluster [116], has a square pyramidal

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FIG. 3 Mechanistic pathways for the reduction of N2 on Ru5-TiO2 nanocluster [115].

arrangement in the global minimum conformation with singlet ground-state spin multiplicity [117–119]. On the other hand, the global minimum structure of Ru6 is trigonal prism, having quintet spin multiplicity [117]. The study of Ghoshal et al. [115] demonstrates that simultaneous activation of N2 and H2 and thereafter proton transfer from activated dihydrogen requires a huge energy barrier. However, due to strong hydroscopic nature of both Run (n ¼ 5, 6) clusters and TiO2, the adsorbed water molecule participates in proton transfer reaction thereby forming different hydrogenated products, including diazene (N2H2), hydrazine (N2H4), and ammonia (NH3). Noteworthy, water-driven NH3 synthesis is assigned as thermodynamically nonspontaneous process [100]. On the other hand, bare TiO2 (110) surface with abundant oxygen vacancies as well as single Ru atom decorated TiO2 nanosheet exhibit N2 photofixation [93, 95, 96, 120]. However, thermal conversion of N2 into NH3 in association with water splitting over Ru cluster is unsuitable because the metal cluster reforms into oxide cluster by capturing oxygen from water molecule [121]. Furthermore, more water adsorption on Ru-center yields complicated ruthenium oxide nanoclusters from which oxygen liberation is really difficult [122]. So, the water-catalyzed N2 fixation is of fundamental importance. On the basis of DFT calculations, Ghoshal et al. [115] demonstrate that partial reduction of N2 into NH3 or N2H4 could be achieved by two H2O oxidation reactions, while additional H2 is necessary for making the Run cluster as oxide free. The mechanistic study reveals that two H2O oxidation over Ru5-TiO2 cluster can liberate 1 mol of NH3 from N2 via distal mechanism, while N2H4 could be formed as favorable product by alternating type of mechanism (Fig. 3). However, the enzymatic pathway is not effective to produce neither NH3 nor N2H4.

3. H2 activation Kumar et al. [123] explored the potentiality of Al nanoclusters, supported over defective graphene sheet, for dissociative adsorption of molecular hydrogen. The defect-induced graphene sheet was shown to be an excellent material for anchoring small-sized Aln nanoclusters with high adsorption energies, while the adsorption energy of a molecular hydrogen onto the supported Aln nanoclusters was low enough, rendering the excellent potentiality of the material for room temperature hydrogen desorption. This is an important step forward in fuel cell technology because storing gaseous hydrogen with moderate density is a challenging task. The US Department of Energy has proposed the following criteria for efficient hydrogen storage materials: (i) storing gravimetric density of at least 7.5%, (ii) volumetric density of hydrogen, 0.07 kg L1, and (iii) reversible adsorption and desorption at an ambient temperature and moderate pressure. Storing molecular hydrogen by porous materials is an attractive method because of their low density, large surface area, and maintaining reversibility

252 Atomic clusters with unusual structure, bonding and reactivity

of hydrogen adsorption and desorption equilibrium with high storage capacity at low temperature. However, the storage capacity of those materials is dramatically reduced at normal temperature and pressure [124, 125]. On the basis of thermodynamic calculations, it has been estimated that for an efficient cyclic adsorption/desorption operation at room temperature and moderate pressure, the adsorption energy should lie within the range of 0.2–0.6 eV per molecular hydrogen. The estimated energy is in between the typical physisorption (0.6 eV). Although hydrogen adsorption and storage on porous carbon substrates is a promising technology, they could not fulfill the targeted demands [126]. However, further developments are achieved by doping transition element like Pd in carbon substrates, which substantially enhance the storage capacity [127–129]. In the same way, metal clusters could also be doped in carbon hosts for enhancing the storage capacity [130, 131]. However, the major problem is that the metal atoms or clusters, deposited onto the carbon support, try to aggregate more via atomic migration due to stronger metal-metal interactions in comparison to that of metal-carbon interactions [132–134]. This results in reduction of the hydrogen storage capacity of such materials. People have tried to recover such problem by introducing defects in the carbon-based materials [135]. Defective graphene has been shown as an excellent support for various TMs [136–138] along with light metal like Al [139]. Since Al is high abundant and cost-effective metal, defective graphene-supported Aln nanoclusters are supposed to be very exciting for activating and storing molecular hydrogen. It has been revealed that Aln nanoclusters are not stable on the pristine graphene surface but they are stabilized over defective graphene surface and the resultant materials are very promising for hydrogen storage and management in fuel cell [123]. The optimized geometries and charge-transfer interactions between defective graphene-supported Aln nanoclusters and molecular hydrogen are shown in Fig. 4. The H2 molecules undergo dissociative adsorption on the supported Aln nanoclusters and as evidenced from the CDD plot, there involves electrostatic interactions between them, which arises as a result of charge transfer from the Aln nanocluster to the H2 molecule.

FIG. 4 Adsorption of H2 on defective graphene-supported Aln (n ¼ 4–8 and 13) nanoclusters. Charge density differences (CDD) of the H2 adsorbed complex nanostructures are shown in right panel (yellow and cyan colors (please see online version) represent charge accumulation and depletion, respectively) [123].

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4. Activation and reduction of CO2 The greenhouse effect, arising due to excessive consumption of fossil fuels and the destruction of forests, has now become a major threat to the human civilization. So, it is our essential duty to maintain the balance of the main greenhouse gas, CO2 for further sustaining the modern society. Over the last few decades, a number of technological developments have been performed, aiming to capture and/or convert CO2 into others. The technologies involved in such processes include electrochemical [140–144], thermochemical [145–148], photochemical [149, 150], and biochemical [151, 152] processes; among them, electrochemical reduction of CO2 is probably the most studied one and it is, of course, the most practical solution, which converts CO2 into valuable chemicals [140]. Electrochemical reduction of CO2 to various fuels such as HCHO, CH3OH, CH4, C2H4, etc. has been investigated both experimentally and theoretically using various TMs as electrode [140, 153–155]. The surface orientation of electrode material plays a significant role in the product selectivity [156]. Computational studies have been performed for investigating the electrochemical reduction mechanisms for the conversion of CO2 to CH4 and longer-chain hydrocarbons at Cu electrodes [154, 157, 158]. It has been shown that methane formation passes through the –CHO intermediate thereby scissoring of the C–O bond; on the other hand, ethylene is formed in a pathway, where the first step is the formation of a CO dimer, followed by the formation of a surface-bonded enediol or enediolate, or the formation of an oxametallacycle. Thus, the product selectivity depends on the catalytic surface orientation and the mode of adsorption of CO2. Of course, the electronic effect, that is, the type of metal must have crucial role. However, the major problem in electrochemical reduction of CO2 is that it requires a very high overpotential due to extraordinarily stability of C¼O bond (bond energy ¼ 806 kJ mol1). At the same time, the competing hydrogen evolution reaction (HER) in the electrolyte also imposes adverse effect in the Faradaic efficiency of the desired products. Apart from these, electrochemical CO2 reduction (ECR) suffers from low product selectivity, arising due to improper adsorption configurations on to the activating materials. A suitable catalyst can, however, solve these issues by suppressing the HER possibility, dictating proper adsorption sites, and minimizing the reaction overpotential. Over the years, people have searched for such a “suitable catalyst” under both homogeneous and heterogeneous conditions, which include noble metals, TMs, and even metal-free conditions. Of course, nanomaterials are superior in this aspect because of their higher surface areas, providing abundant active sites for CO2 adsorption. In fact, one can effectively improve the activity and selectivity on a catalyst by controlling the shape and size of the nanostructured materials. Batista et al. [159], on the basis of DFT calculations, investigated the adsorption properties and activation of CO2 on 13-atom transition-metal clusters (TM13, TM ¼ Ru, Rh, Pd, Ag), which is supposed to be the key step for the development of nanocatalysts CO2 hydrogenation. Among the TM clusters, Run clusters draw spatial attention because of their susceptibility to activating small molecules [116, 160–162]. Using ion-trap mass spectrometry in conjunction with first-principles DFT calculations, Lang et al. [160] identified three fundamental properties of monopositive Run clusters, which determine their selectivity and catalytic activity. First, the Ru+n clusters provide better reactivity toward CO in comparison to that of CO2. Interestingly, those clusters are not affected by CO adsorption, instead H2 coadsorption occurs cooperatively. Most importantly, surface Ru atoms, having lower number of Ru–Ru bonds, are found to be very active for H2 coadsorption. So, this provides a mechanistic aspect for the adsorption and hydrogenation of CO. As 2e transfer to the activated CO2 provides CO, the cooperative H2 adsorption on modified Run clusters might be utilized for complete hydrogenation of CO2. Structure and electronic properties of Run clusters have been widely studied by using first-principles calculations [118]. It has been revealed that the basic unit for the Run clusters is the Ru4 square unit. An even-odd oscillation is observed for the stability of the clusters with their sizes; however, the clusters comprising integral number of square units are highly stable and are known as magic clusters. Although the exact ground-state geometry and magnetic moment of the Run clusters are still debated issue, studying such clusters for ECR is an interesting field of research. The reactivity of the Run clusters could be enhanced by modifying the shape, size, and also by incorporating doping effect. It has been revealed that bimetallic Ru-based clusters exhibit a significant enhancement in the electronic and catalytic properties in comparison to that of the pure Run clusters [163, 164]. Rh doping plays a spatial significance here because Rh has substantial role in many chemical transformations. For instance, Rh clusters are shown to be very efficient in the decomposition of nitrous oxide [165, 166]. Furthermore, rhodium nanocatalysts show great chemical stability, high reactivity selectivity in various hydrogenation processes [167]. The magnetism of Rh nanoclusters is also of great concern; they are superparamagnetic at very low temperature [168]. So, studying Rh-doped Run nanoclusters of various sizes with the aim of their catalytic activities toward CO2 hydrogenation reactions is of paramount importance. Our group is presently involved in studying different catalytic routes for the hydrogenation of CO2 over pristine and Rh-doped subnanometer Run

254 Atomic clusters with unusual structure, bonding and reactivity

O

O C

O

H C

O

H H

(*CO2*H2)-I via CO2 dissociation

O C

O

H

H

*COOH*H

via carboxyl intermediate

O

H

C

O

O C

H

–H2O O

H

C

O

*CO*H2O

*CO RWGS pathway

H H

*CO*O*H2

*CO*OH*H H

C O

H

C

O

O

H C

O

O

O

H H

H

(*CO2*H2)-II

*HCOO*H

Formate pathway *H2COO

FIG. 5 Schematic representation showing possible pathways for hydrogenation of CO2.

clusters to form different stable products such as HCOOH, H2CO, and CH3OH either via reverse water gas synthesis (RWGS) or via HCOOH formation pathway as shown in Fig. 5.

4.1 Specific role of metal hydride for the reduction of CO2 While discussing the pathways of CO2 hydrogenation, unambiguously, metal hydrides are the key intermediates. In most of the cases, the reactions proceed through the formation of metal-formate anionic adducts. The metal hydrides may be bare, as in gas phase or they are ligand-passivated in solution phase. In any case, the critical reduction step involves the insertion of one of the C–O groups of CO2 into the M–H bond [169–172]. To unravel the mechanistic pathways, extensive theoretical and experimental studies have been carried out on the activation and reduction of CO2 in gas phase [169, 170]. Gas-phase infrared photodissociation studies on different M–CO2 anionic complexes reveal that both physisorbed and chemisorbed anionic complexes are involved in the rate-determining step, depending upon the nature of the metal center (M). By using anion photoelectron spectroscopy together with theoretical calculations, Zhang et al. [173] claimed that Cu and Ag exhibit chemisorption and physisorption, respectively, while Au shows both chemisorption and physisorption characteristics in the M–CO2 anionic complex. Noteworthy, the mode of activation and electron-donating capability of the metal hydride determines the product selectivity, that is, whether it will form HCO 2 or CO [169, 170]. Recently, high-level DFT calculations have been carried out to show unidirectional hydrogenation of CO2 by coinage metal hydride anions (MH, M ¼ Au, Ag, and Cu) [174]. It has been claimed that those metal hydrides are excellent for specific hydrogenation of CO2 to HCO2. Although initial activation proceeds through multiple pathways (attacking CO2 from both M and H sites), both of them lead to the same hydrogenated product, HCO2 as shown in Fig. 6. In this case, the product selectivity is determined by the electronegativity of the metals, vertical detachment energy, hydride donor ability, and most importantly, cooperative charge-transfer interactions of the metal hydrides. As a combined effect of these, the thermodynamic and kinetic control over CO2 reduction leads to the same product, HCO2. It has been shown that direct activation of CO2 through H–C bond formation is almost barrier less, which could form HCO2. On the other hand, the activation through M–C bond formation leads to the same hydrogenation product via a hydride insertion in between the M–C bond. The thermodynamic calculations reveal that the activation barrier of such process is low enough, especially for Ag–H, thus, making AgH as the most

Designing nanoclusters for catalytic activation of small molecules Chapter

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255

FIG. 6 The calculated reaction pathways for CO2 activation followed by complete hydrogenation by AgH. The horizontal line corresponds to the sum of energy of the reactants AgH and CO2, fixed at zero. The relative free energies (in kcal mol1) of all the intermediates, transition states and products are shown with respect to the reactants. Both Path A (activation by the metal ion, brown line followed by green line (please see online version)) and Path B (activation by the hydride ion, green line) lead to the complete hydrogenation of CO2. Ease of formation of the hydrogenated product HCO 2 is facile for path A. The overall reaction for the formation of HCO 2 is exergonic in nature. The calculations are performed by using the B3LYP/6-311G++(3df,2p)/ SDD level of theory [174].

effective species for the specific hydrogenation product. The overall hydride transfer reaction is also found to be exergonic in nature. The intermediate electronegativity of Ag, in comparison to that of Cu and Au, and its relatively higher atomic radius make AgH the most active species.

5. Activation of O2 and oxidation of CO on Aun nanoclusters In this section, we focus on the effectiveness of gold nanoclusters for different small molecules activation. Small gold nanoclusters have attracted significant research attention because of their unique cluster size effects on activating such small molecules [38, 175–177]. Although the noble metal gold is mostly known for its chemical inertness, the nanogolds, especially the small gold nanoclusters (constituting 3–10 Au atoms) are reported to be excellent catalysts [178, 179]. The origin of catalytic activity of Au is still a debated issue; however, very small gold nanoparticles can adsorb (chemisorb) reactant molecules, which are essential parameter in heterogeneous gold-based catalysis [180]. Although the overall interactions of Au with small molecules, such as CO, O2, H2, etc., are very weak, they are crucial for initiating the reactions. Moreover, it should be kept in mind that one essential aspect of catalysis is the formation of surface intermediate species of moderate stability [181]. As the catalytic activity of Au nanoclusters is dependent upon the shape and size of the clusters, it is essential to have concrete idea on the geometry and ground-state electronic natures of the clusters. A large number of research works have done for understanding a general trend in the structural and electronic properties of gold nanoclusters [177, 182–185]. The structure and reactivity of small-sized Au nanoclusters are highly influenced by the strong relativistic effect of Au [186]. One of the most intriguing properties of gold clusters is that they try to retain their planar structure up to a significantly large size. Some people report that gold nanoclusters with 4–12 atoms exhibit two-dimensional (2D) planar structures [55, 187]. This planarity of Au nanoclusters may be attributed to the strong relativistic effects of gold. Although the actual “transition

256 Atomic clusters with unusual structure, bonding and reactivity

point” from planar to three-dimensional (3D) crossover is still in ambiguity, a sufficient large cluster-like Au6 is determined to be very stable in its planar triangular atomic arrangement [186, 188, 189]. Adsorption of molecular oxygen on gold nanocluster is technologically very important because the activated oxygen molecule, adsorbed on gold nanocluster, could be utilized for oxidizing poisonous gas CO. Sufficiently activated molecular oxygen over gold nanocluster can catalyze the oxidation of CO to forming CO2 at temperatures far below the room temperature [190–192]. Theoretical studies have demonstrated that reactivity of pristine gold clusters with molecular oxygen depends on several factors like shape and size of the cluster, surface passivating ligands, charge state, and supportive materials of the cluster [193–196]. It has been shown that anionic clusters can also bind molecular oxygen even more strongly, promoting electron transfer from the gold cluster to the oxygen molecule. It has been demonstrated that charge and/or impurity doping in gold clusters dramatically change their structural and electronic properties [187, 197–199]. Jena et al. [200] studied the effect of hydrogen doping on the reactivity and catalytic activity of a neutral gold cluster. The authors demonstrated that hydrogen doping in gold cluster preferentially activates molecular oxygen, which subsequently reduces the CO oxidation barrier. It has been shown that anionic gold clusters constituting 16–18 atoms possess cage-like structures and they can act as efficient catalysts due to their large surface-to-volume ratio [201, 202].

5.1 Effect of doping in Aun nanoclusters Doping of nontransitional elements, especially s- or p-block elements in gold nanocluster has a remarkable impact on the structural evolution. s-block element Na, while doped in Aun nanocluster, provide a higher polarization and more directionality in the Au–Au bonds [203]. Sahoo et al. [187] showed that while Aun nanoclusters favor planar isomers up to n ¼ 13, p-block element Sn has a remarkable influence on the ground-state geometries of AunSn nanoclusters. For the latter case, it follows 3D structure for n ¼ 3 and 4, a quasiplanar geometry is observed for n ¼ 5–11 and again 3D isomers are the most stable ground states for n ¼ 12 onward (see Fig. 7). This peculiar behavior of the ground-state geometries may be ascribed as the enhanced contribution of Au p-orbital and significant charge transfer from Sn to the Au atoms of the AunSn nanoclusters. Another p-block element, Si doping in gold nanocluster is a fascinating research as such clusters are reported to be the gold-hydrogen analogy [204, 205]. Pal et al. [206] have carried a systematic study on the structural evolution of silicon-doped gold clusters. The authors have demonstrated that silicon doping on gold clusters leads to a

FIG. 7 Lowest-energy structure of neutral AunSn clusters (n ¼ 2–13) at B3LYP/lanl2dz level of theory [187].

Designing nanoclusters for catalytic activation of small molecules Chapter

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257

FIG. 8 Reaction pathways for the oxidation of CO on the pristine Au8 (left panel) and doped Au7Si (right panel) nanoclusters [207].

significant change in the structure. An early onset of nonplanar geometries has been reported for AunSi nanocluster. The preference of nonplanar geometry of AunSi clusters has been attributed to the involvement of p-electrons into the bonding, resulting a strong polar covalent bond. Inspired by this work, Manzoor et al. [207] have studied the effect of silicon doping on the reactivity of gold clusters toward molecular oxygen, followed by possible CO oxidation pathway. The authors have also compared the calculated activation barrier for CO oxidation with that of pristine gold clusters. It has been reported that silicon-doped Au7 nanocluster is more effective for binding and activating molecular oxygen in comparison to that of undoped Au8 nanocluster. However, the doping has a marginal effect on the reactivity and catalytic activity of the Au7 cluster. The doped cluster, Au7Si, is reported to show enhanced catalytic activity for the oxidation of CO with a very low activation barrier (Fig. 8). TM doping, in particular, can also lead to the desired structural, electronic, magnetic, and chemical properties of Aun nanoclusters for potential applications [208–210]. As a consequence of these, cluster-molecule interactions are also affected enormously [211]. On the basis of first-principles calculations, our group has recently investigated the global and local minimum structures of M (¼ Ni, Pd, and Pt)-doped Au6 cluster and its possible application in CO oxidation reaction [198]. Furthermore, as hexamer gold cluster has the triangular atomic arrangement as that of Au(111) surface, it could be regarded as a model study for the catalytic activity of Au(111) surface. One interesting observation is that the add-atom Ni transforms the planar geometry of Au6 cluster into a chair-like conformation, while doping of other two same group metal atoms retains its planar structure. It has been demonstrated that NiAu6 cluster adsorbs molecular oxygen in a dissociative addition manner but the tetrameric assembly of this doped cluster, (NiAu6)4 adsorbs molecular ˚ . Further investigation reveals that the oxygen in an undissociated but activated state with O–O bond distance of 1.37 A interaction of activated O2 molecule with two molecules of CO results in cleavage of O–O bond, thereby oxidizing the CO molecules it produces two molecules of CO2 as outlined in Fig. 9. Interestingly, the oxidation process is found to be exergonic in nature.

5.2 Al n anionic nanoclusters: Effect of electron spin Apart from pristine and doped noble metal clusters, the reactivity pattern of small anionic aluminum nanoclusters with oxygen is also point of interest for a long time. It has been shown that spin accommodation plays a deterministic role in the reactivity of anionic aluminum nanoclusters with molecular oxygen. Theoretical investigations have demonstrated that the possibility of spin transfer practically governs the reactivity. More precisely, when spin transfer is not possible, it remains unreactive and when spin accommodation favors, there appears more subtle effects such as the required spin excitation energy, etc. [56]. Molecular oxygen is spin triplet in its ground state, and the lowest two unoccupied orbitals are antibonding in nature. Therefore, reactivity of molecular oxygen requires the filling of those unoccupied orbitals. The filling of these orbitals may be thought of as a spin crossover from triplet to singlet, which could be facilitated by spin-orbit coupling. So, reactivity of molecular oxygen with other light metal having weak spin-orbit coupling effect (such

258 Atomic clusters with unusual structure, bonding and reactivity

FIG. 9 The lowest-energy configurations of the NiAu6 cluster and its cluster assembled nanostructure formed by combining four units (upper panel). The minimum energy geometrical configuration of the CO oxidation on the oxygenated (NiAu6)4 cluster assembled nanostructure (lower panel) [198].

as Al) must retain the spin multiplicity as per the Wigner-Witmer rules of spin conservation [212]. Reber et al. [56] showed that the spin excitation energy controls the reactivity of anionic aluminum nanoclusters in both vivid and subtle ways. The nanoclusters having high-spin excitation energy are highly resistant to etching and are not observably reactive, while odd electron and low excitation energy species react readily. However, it remained a puzzle why the Al n nanoclusters with n ¼ odd tend to react with O2 even more slowly than that with n ¼ even until the seminal paper by Burgert et al. [213] came out in 2008. By using Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) and theoretical calculations, the authors showed that it is the spin conservation which straightforwardly accounts for that unusual reaction trend. It has been demonstrated that the reaction rates of odd-numbered anionic clusters could be increased by replacing  ground-state triplet oxygen with excited singlet oxygen. Conversely, when Al n nanoclusters are replaced with AlnH monohydride clusters, where the additional hydrogen atom shifts the spin state from open-shell to closed-shell (and vice versa), the odd-n hydride clusters react at a faster rate with triplet oxygen. The unusual stability of Al 13 nanocluster is in general ascribed to be the jellium-like shell structure with “magic” electronic number 40 and it has singlet ground state [214]. It was believed that the reactivity of large Aln nanoclusters with O2 could be determined in terms of the energy requirement to remove an Al atom from the nanocluster and the electron affinity of the cluster [215]. However, these could not describe fully the observed odd/even effect of the nanoclusters, while reacting with O2. In fact, the role of the spin had not considered before because spin conservation was estimated to be negligible for multielectron systems like nanoclusters. For such kind of systems, intersystem crossing processes are expected to take place so rapidly that the intermediate products always relax to the lowest possible spin multiplicities. However, Burgert et al. [213] first trembled the idea and pointed out that spin conservation must have an essential influence on reactions of Aln nanoclusters with O2. During the reaction at high temperature, the Aln clusters were observed to degrade due to the formation of Al2O [216]. As observed by FT-ICR MS studies, Al 13 clusters remained almost inert in the presence  of molecular oxygen, producing only trace amount of Al 9 clusters but Al14 cluster (doublet ground state) reacted sponta3  neously with O2, producing Al 10 [213]. The observed phenomena can be described on the basis that as [Al13O2] is in a triplet state but its fragmented products, Al 9 and Al2O are all singlets, the reaction is inherently slow for restoring the spin  2 state. On the other hand, for Al 14, having doublet ground state, can react spontaneously with triplet oxygen to produce Al10 2  and Al2O through the intermediate [Al14O2] , where spin is already conserved. Theoretical calculations reveal that the initial interaction of O2 with the Al n nanocluster leads to an association complex, which thereafter forms a strong Al– O2 bonding in a rate-limiting step. Fig. 10 describes the energy profile diagram for different spin multiplicities. The calculated results indicate that the formation energy of all the initial oxygen adducts are exothermic but the exothermicity    trend (Al 14 >Al13H >Al14H >Al13) closely resembles the trend of their reactivity toward molecular oxygen.

6.

H2O activation

In the previous section, it has been shown that the selectivity of oxygen reactions with anionic aluminum nanocluster can be well described with the help of electronic shell model. However, the generality of the reaction trend is not applicable to

Designing nanoclusters for catalytic activation of small molecules Chapter

0 –0.36

Al13– + 1O2 Al13– + 3O2

[Al13• O2]–

3

? eV, spin transition Multistage process

1

259

FIG. 10 Computed energy profile diagram for the interaction of 1O2 and 3O2 on the Al 13 nanocluster. The transition from 3[Al13O2] to 1Al13O is estimated to be a multistage 2 process in which O2 is first bound side-on, then rearranges to end-on and after that the O–O bond is disrupted, new Al–O bonds are formed (m3) and finally the spin state changes from triplet to singlet. Further degradation to Al 9 and two Al2O is also displayed [213]. Inset: Spin density (blue [dark gray in print version]) plot of Al13O 2 cluster [56].

E (eV) 1.32

14

–

[Al13• O2]

–1.84

Al9– + 2 Al2O

1

–2.77

Multistage process 1

Al13O2–

–6.37

other small molecule, H2O, having singlet ground-state electronic configuration. Roach et al. [47] demonstrated that the size selectivity of Al n nanocluster with H2O can be attributed to the dissociative chemisorption at specific surface sites. Unlike oxygen reactivity, in this case, the reactivity trend of water depends on geometric parameter rather than electronic shell structure. H2O dissociates into H2 on identical arrangements of multiple active sites of some Al n nanoclusters. The reactivity of the nanoclusters varies sharply with size and most surprisingly, the nanoclusters with n ¼ 16, 17, and 18 preferentially produce H2, while the other nanoclusters just only adsorb H2O. The reason behind this is that it requires the cleavage of an OH bond of H2O to be adsorbed onto the gas-phase Al n nanoclusters by the expense of thermal energies. Now, some specific surface sites, which act appropriately as a Lewis acid and a Lewis base to accept the fragments of H2O, facilitate this process. So, the nanoclusters with those specific surface sites are more reactive than others. The presence of an active site arises due to irregular charge distribution on the cluster surface, which in turn is related to some specific defect formation in the surface. As shown in Fig. 11A, Al 17 has a structure having two sets of adjunct Al atom dimers located on opposing vertices of a 13-atom icosahedral core. Theoretical calculations reveal that appreciable HOMO density is located on the adjunct dimer, while the LUMO density is projected into vacuum at an adjacent surface site. The presence of adjacent HOMO and LUMO densities results in an energetically accessible transition state for cleaving water [47]. The complete dissociation channel of

0.0

B

Al17–

C

A

Energy (eV)

–0.5 –1.0 –1.5

D

F

I

E

–2.0

J

–2.5 –3.0

G

H

+H2

FIG. 11 Computed energy profile diagram for the formation of H2 from Al 17 and two molecules of H2O. Al, O, and H atoms are represented by yellow-brown, red, and white colored balls (please see online version), respectively. Al atoms representing complementary active sites are shown with the Lewis acid site in orange and the Lewis base site in purple. (A) The LUMO charge density. (B) One water chemisorbed to the cluster with HOMO charge density. (C) The transition state to dissociative chemisorption. (D) The resulting complex with LUMO+2 charge density. (E) The second water chemisorbed to the adjacent Lewis acid-Lewis base pair. (F) The transition state. (G) A complex with two dissociative chemisorbed waters. (H) The complex rearranges to more efficiently release H2. (I) The transition state for H2 release. (J) The cluster after H2 is released [47].

260 Atomic clusters with unusual structure, bonding and reactivity

H2O to H2 is shown in Fig. 11. It has been proposed that the intermediate species, 4G contains a sufficient amount of thermal energy from the dissociative chemisorption of water, which thereafter allows the recombination of the individual surfacebound hydrogen atoms for releasing H2. The mechanism is in consistence with the Langmuir-Hinshelwood processes, which are observed on extended surfaces [217].

7.

C–X and C–H bonds activation

7.1 C–X bond activation on Aln nanoclusters In organic synthesis, C–C cross-coupling reaction is very challenging because of its high-energy demanding and very slow kinetics [218]. However, a suitable catalyst can bring down the energy barrier and make the reaction feasible. In most of the cases, first-, second-, or even third-row TMs (such as Fe, Ni, Cu, Pd, Au, etc.) are used as catalyst [219–222]. Nevertheless, TM-free cross-coupling reactions are gaining severe attention, nowadays [223, 224]. In this case, catalytic cleavage of C–X (X ¼ heteroatom) bond is a one step forward, especially when lighter atoms are used for this purpose. Aln nanoclusters are well known for their reactivity, especially small-sized clusters are reported to activate small molecules such as H2, N2, O2, H2O, etc. [53–55]. Bergeron et al. [225] showed that small-sized aluminum cluster anions could activate C–I bond in methyl iodide. A similar observation was also reported over Al(111) surface on the basis of both experimental and theoretical results [226]. Inspired by those studies, Sengupta et al. [227] performed a detailed theoretical study to understand the stabilities and reactivities of different-sized Aln nanoclusters toward C–I bond. The authors have presented the thermodynamic and kinetic details for the dissociation of C–I moiety on Aln nanoclusters. Besides these, an in-depth reaction mechanism, detailed structural analysis and effect of shell structures of the clusters on the reactions have also been properly accounted with molecular dynamics simulations and on the basis of charge analysis. It has been revealed that C–I bond undergoes oxidative addition to the Aln nanoclusters. Note that oxidative addition is the process by which a chemical bond dissociates and two separate bonds are created at the adsorbing moiety. The most important observation of the present investigation is that although the p-block element Al is ineffective to dissociate C–X bond in bulk phase, the Aln nanoclusters do the same thing efficiently. In fact, Aln nanoclusters are found to be more efficient in activating C–I bond in comparison to that of Aun nanoclusters. Furthermore, the computed activation barrier for C–I bond activation is in the comparable range with that of the reported most versatile and efficient catalyst, Pd. It has been demonstrated that the reactivity of Aln nanocluster could be enhanced by increasing the cluster size or even depositing the cluster on solid support. Computed energy profile diagram along with the optimized geometries of reactants, products, and transition states for the reactions of different alkyl iodides with some selective Aln nanoclusters are shown in Fig. 12.

7.2 Competitive H–X elimination on alumina nanoclusters Similar to aluminum nanoclusters, a wide range of catalytic activities are also shown by nanocrsytals of alumina (Al2O3). Alumina crystallizes in different forms such as g, y, E, a, etc. Among them, g phase is supposed to be metastable in nature, which exhibits the highest catalytic activity both in heterogeneous and homogeneous conditions [228]. The catalytic activity of alumina is related to its calcination temperature, which generally follows the sequence g > y > E > a [229]. Now, with increasing the calcination temperature, the density of Lewis acid sites (Al3+) on the surface of alumina decreases, which practically enhances the catalytic activity of g-alumina [230, 231]. Comas-Vives et al. [232] showed that g-alumina catalyzes the C–F bond activation. Apart from this, a number of experimental and theoretical works have been performed for demonstrating the catalytic activities of g-alumina [233, 234]. Theoretical models have been proposed to mimic the active sites of g-alumina. In this vein, cluster model is very promising and hence people have developed appropriate cluster models of alumina for investigating the mechanistic aspect and specific role of active sites in the alumina surface. Al8O12 nanocluster is one such model, which has been largely investigated as a model structure of g-alumina. This is primarily because of the fact that Al8O12 nanocluster is highly stable against structural relaxation and second, the nanocluster reproduces well the surface properties as obtained from experimental measurements and periodic slab models from theoretical studies [235–237]. This particular cluster model has successfully implemented for studying the mechanism of alcohol dehydration, dehydrogenation, and different kinds of condensation reactions [235, 236]. Noteworthy, although Al8O12 nanocluster achieves a relaxed geometry, resembling the bulk structure of g-alumina, many authors have reported that one as a model structure for a-alumina too [237]. In reality, an ambiguity regarding the exact structure of the gas-phase cluster is still there; nevertheless, Jaroszynska-Wolinska et al. [238] recently conclude that both cationic [Al8O12]+ and neutral [Al8O12] systems could be most appropriately described as molecules, rather than clusters. Apart from Al8O12, several other small-sized alumina nanoclusters, both in the neutral and ionic forms, are investigated for showing catalytic activities. Neutral as well as both cationic and

Designing nanoclusters for catalytic activation of small molecules Chapter

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261

FIG. 12 Computed energy profile diagrams for the reactions of three different alkyl iodides over Al13 and Al20 nanoclusters. Orange and green arrows (please see online version) indicate the corresponding activation barrier (DG‡) and exothermicity (DH) of the reaction. Computations are done with M06-2X functional [227].

anionic Al2O3 nanoclusters are shown to be very efficient for activating C–H bonds [239–245]. It has been demonstrated that the strong ionic nature of the Al–O bond is responsible for capturing the hydrogen atom of the weakly polar C–H bond. On the basis of computational analysis, it has been demonstrated that the presence of high-spin density at the terminal oxygen atom of the Al2O3 nanocluster is very crucial for that activation process [246]. g-Alumina is used as catalyst for the industrial preparation of ethylene by the dehydration of ethanol [247]. However, the reaction is inhibited by the surface hydroxyl groups, formed during the course of the reaction, by blocking the C–H bond cleavage which must be occurred for the dehydration process [233]. Another possibility of activating C–H bond of ethanol over alumina catalyst is the dehydrogenation, forming acetaldehyde as the major product. In both cases, it has been demonstrated that the cleavage of C–H bond is the rate-determining step. Noteworthy, although the catalytic activity of g-alumina for alcohol dehydration is being known for a long time, the exact mechanism and role of different active sites are long-standing debates. Theoretical calculations demonstrate both E1 and E2 concerted mechanisms for the dehydration process [234, 236, 248, 249]. On the basis of experimental and back-to-back theoretical studies, Roy et al. [236] reported that the tricoordinated Al surface atoms, which are strong Lewis acidic in nature, are the active sites of the catalyst. Of particular note, the authors used Al8O12 nanocluster as a model of g-alumina. Biswas et al. [45] in a recent study demonstrate that alkyl halides (R–X, X ¼ F, Cl, Br, I) undergo dehydrohalogenation on alumina nanoclusters at room temperature thereby producing alkenes. Although dehydrohalogenation of alkyl halides is a classic transformation in organic chemistry, which are generally occurred by strong Br€onsted acid or base at high temperature [250], the present study opens up a new possibility of preparing alkenes at harsh-less condition. Note that Al(100) surface-mediated decomposition of alkyl iodides was performed long time back but it required very high temperature [251]. Besides this, Bissember et al. [252] exploited an undesired b-hydride elimination rout for dehydrohalogenation of alkyl bromide in the presence of Pd-catalyst. Porous alumina has also been utilized for dehydrofluorination of 1,1,1,2-tetrafluoroethane. These studies demonstrated that weak Lewis acid cites of the alumina surface practically have

262 Atomic clusters with unusual structure, bonding and reactivity

the most effective catalytic performance for the dehydrohalogenation reactions [228]. It has been reported that the y-alumina, calcined at 950°C, shows the highest catalytic activity as a result of its weak Lewis acidic strength and having relatively fewer number of strong Lewis acid sites in comparison to other aluminas. In this perspective, theoretical work of Biswas et al. [45] has a significant impact. Their quantum chemical calculations demonstrate that alkyl halides decompose on alumina nanoclusters in two different pathways, dissociative addition and elimination. It has been shown that although the dissociative addition product is thermodynamically more stable, the elimination pathway is the kinetically favored one and hence produces olefin at room temperature (Fig. 13). The theoretical results corroborate with the experimental evidence that at room temperature ethylene is formed on activated alumina by the decomposition of methyl fluoride [253]. The dissociative addition reaction follows an SN2 type of mechanism, while the elimination pathway proceeds through an initial activation of the C–X bond and thereafter H–X is eliminated, following an E2 type of mechanism. The DFT calculations also predict that, unlike under normal acid/base catalytic conditions, the alumina-catalyzed elimination rate decreases going down the periodic table.

7.3 Selectivity of alumina nanoclusters during elimination The catalytic performance and extraordinary selectivity of the Al8O12 nanocluster during the elimination reaction have also been studied by Biswas et al. [46]. As obtained from DFT calculations, the proposed mechanism is fundamentally different. The authors have demonstrated that a combined effect of Lewis acid/base interaction and H-bonding force the eliminating group to occupy an unfavorable syn orientation. As a result of that, it can selectively eliminate hydrogen halide in the presence of other leaving group like hydroxyl. Second, as outlined before, the rate of dehydrohalogenation is the fastest for the “poorest” leaving group, fluoride. The eliminations are predicted to occur in a regioselective manner; more precisely, a particular isomeric product is formed predominantly. The most surprising fact is that the preferred syn coplanar orientation of the leaving groups promotes the formation of undesired stereospecified products, which are very difficult to be obtained under normal acid/base catalytic condition. The selectivity nature of the catalyst could be understood on the basis of schematic representation as shown in Fig. 13.

7.4 Selective C–H bond activation Selective activation of C–H bond is a challenging task in synthetic organic chemistry. Activating C–H bong, particularly in CH4, is very difficult because (i) CH4 is thermodynamically the most stable alkane with a standard free energy of formation, DfG0 ¼ 50.8 kJ mol1 [254] and consequently, the C–H bond strength in CH4 is the largest among all sp3-hybridized hydrocarbons (439 kJ mol1) [255]. Moreover, the almost nonpolarity of the C–H bond as a result of very low electron affinity difference between C and H, a wide HOMO-LUMO gap makes C–H bond practically thermally and 1.703 1.782

1.733 1.054

2.651 1.228 2.361 1.471

TSOH

TS2

(A)

TSBr

+

TS1

2.056 1.471

Reactant

2.477 2.303

1.860 2.301 1.710

Elimination Dissociative addition

0.976

(C2)

(B2)

2.299

(B1)

0.977

(C1)

FIG. 13 Left panel: Schematic representation of the energy profile diagram for the two possible reaction pathways showing thermodynamically versus kinetically controlled products [45]. Right panel: Schematic energy profile diagram showing competitive dehydration (left portion) versus dehydrohalo˚ [46]. genation (right portion) from 3-bromopropanol in the presence of Al8O12 nanocluster. Both eliminations are of syn type. Bond lengths are shown in A

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photochemically inactive [256, 257]. Abstraction of H atom from CH4 to generate CH3 is considered to be the decisive step in the oxidative dehydrogenation and dimerization of methane. In spite of so much difficulty, various metal oxides are known which can induce homolytic fission of C–H bond at relatively high temperature [258–261]. Besides these, some simple cationic metal oxides containing V, Fe, Mo, Re, Os, etc. also show the capability for activating methane [262–266]. Apart from TMs, some main group metal oxides, either in the neutral or cationic form, also exhibit methane activation [267, 268]. Group of Sauer and Schwarz nicely demonstrated that alumina radical cationic species undergo thermal activation of methane, where electron spin over the oxide ion has a fundamental role [268]. First-principle DFT calculations reveal that the alumina radical cations have a high-spin density at an oxygen atom, arising as a result of removal of electron from the metal. Thus, it has been conjectured that the doublet nature of the ions makes them reactive toward thermal hydrogen-atom abstraction from methane. As for example, Al8O+ 12 exhibits the abovementioned structural, electronic, and energetic features as shown in Fig. 14. Computational analysis reveals that the hydrogen abstraction reaction proceeds through the formation of a stable Al8O+ 12 /CH4 preactive complex. Thereafter, it passes through a transition state, which is characterized by the transfer of the interacting hydrogen atom to the radicaloid oxygen atom as demonstrated in Fig. 14. This combined experimental and theoretical study is quite significant, being the first example for the thermal activation of methane by a polynuclear main-group metal oxide cluster, because mononuclear metal oxides, which are extremely reactive, could not be considered as ideal model systems for understanding the oxidation catalysis. Apart from alumina nanocluster for direct activation of C–H bond, alumina supported subnanometer-sized platinum clusters, consisting of 8–10 atoms have also been recently utilized for such similar purpose [269]. Those nanoclusters may possess various shapes depending on the processing conditions but their reactivity retains even in conditions other than ultrahigh vacuum [270]. The particles shape is dependent on the calcination temperature; at relatively lower temperature (475 K), tetrahedral particles are preferentially produced, while at higher temperature (575 K) it prefers spherical particles [271]. Most interestingly, the faceted platinum nanoparticles have been shown to retain their shape upon deposition onto alumina supportive surface [272]. So, it is an intelligent way to selectively controlling the shape of the catalyst particles by monitoring the processing conditions and tuning the catalytic activity. Theoretical works have been performed showing the enhancement of catalytic activity and selectivity by regulating the particle size and/or shape of Ptn nanoclusters [273, 274]. Cheng et al. [274], on the basis of first-principles calculations, have explored the methane dehydrogenation pathways on tetrahedral (with exposed Pt [111] facets) and hemispherical platinum nanoclusters, the latter being the mimic of spherical nanocluster deposited onto a support. The authors have pointed out that all steps of methane dehydrogenation on the hemispherical Pt21 nanocluster have high activation barriers, requiring high temperatures for the degradation process. However, the same on the tetrahedral cluster of similar size requires less activation barrier. The detailed potential energy surfaces are shown in Fig. 15. According to their study, the dissociation of the methyl group on tetrahedral Pt20 nanocluster to form methylene and hydrogen has an activation barrier as low as 0.2 eV [274], demonstrating that hydrogen production from FIG. 14 Energy diagram for the reaction of Al8O+ 12 with methane. The values are relative to the entrance channel, corrected for zeropoint energy (ZPE) and given in kJ mol1. For the stationary points of the reaction, the C–H, H–O, and O–Al distances are given in pm (with distances of the unperturbed fragments in parentheses). Inset: spin density for the optimized structure of Al8O+ 12 . Dark vertices: O, light vertices: Al [268].

264 Atomic clusters with unusual structure, bonding and reactivity

Methane dehydrogenation on hemispherical Pt21

Energy of dehydrogenation (eV)

0.6 0.4 0.2

0.291

TS1

0 –0.2 –0.4

–0.098

–0.069

CH4

–0.6

–0.208

–0.202

TS2

TS3

–0.658 CH3+H

–0.8 –1

TS4

–0.796

–0.908

CH2+2H

C+4H

–1.109

–1.2

CH+3H Methane dehydrogenation on tetrahedral Pt20

Energy of dehydrogenation (eV)

0.6 0.4 0.2 0.188

0.0

TS1

–0.2

–0.173

–0.4

CH4

–0.6 –0.8

–0.242

–0.521 CH3+H

TS3

–0.322

–0.491

TS2

TS4 –0.951

–1.0

–0.951

–1.2

CH2+2H

–1.075

C+4H

CH+3H

FIG. 15 Schematic representation of methane dehydrogenation pathway on hemispherical Pt21 (upper panel) and tetrahedral Pt20 (lower panel) nanocluster [274].

methane would proceed at a faster rate. So, it can be concluded that shape of the catalyst is an important controlling parameter for efficient hydrocarbon transformations at low temperature.

8.

Summary and future outlook

In summary, this chapter focuses on some aspects of computationally designed nanoclusters for small molecules activation. It provides reactivity and mechanistic insights based on electronic structure-property relationship. Although catalytic activities depend on several other parameters, such as solvation, pressure, temperature, etc., nevertheless, the computational results sometimes become inevitable for understanding the observed experimental facts, on the basis of which one may think of better catalytic materials. As for example, the effect of electron spin on catalysis could not be thought earlier before it was predicted from computer simulation. In silico design of catalytic materials sometimes save time and cost before their real-world applications. A large portion of this chapter focuses on the main group metal and metal oxide nanoclusters (especially Al) for activating almost inert N–N and C–H bonds, some of which have already been verified experimentally. Computational results on pristine and doped gold nanoclusters provide significant advances in designing catalysts. In spite of that, till date, catalytic performance for the fixation of CO2 and N2 is not up to the mark. It requires further development, addressing the issue of environmental pollution by greenhouse effect of CO2 and energy storage material by fixing N2 and H2. Computation chemistry in strong collaboration with experiment is expected to provide valuable results, which can trigger material design and synthesis for advanced catalysis. For doing this, electronic and photonic interactions must be taken into account and a more realistic model should be accounted by considering explicit solvent and other environmental issues. Another important aspect that must be considered for developing nanomaterials-based catalysis is the time

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period for electron transfer interactions. A real-time domain atomistic simulation may provide valuable insights in this particular field. We do hope that all those issues would be addressed properly in computational nanocatalysis in near future.

Acknowledgments The authors gratefully acknowledge the financial support from BRNS, DAE, Government of India through sponsored research grant (sanction no. 37(2)/14/07/2018-BRNS). P.S. acknowledges UGC, Government of India for providing him with UGC Mid-Career Award, 2021. A.P. is thankful to UGC, Government of India for the financial help through UGC Start-Up-Grant (No. F. 30-557/2021 (BSR)).

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Chapter 15

Molecular electrides: An overview of their structure, bonding, and reactivity Ranajit Sahaa,b and Prasenjit Dasb a

Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Sapporo, Japan, b Department of Chemistry, Indian Institute

of Technology Kharagpur, Kharagpur, India

1. Introduction 1.1 Electrides Electrides are a novel class of stoichiometric chemical entities where the electrons serve as an anion [1,2]. Electrides have loosely bound and diffuse electron(s) which is (are) not directly associated with any atom(s) present in the molecule, cluster, and/or crystal. The electron is considered to be trapped in the hollow which may be the void space in crystals, generated due to the packing or the cavity inside cages [3]. Sophisticated theoretical and computational studies have established that it is not a necessary condition for an electride to hold one single isolated electron, rather than a substantial quantity of it with sufficient localization is adequate [4]. The localized electron (or a portion of it) might be trapped in the channels (solid-state electrides), or inside the void formed due to the crystallization or intercalated between two layers (inorganic electrides), or the cavity of the cavitands, dispersed on higher-level molecular orbitals (molecular electrides) [2,5,6]. The presence of diffuse and loosely bound electrons [3] results in high nonlinear optical (NLO) properties and they are useful as NLO materials or vice versa. Thus, to identify a system as an electride the high NLO properties are a possible indication [7–9]. Other than the utility as NLO materials, electrides are also useful as electron emitter [10], superconductor [11], reversible hydrogen storage materials [12], catalysts [13,14], etc.

1.2 Confinement of the electron The traceback of electron confinement has a long history in the chemical literature. The concept of free electrons, which are not bound to any atoms and/or molecules can be marked as old as 200 years. An unpublished entry from the notebook of Sir Humphry Davy reads, “When 8 grains of potassium were heated in ammoniacal gas - it assumed a beautiful metallic appearance and gradually became of a fine blue colour” [15]. The occurrence of the blue color is explained in terms of the presence of solvated electron as a distinct species in ammoniacal solution. This concept is based on the works of Charles A. Kraus [16,17]. A schematic representation of the e
solvated in NH3 solution is shown in Fig. 1A [18,19]. The model in Fig. 1A shows that the electron is solvated, a form of confinement by the ammonia molecules which are either polarized in ammonia solution [18]. The usefulness of the alkali metal solvated ammonia solution is well known in the reduction chemistry and their mechanistic studies have been performed several times.

1.3 Development of organic electrides The advancement of the studies on the solutions of alkali metals in primary amine-based solvents (such as methylamine and ethylenediamine), [20] as well as the synthesis and progress of crown ethers and cryptands, the route toward the synthesis of the first electride moved forward. In 1983, the research group of Professor James L. Dye from Michigan State University reported the synthesis of first electride Cs+(18-crown-6)2e
[21]. The crystal structure of the same was published by the same group in 1986 [22]. The crystal structure showed that the cesium (Cs) metal center is confiscated and sandwiched between two 18-crown-6 ether molecules as shown in Fig. 1B and the metal center attains unipositive electronic charge on it. The void channel in the crystal holds the stoichiometric electron which serves as an anion to the counter cation. The overall solid is known as an organic electride cause organic cryptand has been used. The reaction process to prepare the Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00018-8 Copyright © 2023 Elsevier Inc. All rights reserved.

275

276 Atomic clusters with unusual structure, bonding and reactivity

FIG. 1 (A) The schematic representation of solvated electron in ammonia solution, The unit cell of (B) first organic electride: Cs+(18-crown-6)2e
, and (C) first inorganic electride: [Ca24Al28O64]4+(4e
).

complex is straightforward, the Cs metal is dissolved in dimethyl ether (or methylamine) solvent where the crown ether is also present in stoichiometric amount. The temperature should be kept constant at
30°C throughout the reaction and crystallization process and the crystal should be grown slowly. Few more organic electrides such as K+(cryptand[2.2.2])e
[23,24], Cs+(15-crown-5)2e
[25], [Cs+(15-crown-5)(18-crown-6)e
]6(18-crown-6) [26], K+(15-crown-5)2e
[2], Li+(cryptand[2.1.1]e
[27], Rb+(cryptand[2.2.2])e
[28], etc. have been synthesized and crystallized by changing the cavitands type, ranging the pore sizes and/or using other alkali metal atoms. The low temperature should be maintained throughout the synthesis and crystallization and should be removed out of moisture. With an increase in the temperatures and air moisture, the electride decomposes via the reductive cleavage of the ether (dOd) linkages. The use of tertiary amine linkage-based cryptands is a solution to this problem as tertiary amines are less susceptible to reductive decomposition. Na+[TriPip222]e
is the first example of this kind of electride, which shows stability up to room temperature [29].

1.4 Development of inorganic electrides The problems associated with the thermal instability and air sensitivity of the organic electrides were blown away with the development of the inorganic electrides. In 2003, Professor Hideo Hosono from the Tokyo Institute of Technology synthesized 12CaO7Al2O3, a cement-like material prepared from lime and alumina where a high concentration of electrons was doped within the cavities [30]. The system is termed as C12A7 electride (alternatively, [Ca24Al28O64]4+(4e
)) as shown in Fig. 1C and it is stable at air and room temperature. This electride is useful as an electron donor, conductor, and catalyst for several reactions [31,32]. After this discovery, various works on inorganic electrides have been carried out by different research groups. Air and water stable electride Y5Si3 was reported in 2016 [33]. This electride can be used as a catalyst for ammonia synthesis. Qu et al. reported two inorganic electrides, [Ba2N2](e
) and [Li2Ca3N6](2e
) which are formed in the presence of dinitrogen ligands [34]. In Ba2N2(e
) system, the ligand is [N2]3
and in the [Li2Ca3N6](2e
) system the ligand is [N2]2
. Dicalcium nitride (Ca2N) can behave as an inorganic electride material having an anionic electron layer and the system is represented as [Ca2N]+(e
) [35]. Apart from the main group element Zang et al. used transition metal element (Y, Y ¼ Yttrium) and reported Y2C electride material which is denoted as [Y2C]1.8+1.8e
[36]. In 2017, Wank et al. reported stable strontium phosphide-based electride Sr5P3 [37]. Recent work on the Yb5Sb3 system shows electride characteristics [38]. The application of high pressure on alkali metal surfaces can induce electride characteristics therein. These electrides are known as a special category, high-pressure electride (HPE) [39]. The high pressure causes compression of the valence electrons and over a certain value of pressure, they get separated from the valence orbitals. These electrons are localized in

Molecular electrides Chapter

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the crystal void. Their presence can be characterized by the presence of nonnuclear attractor (NNA) and maxima of electron localization function (ELF), with a high number of electrons in the corresponding basin. The electronic structure and bonding analysis on HPEs have been studied thoroughly by Professor Roald Hoffman and coworkers [39]. The empty space in the lattice which has been occupied by electron(s) are known as interstitial quasiatoms (ISQs) and they have quantized energy levels like the atoms. The study reveals that with an increase in the pressure, the energy levels of the valence orbital increase and the repulsion between the core and valence electron also increases. After a certain amount of pressure, it is thermodynamically more favorable to occupy electron(s) at the ISQ rather than at the valence orbital of the atoms in that lattice. This model also predicted that Li [40], Na [41], and Al [39] metals are expected to build HPEs below  500 GPa, and elements like Mg [42], Si, Tl, In, and Pb [39] can form the same but higher pressure is required. The electron of [Ca2N]+(e
) [35] electride is considered as “partially delocalized” [43] 2D electron gas between the layers of Ca2N. The electrostatic and electronic study by Walsh et al. has shown that a deep potential well (11 V), positioned at the interlayer void is responsible for the localization of the electrons therein [44]. Furthermore, in ternary M2NX (M ¼ Ca and Sr; X ¼ Cl and Br) compounds, these positions are filled by chemical anions (X in this case), which is also a shred of circumstantial evidence for the localization of electrons in Ca2N [45]. On the other hand, the overlap among these localized electrons results in the “partially delocalized” character. There is a subtle balance between the localized and delocalized character of the electrons in Ca2N. The electride behavior of alkali metals at high pressure (few hundreds of GPa) [39] is the result of the conversion of metallic solid into ionic solid where ISQ acts as the anion, which is different from Ca2N (0.56 GPa) [43]. In the HPE, the stabilization has been achieved by applying hydrostatic pressure and volume change is expected whereas for 2D Ca2N the strain is uniaxial or biaxial. There is also a difference in the electride formation mechanism, due to the high-pressure condition, the energy levels of the s-orbitals of the alkali metals become more unstable than that of the p-orbitals, and electron confinement into an ISQ from the s-orbitals occur. The inorganic electrides are capable to catalyze various chemical reactions of synthetic and industrial worth, like dinitrogen activation, followed by ammonia production [12,46], carbon-dioxide activation, followed by decomposition of the C]O bond and formation of carbon monoxide and oxygen atom [47], chemo-selective hydrogenation of keto group (aldehydes and ketones) and nitroarenes [48], pinacol coupling [49], and many more. For example, the C12A7 ([Ca24Al28O64]4+(4e
)) electride loaded with the Rh metal nanoparticles on its surface can activate the dinitrogen followed by the synthesis of ammonia. The trapped electrons inside the cavities of C12A7 material have been shoved into the p* molecular orbitals of N2 via the Rh-atoms on the surface and this step leads to the lengthening of the NdN bond which breaks in the subsequent steps [50]. This electron donation mechanism is true for almost every catalytic process that includes electrides. The high-throughput ab initio calculations and screening presented by Prof. H. Hosono and coworkers have described that features like the oxidation state, chemical formula, and/or the correspondingly assigned formal charges cannot be used as a direct and suitable method to find an electride [51]. These features can give us a hunch for several candidates who might (or might not) act as 2D inorganic electride materials, but complete electronic structure-based calculations are required. Features like the Fermi velocities of anionic electrons, the number ratio of anionic electrons, bandwidths, the ionic ratio, and energy difference with respect to the existing stable compounds should be carefully evaluated and discussed to identify a 2D inorganic electrides [51].

1.5 Toward the molecular electride In 2005, computational work has shown that the host-guest complex, Li@calix[4]pyrrole behaves like an electride. Here, one Li-atom is intercalated within the calix[4]pyrrole cage and the point group of symmetry of the resulting complex is C4v. The four lone pairs of electrons on N-atoms push away the electron density from the 2s-orbital of the host Li-atom. The resultant complex is an electride and the molecular formula can be expressed like Li+calix[4]pyrrolee
[52]. The comparison of the average polarizability and first hyperpolarizability values of the parent calix and resultant complex show that there is a tremendous increase due to the Li atom incorporation. It is the first example that a single molecule is showing electride property and such systems are known as molecular electrides [53]. Other examples in these types are Li+B10H14e
[54] and its fluoro derivatives [55], M+2 TCNQe
(M ¼ Li, Na, K, and TCNQ ¼ tetracyanoquinodimethane) [56], M+(e
@C20F20) and M3O+(e
@C20F20) (M ¼ Na, K) [57], e
@C60F60 [58], Mg2EP (EP ¼ Extended (3.1.3.1) Porphyrin) [59], Mg2@C60 [60], Li+6 [61], Be6 [61], and many more. All of the abovementioned examples of electrides contain at least one localized electron (or a significant share of it) in spatial regions or inside the cavity. Recently, a novel prototype of molecular electrides called metal cluster electrides has been suggested where delocalized polyattractor character is found [61]. This computational work illustrates that chemical moieties like octahedral 4A1g Li+6 and 5A1g Be6 have a large number of nonnuclear attractors along with a highly delocalized valence electron density.

278 Atomic clusters with unusual structure, bonding and reactivity

2.

Norms and conditions of being a molecular electride

Questions like “What are the geometric and electronic signature of molecular electrides?” or “What should be the minimum but essential criteria to identify a system as a molecular electride?” have been raised several times. One thing for sure is known for electrides that they should have localized electrons and it “does not ‘belong’ to any particular atom, molecule, or bond” [2]. But quantification of the amount of localization of the electrons by the experimental procedure is still unknown. The signature of the localized electrons in the electrides is not powerful enough that they can be mapped out using X-ray spectroscopy and/or by other experimentally known methods or techniques. Here, the indirect quantum chemical tools are very useful to identify and locate the position of the localized electrons and to quantify them in the molecular electrides. The computational works by Sola` et al. and Dale et al. using high-level calculations have demonstrated the criteria to identify and describe the electrides and differentiate them from other compounds (that show equivalent properties) [4,62,63]. The computational criteria that a molecular electride should possess can be detailed as: (a) Presence of the NNA of the electron density The nonnuclear maximum (NNM), is widely known as nonnuclear attractor (NNA), is a more used term in the chemical literature, which stands for the local maximum of the electron density [r(r)] in molecules or clusters [64,65]. As per Bader’s “Quantum Theory of Atoms in Molecule (QTAIM)”, the NNAs are (3,
3) types of critical points (CP) without atomic nuclei associated with them [66,67]. The NNA attracts its own density gradient [rr(r)] vector field forming a basin that is equivalent to the regular atomic basins and is also known as “pseudoatoms” [68]. In the molecular electrides, these NNAs accumulate electron density and act as anions. The population of electron(s) in the NNA basin, the volume of the NNA basin, and the amount of localization of the electron in NNA are possible to compute for the characterization of molecular electride. This atomic nature of localized electrons in the molecular electrides is a necessary condition and that is possible to prove by the presence of the NNA [61]. For example, Li@calix[4]pyrrole [52] and Li@B10H14 [54], both uni-lithiated complexes are argued as an electride, but the presence of NNA is confirmed in the former one and hence it is an electride [4]. Furthermore, the strength of the electride character can also be evaluated from a detailed analysis of the NNA basin. Matio and coworkers [69] have answered several questions like “How many electrons does a molecular electride hold?”, “Is it possible to connect the formal oxidation state of the NNA basin with the number of electrons present therein?”, “What is the probability of finding at least one electron in the NNA basin?” and connected them with the strength of the electrides. The probabilities of finding zero, one, and two electrons in the associated NNA basin are P0, P1, and P2, respectively. The probability of finding at least one electron is (1
P0) and if it is greater than P0, i.e., (1
P0) > 0.5, then that molecular electride holds at least one electron at the NNA basin. The study over nine molecular electride showed that Li2TCNQ, Mg2EP, and Mg2@C60 are one-electron electride, as, (1
P0) > 0.5. Others fall between zero- and one-electron electrides (P0  0.5, viz., Na2TCNQ, Na3TCNE, and Na4TCNE) and low-electron-number electrides (0.5 ≫ (1
P0) ≫ 0), viz., Li@calix[4]pyrrole, Na@calix[4]pyrrole, and e
@C60F60. The probability of finding two electrons (P2) at the NNA basin of the electrides are negligible (P2  0), except for Mg2EP and Mg2@C60. The effective oxidation state (EOS) calculations have shown that neither of the partition schemes applied can assign a
1 formal oxidation state of the NNA for these two Mg2EP and Mg2@C60 electrides. (b) Negative value of the Laplacian of electron density [r2r(r)] at the NNA This criterion can be considered as interdependent to the previous one of having an NNA in the molecular system. As per the QTAIM, the NNAs are (3,
3) type of CPs, so all of the three curvatures are certainly negative and all three components of the r2r(rc) are negative, which results in a total negative r2r(rc) value at the NNA [4,7,61]. (c) Existence of the ELF basin near the NNA The ELF is described as the probability density for finding an electron in a space in the presence of another like-spin electron near the reference point [70]. The ELF value ranges between 0.0 and 1.0, representing the lowest and highest probability of finding another like-spin electron, respectively. In the molecular electrides, a high value of ELF (near 1.0) is an indication that a localized electron is present and a plot of ELF shows the location of the localized electron. One should carefully differentiate between the localized free electrons with the lone pair during the ELF analysis. (d) High NLO properties The electrides are known to have high NLO properties due to the presence of loosely bound and polarizable electrons. The diffuse nature of these electrons leads to large NLO properties. The values of the average linear polarizability (a), first

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hyperpolarizability (b), and second hyperpolarizability (gk) were calculated (details are in a later section). A sharp increase in the NLO properties can be seen after the incorporation of the alkali metal atom(s) in the organic molecules or cavity, e.g., the TCNQ molecule is symmetrical and the first hyperpolarizability is zero. The complexion with Li2, Na2, and K2 in M+2 TCNQe
[56] results in a dramatic increase in the b value. The second hyperpolarizability (gk) values also show a large increase due to the incorporation of the alkali metal dimers. The dynamic NLO properties (up to the second hyperpolarizability) are also important to be considered as well as the static one [71]. Sometimes, such a choice of analysis is necessary to overcome the resonance enhancement effects [7]. The NLO properties of the molecular electrides are studied through the frequency-dependent second harmonic generation (SHG), the electro-optic Pockels effect, electric field-induced SHG, etc. [72–74]. These analyses are helpful for experimental validation in future. (e) Noncovalent interaction index The noncovalent interactions (NCI) can be mapped and the location of the localized electron in molecular electride is possible to know using the NCI indices [75,76]. The generation of the green surface in the interspatial cavities indicates that the electrons are localized therein. The computations of molecular electrostatic potential (MESP) and their plots were considered to be useful to trace the localized electrons in the system [77]. In the study, it was also mentioned that the MESP cannot differentiate between the lone pairs and the localized electrons in the electride. Thus, one should be careful using the MESP to identify the electrides. Thus, MESP computations are not considered a good indicator to characterize the molecular electrides. The systems that fall under the molecular electrides are found to show a tremendous variety of bonding nature, e.g., in the Li+calix[4]pyrrolee
, the electron density from the 2s-orbital of the Li-atom is pushed away by the four N-atoms of the pyrrole rings. Another example of M+2 TCNQe
(M ¼ Li, Na, and K) shows that the electron density from the bonding orbital of M2 has been pushed away by the N-atoms therein. The pushed electron from the molecular electrides can be useful in a similar way to the inorganic electrides to activate small molecules. For example, in the molecular Mg2EP electride, the electron density from the MgdMg bonding orbital is pushed into space which can be donated into antibonding molecular orbitals of small molecules to activate them [59]. Compared to the other kinds of electrides, molecular electrides are not developed that much. The bonding features in molecular electrides and their applicability have rouse interest among computational chemists recently. Several computational studies have been done in recent years and more to come in the upcoming era.

3. Computational methodology To identify the electride characteristics of a chemical system, it is very important to use appropriate computational techniques, otherwise, there will be several flaws in the results. For the crystalline organic and inorganic electrides, computations including the periodic boundary conditions (PBC) are helpful [34,35,62,63,78,79], but they are out of the context of the current discussion in this chapter. For computations on molecular electrides, the optimization (without any constraints) of the desired geometries has been performed using Gaausian09 [80] and Gauian16 [81] program packages. Ab initio methods and/or density functional theories (DFT) have been employed for this purpose. Density functionals like B3LYP (Becke, 3-parameter, Lee-Yang-Parr) [82–85], BP86 (Becke 1988 exchange functional and the Perdew 86 correlation functional) [86–88], M05-2X [Minnesota functional published in 2005 with 56% Hartree-Fock (HF) exchange] [89,90], M06-2X (Minnesota functional published in 2006 with 56% HF exchange) [91], and TPSSTPSS (both exchange and correlation functionals of Tao, Perdew, Staroverov, and Scuseria) [92,93] have been used along with atomic basis sets, like 6-31G(d) [94], 6-311+G(d,p) [95–97], and def2-TZVPP [98,99] depending on the nature of atoms in the structure, thorough literature studies on the benchmark of the geometrical parameters and own calculations to optimize the geometries. For the ab initio calculations, MP2 [100,101] method along with cc-pVDZ, cc-pVTZ, cc-pV(D+d)Z, and cc-pV(T+d)Z basis sets have been used [102,103]. The hessian calculations have been performed considering the optimized geometries to ensure that the geometries are in minima on their respective potential energy surfaces. These optimized geometries are employed to study the nature of bonding, electride characteristics, and their reactivity toward chemical reactions. The natural bond orbital (NBO) [104] analysis is one of the great tools to evaluate the electronic charge on each atom and bond order between a pair of atoms, via, the natural population analysis [105] and the Wiberg bond indices (WBI) [106], respectively. The charge on each atom helps us to understand the direction of charge flow in the system and WBI values are helpful to understand the nature of interactions present therein. The NBO analyses have been performed by using NBO 3.1 [107] as implemented in Gaussian09 and Gaussian16.

280 Atomic clusters with unusual structure, bonding and reactivity

As mentioned earlier that the molecular electrides are identified via the signature of the presence of NNAs and negative Laplacian of electron density (r2r(rc)) at NNA. The atoms in molecules (AIM) [66] analysis helps to look for all CPs present in the system and helps to identify the NNAs. Furthermore, the electron density descriptors associated with the NNAs are evaluated. Moreover, the analysis of the bond CP is also helpful to understand the complete bonding scenario of the systems. The analyses were performed with the help of Multiwfn software [108]. One should know about the limitations of the density functional approximations (DFAs) applied for the calculations on such systems which might induce spurious justifications. Many DFAs possess delocalization errors due to the factitious self-interactions, which results in the overestimation of the electron delocalization in the system. In general, DFAs which have a small percentage of HF exchange are prone to exaggerate electron conjugation and aromaticity in the system. This error might interrupt the presence of NNAs in the electronic structure of the materials. For example, in the case of solid-state inorganic electrides, difficulties arise during the detection of NNAs due to the delocalization error. As it is discussed, the presence of NNA and its quantification depends on the DFAs, but the quality of the basis set should also be kept in mind. Matio and coworkers have shown the importance of the percentage of HF exchange into the DFAs and the quality of basis sets to characterize the NNAs and their properties [69]. As an example, ma-TZVP is the worst choice of basis set for Li2%+ TCNQ%
, as this basis set does not possess functions of angular momentum higher than 1 for Li and hence, this basis set is unable to predict the presence of NNA (conjugated with the CAM-B3LYP and B3LYP functionals). Whereas, large basis sets like ma-QZVP, cc-PVTZ, and aug-cc-PVTZ have functions of angular momentum higher than 1 (d, f, and extra p functions) and this results in a good description of the system with the presence of NNA. The DFAs with a higher percentage of HF exchange, calculate higher values of NNNA. It is suggested to look into the HF and MP2 densities, derived using a good quality basis set, along with the densities that have been calculated with DFAs. The NCIPLOT program package has been used for the calculation of the NCI index. The SCF densities have been used for the NCI analysis as it is more accurate and necessary for the systems under discussion. The NCI analysis involves two different scalar quantities, electron density (r), and reduced density gradient (s). These two quantities are connected as, s¼

1 2ð3p2 Þ1=3

jrrj : r4=3

In the case of the electrides, due to the presence of (3,
3) CP the density gradient (rr) and hence s become zero, which can be observed in the two-dimensional “s vs. r” plot. The gradient isosurface plots help to visualize the location of the trapped electron in molecular electride. The VMD software has been used for this purpose [109]. Though the NCI approach is very useful and popular but it cannot differentiate between the origin of the trough (where the s become zero) is due to the (3,
1) type of BCPs regarding the weak noncovalent type of interactions between two atomic pairs or due to the presence of NNA. For the evaluation of the NLO properties [7,110], the average linear polarizability (a) and first hyperpolarizability (b) and second hyperpolarizability (gk) have been computed as shown in the following equations: 1 X
a a¼ 3 i¼x, y, z ii !12 X 2 b¼ bi i¼x, y, z

 1 X bijj + bjij + bjji 3 j¼x, y, z  1 X  giijj + gijij + gjjii and gk ¼ 15 i, j¼x, y, z

where bi ¼

One should note that for the calculation of these NLO properties, the selection of appropriate computational methods and basis sets are necessary [111]. The application of sophisticated quadratic configuration interaction including single and double excitation (QCISD) [112] method produces satisfactory results. The computational cost is high for these kind of calculations. The second-order Møller–Plesset (MP2) [113] perturbation method can be used instead of QCISD as it takes care of the dynamical correlation. But for larger systems, the enhanced computational cost prohibits its use. The density functionals like B3LYP [82–85], CAM-B3LYP [114,115], and M06-2X [91] have been widely used to evaluate such properties as they produce comparable results with affordable computational cost. The diffuse basis sets are required conditions to evaluate such hyperpolarizabilities [116,117].

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4. Examples of molecular electrides 4.1 Alkali metal-doped electrides From the starting era of electrides, the metal atoms from the alkali group are the center of attraction due to their glorious journey in electride chemistry. In all of the earlier examples of organic electrides by Professor J. L. Dye, we have seen the use of alkali metal atoms [2,21–29]. This feature attracts theoretical chemists to study deeper of the field of electrides using alkali metal atoms as dopants and to evaluate their geometrical, electronic, and NLO properties. In 2005, Li-doped (HCN)n (n ¼ 1–3) cluster have been studied thoroughly [118]. Two different kinds of isomers have been considered. The N-atom of the HCN bound to Li-atom and the complexes have been represented as (HCN)n⋯Li. On the other hand, the H-atom of the HCN is bound to Li-atom and the complexes are shown as Li⋯(HCN)n. In the (HCN)n⋯Li systems, the lone pair on the Natom pushes the electron from the 2s atomic orbital of the Li-atom. The singly occupied molecular orbital (SOMO) of HCN⋯Li is shown in Fig. 2A, where the push from N-atom is visible. The Li⋯(HCN)n complexes show a different mechanism, here the electron from the 2s atomic orbital of the Li-atom has been pulled by the H-atom. Fig. 2B shows the SOMO of Li ⋯ HCN, where the 2s electron of Li-atom is pulled toward the H-atom. Depending on this mechanism, the electrides are termed as pushed electrides and pulled electrides for the former and latter examples, respectively. The average polarizabilities and first hyperpolarizabilities for all of these systems have been computed at the MP2/6-311 ++G(3df,3pd) level of theory (see Table 1). In both of the cases, an enormous increase in the NLO properties has been observed due to the incorporation of Li-atom. Another observation is that the pushed electrides show higher values of the NLO properties as compared to that of the pulled electrides. The complexant, calix[4]pyrrole has been employed to host one Li-atom near the central position of its cavity. The resultant Li@calix[4]pyrrole complex has a perfect C4v point group of symmetry. The four lone pairs of electrons on N-atoms give a strong push toward the 2s valence electron on the Li-atom as shown in Fig. 3A. The SOMO of Li@calix[4]pyrrole is the diffuse s-orbital of the Li-atom as the result of the push from the N-atoms and can be held responsible for the high NLO properties. A schematic representation of the “push” has been depicted in Fig. 3B. The system shows high average polarizability and first hyperpolarizability values which have been much increased as compared to the alone calix[4]pyrrole. The system fulfills all the criteria to be an electride and it can be designated as molecular electride with a formula as, Li+calix[4]pyrrolee
. Another well-known example of lithiated electride is Li@B10H14 (see Fig. 4A). Here the Li-atom is bound to the open hole of the B10H14-basket. The electron-deficient B-atoms of the basket pull the valence electron density from the Li-atom inside the B10-basket. This results in the oxidation of the Li-atom and the Li acquires a unipositive electronic charge on it. The system is represented as Li+B10H14e
. The fluoro-derivatives of the B10H14 have been studied to emphasize their electride nature and NLO properties [55]. The H-atoms of the B10H14 basket have been consecutively replaced with the F-atoms. The resultant lithiated complexes show an increase in the NLO values upon replacement of H-atoms with F-atoms. Multiple metal atoms incorporation and the resultant complexes have also been studied and they show promising results in the electride field. Alkali metal dimers (M2; M ¼ Li, Na, and K) can bind to one of the dC(CN)2 sides of the TCNQ, which is a strong electron acceptor and produce the resultant complexes as, M2%+TCNQ%
[56]. The optimized systems are found to be in the triplet ground state. The Li2%+TCNQ%
is shown in Fig. 4B as a representative of the M2%+TCNQ%
systems. The M2 dimer incorporation causes a drastic increase in the NLO properties and the charge transfer into the system is being responsible for that. The lone pairs of electrons on the N-atoms push the electron density from the MdM bonding

FIG. 2 The geometry and singly occupied molecular orbital (SOMO) of (A) HCN⋯ Li and (B) Li⋯ HCN. The color code follows as, gray indicates C, white indicates H, blue (dark gray in print version) indicates N, and purple (light gray in print version) indicates Li atoms.

282 Atomic clusters with unusual structure, bonding and reactivity

TABLE 1 Average linear polarizabilities (a), first hyperpolarizabilities (b), second hyperpolarizabilities (gk), and the summary of the criteria used to characterize molecular electrides.

NLOP

Li@calix[4] pyrrolea

Li@B10H14

HCN ⋯ Li

Li ⋯ HCN

Li2TCNQ

Na2TCNQ

✓

✓

✓

✓

✓

✓

a

381.6

150.3

238.6

181.8

371.9

389.7a

b

1.04  104a

7.25  103a


1.57  104b


3.39  103b

3.65  104a

2.16  104a

gk

4.8  106a

4.8  105a

n.a.

n.a.

1.4  106a

1.3  106a

NNA

✓







✓

✓

a

a

b

b

c

N(O)NNA

0.15

(r2r(rc))

✓

c







✓

✓

✓

c

N(O)ELF

0.63

Molecular electride

Yes

✓

✓
3c


3.67  10 ✓

0.23c

0.52


3c

ELF

a


7.37  10

6.14  10
3c

✓

✓ c

No

No

No

0.92

0.26c

Yes

Yes

N(O) represents the electron population at the respective basin. a Values are taken from Ref. [7]. b Values are taken from Ref. [118]. c Values are taken from Ref. [4].

FIG. 3 (A) The geometry of Li+calix[4]pyrrolee
and (B) schematic representation of the “push” by the lone pairs on N-atoms toward the 2s electron of Li-atom, two pyrrole rings are hidden for clarity. The color code follows as, gray indicates C, white indicates H, blue (dark gray in print version) indicates N, and purple (light gray in print version) indicates Li atoms.

FIG. 4 The geometries of (A) Li@B10H14, (B) Li2%+ TCNQ%
, and (C) e
@C60F60. The color code follows as, pink (light gray in print version) indicates boron, white indicates H, gray indicates C, blue (dark gray in print version) indicates N, green (light gray in print version) indicates fluorine, and purple (light gray in print version) indicates Li atoms.

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orbital to space. The NLO properties increase on going from Li to K, via Na (see Table 1). The bigger size and incorporation of the loosely bound electron are responsible for this observed phenomenon. The four H-atoms connected to the N-atoms (the dNH group of the pyrrole) in the Li+calix[4]pyrrolee
have been substituted with four Li-atoms and the electride characteristics have been explored computationally. It is found that the electron population into the NNA and ELF basins are 0.74 e and 1.02 e, respectively which indicates a sole electron is residing separately and serving as anion to the system. Multiple Na-atoms decoration and their effect on electride property have been carried out. The porphyrin as the host system has been considered and the study has shown that a maximum of 10 Li-atoms can be incorporated into one porphyrin ring [119]. The resultant Li10Pr (Pr ¼ porphyrin) system has fulfilled all the computational checkpoints and established itself as a molecular electride. Another example has been provided where a maximum of four Li-atoms are possible to incorporate within tetracyanoethene (TCNE) [120]. The binding energy analysis shows the stability of the complexes. But complete analysis shows that only Li3@TCNE acts as a molecular electride where the others are lithiated salt. All of the above-discussed electrides have their excess electron density near the alkali metal atoms. The e
@C60F60 is an example where the C60F60 system stabilizes one electron inside it and behaves as an electride [48] without the presence of an alkali metal atom. These types of systems are known as nonalkali electride. The geometry of the e
@C60F60 is shown in Fig. 4C. The electron density analysis shows an NNA with a population of 0.19 e and with 18% localization is present at the center of the cage. A corresponding ELF basin is also present therein. Although these studies have provided information about the electrides, still identification of a true electride is very essential. The criteria mentioned in Section 2 should be strictly followed to tag a system as an electride. For example, Li@calix[4]pyrrole and Li@B10H14 (see Table 1), both systems were known as electride for a long time. But the thorough computational works [4] have revealed that the Li@B10H14 system does not have any pseudoatoms present therein. Thus, the atomic behavior of the electron and hence “electride” property cannot be established for Li@B10H14. A similar fate is true for HCN ⋯ Li and Li ⋯ HCN systems (see Table 1). Though they full fill other criteria, but due to the lack of the presence of NNA, they are excluded from the list of being electride [4]. Similar to the alkali metals, complexes of alkaline earth metals also show the presence of NNA. The presence of NNA in dimeric magnesium (Mg) complex, {Mg(DippNacnac)}2 (DippNacnac ¼ [(Dipp)NC(Me)]2CH, Dipp ¼ 2,6-diisopropylphenyl) [121], has been confirmed by high-resolution X-ray diffraction data and DFT-based calculations [122]. The position of the NNA in this complex is in the middle of the Mg(I)dMg(I) bond. Motivated from this result, a new class of molecules, complexes, or clusters have been searched where the electron density from the NNA can be pushed in the space. The first example of this kind is the Mg2 dimer complex with an extended porphyrin.

4.2 Mg2EP, molecular electride and small molecule activation The binding mode of the DippNacnac (DippNacnac ¼ [(Dipp)NC-(Me)]2CH, Dipp ¼ 2,6-diisopropylphenyl) ligand is mimicked in the extended porphyrins. The normal porphyrins are not able to accumulate two Mg-atoms inside the periphery [123]. The extended (3.1.3.1) porphyrin [124,125] (denoted as EP) is considered for the study and it is a good choice to 2
stabilize the Mg2+ anion. The formation of the Mg2EP [59] complex is thermodynamically feasible as 2 ion using the EP the changes in the Gibbs free energy values are negative. The Mg2EP complex has a nonplanner geometry, where the two Mg atoms are slightly out of the plane, have been shown in Fig. 5A. The geometry of Mg2EP is in the C2v point group of ˚ , respectively which have been computed at the symmetry. The MgdMg and MgdN bond lengths are 2.787 and 2.062 A M06-2X-D3/6-311G(d,p) level of theory. Ab initio molecular dynamics simulation shows the inversion of the “Mg2” fragment from one side of the plane to the other side occurs at 500 K temperature. The barrier of the inversion is computed to be 5.8 kcal/mol. The complex has one Mg(I)dMg(I) s-bond. The HOMO (see Fig. 5B) corresponds to this MgdMg s-bond. The overlap of the 3 s-atomic orbitals of Mg atoms results in the formation of this Mg(I)dMg(I) s-bond. One NNA is present near the middle point of the two Mg-atoms. The AIM analysis shows that there is no direct MgdMg bond path (BP), rather than the two Mg-atoms are connected as Mg-NNA BP and then another NNA-Mg BP (see Fig. 5C). This picture is quite common and a signature for the dimeric Mg complexes. The r2r(rc) value is negative at NNA and an ELF basin is present near the NNA. The population of electrons at the NNA basin as per the electron density analysis is 1.02 e with 46% of localization. On the other hand, the ELF basin computations show 1.78 e population with 61% of localization in it. The a, b and gk values of the Mg2EP are 493.7, 3.62  102, and 1.1  105 au, respectively. A comparison between the Mg2EP and other known electrides has been carried out in terms of the NLO parameters shows that the avalue of the Mg2EP is much higher as compared to the other systems whereas the b and gk values are comparable or lower. As all of the criteria have been satisfied, the Mg2EP is termed as an electride.

284 Atomic clusters with unusual structure, bonding and reactivity

FIG. 5 (A) The optimized geometry of Mg2EP, (B) the HOMO of Mg2EP, and (C) the molecular graph of Mg2EP. The color code follows as, gray indicates C, white indicates H, blue (dark gray in print version) indicates N, and green (light gray in print version) indicates Mg atoms.

FIG. 6 (A) Relative Gibbs free energy (kcal/mol) profile of H2 activation with the help of Mg2EP and (B) relative Gibbs free energy (kcal/mol) profile of CO2 activation with the help Mg2EP. Both are calculated at the M06-2X-D3/6-311 + G(d,p) level of theory. The color code for both figures are as follows: gray indicates C, white indicates H, blue (dark gray in print version) indicates N, red (dark gray in print version) indicates O, and green (light gray in print version) indicates Mg atoms.

A step ahead, the loose electron can be donated into the antibonding molecular orbital associated with several bonds like HdH (in H2), CdO (in CO2), NdO (in N2O), and aliphatic and aromatic CdH (in CH4 and C6H6, respectively) [59]. The fate of this donor-acceptor interaction leads to the activation followed by the cleavage of the abovementioned bonds. The H2 molecule reacts with the Mg2EP and at the end produces Mg2EP(m-H)2. The reaction profile at the M06-2X -D3/6-311 + G (d,p) level of theory has been shown in Fig. 6A. The barrier height for the process is 36.1 kcal/mol. The two H atoms are connected to both of the Mg-atoms, via m-H connectivity, i.e., banana bond formation. The activation followed by the

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dissociation of the CdH bonds in CH4 and C6H6 follow the similar mechanism as shown for the H2 molecule and result in the formation of Mg2EP(m-CH3)(m-H) and Mg2EP(m-C6H5)(m-H) complexes, respectively. The reaction barriers are 43.5 and 36.2 kcal/mol for the aliphatic and aromatic CdH bonds, respectively. The mechanism for the activation of the CdH and C]O bonds does not follow a similar mechanism. The mechanistic pathway for the CO2 is given in Fig. 6B which has been computed at the M06-2X-D3/6-311 + G(d,p) level of theory. The activation barrier for the CO2 is 5.9 kcal/mol, which is lower than the other Mg(I) complexes [126]. This low activation barrier makes Mg2EP an attractive catalyst for CO2 activation. As the reaction progresses, one C]O bond breaks and produces Mg2EP(m-O) and CO. The resultant complex, Mg2EP(m-O) has a bridged oxo (m-O) group bound to both of the Mg atoms. The CO extrudes out of the system and the Mg2EP(m-O) further binds to another CO2 molecule and results in a carbonate bridged complex as shown in Fig. 6B. The reaction profile of the N2O with Mg2EP follows a similar way as CO2 activation. The resultant complex is Mg2EP(m-O) along with the extrusion of the N2 molecule.

22 4.3 Bonding in [Mg4(Dipp complex and its electride nature HL)2]

A modified version of the DippNacnac ligand has been designed as Dipp MeL, keeping the b-diketiminate fragment intact for the binding to Mg atoms. Moreover, the denticity of the ligand supports forming a complex which can hold four Mg-atoms simultaneously [127]. The studied complex ([Mg4(DippHL)2]2
) is shown in Fig. 7A. The four Mg-atoms generate two equiv˚ at the M06-2X-D3/6-31G(d) level of theory. The complex alent MgdMg bonds where the MgdMg distance is 2.885 A does not show presence of any 4c-2e bond. A similar synthetic route as {Mg(DippNacnac)}2 has been proposed for the [Mg4(DippHL)2]2
and the thermodynamic parameters show that the reaction is exergonic in nature. A counter-cation, [K@CE]+ (CE ¼ crown-6-ether) is considered to neutralize and stabilize the system and this process is also exergonic. 2
Thus, both of the complexes, [Mg4(Dipp and [Mg4(DippHL)2]2
2[K@CE]+ (see Fig. 7B) are viable and possible HL)2] to synthesize in the future. The molecular orbital and natural orbital pictures show presence of two simultaneous Mg(I)dMg(I) bonds. The electronic charges on the Mg atoms are 0.95 j ej which is close to +1. The inspection of the Mg(I)dMg(I) bond orbitals shows the overlap between the 3s-atomic orbitals. The AIM analysis reveals the presence of two NNAs near the central point of the two Mg(I)dMg(I) bonds. Fig. 7C shows the presence of the two NNAs in [Mg4(DippHL)2]2
. The electron population at the NNA is 1.18 je j with 52% of localization in it as computed at the MP2

2
FIG. 7 The optimized geometries of (A) [Mg4(Dipp and (B) [Mg4(DippHL)2]2
2[K@CE]+, The color codes for both figures follow as: gray indicates C, H L)2] white indicates H, blue (dark gray in print version) indicates N, green (light gray in print version) indicates Mg, red (dark gray in print version) indicates O, 2
and purple (light gray in print version) indicates potassium atoms, (C) the molecular graph of [Mg4(Dipp and (D) the plot of Laplacian of electron density H L)2] 2 (r r(r)), the magenta colored solid and blue (dark gray in print version) colored dashed lines represents the regions with r2r(r) > 0 and r2r(r) < 0, respectively and (E) two-dimensional plot of ELF.

286 Atomic clusters with unusual structure, bonding and reactivity

method. The ELF basin calculations have also revealed high electron localization therein. The negative value of the r2r(r) 2
and plot of the ELF basin for the [Mg4(Dipp is shown in Fig. 7D and E, respectively. These positions of the localized HL)2] 2
2
+ Dipp electron in both of the [Mg4( HL)2] and [Mg4(Dipp HL)2] 2[K@CE] is shown by the plots of the NCI. The a, b and gk 2
5 Dipp values of the Mg4( HL)2] are 891.7, 0.0, and 8.9  10 au, respectively. This high a value is attributed to the presence of two Mg(I)dMg(I) bonds. The b value is zero due to the symmetric structure of the Mg4(DippHL)2]2
. The Mg4(DippHL)2]2
complex passes all of the criteria as an electride.

4.4 Mg2@C60 and its electride characteristics The C60 cage [128] can take gas molecules, metal atoms inside its cavity and can form endohedral fullerene [129–137]. When the encapsulated guest species are metals, the systems are called endohedral metallofullerenes [138–141]. Encapsulation of dimeric Mg2 molecule at the cavity of the C60 cage has been designed and the resultant system is Mg2@C60 [60]. Three isomers of the designed system have been generated having the point group of symmetry C2h, D2h, and Ci but the minimum energy isomer has C2h point group of symmetry, which is shown in Fig. 8A. The difference in energy between the singlet and the triplet state is low. Due to the confinement effect by the C60 cage, the MgdMg bond length decreases as compared to the free Mg2 molecule and Mg2+ 2 ion, and the vibrational frequency of the MgdMg bond increases. The ˚ for the singlet state, whereas that of the triplet state is MgdMg bond distance in the confined system is 2.598 A ˚ 2.594 A, both have been computed at BP86-D3/def2-TZVPP level of theory. The Gibbs free energy change for this encapsulation process is negative, which indicates the formation of this endohedral system is spontaneous. The atom-centered density matrix propagation (ADMP) simulation [142–144] at 500 K temperature shows the movement of the Mg2 unit inside the C60 cage while no rupture of the cage during the simulation. The system contains one Mg(I)dMg(I) bond which is confirmed by the NBO and MO analyses. The charges on both Mg atoms are 0.95 jej which is close to +1. The overlap of the 3s atomic orbitals of both Mg atoms is responsible for the formation of the MgdMg bond. The charge transfer takes 2
place from the Mg2 moiety to the C60 cage by 2 unit and the system is represented as Mg2+ 2 @C60 . The energy decomposition analysis (EDA) also supports the statement. The AIM analysis shows that the system has an NNA at the center of the MgdMg bond as shown in Fig. 8B. In the singlet state of the system, the NNA population is 0.56 je j with 23% localization of electron density. While in triplet state the population of the NNA is 0.60 je j with 27% localization of electron density. An ELF basin is present near the NNA region and the basin population for singlet and triplet states are 0.89 jej with 28% localization and 1.80 je j with 54% localization of electron density, respectively. At the NNA, the values of r2r(rc) are negative (see Fig. 8C). The NCI plots reveal the position of localized electrons (Fig. 8D) in both singlet and triplet FIG. 8 (A) The optimized geometry of Mg2@C60, gray and green (light gray in print version) indicate C and Mg atoms, respectively. (B) the molecular graph of Mg2@C60 showing the position of NNA, (C) the plot of the Laplacian of the electron density [r2r(r)], the magenta colored solid and blue (dark gray in print version) colored dashed lines represents the regions with r2r(r) > 0 and r2r(r) < 0, respectively, and (D) the NCI plot for Mg2@C60.

Molecular electrides Chapter

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spin states of the system. The a, b and gk values of the system are 586.6, 0.0, and 1.9  107 au, respectively. The symmetrical geometry of the system causes the b value to be zero. The designed system obeyed all the necessary conditions and is termed as an electride.

4.5 Binuclear Sandwich complexes of alkaline earth metals as electrides


Four isoelectronic cyclic ligands (L ¼ C5H
5 , N5 , P5 , and As5 ) have been considered for the bonding with Be and Mg metal dimers [145]. These ligands can bind with the metals through n (n ¼ 1–5) modes. The 5 mode of binding is most stable among all of the binding modes under consideration. This 5 mode of binding results two conformations for all of the complexes, viz., eclipsed, and staggered. These two conformations correspond to the almost same energy conformers. Only staggered conformations have been discussed here as the eclipsed conformations give similar results under the bonding context. Fig. 9A shows the minimum energy structures of the studied complexes along with the optimized bond lengths. The complexes show high positive values of change in the Gibbs free energies (DG) and zero-point corrected dissociation energies (D0) toward dissociation processes which indicates that all of the studied complexes are stable at room temperature. The complexes of Be show higher thermodynamic stability as compared to their Mg analogues. The MO analysis and the NBO analysis show the presence of M(I)dM(I) s-bonds in all cases. The MdM bonds are covalent in nature, whereas the MdL bonds are ionic. The charges on Be atoms are 0.83, 0.85, 0.61, and 0.55 j ej in Be2(5-C5H5)2, Be2(5-N5)2, Be2(5-P5)2, and Be2(5-As5)2 complexes, respectively, and the charges on Mg atoms are 0.91, 0.92, 0.82, and 0.79 je j in Mg2(5-C5H5)2, Mg2(5-N5)2, Mg2(5-P5)2, and Mg2(5-As5)2 complexes, respectively. The BedBe and MgdMg bond orbitals are formed by the overlap of 2s and 3s atomic orbitals on Be and Mg atoms, respectively.
The EDA confirms that M2+ are the bonding fragments. The electron density descriptors show that each 2 and 2L MdM bond contains one NNA near the center of that bond (see Fig. 9B). The r2r(rc) values at the NNAs are negative for all the complexes. The NNA populations and the percentage localization of electron density at the NNAs for the designed complexes are given in Table 2. An ELF basin is present at the middle of the M(I)dM(I) bond (M ¼ Be and Mg) indicating localization of electron density therein (see Fig. 9C). The a, b, and gk values for all the complexes are also given in Table 2. The symmetrical geometries of all the complexes cause the b values to be zero. All the complexes under study fulfilled all the necessary conditions and hence they act as electrides.




FIG. 9 (A) The optimized geometries of M2(5-L)2 (L ¼ C5H
5 , N5 , P5 , and As5 ), The color code for both figures follows as, gray indicates C, white indicates H, blue (dark gray in print version) indicates N, saffron indicates P, purple (light gray in print version) indicates As, light green (light gray in print version) indicates Be, and deep green (light gray in print version) indicates Mg atoms. (B) the molecular graph of M2(5-L)2 showing the position of NNA and (C) the plot of the ELF for M2(5- P5)2.

288 Atomic clusters with unusual structure, bonding and reactivity

TABLE 2 NNA population (N(V)), localization index (LI), and percentage of localization index (% LI), average linear polarizability (a), first hyperpolarizability (b), and second hyperpolarizability (g k) of all the studied complexes. Complexes

N(V)

LI

%LI

a

b

gk

5

Be2( -C5H5)2

1.34

0.72

54

131.7

0.0

6.3  104

Be2(5-N5)2

1.39

0.86

62

82.8

0.0

1.3  104

Be2(5-P5)2

1.32

0.65

49

257.4

0.0

1.5  105

Be2(5-As5)2

1.33

0.65

49

317.1

0.0

2.9  105

Mg2(5-C5H5)2

1.02

0.49

48

176.6

0.0

1.1  105

Mg2(5-N5)2

0.98

0.47

48

115.5

0.0

2.1  104

Mg2(5-P5)2

0.95

0.42

44

309.6

0.0

1.8  105

Mg2(5-As5)2

0.95

0.41

43

374.9

0.0

3.0  105

4.6 Li3@Cg (Cg 5 B40 and C60) and their electride nature Cages are being used for the encapsulation of various entities ranging from gas molecules to metal atoms. Due to the differences in the sizes of the cavity, different cages can uptake a variety of molecules and/or clusters inside their cavity. The Li3 cluster has been encapsulated in the cavity of B40 [146] and C60 [128] cages and the resultant systems are Li3@B40 and Li3@C60, respectively [147]. The geometries of these systems are given in Fig. 10A and B for Li3@B40 and Li3@C60, respectively, computed at BP86-D3/def2-TZVPP level of theory. The three LidLi bond lengths are not equal for a given Li3@Cg system. The LidLi bond distances for these two confined systems show smaller values as compared to that of the ˚ for Li3@B40 and 2.757, 2.663, and 2.664 A ˚ for free Li3 cluster. The LidLi bond lengths are 2.293, 2.294, and 2.467 A Li3@C60. The shorter LidLi bond lengths in Li3@B40 as compared to Li3@C60 can be rationalized by the fact that B40 has a smaller cavity which is responsible for higher pressure as compared to that of the C60 cage. Due to the confinement effect of the cages, an increase in the LidLi bond vibrational frequencies in these systems is observed and the same is again higher in the case of the Li3@B40 system. The changes in the Gibbs free energy are negative for these encapsulation processes which indicates the formation of these endohedral systems is spontaneous. The ADMP simulation of the Li3@Cg at 300 and 500 K temperatures supports the stability of the systems. Several boron clusters show a special property, known as “molecular Wankel motor”, where an inner ring moves within the peripheral of another large outer ring [148–151]. As for example, IrB
12 cluster, where the inner B3 ring rotates within the peripheral B9 ring behaving as a Wankel motor [150]. A close inspection of the trajectories obtained for both Li3 encapsulated cages shows that the movement of the Li3 cluster inside the two cages is not like that is observed in IrB
12 complex. Hence, Li3@Cg systems cannot be considered as a Wankel motor. The natural charges on the Li atoms are 0.60, 0.73, 0.74 jej and 0.60, 0.60, 0.55 je j for Li3@B40 and Li3@C60, respectively. In the case of the Li3@B40 system greater amount of charge transfer takes place from the Li3 cluster to the B40 cage. The AIM analysis confirmed that both systems contained an NNA at the middle position of the Li3 cluster (see Fig. 10C and D). The r2r(rc) values at the NNAs are negative for both these systems. The plot of the r2r(rc) for Li3@C60 system is given in Fig. 10E. The systems have an ELF basin at the center of the Li3 cluster where NNA exists. Fig. 10F corresponds to the ELF plot for the Li3@C60 system. Li3@B40 system shows similar r2r(rc) and ELF plots. For the Li3@B40 system, the population of NNA is 0.17 j ej with a 12% localization of electron. However, in the case of the Li3@C60 system, the NNA population is 0.59 je j with 46% localization of electron density. Li3@C60 system shows a higher NNA population than that of the Li3@B40 system. The presence of electron-deficient boron atoms causes a lowering of the NNA population in the Li3@B40 system. The electron density analysis of the ELF basins shows the populations of 0.58 je j with 22% localization and 1.01 j ej with 56% localization for Li3@B40 and Li3@C60, respectively. The a, b, and gk values are 554.2, 129.4, 3.6  105 au and 584.7, 79.9, 2.1  105 au for Li3@B40 and Li3@C60 systems, respectively. So, both these designed systems have fulfilled all the necessary conditions to be identified as electrides.

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FIG. 10 (A) The optimized geometry of Li3@B40 and (B) the optimized geometry of Li3@C60. The color code for both (A) and (B) follows as: gray indicates C, pink (light gray in print version) indicates B, and purple (light gray in print version) indicates Mg atoms. (C) The molecular graph of Li3@B40, (D) the molecular graph of Li3@C60, (E) the plot of the Laplacian of the electron density (r2r(r)) of Li3@C60, the magenta colored solid and blue (dark gray in print version) colored dashed lines represent the regions with r2r(r) > 0 and r2r(r) < 0, respectively, and (F) twodimensional plot of ELF of Li3@C60.

5. Conclusion The present chapter has shed light on the geometrical, electronic, and bonding properties of several electrides including alkali and alkaline earth metals. It is always interesting to study the electronic description of such systems where the loosely bound electrons play a great role in determining the electronic properties and hence reactivity of the systems. The molecular electrides discussed in this chapter have revealed a huge variety of bonding galleries. For example, the electron density from the bonding molecular orbital is possible to “push” into space with the help of N-atoms present in the ligands. Later on, that loose electron is useful for the activation of small molecules. In another case, the cage is capable of holding a portion of electrons at the center of the cage. Due to their promising future applications, the research area of molecular electrides is evolving rapidly. The interplay between the metal atoms (the guests) and multidentate and peripheral ligands and cavitands (the host) can form a wide rainbow of electrides. Under this rainbow, unusual bonding nature, geometrical structures, and thermodynamic properties can be observed and studied thoroughly. Moreover, the loose electron can act as an electron donor toward other systems which is a significant step toward catalysis. Several economically demanding

290 Atomic clusters with unusual structure, bonding and reactivity

reactions like N2 activation, CO2 activation, and water splitting, are possible to achieve in near future. On the other hand, several organic reactions can also be catalyzed toward a better synthetic route. The design and development of NLO materials are also an intriguing part of molecular electride research. In future days, more fruitful addition of the molecular electrides based on their applicability as NLO material will be observed.

Acknowledgments RS is thankful to Professor Pratim Kumar Chattaraj, Professor Gabriel Merino, and Dr. Sudip Pan for the invitation to contribute the chapter, entitled, “Molecular electrides: An overview of their structure, bonding, and reactivity” to this book, entitled, "Atomic Clusters with Unusual Structure, Bonding and Reactivity" published by Elsevier. RS acknowledges the University Grants Commission (UGC), New Delhi, India for his research fellowship. He also acknowledges the Department of Chemistry, Indian Institute of Technology Kharagpur (IITKGP), India where most of these works (mentioned in this chapter) have been carried out. PD thanks UGC, New Delhi, India for the Research Fellowship.

Authors note The authors declare no conflict of interest.

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[146] H.-J. Zhai, Y.-F. Zhao, W.-L. Li, Q. Chen, H. Bai, H.-S. Hu, Z.A. Piazza, W.-J. Tian, H.G. Lu, Y.-B. Wu, Y.-W. Mu, G.-F. Wei, Z.-P. Liu, J. Li, S.-D. Li, L.-S. Wang, Observation of an all- boron fullerene, Nat. Chem. 6 (2014) 727–731. [147] P. Das, P.K. Chattaraj, Comparison between electride characteristics of Li3@B40 and Li3@C60, Front. Chem. 9 (2021), 638581. [148] D. Moreno, S. Pan, L.L. Zeonjuk, R. Islas, E. Osorio, G. Martı´nez-Guajardo, P.K. Chattaraj, T. Heine, G. Merino, B2
18 : a quasi-planar bowl member of the Wankel motor family, Chem. Commun. 50 (2014) 8140–8143. [149] J.O.C. Jimenez-Halla, R. Islas, T. Heine, G. Merino, B
19: an aromatic Wankel motor, Angew. Chem. Int. Ed. 49 (2010) 5668–5671. [150] L. Liu, D. Moreno, E. Osorio, A.C. Castro, S. Pan, P.K. Chattaraj, T. Heine, G. Merino, Structure and bonding of IrB
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Chapter 16

Hydrogen trapping potential of a few novel molecular clusters and ions Sukanta Mondala, Prasenjit Dasb, and Santanab Giric a

Department of Education, Ashutosh Mukhopadhyay School of Educational Sciences, Assam University, Silchar, Assam, India, b Department of Chemistry,

Indian Institute of Technology Kharagpur, Kharagpur, India, c School of Applied Sciences and Humanities, Haldia Institute of Technology, Haldia, India

1. Introduction In the name of progress and to live a comfortable easy life human civilization has been destroying the balance of our Nature in many ways. Primarily, by the emission of greenhouse gasses bringing the global warming, climate change, and unusual weather [1–5]. CO2 alone contributes 82.2% of the total emitted greenhouse gasses in the United States, which comes from the usage of fossil fuels, which is overall a 5.6% increment from 1990 to 2015 [6]. So, fossil fuels are mainly responsible for the emission of greenhouse gasses. On the other hand, fossil fuels accounted for the production of
85% of the total energy produced globally in 2017 [7]. We have reached to the point where it is almost impossible to maintain the stability of nature and our need and comfort, together. To balance the supply and demand of energy, keeping in mind the health of the environment it is a need of the hour to have sustainable energy resources, which would replace the fossil fuels. Hydrogen sounds as logical alternative energy resource due to its huge abundance in the form of surface water in the earth and environment friendly combustion. Moreover, hydrogen is having striking importance owing to its significant role in the hydrogen-oxygen fuel cells. Hydrogen storage has been receiving a great importance to dodge with the problem of energy scarcity. It is important to mention here that the physical storage of H2 under high pressure or in the form of liquid H2 in cryogenic condition is neither economic nor safe [8]. In addition, due to the low density of hydrogen
0.09 kg/m3 at room temperature and pressure, its storage is most thought provoking [9]. It is also important that the processes of achieving hydrogen have to be from the renewable energy sources and of minimum cost and minimum pollution [10]. Although, there are numerous numbers of research groups experimental and theoretical as well, working on the hydrogen storage materials but finding the potential host is not over yet. To have a grip over the vast hydrogen storage materials, one can classify them into chemical H2 storage materials and physical H2 storage materials. Particularly, due to the low energy density of hydrogen, it is well known that the transportation of pure H2 is difficult. This problem directs the research minds to dwell into chemical/physical H2 storage. The term chemical hydrogen storage system indicates the group of materials/ molecules that store H2 via covalent bonding. Usually, such materials possess maximum H2 density. The process of liberation of H2 from such materials/molecules is exothermic or in very few cases endothermic. Chemical H2 storage process occurs in the energy range of 47.80–95.60 kcal/mol [11]. In the usage of chemically stored H2, the biggest challenge appears in the process of rehydrogenation of the dehydrogenated products. In many cases, intricacy arises due to the change in dehydrogenated products as well as due to the complexity of rehydrogenation pathways [12]. In the physical storage of pure hydrogen, problem arises due to the cost of cryogenic condition, compression of H2, and risk of hydrogen embrittlement of the cylinder metal components. Here, the physical hydrogen adsorbing materials need to be mentioned. When H2 interacts with the surface of the adsorbent via weak van der Waals dispersion forces, and only monolayer interaction takes place, then physical adsorption of hydrogen occurs. The energy range for such process is 0.24–2.39 kcal/mol for most of the adsorbents [12]. The primary drawback of such method is the gravimetric capacity of the so far developed materials to store H2. Moreover, considering all the possible H2 storage approaches, the energy efficiency is the major challenge [12]. In the search of potential hydrogen trapping agents, metal hydrides, metal organic frameworks (MOFs), covalent organic frameworks (COFs), carbohydrates, synthesized hydrocarbons, liquid organic hydrogen carriers, ammonia-borane complexes, formic acid and imidazolium ionic liquid, molecular sheets, and several types of nanostructured materials have been studied theoretically and experimentally [13].

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00014-0 Copyright © 2023 Elsevier Inc. All rights reserved.

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298 Atomic clusters with unusual structure, bonding and reactivity

Potential of metal hydrides in storing hydrogen has been one of the popular topics of intensive research. For example, the magnesium hydride can store hydrogen with 7.7 wt% capacity. Mg is very economic, and a plenty is available in the earth [14–16]. Although MgH2 was considered as a good hydrogen storage material, its major drawback is the adsorption/desorption rates and the very high desorption temperature [17]. Moreover, there is a question of safety, because MgH2 may react vigorously in contact with water [18]. In comparison to metal hydrides, metal borohydrides are better hydrogen storage materials. Magnesium borohydrides, Mg(BH4)2 exhibit polymorphism under high pressure (d-Mg(BH4)2) with high hydrogen densities [19]. At very high H2 pressure one can synthesize lithium aluminum hydride in the presence of Ti as catalyst by direct hydrogenation of the metals using ball-milling technique. But, it is noted that the achieved gravimetric hydrogen storage ¼ 2.5 wt%, which is low in comparison to other metal hydrides [20,21]. On the other hand, the M[Al(BH4)4] (M ¼ alkali metal/NH+4 ) composites of cheap and abundant aluminum bear high percentage of H2. Thus, M[Al(BH4)4] can be considered as potential agent to store hydrogen [22]. Although, metal borohydrides contain significant wt% of stored hydrogen but the kinetics as well as adsorption/desorption temperatures bring admonishing tone. Like metal hydrides, MOFs also have received great attention for H2 storage purpose. Besides several interesting aspects of the MOFs the most important one is the porosity. And strikingly, the sponginess in MOFs is up to 90% [23]. Moreover, the presence of partial charges in the surface of MOFs due to the constituent organic and inorganic parts, facilitates the adsorption of various gas molecules. Hydrogen storing ability of these MOFs has got the attention of the scientific community. Yaghi et al. reported that per formula unit of metal organic framework-5 (MOF-5), isoreticular metal organic framework-6 and -8 (IRMOF-6 and 8) can store 1.9, 4.2, and 9.1 wt% molecular hydrogen, respectively, under ambient conditions [24]. We can control the hydrogen uptake of MOFs by monitoring different features such as porosity, active surface area, constituent ligand structure, sample purity, and spillover, etc. [25]. Varieties of MOFs are stated in the literature. Even if, research on MOFs advances significantly the reprimanding tones in the application of MOFs originates due to the cost of reagents and solvents [26]. After MOFs, Yaghi et al. prepared the COFs [27,28]. COFs are carboncontaining porous materials [27,28]. Depending on the integrity and shapes of these COFs they are divided into twodimensional (2D) and three-dimensional (3D) types [29]. COFs are very stable as they are composed by strong covalent bonds between C and other light elements (H, B, N, O). These COFs are potential host for hydrogen molecule. In cryogenic condition, at 77 K the H2 storing ability of COFs is reported to be more than 25 wt% [30,31]. High gravimetric wt% of hydrogen can be produced from carbohydrates, as the latter is hydrogen-rich. Specifically, a mixture of starch, water, and enzymes via biosynthetic pathway can produce molecular hydrogen. Zhang et al. gave the following stoichiometric reaction for the same: C6H10O5 (l) + 7H2O (l) ! 12H2 (g) + 6CO2 (g) [32,33]. Important criteria to choose nanomaterials for hydrogen storage are surface area, porosity, thermal stability, and mechanical strength. A material with high surface area, high porosity, optimum thermal stability, and mechanical strength can be considered as potential hydrogen storing agent. Nanomaterials are having better H2 trapping ability in comparison to macroscopic bulk materials. Particularly, in the nanomaterials a reduction in the diffusion distance for H2 leads to the fast hydrogen exchange. In hydrogen storage, the carbon-based nanomaterials and metal-doped carbon-based nanomaterials are well known [34]. Due to weak bonding interaction (van der Waals forces) between C-atoms of carbon-based nanomaterials and H2 molecule, the adsorption process takes place at the surface of the material. Carbon nanotubes (CNTs) can store hydrogen with the gravimetric capacity of 6 wt%. The reasons behind such a good storage capacity of CNTs are microporous tubular structures, single or multiple-wall skeleton exposes multiple adsorption sites. In order to increase the H2 binding ability of these CNTs they are modified by alkali metal or transition metal doping [35]. Metal-incorporated CNTs suffer from material instability and due to this their usage is limited [35]. In comparison to CNTs, the graphene is not only found to be more efficient in hydrogen storage but also, they are cheap, safe, and easy to prepare. Besides, one can tune the H2 storage capacity by changing the distance between successive layers or by monitoring the graphene sheet curvature or by functionalizing with suitable chemicals [36]. Research for a suitable hydrogen storage material possessing the binding energies in between strong chemisorption and weak physisorption keeping in mind the Department of Energy (DOE), United States, target of high gravimetric storage capacity as well, has become a challenging task. Earlier DOE target was to achieve materials having 0.081 kg/L volumetric density and 9.0 wt% gravimetric density of H2 storage potential by the year 2015. The recently set target for onboard H2 storage material is 0.040 kg/L by the year 2025 with the ultimatum 0.050 kg/L. The corresponding gravimetric data are 5.5 wt% by the year 2025 with the ultimatum 6.5 wt% [37]. He et al. shown that MXene, a novel class of compounds having graphene-type 2D forms can store hydrogen satisfying the mentioned criteria at its maximum [38]. These MXenes are nothing but layered ternary transition-metal carbides or nitrides. Their general chemical formula is Mn+1AXn, here M indicates the transition metals, which are generally IIIA or IVA group elements, X is C and/or N, and the values of n ¼ 1, 2, or 3 [39–42]. He et al. studied the H2 storage potential of 2D Ti2C as a representative MXene via first-principle calculations.

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They found that the hydrogens were stored via three different ways, namely, chemisorption, physisorption, and Kubas-type interaction. Most interestingly, they found that for those hydrogen molecules that are bound via Kubas-type interaction, their binding energy falls in the particular range suitable for reversible hydrogen storage under atmospheric condition. Furthermore, they reported the maximum hydrogen storage capacity of the Ti2C layers to be 8.6 wt%, which was much above the DOE set target of gravimetric hydrogen storage capacity by the year 2015 [38]. Rud et al. prepared Mg/MAX phase under H2 gas atmosphere. Moreover, they reported the hydrogen storage potential of this synthesized composite. Dehydrogenation processes of the synthesized Mg+ 7 wt% MAX phase were revealed [43]. Storage and transportation of liquid hydrogen are one of the most expensive ways of utilization of hydrogen as a fuel. Ammonia and liquid organic hydrogen carriers (LOHCs) can serve as a good substitute as they have reasonable storage condition in comparison to liquid hydrogen [44–46]. Contrary to this advantage, the generation of H2 from ammonia and LOHCs limits their usage in practical purposes [45]. The simplest examples to discuss LOHCs are cyclohexane and its alkylated derivatives. It is reported that dehydrogenation of these LOHCs would yield corresponding aromatic compounds at high temperature in the presence of Pt-based catalyst [46]. Due to the needed high temperature and Pt containing expensive catalyst its utilization is limited and that’s why it is a very challenging field of research. Crabtree et al. have shown via DFT calculations that replacement of ring carbon in the cycloalkanes by nitrogen as well as incorporation of N-containing groups in the side chain facilitates the release of molecular hydrogen. Moreover, they mentioned that five-membered rings are having better propensity in a comparison to six-membered ring containing cycloalkanes to release H2 [47]. Chen and coworkers reported the potential of dilithiated ethylenediamine (Li2EDA) as a hydrogen storage material. Moreover, they insighted that the selective dehydrogenation of Li2EDA follows an a,b-LiH elimination path [48]. The same research group has shown that the dehydrogenation of Li2EDA takes place via the energy barrier of 43.26  1.91 kcal/mol. In contrary, the partially lithiated ethylenediamines undergo polymerizations via the liberation of ammonia [49]. In the same field, Katikaneni et al. reviewed the cyclic and polycyclic alkanes and mentioned about the incompleteness of the catalyzed dehydrogenation process [50]. Generation of hydrogen from ammonia and further use of hydrogen as a fuel, are carbon-free approaches to energy. The production of hydrogen from ammonia or ammonia-containing chemicals (liquid hydrogen carrier) has not been considered yet as a potential technological way of H2 generation, mainly, due to the high-temperature requirement and cost of catalysts [51]. Nevertheless, there are many catalysts for the decomposition of ammonia, a few particular alkali and alkaline earth metal-doped ruthenium supported on oxides and carbonous species are somewhat potential. But still, at low temperature these catalysts are not suitable and moreover cannot yield H2 in large scale [52]. The next section contains a detailed account on the used theoretical procedures as well as different methodologies.

2. Theoretical background Among the various theoretical methodologies and multiscale computational techniques used by Chattaraj and coworkers, here we discuss only those that were used to study the outlined systems. In a broad spectrum comprising various branches of chemistry and material science the usage of density functional theory (DFT) and conceptual density functional theory (CDFT) is becoming essential [53–57]. Many parameters have been developed from DFT to study properties of chemicals, materials, and their changes, these are called CDFT-based reactivity descriptors. Among these, electronegativity (w), hardness (
), and electrophilicity (o) help to disclose the overall property of a molecule, let it be stability or reactivity, whereas Fukui functions (fk) deal with the reactivity of a particular site/center of the molecule [58–67]. w of an N electron system can be written as:
 ∂E (1) w¼ ∂N vðrÞ Chemical potential (m) of a system can be expressed as:




∂E ∂N

m¼

 (2) vðrÞ

Expression of hardness (
) is:

¼

∂2 E ∂N 2

 (3) vð r Þ

300 Atomic clusters with unusual structure, bonding and reactivity

The equation of electrophilicity (o) is: o¼

m 2 w2 ¼ 2
2


(4)

In the above equations, v(r) is the external potential [57–65,67]. Using finite difference method electronegativity and hardness can be expressed as: w ¼ ðI + AÞ=2

(5)


¼ ð I  AÞ

(6)

Here, I and A indicate the ionization potential and electron affinity of a system, respectively, and they can be calculated using Koopmans’ theorem [68], I   EHOMO and A   ELUMO  1 EHOMO + ELUMO 2  
¼  EHOMO  ELUMO

w¼

(7) (8)

In the above two equations, EHOMO and ELUMO indicate the energies of highest occupied and lowest unoccupied molecular orbitals. Using the D SCF technique, the I and A of a system can be calculated via calculation of energies of the N and N  1 electronic systems in the following way: I  Eð N  1 Þ  Eð N Þ

(9)

A  Eð N Þ  Eð N + 1 Þ

(10)

Here, E(N  1), E(N), and E(N + 1) represent the single point energies calculated using the optimized geometry of the system considering (N  1), N, and (N + 1) electrons, respectively. The Fukui function estimates an alteration in the electron density when an electron is added to or removed from a system at constant external potential [67].   ∂rðr Þ f ðr Þ ¼ (11) ∂N vðrÞ Via the calculation of electron population, one can calculate Fukui function. f ðr Þ can also be written as [69], f +k ¼ pk ðN + 1Þ  pk ðN Þ, for nucleophilic attack,

(12a)

f k

¼ pk ðN Þ  pk ðN  1Þ, for electrophilic attack,

(12b)

½pk ðN + 1Þ  pk ðN  1Þ , for radical attack: 2

(12c)

f 0k ¼

To fetch information about the site selectivity for a specific atomic position, we can calculate local philicity (o∝ k ). For the kth atomic location, the o∝ k is ∝ o∝ k ¼ o  fk

(13)

To represent the nucleophilic, electrophilic, and radical attacks, the value of ∝ would be +, , and 0, respectively. Maximum hardness principle (MHP) [70–72] is: “There seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible.” Minimum electrophilicity principle (MEP) [73–76] is: “Electrophilicity will be a minimum (maximum) when both chemical potential and hardness are maxima (minima).” Extremal values of electrophilicity (o) and stability of a molecular system are correlated. In the assessment of this, it was noted [73–76] that electrophilicity extremum occurs in chemical reactions, molecular vibrations, and rotations at particular points, which obey the following condition   ∂m m ∂h (14) 5 ∂l 2h ∂l here l signifies reaction coordinate for chemical changes, bond length for bond stretching/compression, bond angle in the case of bending, or dihedral angle when internal rotations are there. It is noted that the MEP corroborates well with the

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MHP. So far, these electronic structure principles are used by many researchers to assess stability, reactivity, and aromaticity of various molecular systems. Aromaticity is mostly known by the property of p molecular structures associated with substantial extra stability in a comparison to equivalent molecular motifs. Faraday first isolated the benzene by compressing illuminating gas. And in the same year, 1825, he noted the simplicity of benzene molecule to illustrate electron delocalization [77]. Later, in 1865 Kekule` came up with significant brilliant ideas and suggested alternating single and double bonds in benzene [78–81]. More than one and a half century has passed, many unusually stable molecular systems are studied by applying the concept of aromaticity. Important features of a molecule to be aromatic are as follows: the molecular motif must possess (4n + 2) numbers of p electrons (n ¼ 0, 1, 2, 3, etc.), the skeleton of the molecule must be cyclic and planar. The (4n + 2) p electrons count was first proposed by H€ uckel, as the mandatory criterion of aromaticity. Among the available several methods for the measurement of aromaticity, a few mainly used by the chemistry community are harmonic oscillator model of aromaticity (HOMA) [82–84], nucleus-independent chemical shift (NICS) [85], electron localization function (ELF), multicenter bond index (MCI), etc. In the work presented in this chapter we used NICS. The concept of NICS (in ppm) was first proposed by Schleyer et al. in studying aromaticity [85]. NICS is calculated based on the use of magnetic shielding tensor for a dummy magnetic dipole considered at the center of an underexamination aromatic ring. It is called nucleus-independent chemical shift, because there is no nucleus at the center of the molecular ring to experience the effective magnetic field. When the NICS value is calculated at the center of an aro˚ above the ring then denoted as NICS(1). Negative and positive matic ring then it is denoted as NICS(0) and when at 1 A values of NICS are obtained for aromatic and antiaromatic systems, respectively. Many molecular cages show negative NICS values calculated at the center. Such values may be correlated with the ordinary aromaticity, and moreover, with 2(n + 1)2 valence electrons. These cages are said to exhibit spherical aromaticity [86]. This rule of aromaticity is universally applicable for organic and inorganic symmetric cage-like molecules bearing conjugated p-network. Moreover, properties of cage-like molecules having open faces (or we may call open cage molecular systems) can be better understood via the concept of “Open-Shell Spherical Aromaticity” introduced by Sola` et al. [87]. An open cage system must possess (2N2 + 2N + 1) numbers of p-electrons to exhibit the open-shell spherical aromaticity [87].

3. Computational details Different molecular systems reported in this chapter were modeled either from scratch using the GaussView 3.0 (or GaussView 5.0.8) [88] or by using earlier reported coordinates. Moreover, associated resulting structures were also understood using the same graphical software. The results of the computations stated here were mostly done by using Gaussian 03 and Gaussian 09 suites of programs [89,90]. First, the modeled geometries and adapted coordinates were optimized to achieve the stationary points. Harmonic vibrational frequency analysis was done to verify the stationary points, and local minima forms were identified (imaginary frequency ¼ 0). Various levels of theories were used: MP2, B3LYP, DFT-D-B3LYP, M052X, M06, in conjunction with different basis sets: 6-31G(d), 6-31G(d,p), 6-311 +G(d,p), 6-311 +G (d), cc-PVDZ, according to the molecular system. Necessary computations for the periodic systems were done using the Vienna ab initio Simulation Package (VASP) [91–94], and they were modeled using the graphical software XCrySDen [95]. Projector Augmented Wave (PAW) potentials were used with a kinetic energy cutoff of 550 eV for the considered elements [96,97]. Generalized Gradient Approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [98] was used to calculate the exchange-correlation energy density functional Exc[r]. The self consistent field iterations were continued with the relaxed electronic degrees of freedom until the change in the energy became less than 1  106 eV. The sampling of Brillouin zone was done via the automatic generation of 1  1  6 Monkhorst-Pack set of k-points [99]. In order to understand the effect of electric field on few molecular systems, particularly, to study the effects on the hydrogen adsorption and desorption processes, electric field had been applied along the x direction of the hydrogen adsorbed molecular motifs [100,101]. The energy change of the molecular system due to the application of the electric field was calculated using the following equation: h i DEF ¼  EFH2 @system  EH2 @system

(15)

EHF2@system is the total energy of H2 adsorbed system under the electric field, and EH2@system indicates the same but without the electric field.

302 Atomic clusters with unusual structure, bonding and reactivity

4.

Atomic and molecular clusters

4.1 Mg and Ca clusters Various theoretical studies surfaced the fact that, researchers have been using cage-like molecules and molecular motifs for H2 storage purpose. Giri et al. studied the hydrogen trapping ability of Mg and Ca cages along with the aromaticity of their H2 bound analogues [102]. The optimized structures of H2 encapsulated Mgn and Can cages (n ¼ 8–10) are presented in Fig. 1. The bare cages are unstable, whereas upon encaging H2 inside them they gained stability. In the H2Ca10 cluster, the adsorbed hydrogen was in atomic form while in other cages the hydrogens were in molecular form which is shown in Fig. 1. In these H2-entombed Mgn cages, the NICS(0) and NICS(1) values of both the upper and lower Mg4 rings and the Ca4 rings in Can clusters were negative indicating the diatropic ring current. The charges on the metal sites in their H2 trapped analogues were mostly positive. However, in H2Mg9, H2Mg10, and H2Ca10 clusters few sites show negative charges. Computed charge on the atoms of Mgn, Can (n ¼ 8–10) moieties discloses the partial (+)Ve nature of the upper and lower frame metal components. Thereby, they are susceptible to nucleophilic attacks. On the other hand, the metal centers of H2Mg8 as a whole as well as few metal atoms in H2 bound other Mg and Ca cages contain ()Ve charges, and they are prone to electrophilic attack. Readers may see the values of calculated local reactivity descriptors from the reference [102] for further insight.

4.2 B2Li and B2Li2 moieties Boron-Li clusters showed great attention in this field on hydrogen storage [103,104]. Bandaru et al. studied the stability and the H2 adsorption ability of neutral B2Li and B2Li2 moieties based on CDFT [105]. The optimized geometries of the bare B2Li and B2Li2 moieties and their corresponding H2-bound analogues are given in Fig. 2. The B2Li cluster can trap a maximum of four hydrogens in the molecular form. However, the B2Li2 cluster adsorbed a maximum of eight H2 units. The binding energy values per H2 molecule (Eb) in the B2Li cluster are in the range of 1.55–2.41 kcal/mol and in the B2Li2 cluster, the Eb values are in between 1.46 and 2.33 kcal/mol. The charges on Li atoms in the B2Li2 cluster and their H2-bound analogues are higher than that in the B2Li cluster and their hydrogen-bound forms.

FIG. 1 Geometries of H2Mn (where M ¼ Mg, Ca; n ¼ 8, 9, 10) clusters. (Reproduced from S. Giri, A. Chakraborty, P.K. Chattaraj, Potential use of some metal clusters as hydrogen storage materials—a conceptual DFT approach, J. Mol. Model 17 (2011) 777–784 with permission from the Springer Nature.)

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FIG. 2 Geometries of (A) B2Li and their hydrogen-trapped analogues; (B) B2Li2 and their hydrogen-trapped analogues. (Reproduced from S. Bandaru, A. Chakraborty, S. Giri, P.K. Chattaraj, Towards analyzing some neutral and cationic boron-lithium clusters (BxLiy x ¼ 2-6; y ¼ 1, 2) as effective hydrogen storage materials: a conceptual density functional study, Int. J. Quantum Chem. 112 (2012) 695–702 with permission from the John Wiley and Sons.)

The charges on Li centers in both clusters decrease upon gradual adsorption of H2. This is because of the shifting of electron density from the neutral H2 molecule to the Li centers. Although the
values are not much changing on gradual adsorption of H2 to the clusters but the hardness values of the bare B2Li cluster and their H2 bound analogues are higher than that of the bare B2Li2 and their H2 adsorbed analogues. These results indicate the greater stability of the H2-bound B2Li clusters than the H2-bound B2Li2 clusters. Again, the H2 trapping by these clusters shows negative values of reaction electrophilicity (Do), indicated that all the H2 trapped systems obey the MEP that is used for predicting molecular stability. Furthermore, the binding of H2 molecules to the Li center is discussed in terms of average dissociative chemisorption energy (DECE). The positive DECE values for the H2 trapped systems specify the favorable binding of H2 at the Li centers. So, these neutral boron-lithium clusters can be used as hydrogen trapping materials.

4.3 C12N12 cage Mondal et al. reported different cage-like structures of C12N12 and their possibility of utilization as hydrogen storage material as well as high-energy-density materials [106]. Reported possible forms of C12N12 are: C12N12-A, C12N12-B, and C12N12-C. The point group of the isomer C12N12-A is D6d, whereas of C12N12-B and C12N12-C are C3 and C2v, respectively, please see Fig. 3. In the text, these isomers are further indicated as A, B, and C, respectively. Although the structures of these isomers can be perceived from Fig. 3, a brief discussion would help the readers to get further insight. The A form is having two planar six-membered C6 units at the ends of a drum-like cage, where the cage wall is composed by five-membered C2N3 units. The isomer B possesses two chair-like C3N3 hexagonal units, three C2N3 units, and three C3N2 units. Whereas the third isomer, C, appears like a channel segment, formed by eight C3N2 pentagonal units. These forms and their H2-embedded analogues were obtained computationally by using the DFT-D-B3LYP/6-31G(d) level of theory. It was noted that the C isomer is 55.41 kcal/mol more stable than the B isomer whereas the latter is 66.11 kcal/mol more stable than the isomer A. NICS(0) values were calculated for these three isomers, and negative magnitudes (2.601, 5.876, and 4.639 ppm, respectively) indicate their aromatic nature. But, as the isomers A and B contain 24p electrons, they are not obeying 2(N + 1)2 p rule of spherical aromaticity [86,107]. The open cage form, C, also possesses 24p electrons and does not obey the (2N2 + 2N + 1) p electron rule of open-shell spherical aromaticity [87]. These isomers can interact with molecular hydrogen exohedrally. Possible particular site of hydrogen binding was also explored and they mentioned that among the three situations, (i) atop N atom, (ii) atop C atom, and (iii) atop the midpoint of CdN bond, only in the first case local minima were found. It was noted that each of the isomers can hold up to 12 hydrogen molecules. All the isomers and their hydrogen adsorbed analogues (up to 12 H2) were found to be local minima on the corresponding potential energy

304 Atomic clusters with unusual structure, bonding and reactivity

FIG. 3 Modeled structures of C12N12-A, C12N12-B, and C12N12-C. Relative energies with respect to the lowest energy minimum are given in parentheses (in kcal/mol). Besides the relative energies, NICS(0) values calculated at the centers of the respective C12N12 isomers are also given. (Reproduced from S. Mondal, K. Srinivasu, S.K. Ghosh, P.K. Chattaraj, Isomers of C12N12 as potential hydrogen storage materials and the effect of the electric field therein, RSC Adv. 3 (2013) 6991–7000 with permission from the Royal Society of Chemistry.)

surfaces excluding 11 and 12 H2 molecules adsorbed C isomer analogues. Interaction of hydrogen molecules with the C12N12 isomers was estimated via the calculation of binding energy per H2 molecule using the following equation: DE ¼ ðEnH2 @C12 N12 EC12 N12 nEH2 Þ=n

(16)

here, Esystem indicates the energies of different systems and n marks the number of hydrogen molecules. Gravimetric hydrogen storage was calculated to be 7.2 wt% by all the isomers. Change in the values of free energy at different temperature and pressure was calculated for the hydrogen adsorption process by the A isomer and the feasible region was located. Hardness values were calculated for all the systems. The data indicate nonlinear gradual increment for nH2-loadedA and C isomers. Whereas for the B isomer random variations in the hardness values were noted. But the variations of hardness and binding energy values are correlated. NICS(0) values computed at the cage center of all the three isomers and their hydrogen adsorbed analogues reveal aromaticity of all the systems. It was noted that as the number of adsorbed H2 molecules increases the increment in negative NICS(0) value is greater in the cases of A isomer in a comparison to B and C isomers. Variation in the chemical shift values of hydrogen loaded B isomer is in line with the A isomer analogues. Moreover, the variations of NICS(0) values of the H2-loaded systems of A isomer and the variations of binding energy as well as hardness follow the same trend. Application of electric field can influence weak interaction between atoms and molecules. From the work of Zhou et al. it became clear that under electric field, the interaction between the host and the guest hydrogen molecules improves to yield higher hydrogen storage [100]. It was noted by studying the effect of electric field (of strength 5  103 a.u.) on the A isomer analogue that the binding energy improves by 0.46 kcal/mol per H2 molecule. Moreover, the removal of electric field facilitates the desorption process. Using PBE method C12N24 nanotube (higher analogue of C12N12-A isomer) was modeled. In its skeleton a zigzag N12 frame is covered axially by two hexagonal C6 rings. Noted bond lengths are 1.538 (CdC), 1.475 (CdN), and 1.475 (NdN) ˚ . Bond lengths reflect the sp3 hybridizations of the involved atoms. The bond angles between the C atoms of the hexagonal A

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FIG. 4 (A) Top view, (B) the 1  1  3 super cell structure of the optimized C12N24(H2)24 unit cell. (Reproduced from S. Mondal, K. Srinivasu, S.K. Ghosh, P.K. Chattaraj, Isomers of C12N12 as potential hydrogen storage materials and the effect of the electric field therein, RSC Adv. 3 (2013) 6991–7000 with permission from the Royal Society of Chemistry.)

rings are 120 degrees. Whereas the other measured angles are 107.5 degrees (NNN), 109.2 degrees (NNC), 106.6 degrees (CCN), and 110.4 degrees (NCN). In the search for a minimum energy location of H2 molecule, it was found that the positions where the H2 unit can interact with two nitrogen atoms of the tube are suitable. In such locations, above the CdC bonds, at first 12 H2 molecules were placed per unit cell. The interaction energy of adsorption was calculated to be 1.73 kcal/mol and the gravimetric H2 storage capacity was 4.76 wt%. Further, the optimization of another system with the same host containing 24 H2 molecules (Fig. 4) per unit cell revealed 9.10 wt% gravimetric storage with the binding energy of 1.44 kcal/mol per hydrogen molecule.

5. Ionic clusters 5.1 N4Li2 and N6Ca2 clusters Several theoretical studies showed the importance of polynitrogen molecules to be used as high-energy-density materials 2 [108–112]. Duley et al. studied the aromaticity of N4 6 and N4 planar cyclic polynitrogen rings and the possible hydrogen trapping ability by their corresponding N4Li2 and N6Ca2 forms [113]. Investigation of aromaticity of the rings considered 2 was done by NICS. The computed NICS(0) values of N4 6 and N4 planar cyclic rings are comparable with that of the benzene and the cyclobutadiene, respectively. The NICS scan plots show that the N4 6 ring gives a similar pattern as that 4 of the benzene, but the pattern of the N2 4 ring and the cyclobutadiene rings are different. The N6 ring with 10 p-electrons 2 shows aromatic behavior. However, the N4 ring with six p-electrons shows simultaneous existence of s-antiaromaticity and p-aromaticity, called conflicting aromaticity. The s-antiaromaticity and p-aromaticity were predicted in terms of NICS(0) > 0 and NICS(1) < 0, respectively. The N2 4 ring shows aromatic behavior or diatropic magnetic shielding at ˚ above the ring and it gradually decreases with increasing the distance above the ring. Using suitable counterions, 1A 2+ + 2 2 the N4 ion stabilizes the N4 6 and N4 rings may be stabilized. The Ca 6 ring and two Li ions stabilize the N4 ring through cation-p interaction and the resulted complexes are N6Ca2 and N4Li2, respectively. The optimized geometries of N6Ca2 and N4Li2 complexes and their corresponding H2 trapped analogues are presented in Fig. 5. When Ca2+ is attached + 2 to the N4 6 ring the aromaticity of the ring increases but on complexation of two Li ions with N4 ring it still shows conflicting aromaticity. The metal centers containing highly partial positive charges enables the binding of H2 molecules to them. The hydrogen trapping on Li and Ca centers of the studied complexes can be justified from the negative values of the hydrogen adsorption energy (DEads). For the N4Li2 complex the Li atoms interact with eight H2 molecules; four on each Li center with 1.2 kcal/mol adsorption energy. However, for the N6Ca2 cluster, each Ca binds a maximum of six H2 molecules with 1.3 kcal/mol adsorption energy.

306 Atomic clusters with unusual structure, bonding and reactivity

FIG. 5 (A) Optimized geometries of N6Ca2 and N4Li2 and their corresponding hydrogen-trapped analogues; Color code: blue (gray color in print version) for N, pink (light gray in print version) for Li, green (dark gray in print version) for Ca, and white for H atoms. (B) NICS-scan plots for 2 N4 6 , N4 , benzene (Bz), and cyclobutadiene (Cb). (Reproduced from the S. Duley, S. Giri, N. Sathymurthy, R. Islas, G. Merino, P.K. Chattaraj, Aromaticity and hydrogen storage capability of planar N64- and N42-rings, Chem. Phys. Lett. 506 (2011) 315–320.)

5.2 Li+3 and Na+3 ions Aromaticity brings thermodynamic stability in a system and thereby plays an important role in the designing of new stable materials for hydrogen storage. Giri et al. studied the hydrogen trapping ability of cationic trigonal Li+3 and Na+3 clusters and the effect of hydrogen trapping on aromaticity [102]. The optimized structures of the H2-bound analogues of Li+3 and Na+3 clusters are presented in Fig. 6. The studies show two different H2 binding positions of the Li+3 cluster, (a) binding of H2 through the trigonal plane; (b) binding of H2 at the vertices of the cluster. However, in the case of the Na+3 cluster H2 binding occurred at the vertices of the cluster. The adsorbed H2 is in molecular form at the vertices of both the clusters. But the absorbed H2 is dissociated into atoms while interacting through the plane. The binding energy per H2 molecule (Eb)

FIG. 6 (A) Minimum energy structures of H2-bound Li+3 (B) Minimum energy structures of H2-bound Na+3 . (Reproduced from S. Giri, A. Chakraborty, P. K. Chattaraj, Potential use of some metal clusters as hydrogen storage materials—a conceptual DFT approach, J. Mol. Model. 17 (2011) 777–784.)

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in a dissociative manner is about 32.2 kcal/mol in the H2Li+3 cluster, while in higher H2-bound clusters the values are in the range of 3.6–3.7 kcal/mol. On the other hand, the Eb values are in the range of 2.2–2.5 kcal/mol for H2 binding to the Li+3 cluster in the molecular form. However, in the case of the Na+3 cluster, the Eb values are comparatively lower and in the range of 0.2–0.7 kcal/mol. The bare clusters and the H2-bound clusters show aromatic behavior as predicted from the negative values of NICS(0). In comparison, the NICS(0) values in bare Li+3 cluster and in H2Li+3 are 8.75 and 14.57 ppm, respectively. The Li+3 cluster adsorbs a maximum of six H2 molecules through all the vertices and a maximum of eight H2 molecules using both vertices and the plane. However, the Na+3 cluster adsorbs a maximum of 10 H2 molecules through all the vertices. The electrophilicity values of the clusters were decreasing on increasing the number of adsorbed H2 molecules. This trend indicates that the stability of the H2 trapped Li+3 and Na+3 clusters increases with increasing cluster size.

5.3 B2Li+ and B2Li+2 ions The hydrogen trapping ability of neutral B2Li and B2Li2 moieties is already discussed in Section 4.2 [105]. Along with the neutral B2Li and B2Li2 cluster, Bandaru et al. studied the hydrogen trapping possibility of B2Li+ and B2Li+2 ions [105]. The optimized structures of the bare ions and their H2-trapped analogues are presented in Fig. 7. Similar to the neutral analogues, the B2Li+ cluster ion binds with a maximum of four H2 molecules while the B2Li+2 ion cluster traps a maximum of eight H2 molecules. The trapped hydrogens on these ionic clusters are in molecular form. The binding energy per H2 molecule (Eb) is in the range of 3.15–4.61 kcal/mol and 3.09–11.94 kcal/mol for B2Li+ and B2Li+2 ions, respectively. The charges on Li atoms are comparatively higher as compared with their neutral analogues, and for this reason, the H2 binding ability of the cationic clusters is higher than that of the neutral counterparts. Again, the charge on the Li atom in the B2Li+2 ion is higher than that in the B2Li+ cluster, indicating the greater binding ability of H2 molecules. Here also the charges on Li atoms decrease on adding H2 molecules gradually because of electron density shifting from H2 molecules to Li centers. The reaction electrophilicity for H2 trapping in these clusters is negative, which indicates that the H2-bound cationic clusters obey the MEP principle similar to those neutral systems discussed in Section 4.2. The chemical hardness values for the H2-bound B2Li+ and B2Li+2 clusters are almost comparable but greater than their neutral analogues.

FIG. 7 Geometries of (A) B2Li+ and their hydrogen-trapped analogues; (B) B2Li+2 , and their hydrogen-trapped analogues. (Reproduced from S. Bandaru, A. Chakraborty, S. Giri, P.K. Chattaraj, Towards analyzing some neutral and cationic boron-lithium clusters (BxLiy x ¼ 2-6; y ¼ 1, 2) as effective hydrogen storage materials: a conceptual density functional study, Int. J. Quantum Chem. 112 (2012) 695–702.)

308 Atomic clusters with unusual structure, bonding and reactivity

5.4 M5Li+7 (M 5 C, Si, Ge) clusters It is well known that the Li centers in a Li-containing molecule or ion play key role in the adsorption of molecular hydrogen. With this concept Pan et al. chose the stable and aromatic C5Li+7 and Si5Li+7 star-like systems for the exploration of their hydrogen trapping potential [114]. The higher congener of these systems, Ge5Li+7 was also considered for the same. The global minima form of C5Li+7 and Si5Li+7 systems was already reported by that time [114]. Both of these forms are D5h symmetric and possess p-aromaticity. It is worthy to mention that C5Li+7 is s-nonaromatic and Si5Li+7 shows s-aromaticity [114]. Using M06/6-311 + G(d,p) level of theory Pan et al. modeled the D5h forms of C5Li+7 , Si5Li+7 , and near D5h geometry of Ge5Li+7 . Due to the difference in electronegativity of C, Si, and Ge from Li, the CdLi, SidLi, and GedLi bonds are polar in nature. To find out the natural population analysis (NPA) charges on different atoms of these star-like systems requisite calculations were done. In the C5Li+7 system, the NPA charges were found to be +0.78e and +0.71e on each of the equatorial and axial Li centers, whereas 0.87e was noted for each of the C atoms. Substantial partial positive charges were also revealed by the Li centers of Si5Li+7 and Ge5Li+7 moieties. Each of the Li centers in the C-centered cluster ion can bind up to three H2 molecules with the interaction energy of 2.8 kcal/mol per hydrogen molecule. Structure of C5Li+7 is depicted in Fig. 8. Maintaining the mentioned ratio of host Li center and guest H2 molecule, up to 21 H2 molecules were placed around the C5Li+7 . One can note that the hardness and electrophilicity values for this gradual H2 loading increase and decrease, respectively, thereby obeying the MHP and MEP. They inferred from the fact of lengthening of HdH bond on adsorption and reduction in the values of partial charges on Li centers that a transfer of electron density occurs from the s-bond of H2 to the Li centers. For the Si5Li+7 system they noted that all the Li atoms can interact with and bind up to three H2 molecules. And for the same system, energy of interactions between axial Li and H2 is 2.2 kcal/mol, whereas between equatorial Li and H2 is 3.3 kcal/mol. But, for Ge5Li+7 they noted that the axial Li atoms can bind up to two H2 and equatorial Li atoms can hold up to three H2 with the interaction energy of 2.5 and 3.4 kcal/mol, respectively. Structures of M5Li+7 (M ¼ Si, Ge) are provided in Fig. 8. Although, a similar trend (like C5Li+7 ) of electrophilicities was noted for both the Si-centered and Gecentered ions for their gradual H2 adsorption, the variation in hardness is reverse and the authors mentioned that for such cases scaled hardness per atom (or per electron) may serve as a better parameter [114]. The nature of bonding and change in bonding due to the H2 adsorption by the Si5Li+7 , Ge5Li+7 systems were found to be similar like the carbon-centered moiety. Gravimetric hydrogen storage capacity of Si5Li+7 and Ge5Li+7 was calculated to be 18.3 and 9.3 wt%, respectively. Application of electric field further improves the hydrogen binding ability of these star-like systems, M5Li+7 (M ¼ C, Si, Ge).

6.

Conclusion

Interaction between different neutral and ionic clusters with H2 can lead to significant adsorption of molecular hydrogen. Hardness and electrophilicity in conjunction with maximum hardness principle (MHP) and minimum electrophilicity principle (MEP), along with aromaticity and local reactivity descriptors can help in understanding the stability and involved chemical bonding in such type of hydrogen adsorption processes. Mgn and Can (n ¼ 8–10) clusters can store H2 molecule endohedrally. In small Li-containing neutral molecular systems, B2Li and B2Li2, each of the Li centers can bind up to four H2 molecules with the binding energy range of 1.46–2.41 kcal/mol per hydrogen molecule. The C12N12 cages can bind up to 12 H2 molecules. Out of the three isomers of C12N12 the structure having D6d point group revealed the ability of storing hydrogen with gravimetric density of 7.2 wt%. Moreover, the anticipated C12N24 nanotube-like structure can deliver gravi2 2+ metric density of 9.1 wt%. The N4 and Li+ ions, respectively. 6 and N4 rings get stabilized through the binding with Ca 4 2+ On binding of Ca ion through cation-p interaction, the aromaticity of the N6 ring increases. But, both the N2 4 ring and

FIG. 8 Structures of C5Li+7 , Si5Li+7 , and Ge5Li+7 . (Redrawn from S. Pan, G. Merino, P.K. Chattaraj, The hydrogen trapping potential of some Li-doped star-like clusters and super-alkali systems, Phys. Chem. Chem. Phys. 14 (2012) 10345–10350.)

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N4Li2 show conflicting aromaticity. N4Li2 and N6Ca2 clusters show hydrogen trapping potential. Aromatic behavior of Li+3 and Na+3 clusters as well as their H2 adsorbed analogues is revealed. Local reactivity descriptor discloses the reactivity of the bare and H2-adsorbed Li+3 and Na+3 moieties. The binding energies and reaction electrophilicities provide valuable evidence disclosing favorable interaction of the H2 molecules with the monocationic boron-lithium clusters. Gravimetric hydrogen storage capacity of star-like C5Li+7 , Si5Li+7 , and Ge5Li+7 ions is 28.0, 18.3, and 9.3 wt%, respectively.

Acknowledgments SM thanks University Grants Commission, New Delhi, for UGC-BSR Research Start-Up-Grant (No. F.30-458/2019(BSR)) and his coworkers whose work is presented in this book chapter. PD thanks UGC, New Delhi, India, for his research fellowship.

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[103] X. Wu, Y. Gao, X.C. Zeng, Hydrogen storage in pillared Li-dispersed boron carbide nanotubes, J. Phys. Chem. C 112 (2008) 8458–8463. [104] S. Er, G.A. de Wijs, G. Brocks, DFT study of planar boron sheets: a new template for hydrogen storage, J. Phys. Chem. C 113 (2009) 18962–18967. [105] S. Bandaru, A. Chakraborty, S. Giri, P.K. Chattaraj, Towards analyzing some neutral and cationic boron-lithium clusters (BxLiy x ¼ 2-6; y ¼ 1, 2) as effective hydrogen storage materials: a conceptual density functional study, Int. J. Quantum Chem. 112 (2012) 695–702. [106] S. Mondal, K. Srinivasu, S.K. Ghosh, P.K. Chattaraj, Isomers of C12N12 as potential hydrogen storage materials and the effect of the electric field therein, RSC Adv. 3 (2013) 6991–7000. [107] Z. Chen, H. Jiao, A. Hirsch, W. Thiel, The 2(N+1)2 rule for spherical aromaticity—further validation, J. Mol. Model. 7 (2001) 161–163. [108] A. Vogler, R.E. Wright, H. Kunkely, Photochemical reductive cis-elimination in cis-diazidobis(triphenylphosphane)platinum(II) evidence of the formation of bis(triphenylphosphane)platinum(0) and hexaazabenzene, Angew. Chem. Int. Ed. Engl. 19 (1980) 717–718. [109] G. Chung, M.W. Schmidt, M.S. Gordon, An ab initio study of potential energy surfaces for N8 isomers, J. Phys. Chem. A 104 (2000) 5647–5650. [110] R. Engelke, Ab initio correlated calculations of six nitrogen (N6) isomers, J. Phys. Chem. 96 (1992) 10789–10792. [111] T.-K. Ha, M.T. Nguyen, The identity of the six nitrogen atoms (N6) species, Chem. Phys. Lett. 195 (1992) 179–183. [112] D.L. Strout, Acyclic N10 fails as a high energy density material, J. Phys. Chem. A 106 (2002) 816–818. 2[113] S. Duley, S. Giri, N. Sathymurthy, R. Islas, G. Merino, P.K. Chattaraj, Aromaticity and hydrogen storage capability of planar N46 and N4 rings, Chem. Phys. Lett. 506 (2011) 315–320. [114] S. Pan, G. Merino, P.K. Chattaraj, The hydrogen trapping potential of some Li-doped star-like clusters and super-alkali systems, Phys. Chem. Chem. Phys. 14 (2012) 10345–10350.

Chapter 17

Polarizability of atoms and atomic clusters Swapan K. Ghosh UM-DAE-Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai, India

1. Introduction Theoretical investigation of quantum systems such as atoms, molecules, clusters, solids, etc., forms a very important step toward understanding and prediction of their properties and design of new materials with tailor-made properties. In particular, the response properties such as those arising under the action of electric or magnetic fields are of special importance from the point of view of understanding as well as applications. In this chapter, we are mainly concerned about the behavior or changes shown by many-electron systems under the influence of an electric field. The electron distribution of the system is polarized leading to induced moments (dipole or multipole), over and above the permanent dipole moment present in some molecular systems, and consequent changes in the energy of the system. If the energy or the dipole moment of the system under the action of an applied field is Taylor expanded using the strength of the electric field as perturbation, the derivative coefficients, which denote polarizability, hyperpolarizability, etc., become central to the description and interpretation of a wide range of phenomena [1]. If the field is uniform, one has only the dipole polarizabilities or hyperpolarizabilities, whereas for nonuniform field, the expansion can be in terms of the field gradients as well, and one comes across the multipole polarizabilities or hyperpolarizabilities. Some of these quantities are of direct concern to intermolecular forces, nonlinear optics, spectroscopy, etc. [2]. While polarizabilities make their appearance in the context of interaction of electric field with the atomic or molecular systems, an analogous situation arises when any atomic or molecular system is approached by other atoms or molecules, since the system is subjected to the interaction potential due to the approach of its partners [3]. This is investigated in the study of chemical binding and reactivity, and many concepts have emerged from time to time to study these topics, leading to rationalization or prediction of various properties of interest. These concepts are known by their general name, the reactivity indices [4] within the framework of conceptual density functional theory (DFT) [5]. Some popular examples are the concepts of electronegativity (identified with electronic chemical potential of DFT), chemical hardness, softness, and so on [6–8]. Since all these quantities are linked with the response of an electronic system to the environmental perturbation, attempts have been rightly made to explore their interconnections. In fact, polarizability has been shown to be closely connected to hardness-softness, size of the system, and other properties [9,10]. Thus, polarizability can play important role in determining the force fields and hence computer simulation, or modeling of many chemical or optical properties, drug design, as descriptors in QSAR approaches and now machine learning as well. Thus, reliable prediction of polarizability quantities of many-electron atomic and molecular systems as well as atomic and molecular clusters is of utmost importance, and hence, it rightly has emerged as a very active and challenging area of research. The simplest quantity is the dipole polarizability [11], a linear response property defined as the second derivative of the total electronic energy or the first derivative of the dipole moment, with respect to the external homogeneous electric field. Since the calculation involves evaluation of expectation value of one-electron operators, electron-density-based theory such as DFT is quite appropriate for this purpose. In what follows, we first discuss the basics of the concept of polarizability in terms of a perturbation approach in the field variable in Section 2, covering the numerical finite field approach for its calculation as well. We consider a DFT-based approach within the framework of Kohn Sham (KS) theory, discussing the equations for polarizability calculation and also briefly mentioning the variation perturbation approach within the KS DFT in Section 3. The simplified equations for spherically symmetric system, including the modified Sternheimer equation, are discussed in Section 4 and are illustrated by considering the jellium model appropriate for a class of atomic clusters (such as metal clusters). Discussion on the interconnection of the reactivity indices and polarizability is presented in Section 5. Illustrative discussion highlighting various trends in the calculated polarizabilities is provided in Section 6. Finally, in Section 7, we offer a few concluding remarks.

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00003-6 Copyright © 2023 Elsevier Inc. All rights reserved.

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314 Atomic clusters with unusual structure, bonding and reactivity

2.

Basics of response properties and polarizability

When a many-electron system such as atom, molecule, or cluster is subjected to an electric field, electric moments will be induced. The induced dipole moment is proportional to the applied electric field, and the proportionality constant is the dipole polarizability. Analogously, the induced quadrupole moment is proportional to the electric field gradient and the proportionality constant defines the quadrupole polarizability. Higher polarizabilities (multipole) are similarly defined from the linear dependence of the induced multipole moments on the higher-order derivatives of the applied electric field or field gradient. Higher-order terms involving nonlinear dependence of the induced moments on the field define the hyperpolarizabilities. In this chapter, we will consider only the linear response for a uniform field and hence only the dipole polarizabilities. One can define the field-dependent energy E(F) as. EðFÞ ¼ Eð0Þ + Si mi Fi + ð1=2Þ Si,j aij Fi Fj

(1)

where Fi (i ¼ 1, 2, 3) denote the three components of the uniform field vector F, mi denotes the ith component of the zero field dipole moment of the system, and aij is the ijth element of the polarizability tensor a, essentially representing the second derivative of the energy, namely
 aij ¼ ∂2 E=∂Fi ∂Fj (2) Alternatively, one can also define the polarizability by considering the expansion of dipole moment as powers of the field, namely mi ðFÞ ¼ mi ð0Þ + Sj aij Fj

(3)

where the polarizability expression can be rewritten as


 aij ¼ ∂mi =∂Fj

(4)

If the energy of the system is calculated using numerical method, the polarizability tensor can be easily obtained by using the so-called finite field method, where one evaluates the energy in the presence of a few values of the field components, and the energy derivative is evaluated numerically. The mean polarizability is then obtained in terms of the trace, namely
 (5) a ¼ ð1=3Þ Si aii ¼ ð1=3Þ axx + ayy + azz The evaluation of polarizability thus requires an efficient and accurate method for the calculation of energy of a manyelectron system. In view of the popularity, computational economy, and conceptual simplicity, the DFT-based approach is discussed here for calculation of the polarizability of atoms, molecules, and clusters [12].

3.

DFT-based approach to calculation of polarizability

In DFT, the energy of an N-electron system, characterized by an external potential vext(r), can be written as a unique functional of the electron density r(r) as [6]. ð Ev ½r ¼ dr rðrÞ vext ðrÞ + F½r (6a) where F[r] is a universal functional of density consisting of the kinetic energy (KE) T[r], classical Coulomb energy ECoul[r], and the exchange correlation (XC) energy, Exc[r], i.e., F½r ¼ T ½r + ECoul ½r + Exc ½r where

ð

(6b)

ð

ECoul ½r ¼ ð1=2Þ dr dr0 rðrÞ rðr0 Þ=|r  r0 |

(6c)

Since the exact density functional forms of the kinetic and XC energies are unknown, approximations have been in use. The most popular level of approximation in which the KE of the actual system T[r] is replaced by the KE Ts[r] of an equivalent noninteracting system of the same density leads to the so-called Kohn-Sham (KS) equations given by (in atomic units) [7].

Polarizability of atoms and atomic clusters Chapter



17

 ð1=2Þr2 + veff ðr; rÞ ci ¼ ei ci , i ¼ 1,…N

315

(7)

which are essentially one-electron Schrodinger-like equations, which need to be solved self-consistently. Here, the effective potential veff(r; r) can be written as veff ðr; rÞ ¼ vext ðrÞ + vint ðr; rÞ with the density-dependent internal potential vint(r; r) given by ð vint ðr; rÞ ¼ dr0 rðr0 Þ=|r  r0 |+dExc ½r=dr

(8a)

(8b)

and the density is given by

The energy expression is given by

rðrÞ ¼ Si ci ∗ ci

(9)

ð ð Ev ½r ¼ Si ei  ð1=2Þ dr dr0 rðrÞ rðr0 Þ=|r  r0 | ð + Exc ½r  dr rðrÞ dExc ½r=drðrÞ

(10)

If this system is subjected to an external applied potential vappl(r), one has to consider the new external potential by adding this extra applied potential to the original external potential. In addition, the internal potential will also be modified due to its dependence on the modified density. The new KS equation will then be. 

 mod  mod ð1=2Þr2 + vmod ci ¼ ei mod ci mod , eff r; r i ¼ 1,…N

(11)

mod ) can be written as where the modified effective potential vmod eff (r; r

    mod mod veff r; r mod ¼ vext ðrÞ + vappl ðrÞ + vint r; r mod

(12a)

with the modified density-dependent internal potential vint(r; rmod) given by ð  mod  ¼ dr rmod ðr0 Þ=|r  r0 | vmod r; r int   + dExc rmod =drmod

(12b)

where the modified density, rmod, is given in terms of the modified KS orbitals, as ∗

rmod ðrÞ ¼ Si ci mod ci mod

(13)

ð ð   X mod e  ð 1=2 Þ dr dr0 rmod ðrÞrmod ðr0 Þ=jr  r0 j Ev rmod ¼ i i ð     + Exc rmod  drrmod ðrÞdExc rmod =dr mod ðrÞ

(14)

The energy expression for this case becomes

If the applied potential corresponds to application of a uniform electric field of strength F0, it can be written as vappl ðr Þ ¼ r  F0

(15)

and one can calculate the dipole moment by evaluating the expectation value of the dipole operator. The dipole polarizability can be evaluated as the coefficient of the first-order change in the dipole moment. For evaluation of the multipole (2L-pole) polarizability, the potential is represented as

316 Atomic clusters with unusual structure, bonding and reactivity

vL appl ðrÞ ¼ r L PL ð cos yÞ FL 0

(16)

with F0L representing the field strength for L ¼ 1 (dipole), field strength gradient for L ¼ 2 (quadrupole), etc. Here PL(cos y) represents the Legendre polynomial of degree L. For dipole polarizability, Eq. (16) becomes vappl ðrÞ ¼ r cos y F0

(16a)

For finite field method, one can calculate the energy in the presence of the field by iterative solution of Eq. (11), and using Eq. (14), and for zero field case, these are Eqs. (7) and (10), respectively. Since one needs only the first-order correction in the KS orbitals, it is useful to use perturbation theory. Thus substituting ei mod ¼ ei + ei ð1Þ F0

(17a)

ci mod ðrÞ ¼ ci ðrÞ + ci ð1Þ ðrÞ F0

(17b) ∗

rmod ðrÞ ¼ rðrÞ + rð1Þ ðrÞ F0 , with rð1Þ ðrÞ ¼ 2 Re Si ci ð1Þ cI

(17c)

where the quantities with superscript (1) denote first-order correction to the respective quantities, one obtains the equation for the first-order correction c(1) i (r) to the ith KS orbital, given by h i   hKS  ei ci ð1Þ ðrÞ ¼ ei ð1Þ  dveff ð1Þ ðr; rÞ ci ðrÞ, i ¼ 1,…N where hKS is defined as

  hKS ¼ ð1=2Þr2 + veff ðr; rÞ

(18)

(19)

the first-order change in energy e(1) i is given by


 ei ð1Þ ¼< ci ðrÞ| vappl ðrÞ=F0 |ci ðrÞ >

and the first-order change in the effective potential can be written as
  modð1Þ  r; rmod dveff ð1Þ ðr; rÞ ¼ vappl ðrÞ=F0 + dvint

(20a)

(20b)

with the modified density-dependent internal potential dvint(r; rmod) given by (with terms retained up to first order) ð  modð1Þ  dvint r; rmod ¼ dr0 rð1Þ ðr0 Þ=|r  r0 | ð   + dr0 rð1Þ ðr0 Þ d2 Exc ½r=drðrÞ drðr0 Þ (20c) Eq. (18) is to be solved self-consistently to obtain {c(1) i (r)}, assuming {ci(r)} for the unperturbed system to be already known. The difficulty arises because all the KS orbitals as well as first-order corrections to them are coupled in the equation. To avoid the difficulty encountered in numerical solution, often these equations are solved using variation method, leading to the so-called variation-perturbation approach [13]. In this method, one writes a variational functional corresponding to the differential Eq. (18), using the procedure applicable for a nonlinear differential equation, given by o  i hn   P ci ð1Þ , ci ¼ Si < ci ð1Þ ðrÞ hKS  ei ci ð1Þ ðrÞ > h i Si < ci ð1Þ ðrÞ ei ð1Þ  dveff ð1Þ ðr; rÞ ci ðrÞ > (21) It may be mentioned that without loss of generality, one can impose the intermediate normalization < c(1) i (r)jci(r)> ¼ 0. For atomic systems, however, numerical solution is simpler and has led to good results for polarizability [14,15]. Once the solutions for the perturbation correction {c(1) i (r)} are obtained, one can easily calculate the dipole polarizability by evaluating an integral, namely

Polarizability of atoms and atomic clusters Chapter

ð h ∗ a2 L ¼ Si dr ci ð1Þ ðrÞ r L PL ð cos yÞ ci ðrÞ i + ci ∗ ðrÞ r L PL ð cos yÞ ci ð1Þ ðrÞ

17

317

(22a)

ci as that can easily be written in terms of the density change r(1)(r) ¼ 2 Re Si c(1)⁎ i ð a2 L ¼ dr rð1Þ ðrÞ r L PL ð cos yÞ

(22b)

which for the case of dipole polarizability (L ¼ 1) simplifies to ð a ð¼ a2 Þ ¼ dr rð1Þ ðrÞ r cos y

(22c)

4. Polarizability of spherically symmetric systems: Atoms and atomic clusters within the jellium model For spherically symmetric systems, such as atoms or metal clusters within the jellium model, the functions c(1) i (r) will have the form ci ð1Þ ðrÞ ¼ fi ðr Þ r L PL ð cos yÞ

(23a)

for a multipole case. For the dipole case, one has ci ð1Þ ðrÞ ¼ fi ðr Þ r cos y

(23b)

which on substitution in Eq. (18) or the corresponding variational functional leads to simplified result in terms of only one r variable, which essentially leads to a modified Sternheimer equation [15], which has been solved for several atoms. Besides atomic systems, there is another class of systems, namely the atomic clusters [16], which although should be treated more elaborately (and correctly) as multicenter problem, can also be treated as spherically symmetric case, if one treats all the nuclei (or nuclei and core electrons) together forming a continuous charge distribution, the so-called jellium model, where the smeared out positive charge density due to the ions can be written as a step function. Within this model, the valence electrons move in the field of a uniform positive charge distribution within a sphere of radius R0 and of course the electron-electron repulsion. For this, the DFT KS equation will be same as Eq. (7), but the external potential vext(r) will not be just Coulombic ( Z/r), but will be given by h i vext ðr Þjellium ¼ ðQ=2R0 Þ 3  ðr=R0 Þ2 , for r < R0 (24) ¼ Q=r , for r > R0 where Q is the total net charge of the jellium sphere [17]. For an n-atom cluster, Q will be (Z  Nc)n, where Nc is the number of the core electrons in the atom concerned. For an n-atom cluster of a monovalent atom, such as Nan cluster, clearly Nc ¼ Z  1 and hence Q ¼ n. Eq. (24) clearly indicates that the potential is a spherical harmonic potential up to the surface of the sphere and Coulombic outside (beyond R0). At r ¼ 0, the potential is (3Q/2R0), while at r ¼ R0, it is  Q/R0. The value of R0 can be related to the density of the cluster, and usually one can assume the metal density in the cluster to be the same as that in the bulk, thus enabling one to fix the value of R0. Thus, the solution of the KS equation (Eq. (7)) can be obtained by following a procedure similar to the one used for an atom, but the sequence and degeneracies of the energy levels will be similar to that for a spherical harmonic oscillator and not a hydrogen atom [18]. This aspect has some important consequences on the extra stability attained by the atomic clusters with selected number of atoms, corresponding to electronic shell closings, analogous to extra stability of inert gas atoms due to electronic shell closings for atoms with certain number of electrons. In fact this has been first observed experimentally by Knight et al. [19], who found regularities in the mass spectra of Nan clusters with n ¼ 2–100. The peaks were found to be large for n ¼ 2, 8, 20, 40, 58, 92, which were ascribed to electronic shell closing representing the so-called magic clusters. The peaks corresponding to n ¼ 18, 34, 68, and 70 are somewhat weaker. The first set corresponds to the electronic shell occupancies for harmonic oscillator potential, whereas the second set appears as additional levels due to slightly deformed potential (intermediate between harmonic oscillator and square

318 Atomic clusters with unusual structure, bonding and reactivity

well potential). The maximum occupancies of the generated levels are (see Fig. 1 in Ref. [18]): 1s(2), 1p(6), 1d(10), 2s(2), 1f(14), 2p(6), 1g(18), 2d(10), 3s(2), 1h(22), 2f(14), 3p(6), etc., in increasing order of energy of the orbitals. This leads to shell closings at 2, 8, 18, 20, 34, 40, 58, 68, 70, 92, etc. Thus, the jellium model predictions of magic clusters are in overall good agreement with experimental results. It is clear that the shell closing based on even simple spherical jellium model is able to reproduce the experimental stabilities for clusters with certain selected number of electrons (magic clusters) quite well [19]. While this is concerned with only the aspect of electronic shell closing, which can be well reproduced by jellium model for alkali and alkaline earth metal clusters, there is also the aspect of additional extra stability corresponding to geometric (arrangement of atoms) origin of magic clusters for many other atoms, which can be accounted for only by considering the clusters with explicit account of positions of the nuclei, as has been discussed in details, for example, by Martin [20]. It has been shown [21] by considering the Lithium clusters as example, an interconnection between the occurrence of magic clusters and its close correspondence to the higher values of chemical hardness parameter, thereby extending the applicability of maximum hardness principle [22] to the domain of metal clusters.

5.

Chemical reactivity indices-based route to polarizability

The approaches to calculate the polarizabilities discussed so far make use of the full KS DFT calculations and all the orbitals. However, simplified approximate approaches have also been possible, as illustrated for atomic systems, with possibility of straightforward extensions to atomic clusters and molecular systems. A simple approach to the calculation of the dipole polarizability of atoms and positive ions has been proposed [23,24] in terms of simple expectation values involving only the frontier orbitals of KS DFT. It thus requires only a single calculation, and the calculated polarizability values have been shown to agree quite well with the standard values. Frontier orbitals can however be thought of approximations to the more general Fukui functions f(r) ¼ (∂ r/∂N)v [8], which is one of the widely used reactivity indices. Polarizability for molecular systems can be calculated by using atomic dipoles. The original atomic dipole model due to Applequist [25], where the overall dipole moment of the molecule in presence of an applied electric field, determines the polarizability. Subsequently, Thole [26] modified the method, leading to improvement by replacing the point dipole interaction by smeared out dipole interaction. An analogous approach, known as Discrete Dipole Approximation (DDA), proposed originally by Purcell and Pennypacker is widely used for the calculation of polarizability and optical properties of nanoclusters [27]. The polarization of the system (molecule or cluster) due to the external electric field not only induces atomic dipoles but also drives interatomic charge transfer. The net induced dipole moment and hence the polarizability can be best calculated by considering both atomic charges and atomic dipoles. For this purpose, one may express the field-induced energy change of a molecule or a cluster in terms of density, within the framework of DFT, by using a perturbation theory. Coarse graining of density can then be implemented by partitioning the density change into atomic domains, and one can express the energy change in terms of the integrated density and its first moment, i.e., the atomic charges and dipoles. One can either minimize the energy or first calculate the effective chemical potential and then demand its equalization among the atoms, obtaining thereby a set of linear equations, which on solution, gives the atomic charges and dipoles, and hence the polarizability [28,29].

6.

Discussion on polarizability values of atomic clusters

There have been many studies on the calculation of metal clusters, using the jellium model, using DFT for the valence electrons and the potential due to the nuclei and the core electrons, given by Eq. (24). (see, e.g., Ref. [17]). Calculations are also reported for the metal clusters using the actual potentials due to the atoms, with the atomic positions obtained through geometry optimization and polarizabilities have been calculated using the finite field method. As illustrative examples, it may be mentioned that Chandrakumar et al. [30] have reported dipole polarizabilities for Nan clusters for n ¼ 1–10. Focus has been on the effect of including electron correlation through the XC potential. They have also attempted to correlate the polarizability values with different properties. For example, the polarizability values are shown to have inverse correlation with the binding energy of the clusters and linear correlation with volume of the cluster. In another work [31], these authors have reported DFT-based results for the polarizability of lithium clusters. The predicted values of polarizability through zero-temperature calculation are however found to be higher in comparison to the experimental results measured at higher temperature. This is rather unusual and can be attributed to underestimation of correlation effects in the XC functional used. One can expect that since correlation potential is negative, increase in magnitude of correlation effect would make the electrons subjected to stronger negative potential and hence more tightly bound, thus

Polarizability of atoms and atomic clusters Chapter

17

319

making the system less polarizable, with a consequent decrease in polarizability value. They have also shown [32] a good correlation between the polarizability and the ionization potential (IP) and softness (reactivity parameter) of the sodium and lithium clusters. The structural evolution of the alkali metal clusters and the size dependence of several reactivity descriptors, such as IP, electron affinity, polarizability, chemical potential, hardness, softness, etc., have been reported [33]. The observed good inverse correlation between the dipole polarizability and the IP of the neutral clusters can have important implications in obtaining the polarizability of metal clusters from their IP values directly. Their softness parameter also correlates strongly with the dipole polarizability of the clusters. In view of the observation of the abovementioned correlations, in particular, the inverse correlation of polarizability with binding energy of the clusters, and higher binding energy of the magic clusters, one expects the polarizability of the magic clusters to be lower than the neighboring clusters. This implies that the minimum polarizability principle [34,35] will be obeyed by the clusters, which is found to be supported by the limited data available on the polarizability as a function of the cluster size (see, for example, the plot of polarizability of Nan clusters in Fig. 3 of Ref. [18], where a clear dip is observed in the values of polarizability per atom for the magic clusters n ¼ 2, 8, 18, 20, etc.), although more studies on higher clusters and clusters of different metals are needed to have a more general and definite conclusion in this regard. Calculation of the static dipole polarizability has not been restricted to alkali metals alone, but has also been extended [36] to moderate-sized silicon clusters. A modified genetic algorithm has been used to identify conformers of silicon isomers. The optimized geometry of the clusters and their relative stability have been reported, and the calculated polarizabilities are shown to correlate well with the experimental data and other available calculated results. Similarly, the clusters Na16, Cu16, and Si20–28, which are special due to their cluster shape transitions [37], have been investigated through DFT calculations. Their polarizabilities, ionization potentials, and electron affinities are found to show characteristic dependence on cluster shape. Polarizability values are found to depend on the cluster volume for all the isomers studied, and interesting relationship emerges for variations in polarizabilities among isomers of the same size, but different shapes.

7. Concluding remarks This work has presented some of the aspects of polarizability of many-electron systems, with special emphasis on atoms and atomic clusters. The quantum mechanical approach within the framework of DFT has mainly been considered here. The general theory, its simplification for spherically symmetric systems, some alternative approaches using the reactivity indices of conceptual DFT have also been briefly indicated. Simplification due to consideration of jellium model for metal clusters has been considered. Illustrative results for simple small atomic clusters have been discussed. The present discussion has however been limited to few of the approaches and also the details have been omitted. Polarizability is a very important property of any system. It is an important ingredient of the conceptual DFT and can play an important role in controlling intermolecular interaction, chemical binding, chemical reactivity, and much of the other chemical phenomena. It is directly linked with hardness/softness parameter. In fact, analogous to the maximum hardness principle [22], the minimum polarizability principle [34,35] seems to be obeyed for clusters as well, as illustrated here with a few examples of magic clusters. However, only a small fraction with few illustrative aspects has been covered in this discussion and much more aspects lie outside the present coverage.

Acknowledgments I express my gratefulness to Professor B.M. Deb and Professor Robert G. Parr, who have guided me over the years. I thank my colleagues Dr. Alok Samanta, Dr. Tapan Ghanty, Dr. K.R.S. Chandrakumar, Dr. Amita Wadehra, and many others for helpful discussions and collaborations. I am particularly thankful to Professor Pratim K. Chattaraj, not only for his kind invitation to contribute this article but also for his enormous patience and encouragement during the delay of its completion.

References [1] G. Maroulis (Ed.), Atoms, Molecules and Clusters in Electric Fields: Theoretical Approaches to the Calculation of Electric Polarizability, Imperial College Press, London, 2006. [2] J. Israelachvili, Intermolecular and Surface Forces, third ed., Academic Press, 2011. [3] S.K. Ghosh, Chemical binding and reactivity parameters: a unified coarse grained density functional view, in: P.K. Chattaraj, D. Chakraborty (Eds.), Chemical Reactivity in Confined Systems: Theory, Modelling and Applications, John-Wiley, 2021, pp. 167–177. [4] P. Geerlings, E. Chamorro, P.K. Chattaraj, F. De Proft, J.L. Ga´zquez, S. Liu, C. Morell, A. Toro-Labbe, A. Vela, P. Ayers, Conceptual density functional theory: status, prospects, issues, Theor. Chem. Accounts 139 (2) (2020), https://doi.org/10.1007/s00214-020-2546-7. [5] P.K. Chattaraj (Ed.), Chemical Reactivity Theory: A Density Functional View, CRC Press, Boca Raton, FL, 2009.

320 Atomic clusters with unusual structure, bonding and reactivity

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864. W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) A1133. R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press and Clarendon Press, New York and Oxford, 1989. R.G. Parr, R.A. Donnelly, M. Levy, W.E. Palke, Electronegativity: the density functional viewpoint, J. Chem. Phys. 68 (8) (1978) 3801–3807. R.G. Parr, R.G. Pearson, Absolute hardness: companion parameter to absolute electronegativity, J. Am. Chem. Soc. 105 (26) (1983) 7512–7516. A. Dalgarno, Atomic polarizabilities and shielding factors, Adv. Phys. 11 (1962) 281–315. G.D. Mahan, K.R. Subbaswamy, Local Density Theory of Polarizability, Plenum Press, New York, 1990. S.K. Ghosh, B.M. Deb, Dynamic polarizability of many-electron systems within a time-dependent density-functional theory, Chem. Phys. 71 (1982) 295–306. S.K. Ghosh, B.M. Deb, A simple density-functional calculation of frequency-dependent multipole polarizabilities of noble gas atoms, J. Mol. Struct. THEOCHEM 103 (1983) 163–176. G.D. Mahan, Modified Sternheimer equation for polarizability, Phys. Rev. A22 (1980) 1780–1785. P. Jena, S.N. Khanna, B.K. Rao (Eds.), Physics and Chemistry of Finite Systems: From Clusters to Crystals, vol. I, Springer Science, 1992. M. Brack, The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches, Rev. Mod. Phys. 65 (1993) 677. W.A. deHeer, W.D. Knight, M.Y. Chou, M.L. Cohen, Electronic shell structure and metal clusters, Solid State Phys. 40 (1987) 94. W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Electronic shell structure and abundances of sodium clusters, Phys. Rev. Lett. 52 (1984) 2141. T.P. Martin, Shells of atoms, Phys. Rep. 273 (1996) 199–241. M.K. Harbola, Magic numbers for metallic clusters and the principle of maximum hardness, Proc. Natl. Acad. Sci. U. S. A. 89 (1992) 1036–1039. R.G. Parr, P.K. Chattaraj, Principle of maximum hardness, J. Am. Chem. Soc. 113 (1991) 1854–1855. T.K. Ghanty, S.K. Ghosh, A frontier orbital density functional approach to polarizability, hardness, electronegativity and covalent radius of atomic systems, J. Am. Chem. Soc. 16 (1994) 8801 (Communication to the Editor). T.K. Ghanty, S.K. Ghosh, A new simple approach to the polarizability of atoms and ions using frontier orbitals of Kohn-Sham density functional theory, J. Mol. Struct. (THEOCHEM) 366 (1996) 139–144. J. Applequist, An atom dipole interaction model for molecular optical properties, Acc. Chem. Res. 10 (1977) 79–85. See also, L. Jensen, P.-O. ˚ strand, A. Osted, J. Kongsted, K.V. Mikkelsen, Polarizability of molecular clusters as calculated by a dipole interaction model, J. Chem. Phys. A 116, 4001 (2002). B.T. Thole, Molecular polarizabilities calculated with a modified dipole interaction, Chem. Phys. 59 (1981) 341. M.A. Yurkina, A.G. Hoekstra, The discrete dipole approximation: an overview and recent developments, J. Quant. Spectrosc. Radiat. Transf. 106 (2007) 558–589. A. Wadehra, S.K. Ghosh, A density functional theory-based chemical potential equalisation approach to molecular polarizability, J. Chem. Sci. 117 (2005) 401–409. S.K. Ghosh, A coarse grained density functional theory, chemical potential equalization and electric response in molecular systems, J. Mol. Struct. THEOCHEM 943 (2010) 178–182. K.R.S. Chandrakumar, T.K. Ghanty, S.K. Ghosh, Static dipole polarizability and binding energy of sodium clusters Nan. n¼1–10.: a critical assessment of all-electron based post Hartree–Fock and density functional methods, J. Chem. Phys. 120 (l4) (2004) 6487–6494. K.R.S. Chandrakumar, T.K. Ghanty, S.K. Ghosh, Ab initio studies on the polarizability of lithium clusters: some unusual results, Int. J. Quantum Chem. 105 (2005) 166–173. K.R.S. Chandrakumar, T.K. Ghanty, S.K. Ghosh, Theoretical studies on polarizability of alkali metal clusters, in: G. Maroulis (Ed.), Atoms, Molecules and Clusters in Electric Fields, 2006, pp. 625–656. K.R.S. Chandrakumar, T.K. Ghanty, S.K. Ghosh, Relationship between ionization potential, polarizability, and softness: a case study of lithium and sodium metal clusters, J. Phys. Chem. A108 (2004) 6661–6666. T.K. Ghanty, S.K. Ghosh, A density functional approach to hardness, polarizability, and valency of molecules in chemical reactions, J. Phys. Chem. 100 (1996) 12295–12298. P.K. Chattaraj, S. Sengupta, Popular electronic structure principles in a dynamical context, J. Phys. Chem. 100 (1996) 16126. V.E. Bazterra, M.C. Caputo, M.B. Ferraro, P. Fuentealba, On the theoretical determination of the static dipole polarizability of intermediate size silicon clusters, J. Chem. Phys. 117 (2002) 11158. X. Chu, M. Yang, K.A. Jackson, The effect of geometry on cluster polarizability: studies of sodium, copper, and silicon clusters at shape-transition sizes, J. Chem. Phys. 134 (2011) 234505.

Chapter 18

Advances in cluster bonding: Bridging superatomic building blocks via intercluster bonds Nikolay V. Tkachenkoa, Zhong-Ming Sunb, Alexander I. Boldyreva, and Alvaro Mun˜oz-Castroc a

Department of Chemistry and Biochemistry, Utah State University, Logan, UT, United States, b State Key Laboratory of Elemento-Organic Chemistry,

Tianjin Key Lab of Rare Earth Materials and Applications, School of Materials Science and Engineering, Nankai University, Tianjin, China, c Grupo de Quı´mica Inorga´nica y Materiales Moleculares, Facultad de Ingenierı´a, Universidad Autonoma de Chile, El Llano Subercaseaux, Santiago, Chile

1. Introduction Aggregation of discrete units growing clusters to bulk matter remains a relevant issue in the rational usage of building blocks to realize well-structured cluster-based materials [1–10]. In recent decades, the rich structural versatility and characteristics of bare and ligand-protected metal clusters have been explored, accounting for their novel physicochemical and size-dependent properties of interest for technological devices [11–15]. Thus, the rationalization of cluster-cluster or intercluster aggregation and their interplay represents a key issue in bringing their characteristics into the overall aggregate or assembly [16–19]. Such individual cluster units are generally stable and isolable species. Their stability is based on the delicate balance between their geometrical and electronic structures. A particular number of cluster electrons have been recognized to lead to a favorable intracluster bonding situation, usually called superatoms [20–23]. Thus, for example, the bonding capabilities between clusters have been denoted in small tetrahedral cluster aggregates of the formula (Zn3Cu)2 and (Mg3Li)2 where four superatomic lone pairs (4c-2e bonding elements) are found via AdNDP analysis, with the interesting addition of two 8c-2e superatomic s- and p-bonds, as reported by Cheng [24]. Another vivid example of superatomic assembly is the Al4 tetrahedra, that has been computationally evaluated to form a new aluminum allotrope based on the extended aggregation of tetrahedral units in a “diamond”-like array, resulting in a metastable ultralight material [25]. Along with that the Au6 core can undergo sp3-hybridization [26], where the 8-cluster electron (8-ce) [Au6{Ni3(CO)6}4]2
species [27] has been ascribed as an analog for archetypical CH4 molecule highlighting the opportunities provided by classical chemical concepts from simple molecules for the design of functional clusters and nanoparticles [24,26,28–30]. To rationalize the chemical bonding between such clusters, the super valence bond (SVB) model was found to be quite useful [24,30–32]. Analysis of the bonding characteristics is usually depicted in terms of delocalized orbitals. In addition, different elements related to bonding patterns in clusters can be elucidated by using the Adaptive Natural Density Partitioning (AdNDP) method proposed by Zubarev and Boldyrev [33–35]. It is a valuable approach to describe multicenter bonding interactions [36,37] and is convenient for evaluating molecular clusters or aggregates of variable sizes and compositions across the periodic table, delivering relevant information about their aromatic/antiaromatic behavior. In addition to isolated clusters [38–43] and molecules [34,36], AdNDP can handle nontrivial cases, such as mechanically trapped systems [44], solvated ions [45], electrides [46], surface defects [47], and periodically extended materials [48–52]. It can also be used to distinguish dative and covalent bonds [53,54]. Importantly, AdNDP is an almost method/basis set-independent tool [39,55,56]. This chapter accounts for the recent bonding characteristics involving novel clusters from transition metals with main group elements leading to ordered architectures. Such features envisage the application of metalloid clusters as covalently bonded building blocks for functional materials [57], besides self-assembled species via supramolecular interactions. Thus, achieving a rational aggregation of different cluster-based building blocks will benefit the opportunities to design functional clusters and related structures.

Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00010-3 Copyright © 2023 Elsevier Inc. All rights reserved.

321

322 Atomic clusters with unusual structure, bonding and reactivity

2.

Intercluster bonding of gold clusters

The anionic Au25(SR)18
cluster is a prototypical 8-ce molecular species with an inner Au135+ core protected by ligands, bearing a closed-shell 1S21P6 electronic configuration leading to a stable structure [58–66]. Interestingly, the electrondeficient 7-ce neutral Au25(SR)18 is prone to form Au38(SR)24 dimeric species [67] at 65°C [68], featuring 14-ce in 1S + 1S and 1P + 1P combinations. This resembles the bonding pattern in single-bonded diatomic halogens such as F2, which is also observed in the doped [M2Au36(SR)24]0 (M ¼ Pd, Pt) (Fig. 1) [28,69,70]. Moreover, the number of cluster electrons can be modified by further manipulating the oxidation state electrochemically [71]. Upon changing the oxidation state, 12- and 10-ce can be achieved, leading to double and triple multiple bonds between the involved core units, similar to simple dimeric molecules such as O2 and N2 [72]. As shown in Fig. 1, the representative frontier orbitals depict a s2p4p*4 configuration for the 14-ce species (* stand for antibonding combinations), leading to an intercluster bonding based on a single superatomic s-bond. For the 12-ce counterpart, a s2p4p*2 configuration is given, resulting in a net bonding contribution from a s- and a p-bond. Analogically, the 10-ce cluster bears a s2p4 configuration with bonding contribution from one s- and two p-bonds. Hence, the formal bond orders are, respectively, one to three, accounting for the single- to triple-bonded characteristics able to be selectively tuned upon different oxidation states [72]. Further aggregation of M13 core has been studied for silver counterparts [73], and linear trimers of PtAg12, which are isolobal to Ne2 and I
3 molecules [74]. In addition, higher angular momenta intercluster bonds forming d-bonds between Au11 assembled units in the overall Au22(dppo)6 cluster have been reported [75]. Besides the linear species, cyclic arrays have also been explored. Thus, a toroid of six Au6 cores capped by six Au atoms was proposed. The D6h-Au42 cluster [76] enables the evaluation and application of classical concepts related to cyclic molecules to cluster arrays. Interestingly, the resulting delocalized orbital pattern resembles a p-orbital manifold involving five occupied p-orbitals, which in turn follows the H€uckel (4n + 2)p rule-bearing 10p electrons (Fig. 2) [77]. The fulfilling of the H€ uckel rule suggests the D6h-Au42 cluster as a two-dimensional aromatic toroid, which is later confirmed by the formation of an induced shielding cone when a magnetic field is applied perpendicularly to the ring. This is in agreement with the shielding cone concept settled from the ring-current effect in benzene and other representative planar molecules [78,79]. Hence, from the magnetic criteria of aromaticity, the toroid shows a shielding region of
76.4 ppm at the center of the ˚ above the center. Furthermore, structure (from NICSzz) of long-range characteristics with values of
18.7 ppm at 9.0 A the charge distribution within the toroid shows a negatively charged region at the center with a positive region around the structural backbone, in line with the benzene charge distribution. Hence, supporting D6h-Au42 as an analog of cyclic aromatic molecules following the H€ uckel (4n + 2)p rule-bearing 10p electrons and their related characteristics.

FIG. 1 Representation of the M2Au36(SR)24 structures, and related frontier orbitals upon different oxidation states for Pd2Au36(SR)24q, and related intercluster bonds. (Adapted from reference A. Mun˜oz-Castro, Chem. Commun. 55 (2019) 7307–7310, with permission from the Royal Society of Chemistry.)

Pd2Au362– 14-ve

Pd2Au360 12-ve

π* π σ

π* π σ

Pd2Au362– Pd2Au360 Pd2Au362+ 14-ve 12-ve 10-ve

Pd2Au362+ 10-ve

Advances in cluster bonding Chapter

π1

π2, π3 π4, π5 π6, π7

π8, π9

Au42

18

323

FIG. 2 Superatomic p-orbitals of the 10p-aromatic D6h-Au42, and related p-orbitals from C9H9
. (Copyright 2016 Wiley. (A)–(C) reproduced with permission from reference A. Mun˜oz-Castro, ChemPhysChem 17 (2016) 3204–3208.)

[C9H9]–

3. Intercluster bonding of Zintl clusters The aggregation of multiple main-group tetrahedral units involving the nido-[E4]4
cluster [80] has been characterized experimentally, resulting in a supertetrahedral aggregate with four E4 units [81,82] or as a threefold structure bearing three E4 blocks [83]. Following a rational strategy of a controlled assembly process involving four [Ge4]4
clusters (Fig. 3) [82], two related heterometallic supertetrahedral clusters [Zn6Ge16]4
and [Cd6Ge16]4
were isolated from the reaction between K12Ge17 and ZnMes2/CdMes2 (Mes ¼ 2,4,6-Me3C6H2) in the presence of 2,2,2-crypt in N,N-dimethylformamide (DMF)/ethylenediamine (en) solutions, respectively. From time-dependent HRESI-MS spectra, it is shown that the larger [Zn6Ge16]4
cluster is raised by the aggregation of a single [Ge4]4
and ZnMes2 units, resulting in the aggregation of four [Ge4]4
units connected by Zn2+ ions, and a consistent approach has been used to achieve a Cd2+ counterpart ([Cd6Ge16]4
). Each nido[E4]4
unit represents a minimal structure bearing a spherical aromatic character [80], featuring a stable unit to act as a building block in the process of cluster aggregates. [Zn6Ge16]4
and [Cd6Ge16]4
clusters feature four [Ge4]4
units at the vertices of the supertetrahedron, with transition metal ions (Zn2+/Cd2+) bridging each edge. ˚ , and 3.50–3.66 A ˚ , respectively, The structure features long Zn-Zn and Cd-Cd distances in the range of 3.30–3.39 A denoting that such contacts do not contribute to the overall stabilization of the structure [82]. To achieve more insights into the bonding characteristics of the supertetrahedron structure, density functional theory (DFT) calculations using a large polarized quadruple-zeta basis with DFT hybrid functional (PBE0/Def2-QZVP level of theory) [84,85], were carried out, showing a large calculated HOMO-LUMO gap of 2.22 eV in both clusters, precluding any second-order Jahn-Teller instabilities, retaining a Td symmetry. The bonding characteristics of such supertetrahedrons exposed by the AdNDP analysis highlight common bonding elements involving 140 valence electrons for both [Zn6Ge16]4
and [Cd6Ge16]4
species, featuring one-center two-electron (1c-2e) elements accounting for five d-type lone pairs located at each Zn or Cd atom and one s-type lone pair located on each Ge atom [82]. The remaining 48 electrons are spread in 12 3c-2e Ge-Ge-Ge s bonds, 3 per Ge4 unit, and 12 3c-2e M-Ge-Ge s bonds, three per Ge4 unit (Fig. 4A and B). Alternatively, the 12 3c-2e Ge-Ge-Ge s bonds within each Ge4 unit could also be considered as more localized elements given by 12 2c-2e Ge-Ge s bonds located over the Ge4 edges with lower occupation numbers (ONs) [82]. Contributions from 3c-2e M-Ge-Ge bonds are relatively high, given by 84% for [Zn6Ge16]4
and 86% for [Cd6Ge16]4
clusters. Moreover, a substantial delocalization over the metal centers involved in such bond, denoted by 0.60 j ej per two 3c-2e bonds, results in an elongation of the Ge-Ge distance in the M-Ge-Ge triangle, indicating that such 3c-2e element exhibits an appreciable covalent character. Thus, the stability of [Zn6Ge16]4
/ [Cd6Ge16]4
clusters manifested via the delocalization in 3c-2e M-Ge-Ge bonding elements, that undergoes a sizable covalent character, instead of a purely ionic M2+-[Ge4]4
cation-anion interaction. Furthermore, the topology of the Electron Localization Function (ELF), Z(r), given as 2D plots (Fig. 4), is in agreement with the covalency of the 3c-2e M-Ge-Ge s bonds, with a clear localization in the M-Ge-Ge region (in the plane of Ge4M fragment, Fig. 4C (right)), with the major contribution coming from the two Ge atoms, with close to zero values at the center of the supertetrahedron (Fig. 4C (left)), denoting the absence of the M-M interactions within the M6 octahedron.

324 Atomic clusters with unusual structure, bonding and reactivity

FIG. 3 (A) Formation scheme of [M6Ge16]4
(E ¼ Zn or Cd); (B) ellipsoid plot (50% level) of the crystal structure of [Zn6Ge16]4
(the same structure for [Cd6Ge16]4
); (C) the experimental and computed geometries of the Ge-Ge2-M unit in [M6Ge16]4
and the average distances of Ge-M ˚ . (Copyright 2020 and Ge-Ge are given in A Wiley, reproduced with permission from reference H. Xu, I.A. Popov, N.V. Tkachenko, Z. Wang, A. Mun˜oz-Castro, A.I. Boldyrev, Z. Sun, Angew. Chem. Int. Ed. 59 (2020) 17286–17290.)

A

4– 4–

Ge 4 Ge

ZnMes2 + 4 K+ 4 [2,2,2-crypt] 4 [K(2,2,2-crypt]+ • + 6 or en/dmf, r.t. CdMes2

Ge Ge

[Zn6Ge16]4– (1) [Cd6Ge16]4– (2)

B

C Ge1

4

Ge4

2

3

Zn1

15

Zn2

16

Zn1

9

Zn4

Zn5

12

FIG. 4 (A) 3c-2e Ge-Ge-Ge s-bonds of [Zn6Ge16]4
show superimposed on the molecular framework (three bonds per Ge4); (B) 3c-2e Zn-Ge-Ge s-bonds of [Zn6Ge16]4
show superimposed on the molecular framework (three bonds per Ge4); (C) ELF distribution in rectangular Ge4Zn fragment (right) and square Zn4 fragment (left). ON denotes occupation number, here and elsewhere. Similar AdNDP and ELF results were obtained for [Cd6Ge16]4
, omitted for clarity. (Copyright 2020 Wiley, reproduced with permission from reference H. Xu, I.A. Popov, N.V. Tkachenko, Z. Wang, A. Mun˜oz-Castro, A.I. Boldyrev, Z. Sun, Angew. Chem. Int. Ed. 59 (2020) 17286–17290.)

14

Cd1

7

2.5

36 2.7

11

60.2° 2.745 59.9° 67-1°

2.5 75 07 2.7 59.6° 2.761 60.2° 66-8°

5

6 [Zn6Ge16]4–

A

63.8° 2.714 58.1° 66-1°

2.5 10 11 2.6 63.0° 2.727 58.5° 65-8°

Ge4

8 Zn6

90

2

Zn3

13

10 11

2.4

9 .5 6

(Exp)

(Calc)

B

Twelve 3c-2e Ge-Ge-Ge s-bonds ON = 1.88 |e|

Twelve 3c-2e Zn-Ge-Ge s-bonds ON = 1.91 |e|

C 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Advances in cluster bonding Chapter

18

325

To test how the four spherically aromatic [Ge4]4
fragments behave in the [M6Ge16]4
clusters, the magnetic response properties were calculated (Fig. 5). To obtain a global view of the possible aromatic character in [M6Ge16]4
, the induced magnetic field (Bind) was calculated, accounting for the isotropic (Iso ¼
(1/3)(sxx + syy + szz)Bext) (orientational average) term, showing four spherical-like shielding regions at each Ge4 fragment, denoting that the spherical aromatic character of the [Ge4]4
parent is retained upon formation of the supertetrahedron structure. Under individual orientations of the applied field, the consequent long-range shielding cones due to four [Ge4]4
spherical aromatic units overlap along with the overall structure, resulting in a global shielding-induced magnetic field. Hence, the reaction of Zintl-ions with transition metals is suggested as a suitable strategy to form controlled aggregates, where the global aromatic character of [M6Ge16]4
species is given by the additive characteristics of the individual spherical aromatic Ge4 fragments displaying a s-aromaticity as depicted by the AdNDP analysis [80]. Another example of the aggregation mediated by transition metals nodes is the formation of Sb3
7 dimers bridged by Au(I) atoms in the characterized structure of [Au2Sb16]4
cluster (Fig. 6) [86]. Such cluster was characterized as a FIG. 5 Magnetic response properties of [Zn6Ge16]4
, given by isotropic term (Bisoind), and under specific orientations of the external field (Bzind, Bxind, and Byind). Isosurfaces at 5 ppm; blue (grey color print version): shielding; red (light gray color print version): deshielding. The same features are found for [Cd6Ge16]4
. (Copyright 2020 Wiley, reproduced with permission from reference H. Xu, I.A. Popov, N.V. Tkachenko, Z. Wang, A. Mun˜oz-Castro, A.I. Boldyrev, Z. Sun, Angew. Chem. Int. Ed. 59 (2020) 17286– 17290.)

ind

B iso

ind

ind

ind

Bx

Bz

By

A

Sb6

Sb9

Sb2 Au2 Sb1

Sb7

B

Sb15 Sb10

Sb4

Sb13

Sb16

Sb8 Sb12 Sb3

Sb5

Au1

Sb11

Sb14

FIG. 6 Thermal ellipsoid plot for [Au2Sb16]4
cluster (drawn at 50% probability), left. Delocalized chemical bonding elements deciphered for [Au2Sb16]4
. Six 5c-2e delocalized s bonds with ON ¼ 1.86–1.99 je j found at the (A) upper AuSb4 (Au1-Sb5-Sb8-Sb11-Sb12) and (B) lower AuSb4 (Au2-Sb4-Sb6-Sb9-Sb10, right. Fragments. (Copyright 2016 Wiley, adapted with permission from reference I.A.A. Popov, F.-X.X. Pan, X.-R. R. You, L.-J.J. Li, E. Matito, C. Liu, H.-J.J. Zhai, Z.-M.M. Sun, A.I.I. Boldyrev, Angew. Chem. Int. Ed. 55 (2016) 15344–15346.)

326 Atomic clusters with unusual structure, bonding and reactivity

[K([2.2.2]crypt)]+ salt from the reaction of K3Sb7, Au(PPh3)Ph, and 2,2,2-crypt, in the en solvent, featuring an Sb16 polyanion in a quasi-C2 point group symmetry. Interestingly, it was previously shown that Sb73
unit undergoes a spherical-like aromatic shielding characteristic [87]. However, in [Au2Sb16]4
cluster, a peculiar s-aromatic character was found [86], where the AdNDP analysis unravels the bonding patterns supporting the overall cluster structure. First, lone pairs (1c-2e) elements were found involving five d-type lone pairs ascribed to each Au atom with ONs range of 1.83–1.99 jej, followed by one s-type lone pair located on each Sb atom with ON ¼ 1.93–1.97 je j, and two p-type lone pair on the two peripheral Sb atoms (Sb1 and Sb16) with ON ¼ 1.71 je j (Fig. 6). Next, multicenter bonding elements were localized as 19 2c-2e Sb-Sb s bonds (ON ¼ 1.93–1.98 j ej), ascribed to be responsible for the stabilization of the cage framework in this cluster, whereas no direct 2c-2e s-bonds between Au-Sb or Sb-Sb were found within the two central AuSb4 units bridging both Sb3
7 units. For each quasi-planar AuSb4 bridging unit, the delocalized bonding elements are given by three 5c-2e s-bonds with ON ¼ 1.86–1.99 je j, involving a contribution from 6s-Au orbitals of 39%. Consequently, each AuSb4 unit exhibits aromatic character as further supported by electronic multicenter indices (Iring and MCIs) [88–91] denoting a delocalization through a zigzag pattern (Sb6-Sb10-Sb9-Sb4 or Sb9-Sb6-Sb10-Sb4 atom groups), similar to Al42
[92]. As an additional example, d10-Cu(I) cations are also experimentally verified to be able to bring together Zintl-ion clusters, as provided by the characterization of a Ge9Mes3
(Mes ¼ mesityl) dimer, in the {[CuGe9Mes]2}4
cluster form [93]. This species was obtained as a [K(2,2,2-crypt)]+ salt (Fig. 7) from the reaction in N,N-dimethylformamide (DMF) solution between K4Ge9, mesityl-copper (CuMes), and 2,2,2-crypt. In this case, effective aggregation of clusters involves the incorporation of Cu(I) atoms into the overall 10-membered cage [93]. The identical CuGe9 units are bridged via a Cu2Ge2 diamond-like structure, supporting the idea that aromatic precursors could be used as building blocks and prone to be connected by d10 ions nodes, which preserves their parent aromatic properties [93]. The observed intracluster Ge6Ge9 and Ge10-Ge13 bond distances at the square face adjacent to the Cu atoms are in the range from 2.6854(6) to 2.7730(6) ˚ [93], in contrast to the elongation from the mesityl-decorated face with Ge-Ge distances of 2.8171(6)–3.337(0) A ˚ within A the squares adjacent to Ge1 and Ge18. This observation is probably due to the electron-withdrawing effect of the mesityl group and the formation of the bridging Cu2Ge2 diamond. In the resulting {[CuGe9Mes]2}4
cluster, several motifs can be recognized showing two-terminal Clar’s sextet p-aromatic phenyl rings from both Mes groups, two twin CuGe9 cages, and a bridging four-membered diamond-like Cu2Ge2 ring. From the bonding analysis provided by the AdNDP approach [93], each monomer unit involves a p-aromatic organic part bound to the metal cage via classical C-Ge s-bonds. In addition, within the 10-membered cage, 24 lone pairs of Cu and Ge atoms were found, with three locally s-aromatic fragments inside the cage, supporting an aromatic behavior FIG. 7 Structure of {[CuGe9Mes]2}4
, thermal ellipsoids are drawn at the 50% probability level (A), and chemical bonding picture of two 3c-2 s-bonds within the central Cu2Ge2 fragment (B). (Adapted from reference Z.-C. Wang, N. V. Tkachenko, L. Qiao, E. Matito, A. Mun˜oz-Castro, A.I. Boldyrev, Z.-M. Sun, Chem. Commun. 56 (2020) 6583–6586, with permission from The Royal Society of Chemistry.)

9

5

A 1

8

4

C1

3 2

Cu1

15

11 10

6 7

Cu2

18

17

12

C2 14

13

B

Two 3c-2e s bonds ON = 1.92|e|

16

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[94]. Upon the formation of the dimer, the number of germanium lone pairs decreases owing to their contribution to two 3c-2e s-bonds (ON ¼ 1.92 je j), which are responsible for providing the binding between the two monomers. Such bonding elements can alternatively be found as two 4c-2e bonds with ON ¼ 1.92–1.93 jej. Moreover, it is worthy to note that interactions between two monomers could also be explained in terms of Wade-Mingos electron counting rules [93]. Lastly, the diamond-like structure of the bridging fragment, the bonding elements, and the number of involved electrons (4 jej) suggest the antiaromatic character of the Cu2Ge2 unit [93], suggesting the total involvement of p-planar aromatic sections provided by the mesityl ligands, locally multiple s-aromatic CuGe9 cages, and an antiaromatic Cu2Ge2 unit within the overall {[CuGe9Mes]2}4
cluster. To further test the multiple aromatic characters of the dimer, the magnetic response properties were calculated (Fig. 8), which supports the aromatic behavior of the CuGe9 cage and the organic mesityl group [93]. The central bridging region retaining the dimeric structure introduces a local antiaromatic Cu2Ge2 fragment as given by its pair of locally 3c-2e s-aromatic islands, leading to a deshielding region above the Cu2Ge2 diamond, which exhibits an enhanced deshielding ind region under a parallel field (Bind and Byind) owing to the z ). Complementary, other orientations of the applied field (Bx spherical aromatic character of CuGe9 cages contribute to the shielding response at Cu2Ge2, leading to a negative averaged NICSiso index [94]. The topological analysis of the electron density for the {[CuGe9Mes]2}4
cluster supports the highly localized structure in the central Cu2Ge2 bridging motif, in line with the antiaromatic character of this fragment. Multicenter indices [88–91] confirm such observation, as denoted by a large Iring value of 0.036 for the p-aromatic ring and a sizably small value of 0.004 for the Cu2Ge2 motif, which denotes its antiaromatic character from other criteria of aromaticity. The bond-order alternation (BOA ¼ 0.23) index of the Cu2Ge2 fragment also supports its antiaromatic description. Note that neither the Laplacian of the electron density nor the multicenter indices indicate a strong aromatic character within each Ge9Cu cage, where multicenter index values of 0.01 for the specific Ge4Cu fragment show a mildly aromatic behavior [94].

x y z ind Biso

Bzind

Bxind

Byind

FIG. 8 Isosurface representation (2 ppm) of the induced magnetic field, accounting for an isotropic and specific orientation of the field. Blue (gray color in print version): shielding; red (gray color in print version), deshielding. (Reproduced from reference Z.-C. Wang, N.V. Tkachenko, L. Qiao, E. Matito, A. Mun˜oz-Castro, A.I. Boldyrev, Z.-M. Sun, Chem. Commun. 56 (2020) 6583–6586, with permission from The Royal Society of Chemistry.)

328 Atomic clusters with unusual structure, bonding and reactivity

The last case of intercluster aggregation of Zintl clusters that we want to mention in this section is the oxidative aggregation of Ge9 clusters. It has been shown experimentally to lead from discrete to extended chains of the form [Ge9-Ge9]6
, [Ge9 ¼ Ge9 ¼ Ge9]6
, [Ge9 ¼ Ge9 ¼ Ge9 ¼ Ge9]8
, and ∞[Ge9]2
[95–98]. Hence, the aggregation of different building blocks is an interesting issue, which, under a rational approach, is able to deliver different numbers of aggregated units within the structure selectively.

4.

Extended networks

The formation of extended networks based on the aggregation of superatomic building blocks has been proposed for several main-group clusters. Al4 units have been evaluated as three-dimensional networks resulting in an optimized cubic face-centered crystal lattice with eight aluminum atoms in a primitive unit cell with a group of spatial symmetry – ˚ (Fig. 9) [25]. The resulting density of the supertetrahedal Fd3̅m (number 227) and a unit cell parameter of a ¼ 13.322 A
3 Al4-based material is 0.61 g cm , which is close to the density of metallic lithium (0.534 g cm
3), sizably lower than the density of metallic aluminum (2.7 g cm
3) owing to the generated cavities between Al4-units. ˚ , with intercluster distances of 2.573 A ˚ . This obserThe obtained Al-Al distances within the Al4 unit amount to 2.609 A vation is given by the 3c-2e bond character within each Al4 unit. In contrast, the shorter distance accounts for a 2c-2e covalent bond character between Al4, which is confirmed via AdNDP analysis from the molecular model Al4(Al4H3)4. The analysis reveals four 3c-2e bonds with 1.908 electron occupation for the Al4 unit, and 2c-2e covalent bonds with 1.993 electron occupation for exo Al-Al bonds [25]. Periodic calculation of the electronic band structure shows that the width of the forbidden band is negligible, leading to a good electrical conductivity and rendering the material as a semimetal. Moreover, the calculated phonon spectrum denotes dynamic stability of the proposed Al4-based material. Similarly, B4 units can form an extended tetrahedral material, following a diamond-like extended array, resulting in a ˚ . The calculated band structure of this supertetrahedral stable structure [99,100], with a unit cell parameter of a ¼ 8.548 A borane shows the absence of the electronic band gap, indicating metallic properties of this system, which is dynamically stable as given by evaluation of the phonon dispersion characteristics [99]. The obtained density of the supertetrahedron borane-based material is 0.93 g cm
3, which is close to the density of water, owing to large cavities between covalently connected B4-units. Moreover, the calculated longest wavelength absorption band falls in the visible region of the spectrum with maxima between 460 and 490 nm.

FIG. 9 Schematic depiction of the supertetrahedral aluminum crystal structure. (Reprinted with permission from J. Phys. Chem. C 121 (40) (2017) 22187–22190. Copyright 2017 American Chemical Society.)

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FIG. 10 Formation of an extended two-dimensional array based on Ge9-building blocks. (Reprinted with permission from J. Phys. Chem. C 121 (40) (2017) 1934–1940. Copyright 2017 American Chemical Society.)

[Ge9R]66–

[M3{Ge9}3]¥

Two-dimensional arrays are also evaluated, where B4H-units are employed to form graphene-like extended aggregates [99], where the unit cell is hexagonal, containing eight boron and two hydrogen atoms, with the B4H-units array showing a chair conformation. Although the estimated phonon dispersion indicates dynamical instability of a single layer of a twodimensional B4H, the multilayer structure was found to be dynamically stable. The calculated electronic band structure suggests that this system is a direct-gap semiconductor with a bandgap of 0.49 eV [99]. In addition, larger tetrahedral structures can also be potentially evaluated as similar three-dimensional extended networks [101,102]. The aggregation of B6 units into the 2D-structure also leads to the formation of stable material with peculiar magnetic properties [50]. Such structure is the first example of a ferromagnetic two-dimensional boron compound that is dynamically and thermally stable up to 300 K. A consequent study involving the aggregation of B4X2-units (X ¼ N, P, As, Sb), into twodimensional extended structures, was also performed [52,103] showing an absence of imaginary frequencies in phonon dispersion spectra supporting the dynamic stability of the array. The calculated band gap amounts to 2.23 eV for X ¼ N, but decreases in X ¼ P counterpart by 0.6 eV, owing to the smaller electronegativity of P in comparison to N. Bonding analysis within the SSAdNDP, as a periodic extension of the AdNDP approach [104], reveals 2c-2e bonds between B4X2-units, and seven 6c-2e within each B4X2 unit, which can be ascribed as a typical example of a 1S21P61D6 superatom. Recently, extended structures based on the consecutive aggregation of superatoms have been proposed [105], leading to the formation of two-dimensional superatomic honeycomb structures from the discrete and periodic fusion of Ge9 clusters by using [Ge9(Si(SiMe3))3]
as a source of clusters (Fig. 10). Thus, encouraging further theoretical evaluation of buildingblocks aggregation toward molecularly conceived extended materials. In addition, Lo´pez Laurrabaquio [106], also theoretically explored the formation of aggregates and crystals from superatoms based on the endohedral M@Si16 clusters, given by [Ti@Si16]n, [Sc@Si16K]n, and [V@Si16F]n structures evaluating linear, planar, and three-dimensional arrays.

5. Conclusions In summary, the intercluster aggregation of different building blocks is a relevant issue to rationalize further and control the obtention of cluster-based material for technological interest. This chapter provided relevant examples exposing the interesting bonding characteristics that enable a robust cluster-cluster bonding, which can be extended to different discrete and extended novel structures. It is important to note that the building block aggregation follows classical concepts of chemistry related to chemical bonding, allowing the introduction of single, double, and higher bond orders to the chemical understanding of cluster-based materials aggregation, which increases the number of shared electrons between the constituents units. Hence, classical concepts widely employed for simple molecules can be applied in the rationalization of different species.

330 Atomic clusters with unusual structure, bonding and reactivity

We believe that further materials can be envisaged by involving the aggregation of multiple cluster-based building blocks, which are bonded together in accordance with the modern concepts of chemical bonding.

Acknowledgments This work was supported by the National Natural Science Foundation of China (21971118 and 21722106) and the Natural Science Foundation of Tianjin City (No. 20JCYBJC01560 and B2021202077) to Z.-M.S. Z.-M.S. thanks the 111 project (B18030) from China. A.I.B. acknowledges financial support from the R. Gaurth Hansen Professorship fund. The support and resources from the Center for High Performance Computing at the University of Utah are gratefully acknowledged. A.M.-C. acknowledges support from FONDECYT/ANID grant 1180683.

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332 Atomic clusters with unusual structure, bonding and reactivity

[100] [101] [102] [103] [104] [105] [106]

T.N. Gribanova, R.M. Minyaev, V.I. Minkin, A.I. Boldyrev, Struct. Chem. 31 (2020) 2105–2128. M.P. Johansson, P. Pyykk€o, Phys. Chem. Chem. Phys. 6 (2004) 2907–2909. J. Li, X. Li, H.-J. Zhai, L.-S. Wang, Science 299 (2003) 864–867. N. Fedik, D.V. Steglenko, A. Mun˜oz-Castro, R.M. Minyaev, V.I. Minkin, J. Phys. Chem. C 125 (2021) 17280–17290. T.R. Galeev, B.D. Dunnington, J.R. Schmidt, A.I. Boldyrev, Phys. Chem. Chem. Phys. 15 (2013) 5022. A. Mun˜oz-Castro, K. Takahashi, J. Phys. Chem. C 121 (2017) 1934–1940. G. Lo´pez Laurrabaquio, M.B. Torres, E.M. Ferna´ndez, L.C. Balba´s, J. Math. Chem. 48 (2010) 109–117.

Chapter 19

Zintl cluster as a building block of superalkali, superhalogen, and superatom Swapan Sinhaa, Ruchi Jhab, Subhra Dasa,c, and Santanab Giria a

School of Applied Sciences and Humanities, Haldia Institute of Technology, Haldia, India, b Advanced Technology Development Center (ATDC), Indian

Institute of Technology Kharagpur, Kharagpur, West Bengal, India, c Department of Chemistry, Cooch Behar Panchanan Barma University, Cooch Behar, West Bengal, India

1. Introduction Almost a century ago, Edward Zintl [1,2] investigated the exciting character of the metalloids and posttransition metalbased polyatomic multianions in liquid ammonia solution, later called Zintl ion. Several works have been done on these multianionic clusters. The most attention on Zintl clusters was given when the synthesis and structural characterization of E5 2
, E9 3
, and E9 4
[E ¼ Sn, Pb, Ge] and also As7 3
, As11 3
, Sb4 2
, Sb7 3
, Sb11 3
, Bi4 2
, etc., were done in macrocyclic polyether (2,2,2-crypt, 18-crown-6) as a cation-sequestering agent [3]. Zintl ion makes Zintl phase when it is encountered with alkali or alkaline earth metals. During the last three decades, systematic investigations on Zintl ions were carried out to develop new Zintl derivatives [4–8]. Zintl cluster is a well-known polyatomic multianionic species, and it is unique because of its bonding pattern and reactivity. In Zintl ion, the bonding between the two atoms is not a familiar 2c-2e bond; rather it has delocalized electrons through the Zintl cage. The structure and stability of the Zintl cluster can be described with the help of well-known electron counting rules such as Wade-Mingos [9] and the jellium electronic shell closure rule [10]. Apart from the synthesis of typical Zintl ions, functionalization of the Zintl cluster is a recent trend in Zintl chemistry. Sevov et al. and some other groups have experimentally synthesized several functionalized Zintl clusters with main group elements via silylation [11,12], stannylation [13,14], phosphanylation [15]. Apart from ligand functionalization, several kinds of research have been done on the Zintl cluster complex with transition metals [16,17]. There have been very few examples of transition metal Zintl compounds, mostly because of the participation of d-orbitals in the bonding description [18–21]. Like synthesis, theoretical works are also performed to know the electronic structure of the Zintl cluster. P. A. Clayborne et al. [22] explained the electronic structure of [Ge9 (Si (SiMe3))3]
with the help of Wade–Mingos and jellium rule. In a recent study, endohedral beryllium-doped Zintl ion [Ge9Be]2
is shown to behave as chalcogen [23], and the corresponding trilithiated endohedral beryllium-doped Zintl cluster [Ge9BeLi3] acts as superalkali [24]. Boron and aluminium centers attached with three functionalized Ge9 4
ligands can show unique reactivity. B[Ge9Y3]3 (Y ¼ H, CH3, BO, CN) complex acts as a Lewis base, whereas Al (Ge9L3)3 (L ¼ H, CH3, CHO, CN) acts as Lewis superacid [25,26]. Recently, Parida et.al [27] have shown the alkylation mechanism of deltahedral Zintl cluster with the help of different conceptual density functional theory (DFT)-based reactivity descriptors. In recent years, it has also come into the picture that the structure and bonding of Zintl clusters can also be understood in terms of aromaticity [28]. The multiple local s aromaticity of the nonagermanide Zintl cluster was also described by Boldyrev et al. [29]. The remarkable magnetic shielding properties of pnictogen nortricyclane Zintl clusters, E7 3
(E ¼ P, As, Sb, and Bi), have also been described [30]. In this chapter, we mainly focused on the designing of Zintl-based superhalogen, superalkali, and superatoms.

2. Computational details In this chapter, the optimization and frequency calculations of all the taken molecules were done with the help of DFT using different functionals (B3LYP, WB97XD) [31–33] and basis sets [6-311 + G(d), 6-31 + G(d,p), 6-311 + G(d,p), and Def2TZVPP]. MP2 calculation [34] was also performed to have better energy values. Natural localized molecular orbitals Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00007-3 Copyright © 2023 Elsevier Inc. All rights reserved.

333

334 Atomic clusters with unusual structure, bonding and reactivity

(NLMOs) [35] have been generated using NBO 7.0 program to know the bonding properly. To see the percentage contribution of the Zintl core and attaching ligand toward the formation of the frontier molecular orbitals of the functionalized Zintl complexes, the density of state (DOS) and partial density of states (PDOS) were generated with the help of the GaussSum package [36]. For analyzing the bonding pattern between the Zintl core with ligand, we have performed the adaptive natural density partitioning (AdNDP) technique [37] in the Multiwfn program [38]. A natural bonding orbital population analysis (NBO) [39] scheme has been taken to evaluate the atomic charges (Qk). The ionization energy (IE) and electron affinity (EA) values were calculated by using the DSCF technique [40]. Based on conceptual density functional theory (CDFT), different types of chemical and reactivity descriptors such as hardness (
) [41], electrophilicity (o) [42], and dual descriptor (Dƒ(r)) [43] were analyzed for some Zintl-based systems [44]. All the calculations were performed in Gaussian 09 program [45].

3.

Zintl superalkali

Alkali metals possess low ionization energies (3.9–5.4 eV) among all the elements in the periodic table. Due to their electronic configuration, they tend to release their valence electron quickly. In 1982, Gutsev and Boldyrev [46,47] proposed that the complexes such as MLk+1 (where M is a central electronegative atom or core and K is the maximum formal valence of M and L is an alkali metal) have lower ionization energy (IE) than the alkali metal. They have coined the term “superalkali” for such complexes. One of the typical examples is Li3O, where oxygen is an electronegative and bivalent atom. As O is a bivalent atom, Li2O will be a stable compound but in Li3O, where the number of Li is higher than the valency of oxygen atom by one will not be stable. Therefore, it easily gives up an electron and becomes stable as Li3O+. This has been experimentally synthesized [48], and the I.E (3.6 eV) is found to be lower than the alkali metal. Thus, Li3O can be termed as a superalkali. Extensive research on superalkali for the past three decades shows different types of binuclear [49], polynuclear [50], nonmetallic [51], organic heterocyclic [52,53], aromatic organometallic superalkalis [54,55]. But Zintl ion-based superalkalis are still not well documented. In 2016, Giri et al. [56] designed P7 3
based organo-Zintl cluster as superalkalis. P7 3
is a well-known Zintl ion having a nortricyclane structure with a C3v point group. The stability of P7 3
is coming from the octet rule. Having 35 valence electrons, P7 needs three extra electrons to fulfill the octet rule and become stable as P7 3
with 38 valence electrons. Experimentally, alkylation on P7 3
Zintl ion results in the complexes such as P7R2
, P7R¯ 3 (where R ¼ Me, Et, tBu, etc.). So, in a controlled way, P7R4 is also possible, which contains 39 valence electrons. To gain stability, the neutral system easily releases one extra electron and stabilizes as a P7R4 +. So, the P7R4 may be considered as a superalkali (Fig. 1). To see the possible superalkali behavior of P7R4, substituents (R) have been taken according to their electron releasing effect dCHO < dH < dMe < dEt < diPr < dtBu. Where dCHO has electron-withdrawing power and dMe, dEt, diPr, dtBu having electron releasing or +I effect. Experimentally it has been observed that the three P atoms in the upper triangle of P7 geometry were suitable for ligand substitution. Now the question is, where will be the fourth substituents attached to P7 3
core. For this, several isomers of P7R4 have been generated and among the four isomers of P7R4, the lowest energy isomer for the cation geometry was considered for further studies. After having the lowest energy isomer, other complexes were designed, and the vertical electron affinity (VEA) was calculated, which are tabulated in Table 1. The calculation of IE of the ligands is also performed to see the effect of ligand on VEA of the complexes. FIG. 1 Ground-state geometries of [P7(Me)4]+ and [P7(CH2Me)4]+ cations. (Adapted with permission from reference S. Giri, G.N. Reddy, P. Jena, Organo-Zintl clusters [P7R4]: a new class of superalkalis, J. Phys. Chem. Lett. 7 (2016) 800–805. Copyright 2016 American Chemical Society.)

Functionalized different Zintl clusters Chapter

19

335

TABLE 1 Calculated ionization energy (IE) of individual ligands (R) and vertical electron affinity (VEA) for organo-Zintl catatonic systems at B3LYP/6-311 + G(d) level of theory. Ligand (R) H (hydrogen) CHO (formyl) Me (methyl) Et (ethyl) i t

Pr (isopropyl) Bu (tert-butyl)

IE, eV 13.66 8.45

System

VEA, eV

+

[P7H4]

5.50 +

[P7(CHO)4]

9.88

[P7(Me)4]

8.24

[P7(Et)4]

+

7.39

i

[P7( Pr)4]

6.79

t

+

4.79 4.62

+

[P7( Bu)4]

6.61

+

4.47 4.27

The experimentally reported ionization energies of alkali metals Li, Na, K, Rb, Cs are 5.39, 5.14, 4.34, 4.18, 3.89 eV, respectively. From Table 1, it has been observed that none of the ligands are superalkali as their IE ranges from 13.66 to 6.79 eV, even for the tBu ligand, which has the lowest IE among all the ligands but has higher IE than the Li atom. The situation has been changed when these organo ligands are attached to P3
7 Zintl core. The VEA values are dramatically decreased. The ionization energy value for individual H is 13.66 eV, but the VEA value for P7H4 is 5.50 eV, which is very close to the IE of the Li atom. From P7(Me)4 to P7(tBu)4, the VEA value changes from 4.79 to 4.27 eV, lower than Li and Na. For P7(tBu)4, the ionization energy is even lower than potassium (K). As a result, P7(Me)4, P7(Et)4, P7(iPr)4, P7(tBu)4 are all superalkalis. However, P7(CHO)4 is not a superalkali as it has a higher VEA value than the alkali metal, i.e., 6.61 eV. The electron-withdrawing effect of the dCHO group increases the ionization energy of the complex. Therefore, the VEA value rises as the electron-withdrawing power of the ligand increases. So, to make organo-Zintl superalkali, the choice of a proper ligand is essential. After the successful design of organo-Zintl superalkalis [56], we wanted to see the possibility of having organo-Zintl superalkali for group 15 members. For this purpose, we have taken a well-known Zintl cluster Ge4
9 . Several studies have been performed on this Zintl ion, and its stability comes from Wade-Mingos and jellium shell closure rule. To decrease the 2
number of substituted ligands on Ge4
9 , we chose the group 15 element as dopant and formed complexes such as [Ge7X2] where X ¼ P, As, Sb, and Bi. The doped deltahedral Zintl cluster is already experimentally synthesized and well characterized. For doped Zintl ion, the stability is also rationalized with the help of the Wade-Mingos rule and jellium shell closer model. To explore the doping of Ge4
9 , two Ge atoms are replaced by two pentavalent group 15 atoms (N, As, Sb, Bi) to fulfill the 40 electrons stable system and make the systems such as [Ge7P2]2
, [Ge7As2]2
, [Ge7Sb2]2
, [Ge7Bi2]2
. Several pieces of literature show that the three caped positions of doped Zintl ion are preferable for ligand substitution according to NBO analysis. Now the three ligands have been attached to [Ge7X2]2
cluster to make the system such as [Ge7X2R3], where R ¼ dCH3, dC2H5, dC3H3, etc. The neutral Ge7X2R3 [(7  4) + (2  5) + (3  1) ¼ 41] contains 41 electrons, so to maintain the stable electronic configuration the system will release one electron and stabilize as Ge7X2R+3 cation [57]. The calculated vertical electron affinity (VEA) of these cationic systems varies from 4.16 to 4.93 eV in the B3LYP/def2-TZVPP level of theory [58], which is lower than the ionization energy of the alkali metals (3.90–5.34 eV), so the neutral system behaves as a superalkali. It has been observed that acyclic organic ligand such as dCH3, dC2H5 shows less vertical electron affinity than the cyclic cyclopropane (dC3H3) ligand. So, the acyclic ligand-based dopped deltahedral Zintl cluster shows more superalkali behavior than the cyclic organic ligand. MP2 level of theory calculation for all dopped deltahedral organo-Zintl clusters shows even lower VEA values. To know the local reactivity of these clusters, a dual descriptor, an important local reactivity parameter, has been calculated. The optimized geometry and the calculated dual descriptor of doped deltahedral Zintl clusters are shown in Fig. 2. The figure shows that unsubstituted Ge atoms are showing nucleophilic region (green color—light gray color in print version), which is a favorable site for electrophilic attack, and the remaining substituted Ge atoms and doped atoms are showing electrophilic region (red color—gray color in print version).

4. Zintl superhalogens In the periodic table, halogens have the highest electron affinity among all elements. In the halogen family, chlorine has the highest electron affinity value (
348.6 kJ mol
1). On the other hand, superhalogens are a class of compounds whose

336 Atomic clusters with unusual structure, bonding and reactivity

FIG. 2 Optimized geometries with calculated dual descriptors (Dƒ(r)) of [Ge7P2(CH3)3]+, [Ge7As2(CH3)3]+, [Ge7Sb2(CH3)3]+, and [Ge7Bi2(CH3)3]+; red (gray color in print version) color indicates electrophilic (Dƒ(r) ¼ +ve), and green (light gray color in print version) color indicates for nucleophilic (Dƒ(r) ¼
ve) region. (Adapted with permission from reference G.N. Reddy, R. Parida, A. Mun˜oz-Castro, M. Jana, S. Giri, Doped deltahedral organo-Zintl superalkali cations, Chem. Phys. Lett. 759 (2020) 137952–137957. Copyright 2020 Elsevier.)

electron affinity is higher than Cl atom. Superhalogens [59,60] designed by Gutsev and Boldyrev in 1981, having the chemical formula MXk+1 where M is a metal atom having maximal valence k, and X is an electronegative atom. Not only superhalogens have been designed by using the MXk+1 formula but also several other kinds of superhalogens have been reported in the literature such as organic [61], aromatic heterocyclic [62], endohedral metallo fullerene superhalogen [63], etc., based on substituting core atoms with ligands or inserting metal to the core. But none of this research reported Zintl-based superhalogens. In 2017, Reddy et al. [64] proposed, for the first time, functionalized deltahedral Zintl complex Ge9R3 (R ¼ CF3, CN, NO2), which acts as a superhalogen. The ground-state structures of [Ge9(CF3)3]
, [Ge9(CN)3]
, and [Ge9(NO2)3]
functionalized complexes are shown in Fig. 3. In this study, Ge94
has been taken as a core and electron-withdrawing substituents (CF3, CN, NO2) have been taken as ligands making the system Ge9R3
(where R ¼ CF3, CN, NO2). Ge94
is a well-known Zintl ion whose stability and reactivity can be explained by the Wade-Mingos and jellium shell closure model. It carries two types of formal charges. Capped positions of Ge94
carry a higher charge than the other positions, which was revealed from NBO/Hirschfeld charge analysis. So, the capped positions are preferable for ligand substitution, which has been examined from the experimental study also. The idea about this study is that the accepting power of the ligand will withdraw the electrons from the Zintl core making the Zintl core electron deficient. The complex will accept extra electrons and show high vertical detachment energy (VDE) to maintain stability. The energy difference between the ground-state anion and the corresponding neutral in the anion geometry is term as VDE. According to their electron-withdrawing power (dCF3 < dCN < dNO2), the VDE also increases for the neutral complexes of Ge9R3. The calculated VDE of Ge9(CF3)3, Ge9(CN)3, Ge9(NO2)3 anionic complexes FIG. 3 Ground-state geometries of [Ge9(CF3)3]
, [Ge9(CN)3]
, and [Ge9(NO2)3]
complexes. (Reproduced from reference G.N. Reddy, R. Parida, S. Giri, Functionalized deltahedral Zintl complex Ge9R3 (R ¼ CF3, CN, NO2): a new class of superhalogen, Chem. Commun. 53 (2017) 13229–13232 with permission from the Royal Society of Chemistry.)

Functionalized different Zintl clusters Chapter

19

337

FIG. 4 AdNDP results of 2c-2e bond in [Ge9(CF3)3]
with the calculated occupation number (ON). (Reproduced from reference G.N. Reddy, R. Parida, S. Giri, Functionalized deltahedral Zintl complex Ge9R3 (R ¼ CF3, CN, NO2): a new class of superhalogen, Chem. Commun. 53 (2017) 13229–13232 with permission from the Royal Society of Chemistry.)

is found to be 3.87, 4.21, and 4.30 eV, respectively, at the wB97XD/Def2-TZVPP [33] level of theory, which is higher than the VDE of Cl atom. The jellium shell closure model can explain the high VDE of these complexes. In Ge9R3, the total number of valence electrons is 39 (four valence electrons from nine Ge atoms and one electron from each ligand), one electron less than a stable 40 electron configuration. So, to gain extra stability, the Ge9R3 complex will need one extra electron and behave like a halogen and depending upon the electron-withdrawing power of the complex will show the superhalogen property. Adaptive Natural Density Partitioning (AdNDP) technique has been adopted to ensure the bonding between Ge and ligands. It was found that Ge94
(Fig. 4) core binds with ligands by a 2c-2e sigma bond with occupation number 1.98 j ej. This indicates the ligands are sharing electrons with the core. As, all the Ge9R3
(where R ¼ dCF3, dCN, dNO2) systems contain 40 valence electrons, and it is a stable electronic configuration according to the jellium shell closure model, we wanted to see the nature of the molecular orbitals. The molecular orbitals are given in Fig. 5. From the molecular orbital study, it has been found that the electronic configuration of the complex according to the jellium shell closure model is 1S2 1P6 1D10 2S2 1F14 2P6. To know the contribution of the core and the ligands toward the formation of frontier molecular orbitals, we have generated the DOS and projected density of states (PDOS). The graphical plots in Fig. 6 showed the contribution of the core is maximum toward the formation of frontier molecular orbital of the complex. FIG. 5 Molecular orbitals of [Ge9(CF3)3]
according to the jellium shell closure model. (Reproduced from reference G.N. Reddy, R. Parida, S. Giri, Functionalized deltahedral Zintl complex Ge9R3 (R ¼ CF3, CN, NO2): a new class of superhalogen, Chem. Commun. 53 (2017) 13229–13232 with permission from the Royal Society of Chemistry.)

338 Atomic clusters with unusual structure, bonding and reactivity

Ge (CF3)3

8

PDOS (eV)

FIG. 6 The calculated PDOS diagrams of [Ge9(CF3)3]
; black, red (gray color in print version), and blue (light gray color in print version) colors indicate the Ge9 core, the ligand (CF3), and the total DOS, respectively. (Reproduced from reference G.N. Reddy, R. Parida, S. Giri, Functionalized deltahedral Zintl complex Ge9R3 (R ¼ CF3, CN, NO2): a new class of superhalogen, Chem. Commun. 53 (2017) 13229–13232 with permission from the Royal Society of Chemistry.)

6

Ge9

LUMO+4

76

24

(CF3)3

LUMO+3

91

9

DOS

LUMO+2

91

9

LUMO+1

100

0

LUMO

100

0

HOMO

97

3

HOMO-1

99

1

HOMO-2

100

0

HOMO-3

99

1

HOMO-4

99

1

HOMO-5

94

6

HOMO-6

99

1

HOMO-7

99

1

HOMO-8

61

39

HOMO-9

60

40

HOMO

LUMO

[Ge9(CF3)3]–

4

2 3.167 0 –15

–10

–5 Energy (eV)

0

5

Further to know the hybridization of the complex, we have generated NLMO (natural localized molecular orbitals). The NLMOs of [Ge9(CF3)3]
and [Ge9(CF3)3] are matched with NH3 and BH3. We assume that the lone pair on N is equal to an extra negative charge of the complex. From Fig. 7, it can be observed that the [Ge9(CF3)3]
is having sp3 hybridization while [Ge9(CF3)3] is sp2 hybridized. After successfully designing deltahedral functionalized Zintl-based superhalogen, we wanted to see whether the same can be obtained with aromatic organic ligands. In 2018, Reddy et al. [65] successfully designed deltahedral organo-Zintl superhalogens with Ge94
as a Zintl core and aromatic heterocyclic compound as ligand. The first-principle calculation shows that such complexes have high VDE, which is higher than halogen, and these kinds of systems can be termed as superhalogens. In this study, the capped positions of well-known Ge94
are attached to an N-center of the imidazole-based organic aromatic heterocyclic molecule. The chosen ligands are C2HBN2F3 and C2HBN2CN3 and their VDEs are 2.38 and 4.23 eV respectively. So, the C2HBN2F3 is not a superhalogen but the C2HBN2CN3 ligand is itself acts as a superhalogen. But both the ligands gain their stability by taking extra electrons. So, it is expected that the ligands can withdraw the electron from the Zintl core and the electron density of the core will be less. So, the system will want to take an electron from outside resulting in high VDE values. The calculated VDEs of [Ge9(C2HBN2F3)3]
and [Ge9(C2HBN2CN3)3]
complex are 3.30 and 4.30 eV, respectively, at the B3LYP/6-31 + G(d, p) level of theory calculation. From the VDE values, it can be concluded that while [Ge9(C2HBN2CN3)3] is a superhalogen, [Ge9(C2HBN2F3)3] does not have superhalogen FIG. 7 Natural localized molecular orbitals (NLMOs) study for the proposed NH3, [Ge9(CF3)3]
, BH3, and [Ge9(CF3)3] clusters. (Reproduced from reference G.N. Reddy, R. Parida, S. Giri, Functionalized deltahedral Zintl complex Ge9R3 (R ¼ CF3, CN, NO2): a new class of superhalogen, Chem. Commun. 53 (2017) 13229–13232 with permission from the Royal Society of Chemistry.)

NH3

[Ge9(CF3)3]–

BH3

[Ge9(CF3)3]

Functionalized different Zintl clusters Chapter

19

339

FIG. 8 Ground-state geometries of (i) [Ge9(C2HBN2F3)3]2, (ii) [Ge9(C2HBN2(CN)3)3]2, (iii) [Ge9(C2BN2F4)3]2, and (iv) [Ge9(C2BN2(CN)4)3]2. (Adapted with permission from reference G.N. Reddy, R. Parida, A. Chakraborty, S. Giri, Deltahedral organo-Zintl superhalogens, Chem. A Eur. J. 24 (2018) 13654–13658. Copyright 2018 Chemistry Europe.)

i

iii

ii

iv

nature. To improve the electron-withdrawing nature of the ligands, the H atom present in C2HBN2F3 and C2HBN2CN is further replaced by F and CN to have the ligands such as C2BN2F4 and C2BN2CN4. The individual VDE for [C2BN2F4]
and [C2BN2CN4]
ligand is 2.63 and 4.99 eV, respectively. With Zintl core, the system such as [Ge9(C2BN2F4)3]
and [Ge9(C2BN2CN4)3]
, the calculated VDE becomes 4.02 and 4.93 eV, respectively. This indicates that both the complexes have higher VDE than the halogen. From the results, it can be concluded that [Ge9(C2BN2F4)3]
and [Ge9(C2BN2CN4)3]
are superhalogens. The ground-state geometries of the organo-Zintl complexes are given in Fig. 8. The presence of a 2c-2e bond between the Ge94
core and the organic heterocyclic ligand, shown in Fig. 9, proves that both Zintl core and ligands are sharing their electrons to make a 2c-2e bond. For a detailed analysis of the bonding patterns in organo-Zintl superhalogen clusters, the DOS and PDOS of the organoZintl complexes have been analyzed. The corresponding plots and percentage contributions of core and ligand toward the formation of FMOs of [Ge9(C2BN2F4)3]
and [Ge9(C2BN2CN4)3]
are shown in Fig. 10. From the PDOS, it is revealed that the Ge9 core contributes more than the ligand in HOMO and LUMO. But ligand contribution is observed for higher/ lower unoccupied/occupied FMOs.

5. Zintl superatom Superatom [66,67] is a group of atoms or clusters, which can mimic the chemistry of one of the atoms in the periodic table. This can be seen as a building block of a modern three-dimensional periodic table, where the cluster molecules can be placed on the

2c-2e ON = 1.97 |e|

2c-2e ON = 1.97 |e| FIG. 9 2c-2e bond results of [Ge9(C2BN2F4)3]
and [Ge9(C2BN2(CN)4)3]
with their occupation numbers using AdNDP technique. (Adapted with permission from reference G.N. Reddy, R. Parida, A. Chakraborty, S. Giri, Deltahedral organo-Zintl superhalogens, Chem. A Eur. J. 24 (2018) 13654–13658. Copyright 2018 Chemistry Europe.)

340 Atomic clusters with unusual structure, bonding and reactivity

Percentage contribution Ge9

6

C2BN2E4 Ge9 DOS

PDOS(eV)

5

HOMO

4

LUMO

3 2 2.90

1 0 –15

–10

–5

0

5

Energy(eV) C2BN2CN4 Ge9 DOS

PDOS(eV)

10 8

HOMO

LUMO

4 2 0 –15

–10

–5 Energy(eV)

79 88 33 27 23 57 62 92 82 98 94 91 98 58 47 91 40 88 36 77

Ge9

12

6

LUMO +9 LUMO +8 LUMO +7 LUMO +6 LUMO +5 LUMO +4 LUMO +3 LUMO +2 LUMO +1 LUMO HOMO HOMO -1 HOMO -2 HOMO -3 HOMO -4 HOMO -5 HOMO -6 HOMO -7 HOMO -8 HOMO -9

0

5

LUMO +9 LUMO +8 LUMO +7 LUMO +6 LUMO +5 LUMO +4 LUMO +3 LUMO +2 LUMO +1 LUMO HOMO HOMO -1 HOMO -2 HOMO -3 HOMO -4 HOMO -5 HOMO -6 HOMO -7 HOMO -8 HOMO -9

13 13 81 79 86 82 92 3 3 3 96 98 98 96 98 95 84 91 8 4

(C2BN2F4)3 21 12 67 73 77 43 38 8 18 2 6 9 2 42 53 9 60 12 64 23

(C2BN2CN4)3 87 87 19 21 14 18 8 97 97 97 4 2 2 4 2 5 16 9 92 96

FIG. 10 Calculated PDOS diagrams and contributions of the core and ligand toward the FMOs of [Ge9 (C2BN2F4)3]
and [Ge9(C2BN2CN4)3]
; black, red (gray color in print version), and blue (light gray color in print version) indicate the ligands C2BN2F4 and C2BN2CN4, the Ge9 core, and the total DOS, respectively. (Adapted with permission from reference G.N. Reddy, R. Parida, A. Chakraborty, S. Giri, Deltahedral organo-Zintl superhalogens, Chem. A Eur. J. 24 (2018) 13654–13658. Copyright 2018 Chemistry Europe.)

position of the atoms of the periodic table. In the initial stage of this field, the designing of superatom was attributed to jellium electron count shell closure rule. This rule states that a cluster having a magic number (2, 8, 20, 40) of valence electrons is said to be exceptionally stable. A classic example of superatom is Al13 having 39 valence electrons, which wants one extra electron to get the magic number, 40, according to the jellium electron shell closure rule, just like a halogen atom. So, Al13 should behave like a halogen [68], and its electron affinity is found to be the same as the Cl atom, i.e., 3.6 eV. Furthermore, jellium shell closure explains the electronic management of the core of the cluster and the ligand attached with the cluster. The organic ligand-based Zintl cluster acting as a superatom was introduced by Reddy et.al [69] in 2017. Two well-known Zintl clusters Ge92
and Ge94
have been taken, which were stable Zintl cluster according to Wade-Mingos rule with closo and nido structure, respectively. Computational study proves that [Ge9(CHO)3] will mimic the chlorine (Cl) atom, whereas [Ge9(CHO)] mimics the iodine (I) atom. The calculated electron affinity of [Ge9(CHO)] is 3.03 eV, which is comparable with iodine having an electron affinity value of 3.01 eV at the B3LYP/6-31 + G(d, p) level of theory and for [Ge9(CHO)3] electron affinity is 3.68 eV. The ground-state geometries of the Zintl complexes are showing in Fig. 11.

Functionalized different Zintl clusters Chapter

19

341

FIG. 11 Optimize geometry of the (i) [Ge9(CHO)3]
, (ii) [Ge9(CHO)3], (iii) [Ge9(CHO)]
, and (iv) [Ge9(CHO)]. (Adapted with permission from reference G.N. Reddy, P. Jena, S. Giri, Organo
Zintl-based superatoms: [Ge9(CHO)3] and [Ge9(CHO)], Chem. Phys. Lett. 686 (2017) 195–202. Copyright 2017 Elsevier.)

The jellium shell closure model can explain this phenomenon. The valence electron of [Ge9(CHO)3] and Al13 is 39. So, to get a stable magic number 40, both systems take one extra electron and stabilize as [Ge9(CHO)3]
and Al13
. Therefore, their corresponding electron affinity is matched with halogen atoms, especially chlorine. From molecular orbital, it is shown that [Ge9(CHO)3]
and Al13
are 40 electron systems and both have the same electronic configuration 1S2 1P6 1D10 2S2 1F6 2P6 1F6 according to the jellium shell closure rule. On the other hand, [Ge9(CHO)] contains 37 valence electrons, less than one electron to get a stable closo structure. So, the system takes one extra electron and stabilizes as [Ge9(CHO)]
. Therefore, their corresponding electron affinity is matched with the iodine atom, and both have a similar type of molecular orbitals (Fig. 12).

eV

(i)

–3 –4 –5 –6 –7 –8

5S2

3d10

5s

3d4 3d4

–11

4p2 4p4

–12 –13

4S [I]–

2

NH3

3d2

–9 –10

4P6

(ii) 2

–14

4s2

[Ge9(CHO)]–

[Ge3(CHO)]–

FIG. 12 The molecular orbitals of [Ge9(CHO)]
and I
according to jellium electronic shell along with the hybridization of NH3 and [Ge9(CHO)]
complex. (Adapted with permission from reference G.N. Reddy, P. Jena, S. Giri, Organo
Zintl-based superatoms: [Ge9(CHO)3] and [Ge9(CHO)], Chem. Phys. Lett. 686 (2017) 195–202. Copyright 2017 Elsevier.)

342 Atomic clusters with unusual structure, bonding and reactivity

From NBO calculation, it has been observed that [Ge9(CHO)3] is having sp2 hybridization with 3.1–3.3 bond order value like BH3 and from NLMO studies, it is observed that the spatial arrangement of [Ge9(CHO)3]
and [Ge9(CHO)]
is like NH3 with sp3 hybridization.

6.

Concluding remarks

A scrutiny of the structures of the Zintl cluster reveals that the three capped positions of the deltahedral Zintl structure are favorable for ligand substitution. Based on the knowledge of electronic effects of the organo-substituent (electron releasing 4
(+I effect) and electron-withdrawing (
I effect)), it is possible to have different functionalized P3
7 and Ge9 core-based superalkali and superhalogens. It has been observed that electron-donating groups can make superalkalis. Whereas superhalogens can be designed by using electron-withdrawing groups. The power of electron-donating/withdrawing ability of the ligands can tune the superalkali/halogen nature. Proper choice of ligand can lead to the formation of superatom. In this chapter, it has been observed that tri and mono formyl substituted Ge4
9 Zintl clusters can mimic the properties of halogen atoms like I and Cl, respectively. So, they can be considered as Zintl-based superatoms.

Acknowledgments We are thankful to the Editors, Professor Pratim Kumar Chattaraj, Professor Gabriel Merino and Professor Sudip Pan, for inviting us to contribute a chapter to the book entitled, Atomic Clusters with Unusual Structure, Bonding and Reactivity to be published by Elsevier. SG and SS gratefully acknowledge the Department of Science and Technology INSPIRE award no. IFA14-CHE-151, Government of India, DST-SERB grant CRG/ 2019/001125, and Haldia Institute of Technology for providing funding and computational facilities.

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344 Atomic clusters with unusual structure, bonding and reactivity

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Chapter 20

Metallic clusters for realizing planar hypercoordinate second-row main group elements and multiple bonded species Amlan J. Kalita, Shahnaz S. Rohman, Chayanika Kashyap, Lakhya J. Mazumder, Indrani Baruah, Ritam Raj Borah, Farnaz Yashmin, Kangkan Sarmah, and Ankur K. Guha Advanced Computational Chemistry Centre, Department of Chemistry, Cotton University, Guwahati, Assam, India

1. Introduction During the last few decades, metallic clusters have attracted immense interest due to their unique structural feature and widespread of applicability [1]. In particular, small beryllium and lithium clusters have called for special attention due to their seemingly simple structural feature. Lithium, with one valence, electron is also the lightest metal. Despite their simplicity, experimentalists and theorists have faced many challenges. Experimentalists often face difficulties in obtaining photo-electron spectroscopy as small masses of Lin clusters can easily reach high speed during the time of flight and are difficult to be decelerated [2]. There is a large body of theoretical study on the ground-state geometry and binding energy of small lithium clusters [3–9]. Theoretical study has also revealed that electron correlation plays a vital role in their groundstate geometry and stability. The common physical laws valid for atoms and molecules may also govern the nature of cluster under quite specific conditions. It is expected that clusters may lead to effects uncommon to stable molecules or solids. Thus, small clusters form a bridge between small molecules and solids, and thus, the unique feature of the electronic feature of these clusters has become one of the important issues in crossing the bridge between a molecule and a solid [1]. Lithium, on the other hand, is also known to form heterogeneous clusters with other elements. The first experimental discovery followed by theoretical validation of the “hyperlithiated” bonding, which violates the formal octet rule in doped lithium clusters [10,11], has triggered many studies on structure and electronic properties of these systems. The first experimental evidence of such doped cluster is the LinO (n ¼ 3, 4, 5) reported by Wu and coworkers [11,12]. Similarly, Li6C was also predicted [13] and later verified experimentally [14]. Even larger polylithiated carbon clusters Li8C and Li10C have also been predicted to be stable [15]. These plethora of studies on pristine and doped Lin clusters have basically focused on the electronic structure and stability. Quantum chemical methods including correlation effects, either in the framework of configuration interaction (if possible, size-consistent one), complete active-space self-consistent field (CASSCF) and its perturbative version (CASPT2) or by means of density functional schemes, incorporating, however, at least the self-interaction corrections should be used for the accurate description of the electronic structure. Recent studies have shown their applications in different directions such as realization of planar hypercoordinate species and multiply bonded main group elements and reversible hydrogen storage. In this chapter, we would like to highlight our contribution on the application of pristine and doped Mn (M ¼ alkali or alkaline earth metals) clusters toward realization of planar hypercoordinate second-row main group elements and multiple bonded species.

2. Planar hypercoordinate main group elements The concept of classical tetrahedral structure of carbon has seen a paradigm shift when the structural prerequisite for stabilizing planar tetracoordinate carbon (ptC) was explored in 1970 [16]. Using the extended H€uckel calculations on planar methane, Hoffmann and coworkers [16] described the bonding situation involving normal set of sp2 hybrids at carbon. Two among the three sp2 hybrids form normal 2c-2e bonds with H atoms while the third hybrid involves in the formation of 3c-2e Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00012-7 Copyright © 2023 Elsevier Inc. All rights reserved.

345

346 Atomic clusters with unusual structure, bonding and reactivity

SCHEME 1 Bonding situation in planar methane.

bonds where the two electrons are now provided by H atoms. The remaining two electrons of carbon atom reside in 2p orbital perpendicular to the molecular plane (Scheme 1). Delocalization of this lone pair to the adjacent atoms or by incorporating them in 4n + 2 p system may help in realizing a planar tetracoordinate carbon (ptC) center [16]. The delocalization can be achieved by attaching electron-withdrawing substituents at the carbon atom as in C(CN)4 [16]. Soon after, Collins and coworkers presented a systematic computational exploration and identified the first ptC molecule with real minima in 1976 [17]. Being p acceptor and s donor, electropositive lithium, as in I–IV, is particularly effective in stabilizing planar form [17]. Moreover, Collins and coworkers [17] showed that multiple substitution by lithium and by three member rings provided further stabilization of the planar form over the tetrahedral form [17]. The first experimental realization of ptC came in the year 1977 [18]. Since then many planar penta- (ppC), hexa- (phC), and heptacoordinate (p7C) carbon were computationally explored [19–38]. For example, in 2008, the first ppC, CAl+5 was experimentally characterized and theoretically found to be global minima [38]. Similarly, the first phC, B6C2 was proposed by Schleyer and coworkers [25]; however, the latter study revealed that carbon avoids planar hypercoordination [26]. The true global minimum containing a phC was recently proposed [39], which contains a planar hexacoordinate carbon atom surrounded by ligands with half covalent and half ionic bonding. Such achievements of novel hypercoordinate carbon (hpC) molecules have created a new dogma in present-day chemistry that highest coordination number of carbon is no longer 4 [39].

These studies have also inspired the quest for other systems containing planar hypercoordinate main group elements. The first molecule containing planar hexacoordinate boron (phB) was predicted in 1991 [27], which triggered further examples of hypercoordinate boron [27–40]. For instance, Zhai et al [31,32] presented experimental and theoretical evi dence that B 8 and B9 anions clusters are perfectly planar molecular wheels with hepta and octa-coordinate boron atom. Yu et al [37] revealed theoretically that B6H+5 cation is aromatic with a planar pentacoordinate boron (ppB) center. Inspired by these, our group has computationally designed a 20-electron ternary cationic beryllium cluster, BBe6H+6 (Fig. 1) containing planar hexacoordinate boron (phB) in singlet electronic ground state [40]. The reported isomer is global H

H Be

1.9

Be

1.91

1

Be B

H Be

Be Be

H FIG. 1 Global minimum structure of 20-electron cationic

BBe6H+6

H cluster.

H

Metallic clusters for realizing planar Chapter

20

347

FIG. 2 Frontier occupied Kohn-Sham orbitals of the global isomer. Energies are in eV.

minimum on the potential energy surface (PES) and possesses double aromaticity owing to s and p delocalization. ABCluster code [41,42] was used to generate 50 isomers at M06-2X/TZVP level of theory [43] and low-lying isomers (within 50 kcal/mol) were then reoptimized at M06-2X/def2-TZVP level of theory and filtered at CCSD(T)/def2-TZVP level of theory. It is important to point that such small clusters may feature multireference character. Hence, perfect choice of the methodology is very crucial for evaluating the wave function of the system. Therefore, the T1 diagonistic value for all the isomers was evaluated and found to be less than 0.02 suggesting monoreference character. The global minimum is shown ˚. in Fig. 1, which reveals a symmetric structure with D6h point group symmetry having BdBe and BedBe distances of 1.91 A Chemical bonding in singlet BBe6H+6 cluster has been analyzed using the canonical molecular orbitals (CMOs). Fig. 2 shows the CMOs. The HOMO is p-type MO being completely delocalized and bonding in the BBe6 core representing 2p aromaticity according to H€ uckel (4n + 2) rule. The HOMO-1, HOMO-2, and HOMO-3 are s-type molecular orbitals, which are composed of B/Be 2p AOs with substantial contribution from H 1s AOs. These three MOs form the s sextet whose spatial distribution closely resembles the p sextet in benzene, thus rendering s aromaticity. Hence, the 20-electron BBe6H+6 cluster features 2p/6s double aromaticity, collectively showing eight-electron counting. The presence of an extra electron in the neutral cluster leads to loss of aromaticity and makes the boron center nonplanar. Further, to check the aromaticity of the proposed cluster, we have calculated nucleus-independent chemical shift (NICS) ˚ above the molecular plane, NICS(1) [44]. A values by placing ghost atoms at the center of the ring, NICS(0), and at 1 A negative value of NICS suggests paratropic ring current and hence aromaticity, while a positive value indicates diatropic ring current and hence, antiaromaticity [41]. It should be noted that the NICS(1) value for benzene, a prototypical aromatic compound, is 10.3 ppm. It is evident from the NICS values that both the core Be6B ring and the BedHdBe ring have significant negative values of both NICS(0) and NICS(1) suggesting the presence of both s and p contributions toward aromaticity (Fig. 3). FIG. 3 NICS(0) (black font) and NICS(1) (red font—gray color in print version) values in ppm.

348 Atomic clusters with unusual structure, bonding and reactivity

˚ ) versus time (ps) obtained from BOMD simulation (M06-2X/TZVP) at 398 K. FIG. 4 Plot of RMSD (A

The proposed molecule is a closed-shell singlet system, which is advantageous for it to prevent dimerization, a common phenomenon in radicals. However, dimerization calculation has not been attempted. Being, a species with uni-positive charge, dimerization is less expected due to repulsion between the positive charges. BOMD simulation up to 15 ps at elevated temperature (398 K) reveals that the species is dynamically stable well above room temperature (Fig. 4) and maintains its phB character during the simulation. In conclusion, a cationic 20-electron BBe6H+6 species is designed with a planar hexacoordinate boron (phB) center. The molecule is well defined in the potential energy surface as the energy difference with the closest isomer is 42.2 kcal/mol. Energetically the planar structure is most stable and hence the best choice. One additional electron in the neutral species makes the boron center nonplanar due to loss of aromaticity. Bonding analyses reveals the presence of 2p/6s double aromaticity in the cationic cluster, which is also found to be dynamically stable.

3.

Planar pentacoordinate nitrogen

In contrast to boron, nitrogen has a greater tendency of localization of the lone pair because of its high electronegativity, which is the cause of frustration in designing planar hypercoordinate nitrogen compounds [45]. The greater localization of the lone pair hiders its delocalization and prompts N atom to stay at the corners rather than occupying the central position. Despite this fact, some planar tetra-, penta-, and hexaccordinate nitrogen centers are reported [45–49]. For example, the first ptN, NSiAl3, and NAl 4 were computationally predicted in 1991 [45]. We have also reported 18-electron ternary cationic NBe5H+4 cluster containing planar pentacoordinate nitrogen (ppN), Fig. 5 [49]. The electronic ground state of the molecule is singlet. The global minimum of the molecule is characterized by the presence of pseudo double aromaticity and is kinetically and thermodynamically very stable. We employed the same theoretical method as was employed for planar hexacoordinate boron compound to arrive at our conclusion. Fig. 6 shows the canonical molecular orbitals (CMOs) of the NBe5H+4 cluster. The highest occupied molecular orbital (HOMO), HOMO-1, and HOMO-3 are the three s-type orbitals that have contributions from 2p orbitals of nitrogen and beryllium with a significant contribution from the H 1s orbital. These three molecular orbitals (MOs) form the basis of s aromaticity. The HOMO-2 is a p-type MO being completely delocalized and bonding in the NBe5 core representing 2p

FIG. 5 Global minimum structure of 18-electron cationic NBe6H+6 cluster.

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FIG. 6 Frontier canonical molecular orbitals (CMOs) of the NBe5H+4 cluster. Orbital energies are in eV.

FIG. 7 NICS 3D grid data of the NBe5H+4 cluster.

aromaticity according to the H€ uckel (4n + 2) rule. Hence, the 18-electron ternary cluster is characterized by double aromaticity arising from 2p/6s. In order to diagnose the aromatic character, we performed nucleus-independent chemical shift (NICS) calculations in 3D grid points (Fig. 7). Calculated out-of-plane NICS values are negative for all of the rings; however, the in-plane NICS value is positive for the NBe2 ring, which suggests that the molecule has pure p aromaticity, but the s aromaticity is not delocalized throughout. We, therefore, prefer to call it “pseudo” s aromatic but pure p aromatic. We also checked the kinetic stability of the global minimum toward isomerization to its closest energy isomers (Fig. 8). It is evident from Fig. 8 that there involves a high barrier for isomerization to the closest energy isomer, which ensures its kinetic stability. In summary, a cationic 18-electron NBe5H+4 species is designed with a ppN center. Interestingly, the planar geometry of the isoelectronic species CBe5H4 is also a local minimum, while CBe5H+5 is a first-order saddle point. The cluster is characterized by the presence of 2p/6s “pseudo” double aromaticity. BOMD calculations reveal that the cluster is dynamically stable. The 18-electron species NBe5H+4 may be readily extended to isovalent NBe5X+4 (X ¼ Li and Cu) with the planar pentacoordinate nitrogen atom.

4. Metal cluster supported multiple bonded second-row main group element Beryllium-beryllium triple bond: Metal-metal multiple bonding is a common phenomenon in d and f block metals. However, such multiple bonds in s-block metals are a rare phenomenon. Among the s-block metals, beryllium possesses the highest electronegativity and ionization energy, and hence, chemically bonded Be-Be interaction is the most studied sblock metal-metal interactions [50–57]. For example, ultraweak and ultrashort BedBe distances with little or no bond have frequently been observed. The weak bond dissociation energy (BDEBedBe) of 0.1 eV in Be2 dimer is responsible for such

350 Atomic clusters with unusual structure, bonding and reactivity

FIG. 8 Isomerization pathway of the global isomer of NBe5H+4 cluster. Relative energies are in kcal/mol.

little or no interaction. Many theoretical strategies were proposed for strengthening the BedBe interaction. Among them the most noticeable proposal was put forward by Liu et al. [56]. In their work, they proposed a unique way of strengthening the BedBe interaction by concerted electron donation from the vertical plane of the BedBe midpoint to achieve true double p bonds. Similarly, Brea et al. [54] have also shown the strategy of using radical ligands, which strengthens the BedBe bond in Be2L2. The calculated Wiberg Bond Index (WBI) value of the BedBe bond in Li4Be2 molecule was found to be 1.963. Different bonding analyses showed the existence of only two p bonds in Li4Be2 without any s bond [56]. This remarkable proposal posed a question into our mind: Does a true BedBe triple bond exist? To answer this question, we have undertaken high level ab initio calculations, which reveal that, indeed, a true triple bond exists between the Be atoms in Li6Be2 molecule [58]. Fig. 9 shows the optimized global minimum geometry of Li6Be2 cluster calculated at M06-2X/def2-TZVP level of theory. As this system may involve multireference character, hence, we checked the T1 diagnostic value at CCSD(T)/ def2-TZVP level on the M06-2X optimized geometry. The T1 diagnostic value of the global isomer was found to be ˚ with a WBI value of 0.021 suggesting negligible multireference character. The BedBe distance in Li6Be2 is 1.943 A ˚ 2.304, which is 0.05 A shorter than that in Li4Be2 cluster.

˚. FIG. 9 Optimized global minimum structure of Li6Be2 cluster. Bond length is in A

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FIG. 10 Frontier Kohn-Sham orbitals of Li6Be2 cluster. Contour value used was 0.03 au. Orbital energies are in eV.

FIG. 11 (A) AdNDP orbitals, (B) contour plot of Laplacian of electron density (red—light gray color in print version—positive region, blue—gray color in print version—negative region) along BedBe axis for Li6Be2 molecule, (C) ELF profile for Li6Be2, and (D) ELF profile of Li4Be2 molecule.

We then turned our attention to investigate the electronic structure of the cluster. Fig. 10 shows the frontier Kohn-Sham orbitals of the cluster. The HOMO is a clearly an s-type orbital formed between two Be atoms while the two degenerate p-type orbitals (HOMO-1) formed between two Be atoms are also clearly evident. The electronic structure has been further analyzed by adaptive natural density partitioning (AdNDP), electron localization function (ELF), and atoms in molecules (AIM) analyses (Fig. 11). There are doubly degenerate p orbitals having occupancy of 1.89 e and a 6c-2e s orbital having occupancy of 2.00 e. It should be noted that unlike Li4Be2, we could not locate any BedBe s* orbital having any occupancy. This implies that the s bond between the Be atoms will remain intact and the exact bonding scenario should be described as having one s and two p bonds between Be atoms. The contour plot of Laplacian of electron density (Fig. 11B) shows symmetrical distribution of valence shell charge concentration zone (VSCC), typical for a cylindrically symmetrical triple bond. The dysnaptic basin V(Be,Be) integrates to 5.4 electrons, quite closer to the expected triple bond, while it integrates to 3.4 electrons Li4Be2 cluster, respectively. Thus, ELF analyses also suggest that the BedBe bond in Li6Be2 is unambiguously a true triple bond comprising one s and two p bonds. The thermodynamic stability of Li6Be2 molecule is accessed from the following reactions in terms of DG at 298 K. All these reactions leading to the formation of Li6Be2 are spontaneous. Li4 Be2 + 2Li ! Li6 Be2 DG ¼ 41:4 kcal=mol Li6 + 2Be ! Li6 Be2

352 Atomic clusters with unusual structure, bonding and reactivity

DG ¼ 91:6 kcal=mol Li4 ðrhombusÞ + 2Be + 2Li ! Li6 Be2 DG ¼ 116:5 kcal=mol In conclusion, quantum chemical calculations predict that a true BedBe triple bond exists in Li6Be2. The unambiguous nature of the triple bond has been established by various bonding analyses strategies. All these analyses confirm the presence of a true triple bond. The thermodynamic stability of this molecule had been analyzed from different reaction possibilities. All the reactions were calculated to be highly spontaneous suggesting its thermodynamic stability. Boron-boron quadruple bond: Boron-boron multiple bond, especially triple bond, has been reported. The first BdB triple bond has been observed in OCBBCO molecule by Zhao et al. in 2002, which was formed in low-temperature argon matrix and characterized by infrared spectroscopy and theoretical calculations [59]. In 2007, Li et al. characterized OBBBO2- anion, an isoelectronic species with OCBBCO molecule, by photoelectron spectroscopy and theoretical calculations [60,61]. Their calculations revealed 2.5 fold BdB bond [55]. The first crystallographic evidence of BdB triple bonded compound was reported by Braunschweig et al. in 2012 [62]. Can boron-boron quadruple bond be ever achieved? To investigate the possibility of boron-boron quadruple bond formation, herein, we have undertaken an in silico study on Li3B 2 and Li4B2 clusters [63]. We adopted similar methodology (M06-2X/def2-TZVP) as applied for Li6Be2 cluster. ABCluster code [41,42] was used to generate 10 isomers and the energies of different isomers were refined at CCSD(T)/def2-TZVP level of theory on the M06-2X optimized geometries ˚ (Fig. 12). The M06-2X/def2-TZVP computed BdB distance in the singlet global minima of Li3B 2 (1A) is 1.531 A while it ˚ in Li4B2. is 1.517 A In view of the relative closeness of isomers, we turned our attention to investigate the lowest-energy isomerization pathway between the two lowest energy isomers. Fig. 13 depicts the energetic involved in the isomerization pathway. Isomerization from 1A to 1B involves a moderate barrier of 15.7 kcal/mol at CCSD(T) level. This suggests that there is finite probability of interconversion between these two isomers. Interestingly, there is insignificant change in BdB distance in 1A, TS1A-1B and in 1B. This suggests that the BdB bond has almost similar character in all of them. We therefore plotted the canonical frontier molecular orbitals of all these species (Fig. 13). Molecular orbital analysis reveals BdB fourfold bonding interaction in all of them. The calculated WBI values at M06-2X/def2-TZVP level in 1A, TS1A-1B and in 1B are respectively 3.17, 2.98, 2.99, which also supports fourfold bonding situation. In summary, quantum chemical calculations reveal that BdB quadruple bond is possible in Li3B-2 and Li4B2 clusters. Owing to the advancement in negative ion photoelectron spectroscopy and experimental technique, we feel that the proposed clusters will be very promising target for their synthesis.

FIG. 12 Optimized isomers of Li3B 2 and neutral Li4B2 clusters. Relative energies are in kcal/mol. ˚ . ZPE stands for zero point Bond lengths are in A corrected.

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FIG. 13 Lowest-energy isomerization pathway between lowest-energy isomers for Li3B 2 and Li4B2 clusters. Relative energies are in kcal/mol. Bond ˚ . Canonical occupied frontier molecular orbitals are also shown. ZPE stands for zero point correction. Displacement vectors of the transition lengths are in A state are shown in blue-colored arrow (gray color in print version).

5. Conclusions and future aspects Small to medium-size metallic clusters are gaining tremendous attention due to their rule-breaking structural motifs as well as their interesting chemical reactivity. Many planar hypercoodinate species have now been proposed and experimentally verified, which are stabilized by the unique bonding situation in metallic clusters. Recent years have also witnessed small molecule activation mediated by small-to-medium pristine or doped clusters. Understanding the bonding situation in small metallic clusters is very crucial as it serves as a bridge between molecules and solids. Many unexpected effects are observed, which warrant rigorous quantum mechanical approach for accurate description of the bonding situation. Correlated methods, which include both static and dynamic correlations, are one of the best theoretical choices. On the other hand, hybrid density functional theory with appropriate choice of Hartree-Fock exchange or methods that largely consider self-interaction corrections have evolved to be one of the economic approaches to tackle such situation. In this chapter, an attempt has been made to elucidate the role of small metallic clusters to stabilize planar hypercoordinate second-row main group elements. Moreover, the interesting role of small metallic clusters in transmuting BedBe triple bond and BdB quadruple bond has been discussed. Since cluster physics opens a new avenue, which is not usually met in the molecular or solid state physics, electronic structure theory with appropriate use of quantum mechanics can be very useful in finding the best probable reason. Electronic structure model has provided many interesting feature of these small to medium-size clusters, which may be very useful in exploring many domains of clusters. The science of cluster of evolving and its dynasty awaits a great future.

Acknowledgment A. K. G. thanks the Science and Engineering Research Board (SERB), Government of India, for providing financial assistance in the form of a project (project no. ECR/2016/001466).

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Chapter 21

Planar hypercoordinate carbon Prasenjit Dasa, Sudip Panb, and Pratim Kumar Chattarajc a

Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur, India, b Institute of Atomic and Molecular Physics, Jilin University,

Changchun, China, c Department of Chemistry, Indian Institute of Technology, Kharagpur, India

1. Introduction The molecules having planar tetracoordinate or hypercoordinate carbons are nonclassical systems [1,2]. It is well known that nonclassical molecules with their exotic electronic structures have potential applications due to the extraordinary electronic, magnetic, and optical properties of the molecules. In organic chemistry and biochemistry, most of the molecules follow two structural rules. The first one is that the maximum coordination number of carbon would be four. The second one is the tetrahedral arrangement of the attached atoms and/or groups to the carbon as proposed by van’t Hoff [3] and Le-Bel [4] independently in 1874. But many molecules do not obey one or both of these rules. The simplest system having more than four coordination of carbon is CH+5 having pentacoordinate carbon (C) that was first detected in 1952 [5]. After this discovery, many systems were reported consisting of central carbon bonded with more than four atoms [6–8]. Before the concept of hypercoordinate planar configurations of carbon, many molecules were characterized without knowing this interesting idea. In 1968, a planar tetracoordinate carbon (ptC) was proposed to be in the transition state of methane (CH4) in a racemization process [9]. But the energy of this planar transition state is very high as compared with the lowest-energy tetrahedral isomer. Ab initio computations proposed that the isolated planar CH4 molecule is not possible, and it cannot be a local minimum structure [10,11]. However, the study on the planar CH4 molecule helps to understand how to stabilize a species containing a ptC center. Although the first experimental example of ptC containing molecule was observed in 1977 in the V2(2,6-dimethoxyphenyl)4 complex (Fig. 1) [12], the original authors did not recognize this interesting fact. In this complex, a triple bond is present in between two vanadium (V) atoms and two ligands have ptC centers. This complex is important in this context because it is the first experimentally characterized ptC containing complex. In the context of planar hypercoordinate carbon chemistry, the boron-carbon systems show fruitful examples. This is because of the strong p-acceptor capacity of electron-deficient boron atoms that causes the delocalization of electron density from the central carbon atom.

2. Planar tetracoordinate carbon (ptC) ptC molecules or ions violate the second rule that we have discussed in the introduction part. The idea of ptC was first introduced by Monkhorst in 1968 [9], after almost a century of the tetrahedral tetracoordinate carbon concept put forward by van’t Hoff and Le-Bel. Hoffmann and coworkers [13] in 1970 suggested the strategies for the stabilization of ptC molecules based on electronic structure analysis of planar methane having D4h symmetry. Methane is the simplest imposed planar ptC system considered. This analysis showed that the planar CH bonds are electron-deficient and one lone pair is present on the central carbon atom perpendicular to the plane. The energy of this planar structure is about 130 kcal/mol higher than the tetrahedral global minimum. So, the suggested strategy by Hoffmann and coworkers was achieved by the incorporation of substituents having simultaneous s donating and p accepting capacity. Using s donating ability, the ligands donate electrons to the electron-deficient central carbon atom. On the other hand, the paccepting capacity helps the delocalization of lone pair on the central carbon atom with the ligand atoms. So, both the simultaneous s donation and p acceptance make the ptC molecules stable. With the help of this electronic approach, many theoretical studies [14–29] on ptC systems and several experimentally detected [30–42] ptC molecules were reported from different research groups. In 1976, Schleyer and coworkers designed the first example of ptC in 1,1-dilithiocyclopropane (Fig. 2A) and 3,3-dilithiocyclopropene molecules (Fig. 2B), which show global minimum structures [43]. In 1991, a theoretical work by Schleyer and Boldyrev showed that cis and trans isomers of the CAl2Si2 system (Fig. 3) have a planar structure with Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00021-8 Copyright © 2023 Elsevier Inc. All rights reserved.

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 1 The molecular structure of V2(2,6-dimethoxyphenyl)4 features a V^V triple bond and two planar tetracoordinate carbon (ptC) centers. The H atoms are omitted for clarity.

A)

B)

C)

Li

Li

Li

Li

P 1.701 (1.685) As 1.824 (1.820) Sb 2.039 (2.040) Bi 2.130 (2.145)

P 2.127 (2.137) As 2.116 (2.120) Sb 2.096 (2.088) Bi 2.082 (2.073) P 1.999 (1.994) As 2.004 (2.001) Sb 2.021 (2.021) Bi 2.023 (2.027)

P 2.531 (2.526) As 2.634 (2.643) Sb 2.840 (2.846) Bi 2.915 (2.929)

P 3.116 (3.124) As 3.072 (3.066) Sb 2.993 (2.977) Bi 2.964 (2.951)

˚ unit. FIG. 2 (A) 1,1-dilithiocyclopropane; (B) 3,3-dilithiocyclopropene; (C) CAl3E (E ¼ P, As, Sb, Bi) clusters. Bond distances are in A

˚ and bond angles are in 0), relative energies (DE, in kcal/mol), and the number of imaginary vibraFIG. 3 Optimized geometries (bond lengths are in A tional frequencies (NIMAG) at the MP2(full)/6-311 +G* level for CSi2Al2 and the MP2(fc)/6-311+G* level for CSi2Ga2 and CGe2Al2.

Planar hypercoordinate carbon Chapter

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a tetracoordinate carbon atom in the middle of the ligand ring and have lower energies than the corresponding tetrahedral isomers [15]. This penta-atomic system has 18 valence electrons. In 1998, Boldyrev and Simons designed CSi2Ga2, and CGe2Al2 systems (Fig. 3) having a ptC at the center of the ligand rings [16]. These two systems also have 18 valence electrons, and the preference of the planar form over tetrahedral one was rationalized by them with the molecular orbital analysis of these complexes. This analysis showed that the presence of 18 valence electrons causes three s and one p bonds between the central carbon atom and the peripheral ligand atoms as well as one ligand-ligand s bond is very crucial for a ptC system. This simultaneous s and p delocalization causes the stability of the complexes in planar form. They also explained that tetrahedral structures of these systems would undergo Jahn-Teller distortion, which would lead to a planar structure. For this purpose, they have compared the occupancy pattern of the valence MOs of tetrahedral CF4 molecule with the tetrahedral structures of their systems. CF4 molecule is a 32-valence electronic system and the occupancy pattern of the occupied MOs is 1a211t622a212t621e43t621t61. They assumed other tetrahedral molecules or nearly tetrahedral structures will follow this occupancy pattern (except for symmetry-imposed degeneracies), and the 18-valence electronic tetrahedral structures show 1a211t622a212t621e2 pattern of occupancy. Due to this partially filled e orbital, the tetrahedral structures of their systems show Jahn-Teller instability and get distorted to a planar structure. Inspired by this work, many neutral systems with a ptC center and 18 valence electrons were studied by different research groups. Merino and coworkers in 2015 designed neutral and 18-valence electronic CAl3E (E ¼ P, As, Sb, Bi) clusters containing a ptC (Fig. 2C) [44]. The designed systems are the global minima, and the C]E bonds are important for the stability of the structures in planar form. In the gas phase, both the thermodynamic and kinetic stabilities are shown by these clusters. Recently, we have studied three neutral 18 valence electronic molecules CGa2Ge2, CAlGaGe2, and CSiGa2Ge (Fig. 4) having a ptC at the center of the ligand rings [45]. These systems are aromatic, kinetically stable, and could be viable in the gas phase. Ionic ptC systems having 18 valence electrons were also reported [46,47]. Zhou et al. theoretically studied the 18-valence electronic mono-anionic CGa3Si system having a ptC [48]. Although this 18-valence electron rule is not mandatory for a system to show a ptC, it can give a guidance to design new ptC systems both computationally and experimentally in the gas phase.

˚ unit. The values in the parentheses are the relative FIG. 4 The optimized geometries of the cyclic rings and the ptC systems. Bond lengths are given in A energies in kcal/mol. The energies of cis isomers are considered to be zero.

360

Atomic clusters with unusual structure, bonding and reactivity

FIG. 5 Optimized ptC global minimum structures of C2v CLiAl2E and CLi2AlE (E ¼ P, As, Sb, Bi) at B3LYP/C, Li, Al, P, As/aug-cc-pVTZ/Sb, Bi/aug˚ unit. cc-pVTZ-pp levels. Bond distances are in A

There are also many systems reported in the literature that showed a ptC without having 18 valence electrons. Recently, a 19-valence electronic CAl4Mg mono-anionic system showed a global minimum with a ptC center [49]. This system shows both s and p delocalization of electron density within the molecule that causes the stability in the planar form. Again, the s/p dual aromaticity supports the stability of the anion in planar geometry. Wu et al. recently showed that a ptC is present in 14- and 16-valence electronic CLi2AlE and CLiAl2E (E ¼ P, As, Sb, Bi) systems (Fig. 5), respectively [50]. In both these types of systems, the C]E bonds are important for the stability of the complexes in planar form. The bonding analysis of the complexes showed the presence of three s and one p bond between the central carbon atom and the peripheral atoms of the ligands, one delocalized s bond among the peripheral atoms. The systems are kinetically stable as predicted from the Born-Oppenheimer molecular dynamics (BOMD) simulations. The same group also predicted the possibility of 12-valence electronic ptC systems CLi3E (E ¼ N, P, As, Sb, Bi) and CLi3E+ (E ¼ O, S, Se, Te, Po) [51]. The 16-valence electronic penta-atomic ptC system CAl3Tl was studied by Ding and coworkers [52]. The global minimum structure of a 15-valence electronic monocationic CB+4 system contains a ptC [53]. Vogt-Geisse et al. in 2015 reported a local minimum structure of two 14-valence electronic Si2CH2 and Ge2CH2 systems having a ptC [54]. Thus, the 18-valence electron counting rule is not a mandatory requirement for the ptC molecules or ions. Apart from the electronic approach, there is another strategy in which the central carbon is forced to stay in planar geometry by creating sufficient strain [55–61]. This strategy is known as the mechanical approach in which small rings, annulenes, and cylindrical cages or tubes are used to create the strain. Although several molecules containing ptC following this approach have been reported theoretically, the experimental realization of a ptC molecule based on this approach is not succeeded. We have presented some previously reported ptC molecules based on this approach in Fig. 6. Initially, the fenestrenes and the aromatic unsaturated fenestrenes, rigid three-dimensional cage, such as octaplane (6d) [62,63] were suggested to stabilize the carbon atom in planar tetracoordinate form. Rasmussen et al. modified the alkaplane 6d structure

(a)

(b)

(c)

FIG. 6 Schematic presentations of various mechanically stabilized ptC molecules.

(d)

(e)

Planar hypercoordinate carbon Chapter

21

361

and formed structure 6e, which is the first in silico characterized successful example of the ptC system using mechanical approach [64].

3. Planar pentacoordinate carbon (ppC) The idea of ptC has been extended to the possibility and characterization of molecules containing ppC center [65]. Bolton et al. in 1995 studied the potential energy surface of the 1,1-dilithioethene molecule (Fig. 7A), which showed the appearance of a local minimum containing a ppC with 7.2 kcal/mol higher in energy than the lowest-energy isomer of the molecule [66]. In 2001, Wang and Schleyer predicted the low-lying local minimum structures of C3B3, C2B4, and CB5 molecules (Fig. 7B) containing a ppC [67]. Using these building blocks, planar aromatic or antiaromatic molecules can be created, and these are known as “hyparenes” (Fig. 7C). The rings formed by atoms other than carbon and boron in the perimeters are also used to design planar ppC molecules. The di-anionic CSi2 5 system shows a ppC in its D5h symmetry (Fig. 8A). The isoelectronic analogues of this system, CSi4P in C2v symmetry (Fig. 8B) and CSi3P2 (Fig. 8C) in its C2v symmetry, show energy minimum structures with a ppC center [2]. However, the systems are not global minimum; rather, they are local minimum structures. After the discovery of the aromatic M5(m-H)5 hydrometal rings (M ¼ Cu, Ag, Au) [68,69], Li and coworkers studied the potential energy surface of the Cu5H5C system and found the local minimum structure containing a ppC [70] (Fig. 9). Further, the Ag5H5C and Au5H5C systems also contain a ppC in their local minimum structures [71]. These ppC systems show large negative nucleus-independent chemical shift (NICS) values above the planar hydrometal rings indicating the aromatic nature of these systems. Tsipis et al. predicted a similar type of molecule U@[c-U5(m2-C)5] containing a ppC [72]. For ptC systems, the 18 valence electrons rule was quite useful to explain the preference of planar geometries over tetra+ hedral one. This rule can also be used to justify the structures of CBe4 5 [73] and CAl5 [74] systems. Zeng and coworkers in 2008 predicted computationally the first global minimum isomer containing a ppC in symmetric CAl+5 system having D5h point group of symmetry (Fig. 10). This global minimum structure has 3.8 kcal/mol lower energy than the nearest structure of this system. The aromaticity character of this ppC system was also confirmed by NICS calculations. The CBe4 5 system showed the local minimum structure containing a ppC (Fig. 11). In this molecule, the ligand (Be4 5 ) acts as both s-donor and p-acceptor and the system shows both s and p aromaticity. Although this system shows a ppC in its planar form, the high negative charge makes the system unstable with respect to the spontaneous dissociation of electrons. This high negative charge density is reduced by complexing with Li+ ions to this tetra-anionic system, and the complexes are in the form n) of CBe5Li(4 (n ¼ 2–5) [75]. More interestingly, this Li+ binding does not disrupt the ppC center in these systems. In all n + these Li capped systems, the ppC acts as both s-acceptor and p-donor. The multicentered s-bonds and the delocalization of 2pz electrons of the central carbon atom through a delocalized p-bond involving peripheral ring make the systems stable in planar form. Similar to the discussed case, H+ can be capped to the CBe4 5 system to reduce the excess charge density on n) the system, and the resulted complexes are CBe5H(4 (n ¼ 2–5) [76]. When two and three H+ are capped to the tetran

FIG. 7 (A) 1,1-dilithioethene; (B) C3B3, C2B4, and CB5 clusters; (C) linking hexanuclear planar species in part B produces hyparenes.

362

Atomic clusters with unusual structure, bonding and reactivity

 FIG. 8 Planar pentacoordinate carbon in CSi2 5 (A), CSi4P (B), and CSi3P2 (C).

FIG. 9 The computed structures of Cu5H5C, Ag5H5C, and Au5H5C, respectively.

a)

+

Al Al

C Al

2.106 Å (B3LYP/aug-cc-pVTZ) (2.136 Å) (MP2/aug-cc-pVTZ)

Al

Al

2.475 Å (2.551 Å) D5h (0.00)

b) +

Al

Al

Al

1.975 Å (1.998 Å)

C Al

2.742 Å (B3LYP/aug-cc-pVTZ) (2.644 Å) (MP2/aug-cc-pVTZ)

Al

2.105 Å (2.124 Å)

C2v (3.80 kcal/mol)

c)

+

Al

C Al

Al

Al

Al

2.023 Å (B3LYP/aug-cc-pVTZ) (2.039 Å) (MP2/aug-cc-pVTZ) 1.996 Å (2.004 Å)

2.861 Å (2.725 Å)

C3v (5.00 kcal/mol) FIG. 10 Global minimum structure of

CAl+5 .

˚ unit. The bond lengths are given in A

Planar hypercoordinate carbon Chapter

21

363

n) n) FIG. 11 (A) Global minimum structures of CBe5Li(4 (n ¼ 2–5); (B) Global minimum structures of CBe5H(4 (n ¼ 2 and 3); (C) The symmetric n n structures of CBe5 and CBe4 . The neutral and tetraanionic forms have the same shape but different bond lengths; (D) The Mg2+ complex CB2Al2Mg 5 features planar pentacoordinate carbon (ppC) bonded to three different donor elements.

ionic system, the planar structure is the global minima having C2v symmetry. But the global minimum structure of the tetra H+ capped system is tetrahedral-like geometry. However, a quasiplanar structure of this system has 1.8 kcal/mol higher energy than the global minimum isomer. When five H+ are capped, a quasiplanar structure corresponds to the minimum energy of the system. The designed protonated complexes are stable due to the presence of the 3c-2e BedHdBe bonds and the aromaticity (both s and p aromaticity). CBe5X+5 (X ¼ F, Cl, Br, Li, Na, K) systems also have ppC or quasi ppC as global minima [77]. Again in these systems, the 3c-2e BedXdBe bonds give stability in planar form. In 2005, Erhardt et al. [78] studied some fluxional wheel-like molecules C2B8, C3B+9 , and C5B+11 in which the internal C2, C3, and C5 units rotate within the boron rings, respectively (Fig. 12A). The interesting point about these molecules is that there is more than one ppC center present in their local minimum structures and the systems showed p-aromaticity.

+ FIG. 12 (A) The boron-carbon clusters C2B8, C3B3+ 9 , and C5B11 are local minimum structures that include conformationally dynamic C2, C3, and C5 units, respectively, within boron rings. (B) The structures of CBe5Al and CBe5Ga feature C bound in a pentagon, two vertices of which are capped by a Be atom.

364

Atomic clusters with unusual structure, bonding and reactivity

We have now discussed some ppC molecules that have 18 valence electrons. As the CAl+5 system showed a global minimum containing a ppC, Wang and coworkers in 2010 designed a neutral and an anionic system containing a ppC in their global minimum structures by isoelectronic substitution on CAl+5 system. They substitute Al by isoelectronic Be to generate neutral CAl4Be (Fig. 13A) and mono-anionic CAl3Be 2 (Fig. 13B) systems having 18 valence electrons [79]. The central carbon takes electrons from peripheral atoms and acts as s-acceptor, which is predicted by the negative charge on the carbon center. Simultaneously, the carbon center donates the lone pair on 2pz orbital (perpendicular to the plane) to the peripheral atoms via back donation. So, in these systems, the central carbon atom acts as both s-acceptor and p-donor. The kinetic stability of these systems was also studied by molecular dynamics simulations. Further, Wu and  coworkers in 2012 designed di-anionic CAl2Be2 3 (Fig. 13C) and mono-anionic LiCAl2Be3 (Fig. 13D) systems [80] by further substitution of Al by Be. The potential energy surface search showed that the global minimum structure of these systems contains a ppC. The vertical detachment energy (VDE) for CAl2Be2 3 is negative indicating the instability toward electron loss. This problem is handled by attaching one Li+ ion with the di-anionic system that resulted in a mono-anionic + LiCAl2Be 3 system without altering the structure of the ppC center. The Li ion is capped to the Be-Be bond in the plane. Further, Castro et al. [81] studied a detailed PES exploration of heptaatomic mono-anionic molecules CBe5E (E ¼ Al, Ga, In, Tl), and they found that for Al and Ga complexes, the ppC structure is the global minima (Fig. 12B), while for the In and Tl complexes, the ppC structures are not global minima rather local minima. However, the local minima containing ppC center are just 1.2 and 1.8 kcal/mol higher in energy than the global minima for In and Tl complexes, respectively. The structures of these clusters showed that the carbon atom is present at the center of the Be4E ring and the additional Be atom is capped to one BedBe bond in the plane. In these complexes, the ppC acts as both s-acceptor and p-donor, and due to the double electronic delocalization, the systems showed stability in planar form.

FIG. 13 Global minimum structures having a ppC.

Planar hypercoordinate carbon Chapter

21

365

4. Planar hexacoordinate carbon (phC) The extension of planar hypercoordination does not stop in pentacoordination, even planar hexacoordinate carbon was reported in the literature. Although many compounds with hexacoordinate carbon were reported, they are not in planar form rather they have three-dimensional structures [82–84] (Fig. 14). The first example containing the phC is CB2 6 (Fig. 15A) studied by Exner et al. in 2000 [85]. This system has a D6h point group. This reported geometry is not the global minimum; rather it is a local minimum and has 143.9 kJ/mol higher in energy than that of the most stable isomer. This system showed a degenerate p set of HOMOs like benzene (Fig. 16). This di-anion is the first system to show six p-aromaticity having a central hexacoordinate atom. The p-aromaticity of this system is confirmed from the negative values of NICS above the planar hexagonal ring. The neutral carbon-boron molecules containing phC are also possible. One D2h isomer (Fig. 15B) and two C2v isomers (Fig. 15C and D) of neutral C3B4 system have a phC at the center of the hexagonal ring and all are the local minima. Please 2 note that the C3B4 system is isoelectronic with the di-anionic CB2 6 system. Wu and coworkers used CB6 cluster and + + performed isoelectronic substitution on it to design mono-cationic CN3Be3 and CO3Li3 systems (Fig. 17) [86]. Both systems correspond to D3h symmetry. The CN3Be+3 system is the local minima and has 25.5 kJ/mol higher energy than the lowest-energy isomer. But CO3Li+3 system is a global minimum structure containing a phC. Several research groups used CB2 6 cluster as a building block and designed new molecules containing phC. Fig. 18 displays the designed phC molecules by Schleyer and coworkers using the CB2 6 cluster as the building block [87]. Further, Li et al. [88] studied K[(6-B6C)Ca]n(6-B6C)K (n ¼ 1, 2, 3), [(6-B6C)Ca]n(6-B6C)2 (n ¼ 1, 2), [(6-B6C)M]i (M ¼ Li, Na, K, and CaCl, i ¼ 1; M ¼ Ca, i ¼ 0), and (6-B6C)(CaCl)2 (Fig. 19) complexes having a phC in the 6-B6C2 ligands. Although in these complexes a phC is present, they are unstable due to the strong tendency of CB2 6 ligands to fuse. The  metal complexes of the CB2 6 cluster were studied by Shahbazian et al. where it was found that [M(B6C)] (M ¼ Li, Na, K, Be, Mg, Ca) can be viable in the gas phase [89]. But the systems are difficult to synthesize because of their instability and conversion into the lower-energy isomers. Recently, Merino and coworkers [90] studied global minimum structures of CE3M+3 (E ¼ S-Te, and M ¼ Li-Cs) series containing a phC (Fig. 15E). They designed the systems by substituting oxygen atoms in the global minimum D3h symmetric CO3Li+3 system by heavier chalcogens. The bonding analysis of the complexes showed that the three chalcogens bonded with central carbon covalently and the three alkali metals bonded ionically with the carbon atom.

5. Higher coordinate carbon The search on planar hypercoordinate carbon is not limited to phC molecules. For the achievement of planar hexacoordinate carbon motifs, boron rings were widely used. In the cyclic boron ring, the size of the carbon is crucial for the planar form. 2 Me

2 H

H

H

H

H

Li Me

Me

Li CLi60/1+/2+

C12H18 H

H

H

Li

2+

CH6

Li

C

Li

Me

2+

H

Li Me

C

Me

C

H

0/1+/2+

Li

C Li

CH3 CH3 Li

Li

C H3

(CH3Li)4

Ru(CO)3

C HB HB

BH

H C B

HB HB

BH BH BH

B H

C2B10H12

(OC)3Ru (OC)2Ru OC

C

Ru(CO)3 Ru(CO)3

Ru(CO)2

CRu6(CO)17

FIG. 14 Some compounds with hexacoordinate carbons.

366

Atomic clusters with unusual structure, bonding and reactivity

+ FIG. 15 (A) Planar hexacoordinate carbon in CB2 6 ; (B–D) isomers of C3B4; (E) Global minimum structures of CE3M3 (E ¼ S-Te, and M ¼ Li-Cs).

FIG. 16 Plots of the important molecular orbitals of CB2 6 with planar hexacoordinate carbon.

CB62– High energy local minimum

C4B3+ High energy local minimum

CN3Be3+ Lowest energy local minimum

CO3Li3+ Global minimum

FIG. 17 Predicted phC motifs in the global minimum CO3Li+3 and low-lying CN3Be+3 .

The first planar heptacoordinate carbon is realized in B7C system having D7h symmetry (Fig. 20A) [91]. The isoelectronic system of B7C anion is designed in neutral form, and the system is B6C2 (Fig. 20B) containing a heptacoordinate carbon at the center of the planar ring [91]. It is expected that the carbon atom will show octacoordination in the center of the B8 ring in the planar geometry. But due to the small size of carbon, it cannot form good bonding with eight boron atoms

Planar hypercoordinate carbon Chapter

˚ FIG. 18 Examples of CB2 6 based phC molecules. Bond lengths are shown in A unit.

21

367

368

Atomic clusters with unusual structure, bonding and reactivity

C6v [(B6C)Li]–

C6v [(B6C)(CaCl)]–

C6v [(B6C)Na]–

C6v [(B6C)K]–

C6v [(B6C)Ca]

D6h [(B6C)2Ca]2–

D6h [(B6C)(CaCl)2] D6h [(B6C)3Ca2]2– D6h [(B6C)2CaK2] D6h [(B6C)3Ca2K2]

D6h [(B6C)4Ca3K2]

FIG. 19 Optimized structures of C6v [(6-B6C)M] (M¼Li, Na, and K), C6v [(6-B6C)Ca], C6v [(6-B6C)CaCl], D6h [(6-B6C)(CaCl)2], D6h [(6-B6C)n+1Can]2 (n ¼ 1 and 2), and D6h [(6-B6C)n+1CanK2] (n ¼ 1, 2, and 3).

FIG. 20 Structures (A) and (B) containing planar heptacoordinate carbon; (C) planar pentacoordinate carbon in B8 ring.

simultaneously. So, the B8C system in D8h symmetry is a transition state and distorted to a stable planar pentacoordinate system having C2v symmetry (Fig. 20C).

6.

Conclusion

The planar hypercoordinate carbon systems violate the fundamental and most widely used structural rules mainly in organic chemistry and biochemistry. The tetracoordinate tetrahedral model of carbon by Van’t Hoff and Le-Bel was violated by the computationally predicted and experimentally realized planar arrangement of four substituents in a tetracoordinate carbon. Later on, this ptC concept is extended to the possibility of higher coordinate carbon in planar form. Mechanical and electronic strategies are helpful to design ptC molecules. But there are no such systematic strategies developed for the designing

Planar hypercoordinate carbon Chapter

21

369

of ppC, phC, and also higher coordinate carbon species. The experimental realization of the in silico studied planar hypercoordinate carbon species is not easy. If the complexes are thermodynamically stable enough, then the experimental characterization could be possible only in the gas phase. Please note that most of the experimentally characterized ptC systems were produced in the gas phase or in matrix isolation. These unusual structures may have interesting unique physical and electronic properties, but due to a lack of good synthetic methods, the applications of these beautiful compounds are not well established.

Acknowledgments PKC thanks DST, New Delhi, India, for the J. C. Bose National Fellowship, grant number SR/S2/JCB-09/2009. PD thanks UGC, New Delhi, India, for the Research Fellowship.

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Planar hypercoordinate carbon Chapter

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Chapter 22

Transformation of nanoclusters without co-reagent Saniya Gratious, Sayani Mukherjee, and Sukhendu Mandal School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram, Trivandrum, Kerala, India

1. Introduction Atomic clusters, consisting of a dozen to a few thousand atoms, have exhibited many unusual size-specific properties that make cluster science relevant to chemistry, biology, physics, and materials science. One of the most exhilarating developments in the field of clusters is the finding of a special class of clusters called “superatoms,” which possess enhanced stabilities and can mimic the behaviors of atoms in the periodic table. This idea offers the potential to create novel materials with tuned properties by using clusters as building units [1–5]. Like the electronic configurations determining the chemical properties of atoms, the filling of electronic shells in clusters is central to the understanding of superatoms. There are several questions that remain unanswered regarding the structure-property relations of this class of nanomaterials. To gain a deeper understanding of this, more variety of nanoclusters (NCs) needs to be synthesized and have their structure elucidated. One of the most widespread methods of synthesizing new NCs is transformation from other existing ones. These transformations are conventionally aided by high thermal energy and the presence of excess ligands, which are structurally very different from the ones protecting the parent NCs. The transformation has helped in obtaining unique NCs, which were not accessible through conventional methods, namely the modified Brust-Schiffrin method and size-focusing methods. Transformation reactions of NCs also aid in knowledge of the gradual structural change that happens to the parent NCs when treated with high temperature and excess free ligands. In recent times, several cluster-to-cluster transformations have been developed, which do not require both the abovementioned conditions and can provide a clearer picture of the NC structure change as there are no external influences. Such co-reactant-free transformations will be discussed in this chapter in further detail.

2. Co-reactant-free transformations Apart from giving rise to new NCs which are conventionally not obtainable, transformations also help in understanding structural evolution among NCs under various reaction conditions. Transformations not aided by any free ligands give us insights into the transformation of NC cores without the interference of any ligands and help us to understand the evolution of metal cores. Recently, several co-reagent-free transformations of NCs have been reported, which occur at ambient temperature aided by conditions such as pH, light irradiation, and oxidizing atmosphere. Even more intriguing are the reports where transformations happen just by thermal treatment without interference from any other reagents. These transformations have been observed to happen between clusters that have related core structures. Below are discussed several instances of transformations under such circumstances.

2.1 pH-induced transformation Olesiak-Banska et al. reported a unique structural transformation of the widely studied Au25 NCs protected by captopril (Au-Capt) ligands induced by pH change in 2018 [1]. The Au-Capt NCs dispersed in water (pH ¼ 7) exhibited absorption bands at 800, 675, 550, 505, 445, and 400 nm (Fig. 1A), like the Au25(PET)18 (where PET—phenylethanethiol) clusters reported by Jin and coworkers [2]. After transferring the NCs to buffers with pH ranging from 2 to 10, no change had been initially observed in the absorption spectra. However, after several hours of keeping the Au-Capt NC at pH ¼ 2, the absorption spectra lost all the characteristic features of Au25 NCs, i.e., absorption bands at 800, 670, and 450 nm. Additionally, a small shoulder at 550 nm developed into a band indicating the formation of a new NC with a different structure Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00022-X Copyright © 2023 Elsevier Inc. All rights reserved.

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Atomic clusters with unusual structure, bonding and reactivity

a)

3.0

pH=2 pH=3 pH=7

2.0 1.5 1.0 0.5 0.0 300

200

pH=2 pH=3 pH=7

180 160

PL intensity

Absorption

2.5

b)

140 120 100 80 60 40 20

400

500

600

700

800

wavelength [nm]

900

1000

0

600

650

700

wavelength [nm]

750

800

FIG. 1 Absorption (A) and emission (B) spectra of Au-Capt measured after several hours of incubation in buffers of various pH values. The samples were excited at 550 nm.

(Fig. 1A). For pH ¼ 3, the leading features of the absorption spectrum were also partly lost, but for higher pH values (between 4 and 10), no changes were observed. The photoluminescence (PL) spectra measured upon treatment with low pH, revealed enhancement in the NCs emission with a maximum at 700 nm (when excited at 550 nm, corresponding to Stokes shift of 0.12 eV). The PL intensity of NCs at pH ¼ 2 was found to be an order of magnitude higher than that reported for NCs at pH ¼ 7 (Fig. 1B). The PL quantum yield (QY) of the NCs was calculated to be 0.37% at pH ¼ 7, whereas at pH ¼ 2, the QY increased  10 times, up to 3.9%. So, it was revealed that the application of low pH in the presence of Cl ions is a new method of transformation from one stable size of NCs to another in water, where the resulting NCs are protected by the same ligand. Thus, it has been suggested that the acidic treatment leads to at least a two-step transformation process: (1) pH ¼ 3, a conversion process of the [Au25SR18] anionic cluster into a charge-neutral cluster [Au25(SR)18]0 and (2) pH ¼ 2, degradation of the NCs with a loss of Au2SR or Au3SR, leading to fluorescent Au23(SR)17 and other sizes in minor content. Although Au25(Capt)18 was stabilized with TOA+ during the whole process, the dramatically low pH could shift the equilibrium in our system. At pH ¼ 2, TOA+ had limited stabilizing activity on protonated NCs, and as a result, the partial decomposition of nanoparticles with Cl anions was possible. A Cl ion, due to its higher electronegativity, acts as an acceptor of electrons and, when in contact with the Au surface, electronic charge transfer from gold ions to chloride ions occurs. Additionally, the protonation of carboxylic groups decreases the electron density in the outer layer of NCs, and thus, chloride-gold interactions become more prominent. On the other hand, NO 3 has never been reported as a decomposition agent, but is known as a strong oxidant, which supports the change of the oxidation state of NCs. The mechanism of size transformation is schematically shown in Scheme 1. These findings of their pH sensitivity and size transformation with the corresponding fluorescence enhancement provide significant information about how a small change in NC solution can trigger a structural change and give rise to new useful physical properties. The Au22(SG)18 (SG: glutathione deprotonated at thiol unit) NCs have been used for bioimaging, pathogenic inversion, and CO2 reduction because of its photoluminescence, biocompatibility, and ability to harvest light. In a recent work by Yang and coworkers, the pH effect on the transformation of the Au22(SG)18 NC has been studied by adjusting the pH of the reaction mixture [3]. Au22(SG)18 were prepared at two different pH values (pH 8 and 2.52) by dissolving the NC solid sample in a NaOH aqueous solution, followed by heating. The effect of the pH on the NC was monitored using UV-vis spectroscopy, and products were analyzed using ESI-MS measurements (Fig. 2). By adjusting the pH from 5.5 to 8, it was observed in the ESI-MS spectra that the NC size was preserved. However, when compared with the sodium adduct species in ESI-MS, an increase in number of the sodium ions at pH 8 was observed compared with the one at pH 5.5. Also, at pH 8, the transformation of Au22(SG)18 NCs to the smaller-sized Au18(SG)14 NCs was observed. The same transformation was observed in three different temperatures by in situ UV-vis, indicating that the transformation is independent

SCHEME 1 Suggested Au25 ! Au23 transformation mechanism. The process is two-stage: in the first, anionic Au25 is oxidized to the neutral form, and in the second step, partial degradation appears with the loss of Au2(Capt).

Transformation of nanoclusters without co-reagent Chapter

3 0

b

pH 5.25

4

2

5

1

6

7

8

5 3 4

6 7

8

2 0

1090

9

14

1

1100

1110

60 min 90 min 120 min 150 min 180 min 210 min 240 min 270 min 330 min

15

1120

1130

400

500

600

700

Wavelength / nm

m/z

c

375

0 min 30 min

pH 8

Absorbance / a.u.

a

22

7–

Au18(SG)14

Intensity / a.u.

Au22(SG)18

8–

6– 7–

1000

1200

1400

1600

1800

2000

m/z FIG. 2 (A) ESI of Au22(SG)18 (pH 5.25 and 8), charge z ¼ 9. (B) In situ UV-vis of the Au22(SG)18 transformation at pH 8. The solution temperature was kept at 59°C. (C) ESI of Au22(SG)18 (pH 8) solution at 330 min.

of temperature. When pH 2.5 was applied to the Au22(SG)18 NC, an immediate formation of larger Au24(SG)20 NC was observed. Kinetic studies performed on this transformation at pH 8 showed a significant decrease in the activation barrier from 30.0  2.2 to 19.3  1.5 kcal/mol and a change to negative entropy . This transformation of Au22(SG)18 NCs at different pH could be induced due to the change in the zwitterionic nature of the glutathione ligands, which was proved by an increase in the number of sodium ions on the ligands in ESI spectra. This transformation thus stands as an example of a surface-induced size transformation mechanism, where the transformation occurs from the exterior surface sites to the interior core.

2.2 Solvent-induced transformation In 2012, Wei et al. had demonstrated that a structural transformation of gold NCs can be achieved by a solvent exchange method [4]. The mixed-ligand protected Au13(PPh3)4(SC12H25)4 NCs with high monodispersity were prepared and the effect of solvent exchange on the clusters was studied. This process was examined by in situ X-ray absorption fine structure (XAFS), UV-vis spectroscopy, TEM, and MALDI measurements. The as-prepared Au clusters in ethanol showed a highly structured absorption spectrum with peaks at 416, 442, and 700 nm, whereas upon dispersing in hexane, all these characteristic peaks were smeared out, making the spectra almost featureless. XANES and XAFS studies on the process showed that after hexane dispersal, the spectra showed significant changes and much resemblance to that of Au foil. The TEM images however revealed that the mean size of the Au NCs has not been altered by hexane dispersal. Hence, it was deduced

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Atomic clusters with unusual structure, bonding and reactivity

that the solvent-induced changes of the UV-vis and X-ray absorption spectra are not due to the size effect but result from some inherent structural transformations of the clusters. The time-dependent Au L3-edge XANES and UV–vis absorption spectra demonstrate the strong variation of electronic structure associated with the rearranged geometric configuration of the clusters. The driving force for this structural transformation was further unraveled by EXAFS studies, where they found a remarkable amplitude reduction in the Au-ligand peak with a considerable increase in the intensity of Au-Au peaks upon hexane dispersal. Desorption of some ligands from cluster surface was also suggested from the quantitative least-squares curve fitting, which showed a reduction in averaged Au-ligand coordination number from 0.9 to 0.4. The DFT calculations indicated that cuboctahedral Au13 can exhibit a metallic behavior, as inferred from the significant density of states (DOS) at the Fermi level. The metallic behavior of the cuboctahedral Au13 cluster was in accordance with a common feature of 1–2 nm-sized fcc nanoparticles whose UV-vis absorption spectra show neither pronounced SPR resonance for larger (>2 nm) nanocrystals nor molecule-like absorption for thiolate-protected clusters. So, it was found that the hexane-induced transformation happens via a quick thiolate desorption step followed by a relatively much slower geometric-structural rearrangement process. This work sheds light on the possibility of tailoring structures for varied applications by simply changing their chemical environment (Fig. 3). Later in 2018, Xie et al. reported an isoelectronic size conversion of [Au23(SR)16] NCs by changing the solvent polarity from ethanol/water to pure water [5]. In this study, they had synthesized [Au23(p-MBA)16] NCs by employing CO-mediated reduction method as reported previously. The as-prepared dark green Au23 NCs in ethanol/water mixture exhibited an absorption maximum at 589 nm and a shoulder peak around 470 nm. When the purified Au23 NC was dissolved in water followed by incubation in a shaker for 2 days at 25°C in 600 rpm, the color of the NC solution changed to reddish brown, which was associated with the changes in the optical absorption spectrum with characteristic peaks at 815, 690, 575, 460, and 430 nm. These changes along with the ESI-MS analysis suggested a complete size transformation of [Au23(p-MBA)16] to the larger-sized [Au25(p-MBA)18] NCs. The solvent polarity-induced size conversion could be attributed to the solvent-dependent stability of SR-[Au(I)-SR]n protecting motifs. It has been known that less polar solvents would prefer to accommodate longer water-soluble SR-[Au(I)-SR]n complexes or motifs. By elevating the polarity of the medium (e.g., changing the solvent from water/ethanol to pure water), the long SR-[Au(I)-SR]3 protecting motif of [Au23(SR)16] would become less thermodynamically favored. As an attempt to minimize the total energy, such long motifs would be tailored into shorter analogues, of which the SR-[Au(I)-SR]2 motif featured in the protecting shell of [Au25(SR)18] is a good candidate. Insights into the reaction dynamics of this size conversion were revealed by MS/MS analysis, where it was found that the dominant fragmentation pathway of [Au23(SR)16] is successive dissociation of [Au(SR)2] and [Au2(SR)], respectively, from the parent and first-generation fragment ions, whereas the dominant fragmentation pathway of [Au25(SR)18] is the dissociation of [Au2(SR)]. Such similar fragmentation behavior of the parent ions of Au25 NCs and first-generation fragment cluster ions of Au23 NCs implies that they share some common and key structure features, which form the structural basis for the size-conversion reaction and thus provide supportive evidence to the surface motif exchange-induced core structure transformation mechanism (Fig. 4). In contrast to the conventionally used thiolates and phosphines, Wang et al. have reported the synthesis of two homoleptic amido protected Ag NCs—[Ag21(dpa)12](SbF6) (Ag21) (dpa ¼ dipyridyl amido) and [Ag22(dpa)12](SbF6)2 (Ag22), where they have also studied the reversible interconversion between these NCs [7]. Both the NCs were synthesized by the reduction of dpa-Ag precursor in the presence of phosphine, AgSbF6, Hdpa, and MeONa, except that Hdpa is not added and the amount of NaBH4 is decreased for synthesizing Ag22. Ag21 showed brown-red color in DCM, while Ag22 was green with characteristic absorption peaks red-shifted compared with that of Ag21. Single crystal structural analysis revealed that

FIG. 3 Schematic illustration of the two-step process of the structural transformation. The first step refers to the quick depletion of the ligands of the surface thiolates from the Au clusters upon hexane dispersal. In the second step, the PPh3-stabilized Au clusters undergo a structural transformation to form fcc-structured clusters.

Transformation of nanoclusters without co-reagent Chapter

a

[Au23(SR)16]–

2.0

589 nm

Normalized absorbance

Normalized absorbance

[Au25(SR)18]–

c

2.0

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1.0 575 nm

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Wavelength (nm) [Au25(SR)18]– 6–

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Peak spacing = 0.02 Charge = 5–

1431

1000 Intensity (a.u.)

Intensity (a.u.)

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5–

1429

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4–

[Au23(SR)16 + x Na – (x + 4) H]5– 7 8 6 5 4 2 3 1 1400

815 nm

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1000

690 nm

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d

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Isoelectronic size conversion

b

470 nm

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1433

1950

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[Au25(SR)18 – 3]

Peak spacing = 0.25 Charge = 4–

1919

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FIG. 4 (A) Schematic illustration of size-conversion reaction from [Au23(SR)16] to [Au25(SR)18] (yellow—light gray color in print version, Au; wine—gray color in print version, S), where SR denotes thiolate ligand. (B, C) UV-vis absorption and D, E electrospray ionization mass spectra of B, D [Au23(SR)16] and (C, E) [Au25(SR)18]. The crystal structures of [Au23(SR)16] and [Au25(SR)18] are drawn according to the reported Au23S16 [6] and Au25S18 [2] skeletons, where all hydrocarbon tails are omitted for clarity. Insets in B, C are digital photos of the corresponding cluster solution. The magenta lines in D, E show simulated isotope patterns of the labeled cluster formulas, which match perfectly with the corresponding experimental data.

Ag21 comprises a monocation cluster [Ag21(dpa)12]+ while the Ag22 cluster contains a di-cation cluster [Ag22(dpa)12]2+ of C3 symmetry, both clusters have SbF 6 counter anions. Both the clusters consist of a centered-icosahedron Ag13 core and were surrounded by 12 dpa ligands. The major structural feature in these two Ag clusters is the various interfacial binding geometries generated by dpa ligands. The flexible arrangement of the three N donors of dpa makes it possible to adjust the coordination positions of dpa, which is an important prerequisite for cluster interconversion. It was found that there is a solvent-dependent equilibrium between Ag21 and Ag22. Ag22 when dissolved in a mixture of EtOH and n-hexane (v:v ¼ 1:4) exhibits a very similar profile to that of the freshly prepared Ag21. The interconversion process can be monitored with absorption spectroscopy because they have significantly different absorption profiles (Fig. 5). The interconversion involves the adding or leaving of an Ag+ ion on the surface structure of the clusters. Moreover, the Ag22 species could be recovered by increasing the ratio of EtOH to n-hexane. Both clusters have identical icosahedral Ag13 cores, and the positions of their dpa ligands are just slightly changed. So, in the interconversion, the surface structure of the cluster only needs to be slightly

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 5 The reversible interconversion between Ag22 and Ag21. (A) Solvent-dependent equilibrium between Ag21 and Ag22. Color codes: purple (dark gray color in print version), green (gray color in print version), and bright green (light gray color in print version) sphere, Ag; blue (black color in print version) sphere, N; gray sphere, C. (B) UV-vis spectra of Ag22 dissolved in mixed solvents with various ratios of EtOH to n-hexane (the total volume of EtOH and n-hexane is kept the same). (C) UV-vis spectra of Ag22 dissolved in a mixture of EtOH and n-hexane (v:v ¼ 1:4) at different temperatures.

adjusted for the taking or releasing of an Ag+ ion. It should be mentioned that both Ag21 and Ag22 carry a valence electron count of 8e and the isoelectronic nature also contributes to the easy and reversible interconversion of these two NCs. In 2020, Zang et al. had reported an Au10 NC protected by levonorgestrel ligands, which has been obtained by the cluster core transformation of an Au8 NC [8]. The Au8 NCs were synthesized using alkynyl-based levonorgestrel as ligands, which is a kind of progestational hormone exhibiting good biocompatibility. Then, the as-prepared Au8 NCs were dissolved in DMSO, and Au10 NCs could be obtained by recrystallization, accompanied by a red shift of the emission peak. Additionally, further conversion from Au10 NCs to Au8 NCs could be achieved in CH2Cl2/CH3CN (Fig. 6). The ESI-TOF-MS spectra showed that Au10 NC broke into fragments once dissolved in CH2Cl2/CH3CN and then reassembled into Au8 NC. Due to the negative zeta potential of Au10 NCs, a cationic polymer, (polyallylamine hydrochloride) (PAH,  17,000), was chosen to prepare monodisperse and stable assemblies with Au10 NC. The as-fabricated positively charged particles with a diameter of 100–200 nm (Au10NC-PAH) displayed a 4–6-fold fluorescence enhancement due to aggregation-induced emission (AIE) owing to the electrostatic interaction between the polymer and the surface ligands of Au10 NC. Notably, when compared to Au8 NC, the emission peak of Au10 NC-PAH displayed a red shift and the lifetime and the quantum yield was increased. The fluorescence intensity was also increased by almost fivefold at an appropriate PAH concentration. Based on the Au10 NC-PAH assemblies, the atomically precise NCs realized effective antibodymediated actin imaging and sustained RA release. Au10 NC-PAH-RA caused a distinctive increase in neurite outgrowth of NG108-15 cells and mostly bipolar differentiation. These results present a very promising approach to easily tune the optical properties of atomically precise metal NCs with charged polymers as cross-linking agents, achieving cell labeling and drug delivery successfully. These improvements might expand the applications of atomically precise metal NCs in bioimaging and disease theranostics.

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A DMSO CH2Cl2/CH3CN

Au10NC

Au8NC

B +

+ +

+ +

+ +

+

PAH – –

– –

–

– –

–

Au10NC

Self-assemble

Au10NC-PAH

FIG. 6 (A) Schematic illustration of the conversion of Au8NCs and Au10NCs; color code: orange (gray color in print version) indicates Au. (B) Selfassembly of Au10NC-PAH. PAH: a cationic polymer, (polyallylamine hydrochloride).

Most recently, Zhu et al. had reported a newly synthesized Ag27H11(SPhMe2)12(DPPM)6 [DPPM ¼ bis(diphenylphosphino)] and studied its spontaneous structural transformation to Ag8(SPhMe2)8(DPPM) NC as well as ligand exchange-triggered transformation into the larger Ag29(SSR)12(DPPM)4 NC (SSR ¼ 1,3-benzene dithiol) [9]. The atomically precise structure of Ag27 was determined by X-ray crystallography, and the locations of 11 hydrides were predicted using DFT calculations. The Ag27H11(SPhMe2)12(DPPM)6 NC was comprised of a highly distorted Ag13H7 kernel that was further stabilized by a ring-like Ag12H4(SPhMe2)6(DPPM)6 and two tetrahedral Ag(SPhMe2)3 motifs. The Ag27 NC was metastable in CH2Cl2 and would undergo a spontaneous transformation into the smaller NC, Ag8(SPhMe2)8(DPPM) (Fig. 7A). Of note is that the metastability of the Ag27 NC was a relative state since its crystals could certainly be obtained. Degradation of both the observed optical absorption and mass signals of Ag27 dissolved in CH2Cl2 had completely disappeared within 12 h (Fig. 7B). The Ag8(SPhMe2)8(DPPM) displayed an intense optical absorption at 425 nm and two shoulder bands at 530 and 810 nm. The excellent match between the experimental and calculated isotope patterns demonstrated the generation of Ag8(SPhMe2)8(DPPM). Considering the existence of both thiol and phosphine ligands in the Ag27 NC, they further performed the ligand exchange on it to prepare a familiar NC, Ag29(SSR)12(DPPM)4. The introduction of H2SSR ligands to the CH2Cl2 solution of Ag27 triggered a rapid cluster transformation, and the 450 nm optical absorption and the mass signals demonstrated the generation of Ag29(SSR)12(DPPM)4.

2.3 Photo-induced transformation The photo-induced size/structure transformation (PIST) from one metal NC to another one with a different size and structure has rarely been studied. To create a library of PIST of a cluster to another one with different size and structure, Li et al. have presented the PIST of phosphine and halide-protected Ag-doped bimetallic cluster [Au37 xAgx(PPh3)13Cl10]3+ (x ¼ 21–26, abbreviated as M37 hereafter) into [Au25 yAgy(PPh3)10Cl8]+ (y ¼ 2–10, denoted as M25) [10]. The M37 NCs could be obtained by the reduction of Ph3PAuCl and Ph3PAgCl by NaBH4 in absolute ethanol, similar to the preparation of M38 and M36 clusters by the Teo group [11–13]. The absorption spectrum of M37 clusters was found to differ in a few minutes under the light in air, which prompted them to test the photostability of M37 clusters. As shown in Fig. 8A, the absorption spectrum of fresh M37 crystals displayed a major peak at 500 nm and a shoulder peak at 330 nm. Upon light irradiation, the new absorption peaks appeared at 329, 422, and 656 nm with reasonable intensities within 10 min, and their intensities further enhanced substantially after 15 min of light irradiation. Besides this, a slight red shift was detected in the peak at 500 nm. These spectral changes observed in time-dependent UV-vis absorption spectra (Fig. 8A) suggested that some new cluster, distinct from the parent M37 cluster has been formed in the solution. In fact, the absorption peaks at 328, 430, 512, and 650 nm in the final product were found to be consistent with the of absorption peaks of the Au13Ag12(PPh3)10Cl8 cluster

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Atomic clusters with unusual structure, bonding and reactivity

B

C Ag8 signal

0min

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10min 1h

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Ag27 signal

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A

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Ag29

Ag8

Ag27

Ligand-Exchange-Triggered Transformation FIG. 7 Transformation of the metastable Ag27H11(SPhMe2)12(DPPM)6 NC. (A) Schematic illustration of the transformation from Ag27H11(SPhMe2)12(DPPM)6 to a size-reduction Ag8(SPhMe2)8(DPPM) (red circle—light gray color in print version) or a size-growth Ag29(SSR)12(DPPM)4 (blue circle—gray color in print version) NC. Color legends: light blue sphere (light gray color in print version), Ag; red sphere (dark gray color in print version), S; magenta sphere (gray color in print version), P; light green sphere (light gray color in print version), hydride; gray sphere, C; white sphere, H. Time-dependent (B) UV-vis and (C) ESI-MS spectra of the transformation from Ag27 to Ag8.

FIG. 8 Time-dependent variation of (A) UV-vis and (B) ESI mass spectra of M37 clusters after irradiation with light.

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Ambient light

[Au25-yAgy(PPh3)10Cl8]

[Au37-xAgx(PPh3)13Cl10]

SCHEME 2 Schematic illustration of the PIST of [Au37–xAgx(PPh3)13Cl10] clusters with a tri-icosahedron M36 metal core to [Au25–yAgy(PPh3)10Cl8]+ with a bi-icosahedron M25 core in the presence of ambient light. Color code: Au or Ag, yellow (gray color in print version); Cl, green (dark gray color in print version); PPh3, pink (light gray color in print version). 3+

[14]. After further extending the reaction time to  40 min, the color of the reaction solution faded completely and black precipitates formed, suggesting the aggregation of the product clusters. As shown in Fig. 8B, only the peaks corresponding to M37 were observed initially in the fresh solution. The signals corresponding to M25 clusters were visible in a few minutes of reaction and became more intense with time. As shown in Fig. 8B, the intensity of peaks assigned to M37 and M25 became almost evenly matched after 10 min. Further, after 20 min, the intensity of peaks corresponding to M37 was found to be very weak, and simultaneously, the peaks of product M25 clusters became dominant, which was accompanied by the fading of color of the reaction solution from purple to faint purple. This suggests that most of the parent M37 clusters have decomposed and subsequently transformed into product M25 clusters. Hence, it was concluded that the light-sensitive M37 clusters undergo size/structure transformation resulting in the formation of new clusters upon shining the ambient light. The photo-induced irreversible size/structure transformation from M37 to M25 cluster was observed in situ by time-dependent UV-vis, ESI-MS, and femtosecond transient absorption spectroscopy. The light near 530 nm serves as the main driving force for the irreversible size/structure transformation, which was further proved by DFT computations (Scheme 2).

2.4 Temperature-induced transformation A pair of Au38(PET)24 (PET ¼ phenylethane thiol) NC isomers-Au38Q and Au38T with same composition and distinctly different structure has been revealed by Jin and coworkers, where these isomers show different optical and catalytic properties and differences in stabilities [15]. The structure of the Au38Q NC consists of a face-fused bi-icosahedral Au23 core capped by a second shell composed of the remaining 15 gold atoms, whereas the Au38T NC is composed of an Au23 core formed by the fusion of an Au13 icosahedron and an Au12 cap by sharing two atoms, and this Au23 core is further protected by gold-thiolate complex units. This Au38T NC showed no obvious spectral changes when stored in toluene at 10°C for almost a month indicating relatively high stability at lower temperatures. However, gradual changes were observed in the optical absorption spectra indicating that Au38T can be transformed to Au38Q at elevated temperatures in toluene, which was supported by TLC and ESI-MS analyses. Under various investigated conditions Au38Q did not transform to Au38T, indicating the higher stability of Au38Q compared with Au38T. It has also been reported that Au38T exhibits remarkably higher catalytic activity than Au38Q at low temperatures in reduction reactions, which might be due to its surface being not as densely protected as the surface of Au38Q. These distinctly different properties of the structural isomers indicate a structure-property correlation, which has future implications in catalytic studies. Another highly unique transformation was reported, where the thermal activation of precursor NCs leads to bimolecular fusion of the paramagnetic [Au25(SR)18]0 to Au38(SR)24 NCs just by heating the former after dissolving in an inert solvent [16]. Not only is this transformation without any co-reagent but is also a much simpler way of making Au38(SR)24 NCs than any reported methods. The transformation reaction was performed by heating a 30 mM solution of Au25(SR)18 NCs in toluene at 65°C and monitoring via UV-vis spectroscopy (Fig. 9). The absorption spectrum of Au25(SR)18 NCs undergoes progressive changes and was virtually identical to that of the Au38(SR)24 NCs. This observation was also supported by ESI-MS and 1H & 13C NMR studies and electrochemistry. NMR evidences the presence of four types of ligands and for the same proton type, double signals caused by the diastereotopicity arising from the chirality of the capping shell. This effect propagates up to the third carbon atom along the ligand chain. Electrochemistry provided a particularly convenient way to study the evolution process and determine the fusion rate constant, which decreases as the ligand length increases.

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Atomic clusters with unusual structure, bonding and reactivity

FIG. 9 UV-vis absorption spectra of samples taken (see text) at different times (see legends) during the reaction of 30 mM [Au25(SC4)18]0 at 65°C in toluene (top graph). The black and the blue (dark gray color in print version) curves show the spectra of [Au25(SC4)18]0 and the purified product, respectively.

The Au25(SR)18 NC structure consists of an Au13 icosahedral core protected by gold–thiolate staple motifs, whereas the Au38(SR)24 NC is having a face-fused bi-icosahedral Au23 core. So, this transformation of Au25(SR)18 NCs to Au38(SR)24 NCs can be viewed as a fusion reaction of two icosahedral cores of Au25(SR)18 NCs to form a bi-icosahedral core of an Au38(SR)24 NC. This is the only case in which a soluble, stable cluster is assembled directly by starting from the fundamental building block Au13, a molecularly guided process that has been sought for many years. Investigation into the mechanistic aspects of this unexpected transformation of very stable Au25(SR)18 NC was carried out, which revealed that the reaction involves a precursor complex formation in which van der Waals interactions between the ligands of the two interacting NCs act as the initial driving force. Since no reaction was observed for the anionic clusters, it was inferred that the radical nature of [Au25(SR)18]0 NCs appears to play an important role. Despite the advances in atomically precise synthesis and structure determination, the concept of reversible isomerization has not yet been explored. Recently, Jin and coworkers reported a thermal stimuli-responsive [Au13Ag12(PPh3)10Cl8](SbF6) (hereafter Au13Ag12 NC), which shows reversible conformational isomerism [17]. It was found that keeping the isomeric mixture of Au13Ag12 NCs at 25°C for 4 weeks led to the formation of S-Au13Ag12 NC, whereas when kept at 10°C for almost 6 weeks, the same isomeric mixture completely transforms to E-Au13Ag12 NC (Fig. 10). Both the isomers were successfully crystallized and characterized. Each of these isomers comprises of two icosahedral Au7Ag6 units fused together by sharing a common vertex of Au. The two Au5 pentagons at the ends of the rod are ligated by 10 phosphine ligands. Two Cl ligands bind with two apical Ag atoms and the remaining six Cl ligands bridge the two icosahedra via bonding with 10 equatorial Ag atoms. The central Au atom connects the two Au7Ag6 icosahedra and serves as the pivot for the rotamerization of the metal configuration of the Au13Ag12. At 10°C, the two Au7Ag6 units are in an eclipsed (E) configuration with D5h symmetry, whereas at 25°C the two Au7Ag6 units rotate by about 36 degrees to form a staggered (S) configuration with D5d symmetry. At the lower temperature (10°C), the Au13Ag12 NC prefers to adapt the configuration with higher symmetry. It was shown that the E-Au13Ag12 and S-Au13Ag12 isomers can be 100% selectively obtained at 10 and 25°C, respectively. To test the reversibility of the isomeric transformation, they monitored the process by UV-vis absorption spectroscopy using the crystal sample of E-Au13Ag12 NCs as the starting material. They first studied the E-Au13Ag12 NC in a dichloromethane solution at 25°C. The characteristic peaks corresponding to E-Au13Ag12 decreased over time and those corresponding to S-Au13Ag12 started appearing. This study shows that the alloying and ligand engineering (i.e., Ag-halide bond) provide a rational strategy to make the framework of metal NCs more flexible for achieving conformational isomerism, which has direct applications in designing intelligent molecular engines. Owing to the quantum confinement effect in the ultrasmall-sized NCs, there are drastic variations in the structural, physical, and chemical properties between NCs differing slightly in composition. Therefore, there is a lack of periodicity among Superatom-like NCs unlike as observed among elements in the periodic table. Over the past few years, however, a few NC series have been discovered in the family of thiolate-protected gold NCs, which show periodicity in their structure and properties [18]. One such series studied recently is the “quantum-box” series comprising highly stable NCs—Au28(SPh-tBu)20, Au36(SPh-tBu)24, Au44(SPh-tBu)28, and Au52(SPh-tBu)32. All these NCs are protected by the same 4-tert-butylbenzenethiolate (SPh-tBu) ligand and follow the general formula—Au8n+4(SPh-tBu)4n+8, where n ¼ 3–6 (Fig. 11). The study of the growth patterns and property evolution energetics in these quantum-box series NCs becomes easier as it depends only on the kernel size and the number of bridging thiolates. So, to expand the scope of this unique class

Transformation of nanoclusters without co-reagent Chapter

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FIG. 10 Thermally responsive transformation of two isomers of the [Au13Ag12(PPh3)10Cl8]+(SbF6) NC (A) E-Au13Ag12 conversion to S-Au13Ag12 at 25°C. (B) The as-obtained S-Au13Ag12 conversion to E-Au13Ag12 at 10°C. E—eclipsed configuration, S—staggered configuration. Color code: Au: yellow (light gray color in print version); Ag: blue (dark gray color in print version); Cl: green (gray color in print version) (C, H, P, and some Cl are omitted for clarity).

Au20

(28, 20)

Au28

(36, 24)

Au36

(44, 28)

Au44

(52, 32)

FIG. 11 Cuboctahedron interpenetration growth patterns in the Au8n+4(SR)4n+8 magic series, n ¼ 3–6 (yellow (light gray color in print version) ¼ S, others ¼ Au, green (gray color in print version) line ¼ S-Au-S-Au-S).

of co-reagent-free transformation, Mandal and co-workers have conducted similar studies on the quantum-box series of NC where they had observed the transformation of Au28(SPh-tBu)20 to Au36(SPh-tBu)24 NC and stability of the latter to heating as well as excess ligands [19]. The fate of the former NC on being heated was studied at various temperatures from 60°C to 100°C. The Au28(SPh-tBu)20 NC showed no change in structure on being heated at 60°C and 80°C. However, on heating at 100°C, the NC showed smooth transformation to its next higher congener in the “quantum box” series Au36(SPh-tBu)24. The transformation had started from the fifth hour and the two NCs were equimolar at the 18th hour. After 36 h of heating the NC mixture at 100°C, the black color of the starting Au28(SPh-tBu)20 solution in toluene had turned green, which is the color of the well-known Au36(SPh-tBu)24. Kinetics studies performed by following the decaying of the UV-vis peaks of the Au28 NC revealed first-order kinetics (Fig. 12). This suggested the breakage of the Au28-NC since no other reagents were utilized for the reaction. Additionally, from the structural pattern, the addition of Au8 unit followed by peeling off bridging thiolates was suggested as the transformation pathway. After observing this interesting phenomenon, we were curious to study the effect

Atomic clusters with unusual structure, bonding and reactivity

1.5

1h 8h 17 h 21 h 28 h 36 h

365 nm

Abs

1.0

480 nm

0.5

Au28(TBBT)20 decomposition

1.5

In [A0] - In [A]

384

Linear Fit of Sheet1 B

1.0 R2 = 0.98 k = 0.042 0.5

580 nm 0.0

0.0 400

500

600

Wavelength nm.

700

0

10

20

30

40

time (hr)

FIG. 12 Time-dependent UV-vis spectra of Au28(TBBT)20 transformation at 100°C. Fitting of decay of 365 nm revealing first-order mechanism.

SCHEME 3 Scheme showing the conversion of Au44(SPh-tBu)28 to Au36(SPh-tBu)a24 NCs. Key: kernel Au atoms, purple (gray color in print version); surface Au atoms, green (dark gray color in print version); S atoms, yellow (light gray color in print version).

of thermal activation on the next congener of the quantum-box series Au36(SPh-tBu)24 NC, which showed remarkable stability and resistance to heating and even excess free TBBT ligands. To further probe the structural transformation in the “quantum-box” series, the next NC Au44(SPh-tBu)28 was studied under thermal treatment (Scheme 3). To that end, detailed studies on the transformation of various concentrations of Au44(SPh-tBu)28 under heating was conducted. It was very surprisingly observed that a structural transformation of Au44(SPh-tBu)28 NC to its lower congener in the quantum-box series instead of the higher one, that is, Au52(SPh-tBu)32 [20]. Kinetic studies of this transformation revealed that it is independent of the reactant concentration with a low activation energy barrier (1.4 eV). This implies that Au44(SPh-tBu)28 NC was deassembled to Au36(SPh-tBu)24 NC via a dissociation mechanism. Theoretical calculations disclosed the fact that staple rearrangement is required to complete the size transformation reaction through a low activation energy pathway, which is the rate-determining step of this reaction. This transformation of Au44 to Au36-NCs is an example of a deassembly mechanism happening through a dynamic surface structure-assisted reduction in kernel binding strength.

3.

Perspectives and conclusions

Understanding more structural transformation among NCs helps us gain insights into the structure-property correlation, which is still very unclear owing to their quantum confinement effect. Several new NCs need to be developed and their transformation probed, to obtain a holistic understanding of their structural evolution. This understanding will fuel the drive to achieve the goal of targeted synthesis of NCs with specific physical and chemical properties for practical applications.

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References [1] M. Waszkielewicz, J. Olesiak-Banska, C. Comby-Zerbino, F. Bertorelle, X. Dagany, A. Bansal, M. Sajjad, I. Samuel, Z. Sanader, M. Rozycka, M. Wojtas, K. Matczyszyn, V. Bonacic-Koutecky, R. Antoine, A. Ozyhar, M. Samoc, Nanoscale 10 (2018) 11335–11341. [2] M. Zhu, C. Aikens, F. Hollander, G. Schatz, R. Jin, J. Am. Chem. Soc. 130 (2008) 5883–5885. [3] B. Zhang, C. Chen, W. Chuang, S. Chen, P. Yang, J. Am. Chem. Soc. 142 (2020) 11514–11520. [4] Y. Li, H. Cheng, T. Yao, Z. Sun, W. Yan, Y. Jiang, Y. Xie, Y. Sun, Y. Huang, S. Liu, J. Zhang, Y. Xie, T. Hu, L. Yang, Z. Wu, S. Wei, J. Am. Chem. Soc. 134 (2012) 17997–18003. [5] Q. Yao, V. Fung, C. Sun, S. Huang, T. Chen, D. Jiang, J. Lee, J. Xie, Nat. Commun. 9 (2018) 1979–1989. [6] A. Das, T. Li, K. Nobusada, C. Zeng, N. Rosi, R. Jin, J. Am. Chem. Soc. 135 (2013) 18264–18267. [7] S. Yuan, Z. Guan, W. Liu, Q. Wang, Nat. Commun. 10 (2019) 4032–4039. [8] M. Xu, T. Jia, B. Li, W. Ma, X. Chen, X. Zhao, S. Zang, Chem. Commun. 56 (2020) 8766–8769. [9] C. Zhu, T. Duan, H. Li, X. Wei, X. Kang, Y. Pei, M. Zhu, Inorg. Chem. Front. 8 (2021) 4407–4414. [10] Z. Qin, J. Wang, S. Sharma, S. Malola, K. Wu, H. H€akkinen, G. Li, J. Phys. Chem. Lett. 12 (2021) 10920–10926. [11] B. Teo, M. Hong, H. Zhang, D. Huang, Angew. Chem. Int. Ed. 26 (1987) 897–900. [12] B. Teo, X. Shi, H. Zhang, Inorg. Chem. 32 (1993) 3987–3988. [13] B. Teo, H. Zhang, X. Shi, J. Am. Chem. Soc. 112 (1990) 8552–8562. [14] M. Zhu, P. Wang, N. Yan, X. Chai, L. He, Y. Zhao, N. Xia, C. Yao, J. Li, H. Deng, Y. Zhu, Y. Pei, Z. Wu, Angew. Chem. Int. Ed. 57 (2018) 4500– 4504. [15] S. Tian, Y. Li, M. Li, J. Yuan, J. Yang, Z. Wu, R. Jin, Nat. Commun. 6 (2015) 8667–8674. [16] T. Dainese, S. Antonello, S. Bogialli, W. Fei, A. Venzo, F. Maran, ACS Nano 12 (2018) 7057–7066. [17] Z. Qin, J. Zhang, C. Wan, S. Liu, H. Abroshan, R. Jin, G. Li, Nat. Commun. 11 (2020) 6019–6025. [18] C. Zeng, Y. Chen, K. Iida, K. Nobusada, K. Kirschbaum, K. Lambright, R. Jin, J. Am. Chem. Soc. 138 (2016) 3950–3953. [19] S. Mukherjee, D. Jayakumar, S. Mandal, J. Phys. Chem. C 125 (2021) 12149–12154. [20] S. Gratious, A. Nair, S. Mukherjee, N. Kachappilly, B. Pathak, S. Mandal, J. Phys. Chem. Lett. (2021) 10987–10993.

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Chapter 23

Application of frustrated Lewis pairs in small molecule activation and associated transformations Dandan Jianga,*, Manas Gharab,*, Sudip Panc, Lili Zhaoa, and Pratim Kumar Chattarajd a

Institute of Advanced Synthesis, School of Chemistry and Molecular Engineering, Jiangsu National Synergetic Innovation Center for Advanced

Materials, Nanjing Tech University, Nanjing, China, b Department of Chemistry and Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur, India, c Institute of Atomic and Molecular Physics, Jilin University, Changchun, China, d Department of Chemistry, Indian Institute of Technology, Kharagpur, India

1. Introduction It is important to replace the toxic and high-cost transition metal (TM) atoms with the environment-friendly main group elements in the field of catalysis, as the main group chemistry sometimes resembles to the chemistry of TMs. A well-known example regarding this is the persistent singlet carbene (R2C:), e.g., N-heterocyclic carbene (NHC), which contains an sp2 hybridized carbon atom with a lone electron pair and an empty 2p orbital [1]. The H2 activation by an alkylamino carbene is demonstrated by Bertrand et al. in 2007 [2]. The carbene activates H2 in a similar way as the oxidative addition of H2 to the TM, i.e., the lone electrons pair of carbene donates the electron density to the antibonding (s*) orbital of H2, and conversely, the carbene accepts the electron density in its empty p orbital of from H2 (see Fig. 1A). Ultimately, the HdH bond is broken as a result of this electron transfer (ET) process. Silylene [3], germylene [4], and stannylene [5] (the heavier congeners of carbene) may also activate H2 in a similar way. Further, ArGeGeAr [6] and ArSnSnAr [7] (the higher analogs of acetylene) can activate H2 by using the frontier orbitals of the unsaturated E-E bonds (E ¼ Ge, Sn) (Fig. 1B). Recently, alkaline earth metals are utilized for acting as catalysts in the hydrogenation of alkene and imine, which performs at a relatively low temperature and moderate H2 pressure. The size and the charge of the metal ions mainly govern the efficiency of the catalysts (see Fig. 2) [8,9].

2. The chemistry of Lewis acids and bases In 1923, Lewis proposed a theory on the properties of acids and bases, where an electron pair acceptor is an “acid” and an electron pair donor is a “base” [10,11]. The Lewis acids (LAs) can accept electron pairs to their lowest unoccupied molecular orbitals (LUMO), and the Lewis bases (LBs) can donate electron pairs from their highest occupied molecular orbitals (HOMO). Thus, a Lewis adduct is formed when a pair of LA and LB comes close to each other. However, any arbitrary combination of LA and LB does not form a Lewis adduct, e.g., lutidine can form a Lewis adduct with BF3, but it cannot form the same with BMe3 [12,13] under similar conditions. Brown et al. described this exception as a result of steric hindrance between the ortho-methyl groups of pyridine and the methyl groups of BMe3, which inhibits the adduct formation (see Fig. 3). Similarly, a ring opening of tetrahydrofuran (THF) takes place between the reaction of BPh3 and THF adduct with Ph3C anion instead of a Ph3C-BPh3 adduct [14] as observed by Wittig and coworkers. Ph3C and BPh3 react with butadiene to form a 1, 2-addition product as noticed by Tochtermann in 1966 [15]. All these examples prove that the presence of bulky groups at the LA and LB centers offers an alternative reaction channel other than the adduct formation.

* These authors are contributed equally in the chapter preparation. Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00023-1 Copyright © 2023 Elsevier Inc. All rights reserved.

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388 Atomic clusters with unusual structure, bonding and reactivity

FIG. 1 Frontier molecular orbital model for activation of H2 by (A) singlet carbene, and (B) E2Ar2 (E ¼ Ge, Sn) compounds.

FIG. 2 Example of hydrogenation reaction catalyzed by calcium complex.

FIG. 3 A few examples of nonclassical reactivity observed by bulky Lewis acids and bases.

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FIG. 4 First examples of heterolytic cleavage of H2 by FLP.

3. Identification of FLP reactivity In 2006, Stephan synthesized a zwitterionic phosphonium hydridoborate, [Mes2HPC6F4BH(C6F5)2], which can release H2 gas upon heating at 150°C giving the ambiphilic phosphinoborane [Mes2PC6F4B(C6F5)2]. Surprisingly, the phosphinoborane may turn back to the phosphonium hydridoborate [16]. This experience can be perceived as a heterolytic cleavage of dihydrogen mediated by a phosphinoborane. In 2007, Stephan et al. further showed the heterolytic cleavage of dihydrogen by a combination of tris-(tertiarybutyl) phosphine (tBu3P) and tris-(pentafluorophenyl) borane (BCF) [17]. Stephan termed this experience as a “Frustrated Lewis pair” (FLP) [18]. A number of FLPs are produced by varying the combination of LAs and LBs whether they are linked or nonlinked. The linked and nonlinked Lewis acid and base pairs are termed as “intramolecular’ and “intermolecular” FLPs, respectively. In the case of “intermolecular” FLP, the LA component becomes neutral boranes, alanes, phosphoniums, cationic silyliums, carbocations, borenium, nitrenium, titanocenes, or zirconocenes [19–26], and the LB component becomes various amines, pyridines, imines, carbenes, phosphines, ethers, carbanions, or silylenes [27–31] (Figs. 4 and 5).

4. Mechanism of H2 activation by FLPs Although the dihydrogen activation by FLP was reported in 2006, a clear picture regarding the mechanism of the activation process appeared later. Earlier, Stephan stated that a side-on interaction of H2 with the B(C6F5)3 LA leads to the formation of an H… 2 B(C6F5)3 adduct and which causes polarization of H2 [17]. In the subsequent step, a phosphine LB (PtBu3) abstracts a proton from the adduct, when it comes closer to it. However, the H… 2 B(C6F5)3 adduct was not detected in the experiment. Initial interaction of H2 with the LB followed by hydride abstraction by the LA may be an alternative path. Again, no H… 2 LB adduct was found in the experiment. Later, a significant mechanistic proposal was introduced by Papai et al. through a quantum mechanical study [32]. They found that the interactions of either B(C6F5)3 or PtBu3 separately with H2 are repulsive under a certain (C6F5)3B…H2 or tBu3P…H2 distances. Therefore, there exists some other reaction channel, where both the LA and LB interact simultaneously with H2. At first, an “encounter complex” (EC) [33] is produced, when the LA and the LB of an FLP are preorganized (see Fig. 6B) and stabilized by weak noncovalent interactions (CdH…F hydrogen bond and dispersion interactions). The existence of weak noncovalent interactions to stabilize the EC was further proven by NMR and molecular dynamics simulation studies [34,35]. Afterward, the Lewis centers of the FLP interact simultaneously with H2, when it enters the cavity of the EC. The LB center of the FLP provides the lone electron pair to the antibonding orbital (s*) of H2 and an empty p-orbital of the LA accepts the bonding (s) electron from H2 (see Fig. 7). This synergistic electron transfer (ET) weakens the HdH bond and ultimately breaks the bond. The HdH bond breaking and the P-H and B-H bond formation take place simultaneously as confirmed by bond order analysis along the intrinsic reaction coordinate (IRC) [36]. This model of H2 activation is termed as the electron transfer (ET) model [37]. On the other hand, a completely different mechanism of H2 activation was proposed by Grimme et al. in 2010 [38]. Although they agreed on the formation of an EC from bulky Lewis acids and bases in the first step as proposed by

390 Atomic clusters with unusual structure, bonding and reactivity

FIG. 5 Examples of (A) LA and (B) LB used to show FLP reactivity. (C) Example of some intramolecular FLPs.

Papai et al., they differed in the opinion as suggested in the generation of an electric field (EF) inside the cavity of the EC. Afterward, an H2 molecule comes into the cavity of the EC and becomes sufficiently polarized. Ultimately, the HdH bond is broken in a barrierless way as a result of polarization. Therefore, in this case, the energydemanding process becomes the entrance of H2 inside the EC. This mechanism is termed as the EF model for H2 activation by FLP.

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a)

b)

FIG. 6 (A) Two proposed intermediates for the reactivity of B(C6F5)3 and tBu3P toward H2. (B) Structure of an “encounter complex” formed by combination of B(C6F5)3 and PtBu3 pair, where the distance between boron (pink; white in print version) and phosphorous (yellow; gray in print version) centers is given in angstrom.

FIG. 7 Schematic picture of the electron transfer (ET) and electric field (EF) model-based interpretations of H2 activation by FLP.

Camaioni et al. analyzed the electronic structures and interaction energy of the H2 activation by a prototype NH3/ BX3 (X ¼ H, F, Cl) Lewis pair to understand the mechanism [39]. The structural reorganization of the precursor complex plays an important role in the activation process as appeared in that study. Although, the EF created by the Lewis pair polarizes H2, but it is not enough to break the HdH bond. It was found that the orbital interactions of the lone pair orbital of nitrogen with s* orbital of H2 and s orbital of H2 with an empty pz orbital of boron at the transition state (TS) are mainly responsible for the HdH bond breaking.

392 Atomic clusters with unusual structure, bonding and reactivity

FIG. 8 Six FLP systems for H2 activation investigated by Papai et al.

Papai et al. reinvestigated the mechanism by taking a set of six FLP systems (see Fig. 8) in order to evaluate the appropriateness of the two different reactivity models (ET and EF) [40]. The HdH distance, bond order, and atomic charges are changed at the TSs implying the activation of H2. Thus, the TS regarding the “entrance” of H2 according to the EF model is confusing. If an H2 molecule enters into a strong EF created by the Lewis pair as given by the EF model, in such a case, the H2 will be ionized not cleaved. On the other hand, a synergistic interaction involving the Lewis pair with H2 takes place at TSs for all the FLP, and the H2 gets polarized because of orbital mixing. Therefore, Papai’s ET model is more acceptable for the activation of H2 and the mode of activation closely resembles the transition metal (TM)-assisted H2 activation. Recently, Pinter et al. [41] clarified the applicability of the two different H2 activation models (ET and EF). The ET model is applicable for high-energy TS, which is characterized as geometrically “late” with large H-H separation and small LB ⋯ H2 distance. LB ! s*(H2) donation is predominant in this case. In contrast, the EF model is applicable to a lowenergy TS, which is characterized as geometrically “early” with small H-H and long LB ⋯ H2 and LA ⋯ H2 distances. Here, the end on LA ⋯ H2 is the predominant interaction.

5.

Thermodynamics on H2 activation by FLP

Besides the kinetics, the thermodynamics of H2 activation by FLP was demonstrated by Papai et al. [42] They described the H2 activation by FLP as a sum of five steps as illustrated in Fig. 9, which involves the separation of LA and LB pair,

FIG. 9 Partitioning scheme of the reaction free energy to interpret the thermodynamic requirement of H2 activation by FLP.

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FIG. 10 Selected examples of small molecule activation by FLPs.

heterolytic cleavage of H2, attachment of proton to the LB, attachment of hydride to the LA, and stabilization by pairing [LBH]+ and [LAH] ions. Among the five steps, the energy consumed for heterolytic cleavage of H2 is independent of the FLP used. FLPs are designed in such a way that the LA/LB separation energy should be small. The stabilizing ion pairing term almost remains the same for different LA/LB combinations as obtained from the computational study. Therefore, it implies that the attachment of the proton to the LB and the hydride to the LA mainly govern the thermodynamics of H2 cleavage by an FLP. Therefore, from the estimation of the third and fourth term in the given figure, one can infer the chances of H2 cleavage by a given FLP. For the case of H+ attachment to LB, experimentally determined pKa values are already available in the literature [43], albeit solvent-dependent. On the other hand, compact experimental data for H attachment to LA are not readily available [44]. However, a recent theoretical study by Heiden and Latham provides a qualitative link between H affinities of LA with other experimental measurements of Lewis acidity [45].

6. Activation of other small molecules Although FLPs play a massive role in the activation of H2, it may activate some other small molecules such as CO, CO2, N2O, NO, SO2, alkenes, alkynes etc., as shown in Fig. 10. Activation of CO2 by an intermolecular or intramolecular P/B FLP produces a novel carbonic acid derivative as demonstrated earlier in 2009 (see Fig. 10) [46]. Afterward, various FLP systems have been utilized for the activation of CO2, which takes place in a similar fashion [47–51]. A density functional theory (DFT)-based study on the activation of CO2 by the intramolecular P/B FLP shows that the activation takes place by cooperative action of the Lewis acid/base pair, and the P-C and B-O bonds are formed simultaneously as indicated by P-C and B-O bond order of 0.23 (at B2PLYP/QZVP level) at the TS (see Fig. 11) [46]. Large P-C and B-O distances and slight bending of CO2 imply an “early” TS of the reaction. In 2018, Liu et al. [52] performed a metadynamics simulation study in order to explore the specific role of the LA/LB pair. They have shown the activation of CO2 by PtBu3/B(C6F5)3 pair as a pseudo-one-step process. At first, the O center of CO2 is captured by B(C6F5)3 and then PtBu3 attacks the C center of CO2. The overall thermodynamics and kinetics are controlled by the strength of the LA as obtained from their study. FLP also adds SO2 in a similar fashion [53]. Cantat et al. [54] performed experimental and computational studies together on the activation of SO2 by N/Si+ and N/B FLPs and compared that with the CO2 activation by the same FLP

394 Atomic clusters with unusual structure, bonding and reactivity

FIG. 11 Transition state for CO2 activation by FLP [45]. Bond distances are given in angstrom.

FIG. 12 Activation of CO2 and SO2 by N/B and N/Si+ FLPs.

(see Fig. 12). The activation of both CO2 and SO2 takes place in a similar way as confirmed by structural analysis. However, the SO2 adduct is thermodynamically more stable than the CO2 as obtained from experimental as well as theoretical studies. The greater stability of the SO2 adduct can be explained by less deformation energy required for bent SO2 molecule in comparison to the linear CO2. Reaction of nitrous oxide with FLP gives an addition compound featuring LB-N ¼ N-O-LA moiety, e.g., the PtBu3/B (C6F5)3 pair reacts with N2O to produce tBu3P(N2O)B(C6F5)3, which contains a PNNOB linkage [55]. Recently, we performed a DFT study on the fixation of N2O by an ambiphilic diazodiboronine [56]. N2O reacts with 1,4,2,5-diazodiboronine (1) to produce three possible products 2, 3, and 4 as depicted in Fig. 13. Although, all the isomers are formed exergonically, the isomers 2 and 4 are formed in the kinetically more favorable way than the isomer 3. Electron density analysis reveals that all the B-N and B-O bonds formed as a result of N2O activation are partly covalent in nature. Very recently, a DFTbased metadynamics study was done on the N2O activation by the PtBu3/B(C6F5)3 pair, which shows that the thermodynamics of the reaction is mainly governed by the strength of LA as in the case of CO2 activation [57]. An ambiphilic borane/phosphine ligand interacts with a TM by accepting electrons from a filled metal d-orbital and by donating electrons to an empty metal d-orbital [58]. These donor/acceptor properties of ambiphilic borane/phosphine system are utilized for the activation of other small molecules, which are analogous to the TM. Thus, an intramolecular P/B FLP (Mes2PCH2CH2B(C6F5)2) attaches CO in a cooperative way to give a cyclic five-membered heterocyclic compound (see Fig. 10) [59]. Similarly, unsaturated vicinal P/B FLP adds tert-butylisocyanide by cooperative addition to it [60]. FLP forms an adduct with NO to give persistent N-oxyl radicals [61]. The reactions of FLP with organic hydrocarbons such as olefins [62] and alkynes [63] give zwitterionic addition products. Earlier, Stephan et al. reported the formation of alkanediyl-linked zwitterionic phosphonium borates as a result of the activation of ethylene and other alkenes by a nonbridged P/B FLP, and the reaction is initiated by the activation of alkenes by Lewis acidic borane followed by the addition of Lewis basic phosphine. Later on, Papai et al. [64] performed a computational study on the concerned reaction and located a TS, where an antarafacial concerted attack of the B(C6F5)3 acid and the PtBu3 base to the ethylene took place (see Fig. 14). The ethylene molecule gets polarized as a result of tBu3P ! p*(C2H4) and p(C2H4) ! B(C6F5)3 donations (see Fig. 15). Further, a concerted 1,2-addition of a bridged P/B FLP to the norbornene C]C double bond takes place as demonstrated by

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FIG. 13 Optimized geometries of 2, 3, and 4 calculated at B3LYP/6-31g(d,p) level of theory.

FIG. 14 Structure of the TS for the concerted attack of the B(C6F5)3 acid and the PtBu3 base to the ethylene. Bond distances are given in angstrom. Color code: B, pink (dark gray in print version); P, yellow (light gray in print version); C, gray; F, violet (black in print version); H, white.

Erker et al. [65], and the reaction is similar to the known “cheletropic” reactions except that the lone pair of electrons and the empty orbitals are present on two different atoms in this case. Recently, Erker et al. reported cyclic P/B FLPs, which may react with organic p-systems [66]. Thus, a bicyclic cycloaddition product is formed upon reaction with ethylene (see Fig. 16). We performed a theoretical study on the activation of ethylene, cyanoethylene, and propylene with that FLP [67]. The results demonstrate that all these reactions take place by the

396 Atomic clusters with unusual structure, bonding and reactivity

FIG. 15 Schematic presentation of cooperative electron transfer between FLP and ethylene.

FIG. 16 Cyclic P/B FLP that undergoes cycloaddition reaction with ethylene. BrEind is the bulky aryl ligand attached to the P center.

concerted attack of the B and the P centers of the FLP to the C]C p-bond with a single TS (see Fig. 17). The P-C and the B-C bonds are formed simultaneously as confirmed by Wiberg bond indices (WBI) and atoms-in-molecule (AIM) analyses at the TSs (see Table 1). Recently, we have shown the plausible activation of allene by the same FLP using DFT study [68]. The deprotonation of terminal alkyne may also take place by a phosphine-borane FLP in an alternative pathway to give a phosphonium alkynylborate compound [65,69]. FLPs react with some other small molecules, which were recognized

FIG. 17 Optimized geometries of the transition states (TS) for the activations of ethylene (TS1), cyanoethylene (TS2a and TS2b), and propylene ˚ unit. Color code: B, pink (dark gray in print version); P, yellow (light gray in (TS3a and TS3b) with the P/B FLP. The bond distances are given in A print version); C, gray; N, blue (black in print version); F, green (light white in print version); H, white.

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TABLE 1 Computed Wiberg bond index (WBI) values and electron density descriptors (in a.u.) at the bond critical points (BCP) of the developing P-C and B-C bonds between the reactant moieties at the TSs. System

Bond

WBI

r(rc)

—2r(rc)

H(rc)

TS1

P-C

0.28

0.039

0.044

0.005

B-C

0.38

0.057

0.010

0.025

P-C

0.47

0.061

0.028

0.013

B-C

0.16

0.025

0.031

0.003

P-C

0.28

0.043

0.041

0.006

B-C

0.35

0.054

0.001

0.021

P-C

0.31

0.044

0.044

0.007

B-C

0.36

0.056

0.011

0.023

P-C

0.24

0.035

0.045

0.004

B-C

0.45

0.067

0.034

0.037

TS2a

TS2b

TS3a

TS3b

later. For example, FLP reacts with N-sulfinyltolylamine (p-TolNSO) yielding a seven-membered cyclic product [70]. A phosphine-borane FLP reacts with 1,3-dienes and a 1,4-addition product is obtained [71]. In contrast, FLP exhibits 1,2-addition to the C]O group of a, b-unsaturated aldehyde [68]. FLP-assisted B-H bond cleavage was also demonstrated, which provides oxygen-ligated borenium cation [72]. As we have mentioned in Section 1, the ring opening of THF-borane adduct by trityl anion was disclosed earlier, it may become possible by other FLP as demonstrated recently [73]. In addition, Stephan and Erker elucidated that FLP reactivity can take advantage of intramolecular cyclization involving sterically prevented amine with olefin or acetylene fragments [74] (see Fig. 18f).

7. Aromaticity-enhanced small molecule activation Trujillo et al. introduced the concept of aromaticity into the chemistry of FLP [75]. Based on the DFT study, they demonstrated that the reactivity of a geminal P/B FLP toward small molecule activation is increased, if a borole fragment is used as the Lewis acid partner instead of BPh2 group in the FLP (see Fig. 19). It was explained as a loss of antiaromaticity of the borole fragment as a result of H2 activation. A five-membered P/B FLPs reported by Erker et al. can activate alkenes to produce cyclic products as discussed in Section 6, it may also cleave the HdH bond in H2 molecule [72]. In this context, we performed a theoretical study on the cleavage of dihydrogen by this FLP (1) and two more designed FLPs 2 and 3 (see Fig. 20) [76]. Fig. 21 illustrates the optimized geometries of the reactant FLPs and the TSs along with the necessary bond distances. The H2 activation is found to be thermodynamically and kinetically more favorable by FLPs 2 and 3 rather than the FLP 1. In order to understand the reason behind the larger reactivity of 2 and 3, nucleus-independent chemical shifts (NICS (0) and NICS (1zz)) are calculated of the C4B ring present in FLPs 2 and 3. Fig. 22 depicts the evolution of NICS (0) and NICS (1zz) of the C4B ring along the reaction coordinate of the concerned reactions from the initial reactants up to the corresponding TSs. It appeared that the antiaromaticity of the C4B ring is gradually decreased and becomes minimum at the TS. Moreover, the decrease in antiaromaticity is faster in case of FLP3 rather than FLP2 as strong electron-withdrawing –C6F5 groups are present around the B center in FLP 3. Therefore, the role of aromaticity is to boost the reactivity of FLP2 and 3. Further, Zhuang et al. [77] demonstrated the role of aromaticity in the activation of CO2 by means of DFT calculation. The aromaticity gain at the TS can reduce the activation barrier remarkably as obtained from their study. Zhu et al. [78] designed some FLPs for the activation of dinitrogen, where the aromaticity plays a pivotal role to make the reactions thermodynamically and kinetically feasible.

398 Atomic clusters with unusual structure, bonding and reactivity

FIG. 18 Other reactions of FLPs.

8.

Catalytic hydrogenation

Since the ability of H2 activation by FLPs can be used as an alternative to the metal-free hydrogenation, it can be achieved by sequential proton and hydride transfer to an unsaturated bond of an organic compound followed by the regeneration of the FLP. Stephan et al. had shown the hydrogenation of nitrile, imine, and aziridine by using Mes2PC6F4B(C6F5)2 as a catalyst after its discovery of it as the foremost FLP [79,80].

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FIG. 19 Geminal P/B FLPs used to study small molecule activation by Trujillo et al.

FIG. 20 Chemical structure of the three FLP systems used in the dihydrogen activation process.

˚ . Color code: B, pink (light white in print FIG. 21 Optimized geometries of the FLPs and the related TSs for H2 activation. The bond distances are in A version); P, yellow (light gray in print version); C, gray; F, violet (black in print version); H, white.

Afterward, they also demonstrated that the LA B(C6F5)3 itself is sufficient for the hydrogenation of sterically hindered imines [81]. Here, the substrate imine itself behaves as a function of LB to fulfill the FLP reactivity. For the case of a less basic imine substrate, a small amount of LB P(C6H2Me3)3 is added to the reaction mixture to increase the reaction rate. In 2014, Stephan et al. [82] and Ashley et al. [83] separately demonstrated the catalytic hydrogenation of carbonyl using B(C6F5)3 as a catalyst in an ethereal solvent. In this case, B(C6F5)3 and ether/1,4-dioxane jointly form an FLP to split H2 and afterward reduce the carbonyl group. Stephan et al. also reported catalytic hydrogenation of aldehydes and ketones into

400 Atomic clusters with unusual structure, bonding and reactivity

FIG. 22 Evaluation of the aromaticity (NICS(0) and NICS(1zz)) along the reaction coordinate of the H2 activation mediated by FLPs 2 and 3 from the start of the reaction up to the corresponding TSs.

alcohols using B(C6F5)3/cyclodextrine FLP in nonpolar solvents, whereas aryl ketone is transformed into deoxygenated aryl compound by similar treatment [84]. Different kinds of polar substrates, e.g., enons, enamines, silyl enol ethers, oximes, etc., are also hydrogenated by FLPs [85–87]. However, a somewhat different hydrogenation process is employed for the hydrogenation of nonpolar substrates such as olefins. Much weaker LB is used with a combination of B(C6F5)3 so that the conjugate acid formed as a result of H2 activation has enough potential to activate the less reactive olefin through protonation. Subsequently, the hydride is abstracted by the olefin from [HB(C6F5)3] in the second step. In 2013, Stephan et al. demonstrated the ability of hydrogenation of alkene by a combination of ether and B(C6F5)3 [88]. H2 activation generates the ions [Et2O…H…OEt2]+ and [HB(C6F5)3], which are employed for hydrogenation as revealed by the computational study. Moreover, the electron-rich olefins are hydrogenated more easily than simple olefins as explored in several studies using different FLP systems. Repo et al. described a simple way of hydrogenation of alkyne to cis-alkene using ansa-aminohydroborane FLP as a catalyst [89]. Hydrogenation of polycyclic aromatic hydrocarbons such as anthracene, tetracene, and tetraphene becomes possible using a combination of weakly basic Ph2PC6F5 and B(C6F5)3 at high temperature (80°C) and pressure (100 atm.) [90]. Hydrogenation of aniline produces N-cyclohexylammonium salt in the presence of H2 and B(C6F5)3 as demonstrated by Stephan et al. [91]. Moreover, some N-heterocyclic compounds such as acridine, quinoline, and phenanthroline are also reduced by B(C6F5)3 catalyst [92]. The hydrogenation of the substrates as discussed above proceeds through some specific mechanisms. It involves the FLP-assisted activation of H2 and then these are transferred to the unsaturated substrate. However, two different mechanisms are proposed depending upon the proton and hydride delivery to the substrates. If the LA component of the FLP is sufficiently strong, e.g., B(C6F5)3, then the resulting borohydride will not be significantly strong to deliver hydride to the unactivated substrate. In that case, substrate activation is necessary by protonation of it or at least through hydrogen bonding interaction to the proton before the hydride delivery. Thus, the protonation step prior to hydride transfer is operative here [93–95]. Sometimes, substrate itself acts as an LB in that case. On the other hand, if hydride transfer prior to proton transfer is operative, in that case substrate activation with another LA molecule is necessary to cause hydride transfer effortlessly [28]. Moreover, concerted proton and hydride transfers to the substrate are also possible as observed in case of hydrogenation of CO2 to formic acid [96]. Jiang et al. [97] studied catalytic hydrogenation of CO2 by different intramolecular P/B FLPs by means of DFT calculation. It reveals two possible ways of reduction of CO2: (i) activation of H2 followed by concerted transfer of activated hydrogens to CO2, (ii) activation of CO2 by FLP followed by metathesis of H2 and then reductive elimination of formic acid. The results obtained from this study reveal that although the reduction of CO2 is difficult by any of these two steps under moderate conditions, it may be accomplished at high pressure and temperature.

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SCHEME 1 Proposed mechanistic cycle for hydrogenation of CO2 catalyzed by A and B, where Lewis base of FLP activates H2.

We performed a DFT-based study on the hydrogenation of CO2 by a bridged B/N FLP [98]. In this case, the hydrogenation of CO2 takes place in a different way than Jiang’s CO2 hydrogenation. In one way, H2 and CO2 are simultaneously activated by the LB and the LA centers of FLP respectively in the first step of the reaction as depicted in Scheme 1. Thus, a formate ion is produced, which is attached to the boron center. The hydrogen of the nitrogen center moves to the oxygen center of the formate moiety in the second step. In an alternative way, H2 and CO2 are simultaneously activated by the LA and the LB centers of FLP respectively in the first step of the reaction as depicted in Scheme 2. Then the COOH moiety present at the nitrogen center changes its orientation so that the hydride ion at the boron center can attack the carbon center of it in order to produce formic acid. Although the reactions related to these catalytic cycles are not kinetically favorable under moderate conditions, they may be accomplished at high pressure and temperature. Therefore, this study provides a guidance for designing effective CO2 hydrogenation catalyst (Fig. 23).

9. Boron-ligand cooperation The term boron-ligand cooperation (BLC) has been introduced in order to describe a distinct approach of the reactivity of a chemical species to activate a chemical bond. In this context, it is required to have a knowledge about metalligand cooperation (MLC). MLC describes a situation, where one of the covalently bound ligands of a TM complex directly participates in the activation of a chemical bond and the covalently bound ligand is converted into a datively bound ligand as a result of bond activation. Fig. 24 illustrates the examples of MLC as demonstrated by Noyori [99] (Fig. 24A) and Milstein [100] (Fig. 24B). There are some distinct classes of intramolecular FLPs, where the bond activation by the FLP is associated with the change of a covalently bound substituent into a datively bound ligand around the LA center of the FLP (see Scheme 3). This is analogous to the H2 activation mediated by TM-ligand cooperation. Therefore, this reactivity of FLPs is described as BLC in analogy to the MLC. The MLC and BLC have a common point, namely both the approaches consist of two reactive centers, which are Lewis acidic and basic in nature. Previous example of BLC includes the H2 and CO2 activation by pyrazolyl borane, which produces pyrazol borane complex as reported by Tamm et al. [101]. Wang et al. [102] performed a computational study on the H2 activation of

402 Atomic clusters with unusual structure, bonding and reactivity

SCHEME 2 Proposed mechanistic cycle for hydrogenation of CO2 catalyzed by A and B, where Lewis base of FLP activates CO2.

a designed substrate to intimate the reactivity of Milstein’s pyridine-based pincer complex (see Scheme 4). The N-B bond is transformed into a dative bond as a result of H2 activation and the product can be considered as a pyridine-borane complex. In 2018, Gellrich et al. reported the reversible activation of H2 by a pyridonate borane FLP [103]. It was claimed that the covalent B-O bond present in boroxypyridine is transferred to a dative bond as a result of H2 activation. We performed a theoretical study to explain the BLC in Boroxypyridine 1 and five more model FLPs 2-6 (See Scheme 5) [104]. The obtained thermochemical results showed that all the reactions are exergonic and they have moderate activation free energy barrier. We examined the changes, which occurred on the B-X bonds (X ¼ O, N, S) before and after the H2 activation. It shows that all the B-X bonds are weakened as indicated by the bond distance and the WBI values of the concerned bonds (see Fig. 25). We performed energy decomposition analysis (EDA) in order to understand the nature of the B-X bonds in the reactant FLPs and in the product. The relative contributions of DEorb and DEelstat toward the total attractive interaction between the B and the X (X ¼ O, N, S) centers show the B+-X formulation to be more acceptable representation in the FLPs. On the other hand, the total interaction between the B and the X centers is decreased in the product i-H2 (i ¼ 1-6) complexes in comparison to the reactant FLPs, which agrees with the weakening of the B-X bonds as a result of H2 activation. Moreover, the decomposition of DEorb into pairwise orbital contribution exhibits to have a predominant interaction channel, which stabilizes the B-X bonds in the products. The deformation densities of the corresponding interaction channel are plotted in Fig. 26. The shapes of these deformation densities clearly indicate that the B-X bonds exist as a result of B X s donation. Therefore, these results reveal the H2 activation by the concerned FLPs to be assisted by BLC working at the B-X (X ¼ O, N and S) bonds. We computed the NICS(0) and NICS(1) of the pyridine ring in the FLPs, TSs and in the corresponding H2 activated products. It shows that the aromaticity of the ring is gradually diminished and becomes minimum at the products. This is due to the loss of aromaticity of the pyridine moiety in the FLPs, which further demonstrates the transfer of the B-X bond into a dative bond as result of H2 activation.

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FIG. 23 A few examples of FLP-catalyzed hydrogenation of polar and nonpolar substrates.

10. Polymerization reaction Zhang et al. reported the polymerization of methyl methacrylate (MMA), a-methylene-g-butyrolactone (MBL), and gmethyl-a-methylene-g-butyrolactone (MMBL) monomers by specific NHC or phosphine/Al(C6F5)3 Lewis acid-base pair [105,106]. The reaction is supposed to proceed through adduct formation between the LA and the monomer followed by the attack of LB to produce a zwitterionic phosphonium or imidazolium enolaluminate active species. Therefore, the

404 Atomic clusters with unusual structure, bonding and reactivity

FIG. 24 Examples of metal-ligand cooperation. H2 activation by (A) Noyori catalysis and (B) Milstein’s pyridine-based pincer complex.

SCHEME 3 H2 activation by a distinct class of intramolecular FLPs, which transforms a covalently bound species into a datively bound ligand.

SCHEME 4 The model system designed by Wang et al. to intimate the reactivity of Milstein’s pyridine-based pincer complex.

SCHEME 5 Chemical structures of six bridged FLP systems.

nucleophilicity of the LB should be high so that the reactive zwitterionic species is generated effortlessly. They performed a computational study on the active zwitterionic species formation, chain initiation and propagation steps for the polymerization of MMA. It reveals that the zwitterion formation largely depends on the strength of the LBs and the bulkiness around the LB center. Now, an MMA is added to the zwitterionic species by one of the two mechanisms as illustrated in Fig. 27. In case of monometallic mechanism, the zwitterionic species attacks an MMA molecule, whereas in bimetallic mechanism the zwitterionic species attacks an MMA molecule, which was activated by Al(C6F5)3 acid. However, the DFT study clearly shows that the bimetallic mechanism is more favorable than the monometallic one. In addition, the bimetallic mechanism is also favorable in the chain propagation step.

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˚ and FIG. 25 Optimized geometries of the FLP 1-6, TSs for H2 activation (TSa), and H2 activated products (i-H2, i ¼ 1-6). Bond distances are in A WBI values are in brackets. Color code: N, blue (dark gray in print version); B, pink (light white in print version); C, gray; O, red (black in print version); H, white; S, yellow (light gray in print version); F, violet (light black in print version). (Continued)

406 Atomic clusters with unusual structure, bonding and reactivity

FIG. 25, cont’d

FIG. 26 Shape of the deformation density Dr(1), which is associated with the main orbital interaction DEorb(1) in i-H2 (i ¼ 1-6) complexes. The isosurface value is 0.003. The color code of the charge flow is red (black in print version) ! blue (light black in print version).

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FIG. 27 (A) Monometallic and (B) bimetallic mechanism for the first MMA addition to the zwitterion.

11. Summary and outlook The present chapter focuses on how the chemistry of FLP continues to develop from the initial H2 and other small molecules’ activation to the catalytic hydrogenation, polymerization reactions, etc. This chapter provides a scrutiny of several experimental works on FLPs and mechanistic insights into their reactivity from electronic structure theory calculations. There exists two mechanistic proposals of H2 activation, which are ET and EF models as revealed by several quantum chemical calculations. These models in turn would help in designing more efficient FLPs for H2 activation. On the other hand, thermodynamics of H2 activation is mainly governed by the proton and hydride affinities of LB and LA, respectively. The small molecules such as CO, NO, CO2, SO2, N2O, alkenes, alkynes, etc., get activated by cooperative action of both the Lewis centers of FLP as demonstrated by different computational analyses. For instance, the activation of alkene by a bridged P/B FLP is through a concerted attack of the Lewis centers to the C]C p-bond of alkene. An antiaromatic environment around an LA center of an FLP can decrease the energy barrier for the activation of H2 and other small molecules in comparison to any other environment around the LA center. This is explained by aromaticity gain of the LA center as a result of these reactions as confirmed by NICS calculations. Thus, the concept of aromaticity can be incorporated with FLP in order to design more effective FLPs. Several unsaturated substrates such as imine, nitrile, enamine, aziridine, aldehyde, ketone, alkene, alkyne, oxime, etc., are hydrogenated by FLP. The hydrogenation takes place by proton and hydride delivery by FLP to the unsaturated bonds of the substrates as demonstrated by several computational analyses. In addition, there are some specific FLPs, where the bond activation by the FLP is associated with the change of a covalently bound substituent into a datively bound ligand around the LA center of the FLP. This reactivity of FLPs is described as boronligand cooperation (BLC) in analogy with the MLC. This concept of BLC can be utilized in the activation of strong bonds like NdN bond in N2 molecule. FLP can also be utilized in the polymerization of a monomer, where a strong LA activates the monomer and the LB partner of the FLP attacks the activated monomer to initiate the polymerization process.

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Stephan, 1,4-Addition reactions of frustrated Lewis pairs to 1,3-dienes. Chemical, Communication (2009) 2335–2337. [72] M.A. Dureen, A. Lough, T.M. Gilbert, D.W. Stephan, B–H Activation by frustrated Lewis pairs: borenium or boryl phosphonium cation? Chem. Commun. (2008) 4303–4305. [73] G.C. Welch, R. Prieto, M.A. Dureen, A.J. Lough, O.A. Labeodan, T. H€oltrichter-R€ossmann, D.W. Stephan, Reactions of phosphines with electron deficient boranes, Dalton Trans. (2009) 1559–1570.

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[74] T. Voss, C. Chen, G. Kehr, E. Nauha, G. Erker, D.W. Stephan, Cyclizations via frustrated Lewis pairs: Lewis acid induced intramolecular additions of amines to olefins and alkynes, Chem. A Eur. J. 16 (2010) 3005–3008. [75] J.J. Cabrera-Trujillo, I. Ferna´ndez, Aromaticity can enhance the reactivity of P-donor/borole frustrated Lewis pairs, Chem. Commun. 55 (2019) 675–678. [76] M. Ghara, P.K. Chattaraj, Can a decrease in anti-aromaticity increase the dihydrogen activation ability of a frustrated phosphorous/borane Lewis pair?: a DFT study, Theor. Chem. Accounts 139 (2020) 183. [77] D. Zhuang, A.M. Rouf, Y. Li, C. Dai, J. Zhu, Aromaticity-promoted CO2 capture by P/N-based frustrated Lewis pairs: a theoretical study, Chem. Asian J. 15 (2020) 266–272. [78] A.M. Rouf, Y. Huang, S. Dong, J. Zhu, Systematic design of a frustrated Lewis pair containing methyleneborane and carbene for dinitrogen activation, Inorg. Chem. 60 (2021) 5598–5606. [79] D.W. Stephan, G. Erker, Frustrated Lewis pair chemistry: development and perspectives, Angew. Chem. Int. Ed. 54 (2015) 6400–6441. [80] D.W. Stephan, G. Erker, Frustrated Lewis pairs: metal-free hydrogen activation and more, Angew. Chem. Int. Ed. 49 (2010) 46–76. [81] P.A. Chase, T. Jurca, D.W. Stephan, Lewis acid-catalyzed hydrogenation: B(C6F5)3-mediated reduction of imines and nitriles with H2, Chem. Commun. (2008) 1701–1703. [82] T. Mahdi, D.W. Stephan, Enabling catalytic ketone hydrogenation by frustrated Lewis pairs, J. Am. Chem. Soc. 136 (2014) 15809–15812. [83] D.J. Scott, M.J. Fuchter, A.E. Ashley, Nonmetal catalyzed hydrogenation of carbonyl compounds, J. Am. Chem. Soc. 136 (2014) 15813–15816. [84] T. Mahdi, D.W. Stephan, Facile protocol for catalytic frustrated Lewis pair hydrogenation and reductive deoxygenation of ketones and aldehydes, Angew. Chem. Int. Ed. 54 (2015) 8511–8514. [85] B. Gao, X. Feng, W. Meng, H. Du, Asymmetric hydrogenation of ketones and enones with chiral Lewis base derived frustrated Lewis pairs, Angew. Chem. Int. Ed. 59 (2020) 4498–4504. [86] P. Spies, S. Schwendemann, S. Lange, G. Kehr, R. Frhlich, G. Erker, Metal-free catalytic hydrogenation of enamines, imines, and conjugated phosphinoalkenylboranes, Angew. Chem. Int. Ed. 47 (2008) 7543–7546. [87] S. Wei, H. Du, A highly enantioselective hydrogenation of silyl enol ethers catalyzed by chiral frustrated Lewis pairs, J. Am. Chem. Soc. 136 (2014) 12261–12264. [88] L.J. Hounjet, C. Bannwarth, C.N. Garon, C.B. Caputo, S. Grimme, D.W. Stephan, Combinations of ethers and B(C6F5)3 function as hydrogenation catalysts, Angew. Chem. Int. Ed. 52 (2013) 7492–7495. [89] K. Chernichenko, A. Madarasz, I. Papai, M. Nieger, M. Leskela, T. Repo, A frustrated-Lewis-pair approach to catalytic reduction of alkynes to cisalkenes, Nat. Chem. 5 (2013) 718–723. [90] Y. Segawa, D.W. Stephan, Metal-free hydrogenation catalysis of polycyclic aromatic hydrocarbons, Chem. Commun. 48 (2012) 11963–11965. [91] T. Mahdi, Z.M. Heiden, S. Grimme, D.W. Stephan, Metal-free aromatic hydrogenation: aniline to cyclohexyl-amine derivatives, J. Am. Chem. Soc. 134 (2012) 4088–4091. [92] S.J. Geier, P.A. Chase, D.W. Stephan, Metal-free reductions of N-heterocycles via Lewis acid catalyzed hydrogenation, Chem. Commun. 46 (2010) 4884–4886. [93] S. Tussing, L. Greb, S. Tamke, B. Schirmer, C. Muhle-Goll, B. Luy, J. Paradies, Autoinduced catalysis and inverse equilibrium isotope effect in the frustrated Lewis pair catalyzed hydrogenation of imines, Chem. A Eur. J. 21 (2015) 8056–8059. [94] L. Greb, S. Tussing, B. Schirmer, P. Ona-Burgos, K. Kaupmees, M. Lokov, I. Leito, S. Grimme, J. Paradies, Electronic effects of triarylphosphines in metal-free hydrogen activation: a kinetic and computational study, Chem. Sci. 4 (2013) 2788–2796. [95] S. Tussing, K. Kaupmees, J. Paradies, Structure–reactivity relationship in the frustrated Lewis pair (FLP)-catalyzed hydrogenation of imines, Chem. A Eur. J. 22 (2016) 7422–7426. [96] A.P. Pulis, E. Rochette, M.-A. Legare, D.W. Stephan, F.-G. Fontaine, Intramolecular B/N frustrated Lewis pairs and the hydrogenation of carbon dioxide, Chem. Commun. 51 (2015) 9797–9800. [97] B. Jiang, Q. Zhang, L. Dang, Theoretical studies on bridged frustrated Lewis pair (FLP) mediated H2 activation and CO2 hydrogenation, Organic Chem. Front. 5 (2018) 1905–1915. [98] M. Ghara, P.K. Chattaraj, A computational study on hydrogenation of CO2, catalyzed by a bridged B/N frustrated Lewis pair, Struct. Chem. 30 (2019) 1067–1077. [99] R. Noyori, T. Ohkuma, Asymmetric catalysis by architectural and functional molecular engineering: practical chemo- and stereoselective hydrogenation of ketones, Angew. Chem. Int. Ed. 40 (2001) 40–73. [100] J. Zhang, G. Leitus, Y. Ben-David, D. Milstein, Facile conversion of alcohols into esters and dihydrogen catalyzed by new ruthenium complexes, J. Am. Chem. Soc. 127 (2005) 10840–10841. [101] E. Theuergarten, D. Schluns, J. Grunenberg, C.G. Daniliuc, P.G. Jones, M. Tamm, Intramolecular heterolytic dihydrogen cleavage by a bifunctional frustrated pyrazolylborane Lewis pair, Chem. Commun. 46 (2010) 8561–8563. [102] G. Lu, H. Li, L. Zhao, F. Huang, P.v.R. Schleyer, Z.-X. Wang, Designing metal-free catalysts by mimicking transition-metal pincer templates, Chem. A Eur. J. 17 (2011) 2038–2043. [103] U. Gellrich, Reversible hydrogen activation by a pyridonate borane complex: combining frustrated Lewis pair reactivity with boron-ligand cooperation, Angew. Chem. Int. Ed. 57 (2018) 4779–4782. [104] M. Ghara, S. Pan, P.K. Chattaraj, A theoretical investigation on boron-ligand cooperation to activate molecular hydrogen by a frustrated Lewis pair and subsequent reduction of carbon dioxide, Phys. Chem. Chem. Phys. 21 (2019) 21267–21277. 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Chapter 24

Ligand-protected clusters Yukatsu Shichibu and Katsuaki Konishi Graduate School of Environmental Science, Hokkaido University, Sapporo, Japan

1. Introduction Ligand-protected gold clusters have played a central role in the recent progress of ligand-protected clusters [1]. Indeed, various properties of the gold clusters including optical, catalytic, and magnetic properties have been explored. Under these circumstances, theoretical studies have constantly contributed to such progress, providing valuable insights for understanding the origins of their properties. In this chapter, we briefly introduce representative theoretical studies of thiolate-protected gold clusters, the most widely investigated metal clusters (Section 2). Then, we review recent advancements in the research of phosphine-protected gold clusters from theoretical point of views in this decade (Section 3).

2. Representative examples of theoretical studies Nowadays, all-thiolate-protected gold clusters are widely known to possess unique surface structures of –SR–(Au–SR)x– (named as staple motifs) [1]. However, before the historical breakthroughs of the structural determinations of the allthiolate-protected Au102 and Au25 clusters [2,3] more than a decade ago, little was known about the geometric structures of the thiolate-protected gold clusters. Under these circumstances, unique structural concepts of “divide-and-protect” [4] and “core-in-cage” [5] about the all-thiolate-protected gold clusters (Fig. 1) were consecutively proposed from the geometry-optimization calculations of Au38 and Au25 clusters. These theoretically predicted structures, both of which include the staple motifs, are in sharp contrast to conventional “metal core–ligand shell” type structures (for a more detailed description, see Section 3.2) and describe the essence of the later determined structures by single-crystal XRD [3,6]. Even now, these concepts still offer informative structural pictures for predicting and understanding the structure and bonding of metal clusters. Based on the crystallographically determined structures of the thiolate-protected gold clusters, theoretical approaches have revealed their various characteristics. For example, superatomic nature of the p-mercaptobenzoic acid (p-MBA)protected Au102 cluster [2], which consists of the bare Au79 core fully protected by –SR–(Au–SR)x– staples, was revealed from the analysis of density of states (Fig. 2) [7]. This demonstrated the importance of the above-mentioned structural concepts (Fig. 1) reflected in the electronic structures of the gold clusters. As other examples, reaction pathways of the CO2 electroreduction on Au25(SR)18 cluster were theoretically probed (Fig. 3A) [8], and geometrical quantification of chirality in all-thiolate-protected gold clusters was performed using a numerical method, the Hausdorff chirality measure (HCM), where degree of chirality was empirically assessed by the HCM value (Fig. 3B) [9]. Such rapid and great advancements were also seen in phosphine-protected gold clusters, the core size of which is basically smaller than that of thiolateprotected gold clusters. In the next section, we introduce recent studies of phosphine-protected gold clusters.

3. Diphosphine-ligated gold clusters 3.1 Jellium models and core shapes For a gold cluster [AunLmXl]z, the number of valence electrons (n∗) derived from 6 s orbitals of the gold atoms is calculated as n∗ ¼ n – l – z, where L is a weak Lewis base ligand (phosphine, arsine, etc.), X is a strong electron-withdrawing ligand (thiolate, halide, alkynyl, etc.), and z is the total charge. According to the spherical jellium model (a spherically constrained free electron model) [10], ultrasmall Aun clusters (n  10) with n∗ ¼ 8 favor spherical geometries and exhibit magic stability due to the shell closure of superatomic S- and P-orbitals (i.e., 1S21P6 configuration) (Fig. 4, middle). When nonspherical distortions of gold cores are substantial, the structural jellium model [11] indicates that the breaking of the degeneracy Atomic Clusters with Unusual Structure, Bonding and Reactivity. https://doi.org/10.1016/B978-0-12-822943-9.00024-3 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 1 Theoretically predicted structures of (A) Au38 and (B) Au25 clusters. € (A) Reproduced with permission Hakkinen H, Walter M, Gr€ onbeck H. J Phys Chem B 2006;110:9927–9931. Copyright 2006, American Chemical Society. (B) Reproduced with permission Iwasa T, Nobusada K. J Phys Chem C 2007;111:45–49. Copyright 2007, American Chemical Society.

FIG. 2 Density of states for (A) Au102(p-MBA)44 and (B) bare Au79 core (58e) without the Authiolate staple layer. Adapted with permission Walter M, Akola J, Lopez-Acevedo O, Jadzinsky € PD, Calero G, Ackerson CJ, Whetten RL, Gronbeck € H, Hakkinen H. Proc Natl Acad Sci USA 2008;105:9157–9162. Copyright 2008, the National Academy of Sciences of the United States of America.

FIG. 3 (A) Electroreduction of CO2 to CO on Au25(SMe)18 and (B) chiral structures and HCM values of the Au79 core, staple array, and their combination for Au102(p-MBA)44. (A) Reproduced with permission Austin N, Zhao S, McKone JR, Jin R, Mpourmpakis G. Catal Sci Technol 2018;8:3795–3805. Copyright 2018, Royal Society of Chemistry. (B) Reproduced with permission Pelayo JJ, Whetten RL, Garzo´n IL. J Phys Chem C 2015;119:28666–28678. Copyright 2015, American Chemical Society.

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FIG. 4 Correlation between the core shapes and valence-electron numbers (n* ¼ 4, 6, 8) of ultrasmall gold clusters.

of the P shell results and nonspherical gold clusters favor prolate or oblate geometries with 1S21P2 or 1S21P4 sub-shell closed configurations, respectively (Fig. 4, left and right). These theoretically constructed jellium models have provided valuable insights into the geometries of gold clusters.

3.2 Geometric studies Compared with the nuclearities of thiolate-protected gold clusters, those of phosphine-protected gold clusters are basically few (nuclearity of around 10), getting the core sizes of them below 1 nm. Until now, numerous subnanometer-sized gold clusters bearing mono- and di-phosphine ligands have been crystallographically well defined. Most of the clusters are spherical (n∗ ¼ 8; [Au13]5+ and [Au11]3+) or oblate (n∗ ¼ 6; [Au9]3+), which match the criteria of the jellium models [12]. However, systematic studies of gold clusters in this decade using various bis(diphenylphosphino) ligands, whose phosphorus atoms are bridged by two to four carbon atoms, have enriched the diversity of their geometrical and electronic structures and highlighted prolate-shaped cores. One example is a dodeca-ligated undecagold cluster ([Au11(dppe)6]3+) protected by the C2-bridged diphosphine, dppe (Ph2P–(CH)2–PPh2) [13]. As shown in Fig. 5A, the Au11 moiety is composed of a toroidal Au9 core plus two exo-attached gold atoms. A unique ligation manner is that each of the exo gold atoms accommodates two phosphorus atoms to form a ligand–Au–ligand substructure on the “Au9 core,” which is like –SR–(Au– SR)2– staples found in thiolate-ligated gold clusters (Fig. 5, blue curves). This is in sharp contrast to conventional “metal core–ligand shell” type gold clusters bearing phosphine ligands, where the atoms placed at the polyhedral-core peripheries are coordinated by one ligand (i.e., on-top coordination). Due to the presence of the two exo atoms on the major axis (z-axis), the Au11 cluster exhibits highly anisotropic shape, which is quite different from the isotropic thiolate-protected Au25 cluster (Fig. 5). Although the n∗ of the Au11 cluster is formally eight, the Au11 moiety can be constructed by the fusion of the next-mentioned [core+ exo]-type Au6 units in four-electron system by sharing one exo atom.

FIG. 5 Skeletal structures of (A) [Au11(dppe)6]3+ and (B) [Au25(SC2H4Ph)18].

414 Atomic clusters with unusual structure, bonding and reactivity

FIG. 6 A series of dppp-protected [core+ exo]-type gold clusters. (A) [Au6(dppp)4]2+, (B) [Au7(dppp)4]3+, and (C) [Au8(dppp)4Cl2]2+. Schematic illustration of a [core +exo]-type gold cluster is given at the bottom.

Such [core+ exo]-type structures were also found in a series of smaller gold clusters protected by the C3-bridged diphosphine, dppp (Ph2P–(CH)3–PPh2). Fig. 6 shows crystal structures of [Au6(dppp)4]2+, [Au7(dppp)4]3+ and [Au8(dppp)4Cl2]2+ [14–16]. The higher-nuclearity Au7 and Au8 clusters were obtained from isoelectronic [Au6(dppp)4]2+ (n∗ ¼ 4) by addition of gold(I) ions, and all three clusters consist of tetrahedron-based polyhedral cores accommodating one or two exo atoms (Fig. 6, bottom), making the shapes of their gold moieties prolate. Interestingly, the AudAu bond lengths perpendicular to ˚ . In addition, the long axis of the clusters (Fig. 6, aqua dotted lines) are shorter than the others by around or more than 0.1 A ˚ the core-to-exo AudAu bond lengths were increased by around 0.1 A with increasing nuclearity. Such obvious contractions and elongations in prolate-shaped gold moieties in the four-electron system indicate the enhancement of prolate nature, supporting the validity of the structural jellium model. For these [core + exo]-type gold clusters, some geometry calculations were performed. The mechanism of the Au(PPh3) Cl-induced size-conversion from [Au6(dppp)4]2+ to [Au8(dppp)4Cl2]2+ was theoretically investigated, and a feasible pathway including Au–P dissociation and ligands’ rearrangements was proposed (Fig. 7A) [17]. In addition, experimentally obtained THz Raman spectra of phosphine-protected Au8 clusters including [Au8(dppp)4Cl2]2+ were reproduced theoretically by using the ligand-simplified Au8 cluster model (Fig. 7B), and a combined experimental and theoretical study on THz Raman spectroscopy was proved to be a powerful tool for the detection of weak gold core–ligand interactions of diacetylenic Au8 clusters (Aup interactions; for details, see Section 3.4) [18].

3.3 Electronic studies Unlike colloidal metal clusters, subnanometer-sized gold clusters exhibit structured absorption patterns, which essentially reflect molecule-like discrete electronic states of individual clusters. Phosphine-protected gold clusters with conventional polyhedral-only gold moieties are known to show UV–vis absorption bands overlapped with monotonously broadened tailing to the near-IR [19]. Accordingly, such patterns (so called tail-and-humps patterns) have been widely used as fingerprints for the identification of gold clusters. However, the nature of the patterns has not been so much focused, partly because most of the colors of conventional gold clusters are brown. In sharp contrast, the abovementioned [core + exo]-type Aun clusters (n ¼ 6, 7, 8, 11) are exceptional in exhibiting various colors and unusual spectral patterns. For example, the [core +exo]-type Au11 cluster shows green and an intense visible absorption band (Fig. 8A, bottom) [13], whereas the absorption spectrum of a toroidal [Au9(PPh3)8]3+, the gold moiety of, which can be taken as a substructure of the Au11 moiety, does not exhibit such an isolated band. This strongly suggests the profound effect of the exo atoms on the unique band. Further insights into the correlation between the absorption property and geometric structure were obtained from the theoretical calculations of the nonphenyl models of the two Au11 clusters ([core + exo]-type and polyhedral-only Au11 clusters) [13]. As compared with the frontier orbitals (HOMO and LUMO) of the polyhedral-only Au11 cluster (Fig. 8B, middle), those of the [core + exo]-type Au11 cluster were energetically rather isolated from their adjacent orbitals (HOMO  1 and LUMO + 1, respectively) (Fig. 8A, middle). This can be partly explained by the breaking of the degeneracy of the P shell in the structural jellium model (Fig. 4). Fig. 8 (bottom) depicts the theoretical absorption spectra of the two models, where the simulated spectra (solid lines) agree well with the experimental spectra of the corresponding Au11

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FIG. 7 (A) Size-growth pathway for the conversion from [Au6(dppp)4]2+ to [Au8(dppp)4Cl2]2+. (B) Experimental (left top) and theoretical (left bottom) THz Raman spectra of Au8 clusters, and representative vibrational modes of the ligand-simplified Au8 cluster model (right). (A) Reproduced with permission Lv Y, Zhao R, Weng S, Yu H. Chem Eur J 2020;26:12382–12387. Copyright 2020, Wiley-VCH. (B) Adapted with permission Kato M, Shichibu Y, Ogura K, Iwasaki M, Sugiuchi M, Konishi K, Yagi I. J Phys Chem Lett 2020;11:7996–8001. Copyright 2020, American Chemical Society.

clusters (dotted lines), especially their shapes. Marked differences were found in the oscillator strengths of the HOMO–LUMO transitions (Fig. 8, bottom, asterisks). The value of the oscillator strength of the [core+ exo]-type Au11 cluster model (0.5917) was about 78 times higher than that of the polyhedral-only Au11 cluster model (0.0076), leading to the generation of the unusually intense low-energy absorption at 663 nm (e ¼ 8.9  104 L cm1 mol1) (Fig. 8A, bottom, dotted line). In addition, for the [core+ exo]-type Au11 cluster, the LUMO distribution was found near the exo gold atoms, whereas the HOMO distribution was located on the shared edges of the toroidal Au9 core and the triangular Au3 units, indicating that the intense visible absorption band of the cluster is mainly due to the core-to-exo intermetal transition. Therefore, the theoretical study proved the critical role of the exo gold atoms in the emergence of the unique absorption band. Such a unique absorption spectral profile (i.e., an isolated visible absorption band) was also observed for a series of the [core + exo]-type Aun clusters (n ¼ 6, 7, 8) (Fig. 9, bottom, dotted lines) [14–16], and theoretical calculations of their corresponding nonphenyl models were performed to investigate the relationship between the absorption band and nuclearity of the clusters [15,20]. The energy level diagrams shown in Fig. 9 (middle) are quite similar to each other, with the HOMO and LUMO largely separated by 0.4–1.2 eV from the HOMO  1 and LUMO + 1, respectively. Because the HOMO and LUMO are mainly composed of Au 6sp atomic orbitals, the lowest-energy excitations of the models (Fig. 9, bottom, asterisks), assignable to the HOMO–LUMO transitions, mostly occur within the 6sp intraband. This was supported by the frontier orbital distributions of the models (Fig. 10), indicating the anisotropic electronic transitions in the core-to-exo direction within the gold frameworks, as was already described for the [core+ exo]-type Au11 cluster. Though the number of the

416 Atomic clusters with unusual structure, bonding and reactivity

FIG. 8 Skeletal structures (top), energy level diagrams (middle), and theoretical absorption spectra (bottom, solid lines) of (A) [Au11(H2P(CH2)2PH2)6]3+ and (B) [Au11(H2P (CH2)3PH2)5]3+. The asterisks indicate the HOMO–LUMO transitions. Experimental absorption profiles of the corresponding Au11 clusters (bottom, dotted lines) are shown for comparison. Reproduced with permission Shichibu Y, Kamei Y, Konishi K. Chem Commun 2012;48:7559–7561. Copyright 2012, Royal Society of Chemistry.

exo atoms differs (i.e., one for the Au7 cluster and two for the Au6 and Au8 clusters), the absorption profile is essentially the same among the three clusters, meaning the presence of only one exo gold atom on a tetrahedron-based core is sufficient to generate unique absorption bands. It should be noted that the experimental and theoretical absorption spectra both showed the blue shift of the absorption bands (or the enhancement of the HOMO–LUMO transition energies) with increasing nuclearity (Fig. 9, bottom). This trend for a series of the [core +exo]-type gold clusters is opposite to that for conventional metal nanoparticles, the lowest-energy optical absorptions of which typically exhibit blue shifts with decreasing object size (i.e., nuclearity) due to the quantum size effect. This contradiction can be resolved by considering the jellium model. The more the number of positive charge (i.e., the number of gold atoms (n) in this case) in the cluster jellium increases, the more efficiently stabilized the valence electrons of the clusters (n∗ ¼ 4) are, becoming a wider HOMO–LUMO gap energy.

3.4 Effects of ligands on geometric and electronic structures Recently, organic ligands on gold moieties have often been found to affect the geometric and electronic structures of ligandprotected gold clusters, leading to generate unique properties. For instance, the two [core +exo]-type Au6 clusters bearing bis(diphenylphosphino) ligands with C3- and C4-briges (i.e., dppp and dppb (Ph2P–(CH)4–PPh2)) showed totally different photoluminescence behaviors (Fig. 11A), which was investigated by the theoretical calculations of the ligand-simplified Au6 cluster models [21]. From the geometry optimization calculations of the models in the ground (S0) and excited (S1 and T1) states, the contraction of the Au4 tetrahedra along the longitudinal direction, triggered by the HOMO-LUMO

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FIG. 9 Skeletal structures (top), energy level diagrams (middle), and theoretical absorption spectra (bottom, solid lines) of (A) [Au6(H2P(CH2)3PH2)4]2+, (B) [Au7(H2P(CH2)3PH2)4]3+, and (C) [Au8(H2P(CH2)3PH2)4Cl2]2+. The asterisks indicate the HOMO–LUMO transitions. Experimental absorption profiles of the corresponding Au6, Au7, and Au8 clusters (bottom, dotted lines) are shown for comparison. (A, C) Adapted with permission Shichibu Y, Konishi K. Inorg Chem 2013;52:6570–6575. Copyright 2013, American Chemical Society. (B) Adapted with permission Shichibu Y, Zhang M, Kamei Y, Konishi K. J Am Chem Soc 2014;136:12892–12895. Copyright 2014, American Chemical Society.

FIG. 10 (top) LUMOs and (bottom) HOMOs of (A) [Au6(H2P (CH2)3PH2)4]2+, (B) [Au7(H2P (CH2)3PH2)4]3+, and (C) [Au8(H2P (CH2)3PH2)4Cl2]2+. Reproduced with permission Konishi K, Iwasaki M, Shichibu Y. Acc Chem Res 2018;51:3125–3133. Copyright 2018, American Chemical Society.

418 Atomic clusters with unusual structure, bonding and reactivity

FIG. 11 (A) Photoluminescence spectra (lex ¼ 586 nm) of (i) dppp- and (ii) dppb-protected Au6 clusters at 20°C (black lines) and 60°C (red lines, gray lines in print version). (B) Geometric relaxations of (i) [Au6(H2P(CH2)3PH2)4]2+ and (ii) [Au6(H2P(CH2)4PH2)4]2+ in the S1-excited states. The black and blue arrows in part b represent directions of motion of each Au atom and conformational changes of the long P–C4–P bridges, respectively. Reproduced with permission Shichibu Y, Zhang M, Iwasa T, Ono Y, Taketsugu T, Omagari S, Nakanishi T, Hasegawa Y, Konishi K. J Phys Chem C 2019;123:6934–6939. Copyright 2019, American Chemical Society.

transitions, was observed in their excitation states, and the degree of the geometric transformations was larger in the S1 state than in the T1 state. Furthermore, in the S1 state, the detachment of an exo Au atom of [Au6(H2P(CH2)3PH2)4]2+ (Fig. 11B (i)) was observed, whereas the original [core + exo] Au6 structures were preserved for [Au6(H2P(CH2)4PH2)4]2+ (Fig. 11B (ii)). In the former case, the shorter P–C3–P chains, which bridge the central Au4 core and exo Au atoms, were conformationally too rigid to follow the contraction of the Au4 core, resulting in the exo detachment and a nonradiative decay to the ground state through an S1/S0 conical intersection. On the other hand, in the latter case, the longer P–C4–P chains on the Au6 moiety were flexible enough to accommodate such a core contraction through the conformational changes. These theoretical results on the excitation-state structural dynamics of the models can reasonably explain the experimentally obtained photoluminescence behaviors of the two Au6 clusters including the inactive/active fluorescence-type emissions in visible region (Fig. 11A), demonstrating that the structural adaptability of surrounding organic ligands notably affects the excitation dynamics of ligand-protected gold clusters. Electronic “gold moiety–ligand” interactions, which induce obvious perturbation effects on various clusters’ properties, are one of the hottest topics in recent research on ligand-protected metal clusters. Such interactions are typically emerged in various experimental data (e.g., UV–vis absorption, IR absorption, atomic distances, NMR, etc.). To support the presence of such interactions, theoretical calculations can be extremely useful tools. For example, an electron coupling between the C^C p-system and the gold moiety of the monoyne-type Au13 cluster ([Au13(dppe)5(C^C–Ph)2]3+), which exhibits different visible absorption from the Cl-type Au13 cluster ([Au13(dppe)5Cl2]3+), was theoretically investigated (Fig. 12A) [22]. From the electronic structure calculations including MO analysis, significant contributions from the p-conjugated units were observed in the HOMO and certain high-energy occupied orbitals, which can reasonably explain the coupling interaction. Another example of C^C concerned interaction was discovered from the experimental and theoretical investigations of diyne-type Au8 clusters ([Au8(dppp)4(C^C–C^C–R)2]2+), which show clear red shifts of the lowest-energy visible absorption bands with respect to the corresponding monoyne-type analogues [23]. Molecular and natural-bond orbital analyses revealed the Au-p attractive interactions between the s-bonded C^C units and the neighboring bitetrahedral Au6 core (Fig. 12B). A more unique but complicated electronic interaction was found in a [core + exo]-type [Au6(mPhDP)4]2+, where mPhDP (1,3-bis(diphenylphosphino)benzene) is a phenylene-bridged bis(diphenylphosphino)

Ligand-protected clusters Chapter

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FIG. 12 (i) Schematic structures and (ii, iii) orbital distributions of (A) [Au13(dppe)5 (C^C–Ph)2]3+, (B) [Au8(dppp)4(C^C– C^C–Ph)2]2+, and (C) [Au6(mPhDP)4]2+. In (i), various types of interactions are indicated in blue (gray in print version). In C(ii) and C(iii), the PPh2 groups of mPhDP are replaced by P(CH3)2 groups. (A) Adapted with permission Sugiuchi M, Shichibu Y, Nakanishi T, Hasegawa Y, Konishi K. Chem Commun 2015;51:13519–13522. Copyright 2015, Royal Society of Chemistry. (B) Adapted with permission Iwasaki M, Shichibu Y, Konishi K. Angew Chem Int Ed 2019;58:2443–2447. Copyright 2019, Wiley-VCH. (C) Reproduced with permission Vicha J, Foroutan-Nejad C, Straka M. Nat Commun 2019;10:1643. Copyright 2019, Nature Publishing Group.

ligand [24]. An unprecedented AuH–C interaction, a kind of “hydrogen bond,” was theoretically proved from an orbital analysis (Fig. 12C) [25]. From a statistical geometry analysis of the crystallographically defined Au13 clusters bearing diphosphine ligands, surface diphosphine ligands were found to induce small torsional twists into the icosahedral Au13 cores, generating intrinsic chirality in the Au13 moieties. Such torsional effects on chiroptical activity were systematically investigated using theoretical calculations of a ligand-simplified Au13 cluster model, i.e., the [Au13(PH3)10Cl2]3+ cluster [26]. By varying the torsion angle y between the two equatorial Au5 rings of the Au13 cores (Fig. 13A), the y-dependent circular dichroism (CD) spectra of the model were computed, where the greatest CD response was obtained at y ¼ 18° (Fig. 13B). Because both a charge translation and charge rotation in an electron transition (i.e., origins of electric and magnetic transition dipole moments (m and m)) yield a helical charge movement (i.e., the origin of a chiroptical response) (Fig. 13C), the chiroptical origins of the model were then investigated profoundly using transition-moment and transition-density analyses. For the dominant excitation components responsible for the positive first and negative second Cotton effects at y ¼ 18° in Fig. 13B (i.e., excitations Ix and Iy, and excitation IIz), the y-dependences of the rotatory strength and the dipole moments were analyzed (Fig. 13D and E), revealing the origin of the abnormally strong charge rotation (or m in Fig. 13E(ii)) arising from the equatorial Au5 rings. Through this systematic theoretical approach based on the simple Au13 cluster model, the unique effect of the circular arrangement of the metal atoms on the chiroptical activities of ligand-protected metal clusters was highlighted.

420 Atomic clusters with unusual structure, bonding and reactivity

FIG. 13 (A) Geometric transformation of the Au13P10Cl2 framework upon variation in y from 36° to 0° via chiral P-helical conformations. (B) Calculated CD spectra of [Au13(PH3)10Cl2]3+ at y ¼ 18°. (C) Components of a helical charge movement. (i) Normalized magnitudes of the rotatory strength (black), electric (red, light gray in print version) and magnetic (blue, gray in print version) transition dipole moments at 36° y 0°, and (ii) the electric (red) and magnetic (blue) transition dipole moments at y ¼ 18° for (D) excitation Iy and (E) excitation IIz of [Au13(PH3)10Cl2]3+. In parts (D)(ii) and (E)(ii), the magnitudes of the moments relative to those for excitation Iy are given near the respective arrow tips. Adapted with permission Shichibu Y, Ogawa Y, Sugiuchi M, Konishi K. Nanoscale Adv 2021;3:1005–1011. Copyright 2021, Royal Society of Chemistry.

4.

Conclusion

In this chapter, we provided an overview of recent theoretical studies of the ligand-protected ultrasmall gold clusters. These studies demonstrate that not only information on the geometric and electronic structures but also insights into the origins of their unique properties (gold moiety–ligand interaction, chirality, photoluminescence, absorption), which are not accessible with experimental-only approaches, can be obtained from theoretical approaches. Combinational theoretical and experimental approaches are valuable to gain physicochemical insights into the origins of novel properties.

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Index Note: Page numbers followed by f indicate figures t indicate tables and s indicate schemes.

A Absorption patterns, gold clusters, 414–415 Absorption spectroscopy, 376–378 Absorption spectrum analysis, 130 Adamantane (Adm), 215, 215f, 219–220 Adaptive density natural partitioning (AdNDP) analysis, 29, 321, 323, 326–328, 333–334, 336–337 aromaticity/antiaromaticity concept, 1 atomic clusters, 1 boron hydrides B3H¯y complexes, 3 BnHn series, isostructural relationships in, 3 deltahedral BnHn2–systems, 5–6 electronic transmutation, 4–5 boron wheels B36 cluster, dynamic behavior of, 13 sandwich structures, design of, 9–12 small boron clusters, dynamic behavior in, 7 Wankel motor family, 7–9 covalent metal-carbon interaction, 1–2 electron pair, concept of, 1 implementation, 2–3 localized and delocalized bonds, 1 natural bond orbital (NBO) analysis, 1–2 wave functions, interpretation of, 1 ADMP-CNN-PSO approach B3LYP/6-311+G (d,p) level, 190 101,101 cluster configurations, 190 20,221 configurations, 190 convolutional neural network (CNN) architecture of, 190, 190f C5 clusters, 201–202 experimental setup, 192 Gaussian 09 program, 190 overall flowchart of, 190, 191f postprocessing, 191 PSO-executed best configuration, 190 Aggregation-induced emission (AIE), 378 Alkali and alkaline earth metals, aromaticity of, 91–92 Alkaline-earth metals barium carbonyl ions Ba(CO)q, 157 Be-Ba, 157 Ca, 157 computational techniques, 158–159 dinitrogen, 157 18-electron rule, 157 energy decomposition analysis, 166–167

triplet M(CO)8 complexes, EDA–NOCV results for, 162–165, 164t triplet M(N2)8 complexes, EDA–NOCV results for, 162–165, 164t triplet M(PH3)8 complexes, EDA–NOCV results for, 162–165, 163t tripletOh symmetric Ca(N2)8 complex, deformation densities of, 165–166, 166f tripletO symmetric Ca(PF3)8 complex, deformation densities of, 165–166, 165f Gaussian 16 suit of program, 158 geometry optimizations, 158 harmonic frequency calculations, 158 “honorary transition metal”, 157 M(Bz)3 complex, 157, 167–169 M(L)8 complexes density functional theory (DFT) calculations, 157–158 structure and stability of, 159–161 molecular orbitals (MOs) and correlation diagram, 157 tripletOh symmetric M(N2)8 complex, 161–162, 161f triplet symmetric Ca(PF3)8 complex, 161–162, 162f octacarbonyl complex, Ba(CO)8, 157 octa-coordinated complexes, 157 pbackdonation, 157 PH3 and PF3, 157–158 scalar-relativistic effective core potentials, 158 zeroth-order regular approximation (ZORA), 158 All-transition-metal aromatic/antiaromatic systems, 93 Aluminum clusters, 186 aromaticity of, 91–92 Aluminum-hydrogen atomic clusters, 5 Ant colony optimization (ACO), 185 Antiaromatic systems, 90 Antimalarial drugs, 80 Aromaticity, 94 history and descriptors all-metal clusters, study of, 90 anti-aromaticity, concept of, 90 Baird’s rule, 89–90 benzene, stability of, 89 energetic and electronic criteria, 90 fluctuation index, 90 generalized population analysis, 90 harmonic oscillator model, 90

magnetic field induced current density, 90 multicenter bond index (MCI), 90 NICS-rate, critical value of, 90 nuclear magnetic resonance (NMR) chemical shift, 90 nucleus-independent chemical shift (NICS) values, 90 probes, 90 quantum mechanical treatment, 89 reactivity parameters, 90 relative aromaticity indices, 90–91 (4n + 2) rule, 89–90 s-aromaticity, concept of, 89–90 of metal clusters alkali and alkaline earth metals, 91–92 aromatic square clusters, 91 metalloaromaticity, 91 organometallic compound, 91 planar pentagonal aromatic clusters, 91 transition metal clusters, 91, 93–94 para-delocalization index (PDI), 89 Aromatic stabilization energy (ASE), 90 Artificial bee colony (ABC), 185 Artificial neural networks (ANNs), 186 Assembly line balancing problem (ALBP), 185–186 Atoms in molecules (AIM), 65–66, 125, 130–132, 280 Aufbau principle, 20–21, 31

B Bader’s theory, 2, 278 Baird’s rule, 89–90 Basin-hopping (BH) algorithm, 43 Basis set superposition error (BSSE), 214, 217t B36 cluster, 13 B182– cluster, AdNDP analysis for, 7, 9f Benzamidin-4-ylidene, 74, 74s Benzene, group III-V hexagonal pnictide clusters electronic properties, 142–143 polymeric growth of electronic properties, 143–145 structural properties, 143 structural properties, 140–142 Berenil, 63 B3H¯y complexes, 3 Biguanides, 79–80 Bimetallic clusters accurate electronic-structure methods, 41

423

424

Index

Bimetallic clusters (Continued) approximate total-energy methods, 41 computational tools energy calculator, 42–43 global structure optimization methods, 43–44 homotops, 41 metastable structures, 41 “nanomaterials,”, 41 properties of, 41 stoichiometry, 41 structural properties of aluminum-hydrogen-oxygen, 44–45 cadmium-selenium, 44 copper-silver, 52–55 nickel-copper, 47–49 nickel-silver, 49–52 potassium/rubidium-cesium, 47 silver-rhodium 257, 55–57 titanium-carbon, 44 zinc-selenium, 46–47 unbiased structure optimizations, 41, 58 Bimetallic gold-silver clusters, 186, 200, 200–201f Binding energy per atom (BE), 107–108, 108–109f Binuclear sandwich complexes, of alkaline earth metals, 287 Li3@Cg (Cg¼B40 and C60) and electride nature, 288 Bis(amino) carbodiphosphoranes, 65, 65s Bis(diphenylphosphino) ligands, 413 Bis(pyridine)-carbone-Pd complex, 67, 68s BLi7@BNF, 125–126, 125f, 126t Boltzmann constant, 238–239 Bond critical points (BCPs), 2, 397t guest@GR/BGR/BNGR moieties, 132, 133t OLi4/NLi5/CLi6/BLi7/Al12Be@BNF, 130, 132t Bond dissociation energies (BDEs), 70, 167–168, 168t Boranes, 61–62 Boric acid (BA), H-storage boric acid cluster (BA20) incorporated fullerene based material, 229–232 boric acid hexamer (BA)6, 225 boric acid pentamer (BA)5, 225 nanoclusters, 222–224 Born-Oppenheimer molecular dynamics (BOMD) simulations, 7, 11, 360 Boron clusters, 186 B5 cluster, 192–200, 192f B6 cluster, 193–194, 193f potential energy surface (PES), 192 Boron—gallium bond, 70–71 Boron hydrides, chemical bonding B3H¯y complexes, 3 BnHn series, isostructural relationships in, 3 deltahedral BnHn2– systems, 5–6 electronic transmutation, 4–5 Boron-ligand cooperation (BLC), 401–402 Boron nitride flake (BNF) finite band gap, 124 monolayer boron nitride flake, 124

OLi4/NLi5/CLi6/BLi7/Al12Be@BNF atoms-in-a-molecule (AIM) results for, 130, 132t electron decomposition analysis (EDA) results of, 132–133, 133t Gibbs free energy change and enthalpy change, 126, 126t IR domain, 130 minimum energy structures of, 125–126, 125f TDDFT results for, 127–130, 129t pristine BNF, 124, 130 Boron wheels, chemical bonding in B36 cluster, dynamic behavior of, 13 sandwich structures, design of, 9–12 small boron clusters, dynamic behavior in, 7 Wankel motor family B182– cluster, AdNDP analysis for, 7, 9f CB18 cluster, AdNDP analysis for, 7, 10f IrB12¯ system, AdNDP analysis for, 4, 10f Borophene, 29–30 Borylene-gold complex, 70–71 Borylenes, monovalent B(I) systems bonding analysis, 70 chemical reactivity and applications, 70–71 crystal structures, 68–69, 68f single-center L!B coordination complexes, types of, 68, 68f synthesis, 69–70 Boustani structure, 28–29 Breslow intermediate (BI), 61–62 Brust-Schiffrin method, 373

C Ca(PF3)8 complex, 159–160 Canonical molecular orbitals (CMOs), 347–349 Carbene-alkylidene adducts, 61–62 Carbene-germylene adduct, 61–62 Carbene-imine adducts, 61–62 Carbocyclic carbenes (CCCs), 61, 68–69, 69s Carbodicarbenes (CDC), 63–64, 64f Carbodiphosphoranes, 63–64, 64f Carbon allotropes, 124 Carbon-based nanostructures, 173 Carbon clusters, 186, 194–198 Carbone-Rh complex, 68, 68s Carbones, divalent C(0) systems bonding analysis, 65–66 catalytic applications, 67–68 chemical reactivity, 66–67 experimentally generated carbones, 2D representation of, 63–64, 64f synthesis, 64–65, 64s bis(amino) carbodiphosphoranes, 65, 65s tetraphenylcarbodicyclopropenylidene and its Lewis acid complexes, 65, 65s Carbon nanotubes (CNTs), 173, 213 CB18 cluster, AdNDP analysis for, 7, 10f C—C cross coupling reactions, 67, 68s CCCs. See Carbocyclic carbenes (CCCs) CDFT. See Conceptual density functional theory (CDFT) CDP-CuCl complex, 67, 67s

Chemical bonding adaptive density natural partitioning (AdNDP) analysis, 1–2 aromaticity/antiaromaticity concept, 1 atomic clusters, 1 boron hydrides (see Boron hydrides, chemical bonding in) boron wheels (see Boron wheels, chemical bonding in) covalent metal-carbon interaction, 1–2 electron pair, concept of, 1 implementation, 2–3 localized and delocalized bonds, 1 natural bond orbital (NBO) analysis, 1–2 wave functions, interpretation of, 1 bond critical points (BCPs), 2 energy decomposition analysis (EDA), 1–2 Lewis chemical bonding model, 1 objective, 1 suicidal clavulanic acid, biosynthesis of, 2 walk rearrangement mechanism, theoretical description of, 2 Clemenger-Nilsson model (CNM), 110 Cluster bonding, intercluster bonds extended networks, 328–329 of gold clusters, 322 of Zintl clusters, 323–328 Coalescence Kick (CK) algorithm, 3 Cocktail decoding method, 185–186 Complete active-space self-consistent field (CASSCF), 345 Conceptual density functional theory (CDFT), 90, 237–239, 299 chemical hardness, 88 Lagrangian multiplier, 88 reactivity parameters, 88–89 relative aromaticity indices, 90–91 Condensed Fukui function, 89 Configuration interaction with singles and doubles (CISD), 240–241 Convolutional neural network (CNN), 186 architecture of, 190–191 C5 clusters, 201–202 Copper (Cu) clusters, 99 Counterpoise (CP) procedure, 214 Covalent organic frameworks (COFs), 297–298 CrB24 sandwich complex, AdNDP analysis for, 11, 11f Crossover process, 43–44 Cubane (cub), 215–218, 215f Cuboctahedron interpenetration growth patterns, 383f Cuckoo search (CS), 185 Cyclic alkyl amino carbenes (CAACs), 61–62, 68–69 Cyclic hydrocopper(I) compounds, 94 Cyclic voltammetry (CV), 70 Cyclohexane (Cyc), 215, 215f, 218–219

D Deep learning, 186 Delocalization index, 89 Density functional approximations (DFAs), 280

Index

Density functional theory (DFT), 70–71, 140, 299 B3LYP/6-31G(d) scheme, 174 conceptual density functional theory (CDFT) chemical hardness, 88 Lagrangian multiplier, 88 reactivity parameters, 88–89 relative aromaticity indices, 90–91 density functional theory-firefly algorithm (DFT-FA), 186 overall flowchart of, 189, 189f planar Al42- structures, energy vs. aromaticity profiles of, 204–206 planarity, effect of, 204 vs. DFT-PSO, 202–204, 203–204t density functional theory-PSO approach, 186, 188, 188f energyvariational principle, 87–88 H3+ and Li3+ clusters aromaticity, 237 CCSD/aug-cc-pVTZ, 240–241 conceptual DFT indices, 242, 242t equilateral triangular geometry, 237, 240t GIMIC distributions, 242, 242f infrared (IR) spectrum, 237 methodology, 238–240 nucleus-independent chemical shift (NICS), 237 quantum mechanics, 237 Hohenberg and Kohn (HK) theorem of, 87 H-storage (see Hydrogen storage) M(L)8 complexes, 157–158 silver and gold clusters, 100, 101t Density functional tight-binding (DFTB) method, 42, 57 Density matrix, 2–3 Density of state (DOS), 333–334, 337 Deterministic algorithm-based methods, 185 1,3-Dienes, hydroarylation of, 68, 68s Dihydrogen endofullerenes (H2@C60), 173–174 Diphosphine-ligated gold clusters electronic studies, 414–416 geometric and electronic structures, ligands effects on, 416–419 geometric studies, 413–414 Au8 clusters, THz Raman spectra of, 414, 415f [Au11 (dppe)6]3+, skeletal structures of, 413, 413f [Au25(SC2H4Ph)18¯], skeletal structures of, 413, 413f dppp-protected [core+exo]-type gold clusters, series of, 414, 414f ligand-simplified Au8 cluster model, representative vibrational modes of, 414, 415f jellium models and core shapes, 411–413, 413f Discrete dipole approximation (DDA), 318 Divalent N(I) compound (nitreones). See Nitreones, divalent N(I) systems Diyne-type Au8 clusters, 418–419 Double (s-n) aromaticity, 94

E Effective coordination numbers (ECNs), 55–56 Effective core potentials (ECPs), 100, 191 18 valence electrons rule, 361–363 Electric field (EF), 389–390 Electrochemical CO2 reduction (ECR), 253 Electrochemistry, 381–382 Electron affinity (EA), 88, 124, 142–143, 333–334 Electron decomposition analysis (EDA), 132 guest@GR/BGR/BNGR, results for, 133–134, 134t OLi4/NLi5/CLi6/BLi7/Al12Be@BNF, results for, 132–133, 133t Electron delocalization, in clusters, 92 in flat clusters, 20–30 heteroatomic cluster science, 19 homoatomic clusters, 19 (pseudo) spherical clusters, 30–35 Electron density, 87–88 Electronic shell model, 123 Electronic structure analysis, 80 Electronic transmutation model, 4–5 Electron localization function (ELF), 75, 130, 239–240 18-Electron rule, 157, 161–162 Electron transfer (ET) model, 389, 392 Electrophilicity index, 88–89 Electrostatic interaction energy (DEelstat), 158, 162–165 Embedded-atom model (EAM), 42–43 Embedding energy (EE), 108 “Encounter complex” (EC), 389 Endofullerenes applications, 173–174 computational techniques, 174–175 dihydrogenendofullerenes (H2@C60), 173–174 fullerene cages, 173–174 La@C60, endohedral metallofullerene complex, 173–174 low ionization energy (IE) species doped endofullerenes, 174, 180–181 alkali atoms, 174–175 Li@C60—PF6endofullerene complex, 177–179 Li@C60vs. Lr@C60endofullerene, 175–177 Li@C60vs. SA@C60endofullerene, 175 SA@C60—BF4endofullerene complexes, 179–180 molecular surgery, 173–174 structures of, 173, 174f waterendofullerenes (H2O@C60), 173–174 Endohedral metallofullerene complex, 173–174 Energy calculator, 42–43 density functional tight-binding (DFTB) method, 42, 57 embedded-atom model (EAM), 42–43 Energy decomposition analysis, 166–167 energy decomposition analysis in combination with natural orbital for chemical valence (EDA-NOCV) analysis, 158–161

425

BP86-D3(BJ)/TZ2P level for M(Bz)3 (D3, 1 A1) complexes, results for, 169, 169t triplet M(CO)8 complexes, results for, 162–165, 164t triplet M(N2)8 complexes, results for, 162–165, 164t triplet M(PH3)8 complexes, results for, 162–165, 163t tripletOh symmetric Ca(N2)8 complex, deformation densities of, 165–166, 166f tripletO symmetric Ca(PF3)8 complex, deformation densities of, 165–166, 165f Energy decomposition analysis (EDA), 65–66, 70, 75, 402 chemical bonding, 1–2 Energy decomposition analysis of the natural orbitals for chemical valence (EDA-NOCV), 32 Exchange-correlation density, 89

F Faradaic efficiency, 253 Finite-difference approximation, 88 Finite field method, 314, 316 Firefly algorithm (FA), 185 convergence criteria, 189 density functional theory-FA, 186 overall flowchart of, 189, 189f planar Al42- structures, energy vs. aromaticity profiles of, 204–206 planarity, effect of, 204 vs. DFT-PSO, 202–204, 203–204t experimental setup, 192 Gaussian 09 (G09) software, 188–189 hyper parameters, 189 patience value, 188 performance improvement of, 189 randomization parameter, 189 FLi2@C60, 175 Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS), 258 Franck-Condon approximation, 125 Frustrated Lewis pair (FLP), 68 alkaline earth metals, 387 alkylaminocarbene, H2 activation, 387 aromaticity-enhanced small molecule activation, 397 boron-ligand cooperation (BLC), 401–402 catalytic hydrogenation, 398–401 electron density, 387 frontier molecular orbital model, 388f hydrogenation reaction, calcium complex, 388f Lewis acids (LAs), chemistry of, 387–388 Lewis bases (LAs), chemistry of, 387–388 mechanism, H2 activation, 389–392 polymerization reaction, 403–406 reactivity, identification of, 389 small molecules activation, 393–397 thermodynamics, H2 activation, 392–393 transition metal (TM) atoms, 387 Fukui function, 88–89, 238–239, 300 Fullerenes alkali metals, doping with, 173–174

426

Index

Fullerenes (Continued) boric acid cluster (BA20) incorporated material, 229–232 endofullerenes (see Endofullerenes) lanthanides and actinides doped fullerene, 173–174 structures of, 173, 174f transition element, doping with, 173–174

G Gauge-including magnetically induced current (GIMIC), 93, 240 Gaussian 09 (G09) program, 125, 174, 187–191 GaussSum package, 333–334 Generalized gradient approximation (GGA), 140 Genetic algorithm (GA), 43–44, 185 Germilones, 63 Giambiagi electron delocalization multicenter index, 24–27 Global minima (GM) energy structures, 186, 206 ADMP-CNN-PSO approach B3LYP/6-311+G (d,p) level, 190 101,101 cluster configurations, 190 20,221 configurations, 190 convolutional neural network (CNN), 190, 190f, 200–202 experimental setup, 192 Gaussian 09 program, 190 overall flowchart of, 190, 191f postprocessing, 191 PSO-executed best configuration, 190 cluster, in multidimensional search space, 187, 187f computational details, 191 firefly algorithm (FA), 185 convergence criteria, 189 density functional theory-FA, 186, 189, 189f, 202–206 experimental setup, 192 G09 software, 188–189 hyper parameters, 189 patience value, 188 performance improvement of, 189 randomization parameter, 189 particle swarm optimization (PSO) algorithm Aun and AunAgm clusters, 200 boron clusters, 192–200 carbon clusters, 194–198 density functional theory-PSO approach, 186, 188, 188f equations, 187 experimental setup, 192 Gaussian 09 (G09) program, 187 global best (gbest) configuration, 187 global coefficient of acceleration, 188 individual coefficient of acceleration, 188 inertia coefficient, 188 local optimum solution, pbest configuration, 187 N42- and N64- clusters, 199–200 randomly generated cluster configuration, 187 single point energy (SPE) of cluster, 187 Global optimization (GO) techniques, 185–186

Global structure optimization methods basin-hopping (BH) algorithm, 43 genetic algorithms (GA), 43–44 Gold (Au) clusters, 322, 420 diphosphine-ligated gold clusters electronic studies, 414–416 geometric and electronic structures, ligands effects on, 416–419 geometric studies, 413–414 jellium models and core shapes, 411–413, 413f thiolate-protected gold clusters, 411 Au38 and Au25 clusters, theoretically predicted structures of, 411, 412f Au25(SR)18 cluster, CO2 electroreduction on, 411 Au79 core, chiral structures and HCM values of, 411, 412f Au102 (p-MBA)44, density of states for, 411, 412f bare Au79 core (58e) without Au-thiolate staple layer, density of states for, 411, 412f (see also Silver and gold clusters) Gold moiety–ligand interaction, 418–419 Gradient embedded genetic algorithm (GEGA) 17 program, 3 Graphdiyne (GDY), 124 Graphene nanoflakes (GR/BGR/BNGR) electron density descriptors at BCP, 132, 133t free energy change and reaction enthalpy change, 126, 128t Laplacian electron density, 126, 127f minimum energy structures, 126, 127f M3O+@ BGR/BNGR moieties electron decomposition analysis (EDA) results of, 133–134, 134t UV-vis domain, 130 M3O+ functionalization processes of, 126 and M3O+@GR/BGR/BNGR moieties, TDDFT results for, 130, 131–132t M3O+@ GR moieties, 130 Group III–V graphene-like materials, potential cluster units custom-built cluster assembled nanomaterials, 145–146 monolayer indium nitride, for thermoelectrics, 146–148 Group III–V hexagonal pnictide clusters benzene electronic properties, 142–143 structural properties, 140–142 computational methods, 140 2D hexagonal aluminum arsenide, 152 2D hexagonal aluminum nitride, 152 2D hexagonal aluminum phosphide, 152 2D monolayer phase of boron nitride, 151 graphene-like materials, potential cluster units custom-built cluster assembled nanomaterials, 145–146 monolayer indium nitride, for thermoelectrics, 146–148 mono- and multilayer thallium nitride, thermoelectrics, 148–151

two-dimensional monolayer hexagonal GaX (X¼N, P, and As), 152

H Haber-Bosch (HB) process, 248–249 H3+ and Li3+ clusters, density functional theory aromaticity, 237 CCSD/aug-cc-pVTZ, 240–241 conceptual DFT indices, 242, 242t equilateral triangular geometry, 237, 240t GIMIC distributions, 242, 242f infrared (IR) spectrum, 237 methodology, 238–240 nucleus-independent chemical shift (NICS), 237 quantum mechanics, 237 Hard and soft acids and bases (HSAB) principle, 88 Harmonic oscillator model of aromaticity (HOMA), 301 Hausdorff chirality measure (HCM), 411, 412f Heck coupling, 67 Heck-Mizoroki reaction, 67 Higher order harmonic generation (HHG), 123 Highest occupied molecular (HOMO-1), 26–27 Highest occupied molecular (HOMO-2), 21, 26–27 Highest occupied molecular orbital (HOMO), 26–27, 36, 90, 92, 92f, 161–162, 259–260, 348–349, 387, 414–416, 417f, 418–419 High-pressure electride (HPE), 276–277 Hohenberg and Kohn (HK) theorem, 87 HOMO–LUMO energy gap (HLG), 142–145, 142t, 147t Host-guest complex, metal clusters. See Metal clusters H€uckel 4n rule, 89–90, 322 of aromaticity, 21–22 Hybrid flow shop (HFS) problems, 185–186 Hybrid PSO (HPSO) algorithm, 185–186 Hydroalkoxylation reaction, 67, 67s Hydroamination reaction, 67, 67s Hydrogen evolution reaction (HER), 253 Hydrogen spillover effect, 213 Hydrogen storage absorption, 213 adsorption, 213 capacity, 213 carbon nanotubes (CNTs), 213 fossil fuel, 213 H-storage in molecules/materials (HSMs), 213–214 boric acid cluster (BA20) incorporated fullerene based material, 229–232 icosahedral (Ih) cages incorporated MOF-5, 228–229 small molecules incorporated MOP-9, 226–228 hydrogenspillover effect, 213 materials, 251–252 in molecular cages in boranes- and alanes-based cages, 220 closo-borane and alane cages, 220, 221s

Index

density functional theory (DFT) approach, 214 nH2@AIXY cages, energetics of, 221–222 nH2@BXY cages, energetics of, 221 in molecular clusters BA nanoclusters, 222–224 boric acid hexamer (BA)6, 225 boric acid pentamer (BA)5, 225 requirements for, 213 road map representation, 214, 215s in small molecules, 214 adamantane (Adm), 215, 215f, 219–220 cubane (cub), 215–218, 215f cyclohexane (Cyc), 215, 215f, 218–219 DFT-based M05-2X/6-31 + G** method, 215, 215f Hydrogen trapping potential atomic and molecular clusters B2Li and B2Li2 moieties, 302–303 C12N12 cage, 303–305 Mg and Ca clusters, 302 carbon nanotubes (CNTs), 298 computational methods, 301 dilithiatedethylenediamine (Li2EDA), 299 ionic clusters B2Li+ and B2Li2+ ions, 307 Li3+ and Na3+ ions, 306–307 M5Li7+ (M¼C, Si, Ge) clusters, 308 N4Li2 and N6Ca2 clusters, 305 Kubas-type interaction, 298–299 liquid organic hydrogen carriers (LOHCs), 299 metal hydrides, 298 theoretical methodologies and multiscale computational techniques, 299–301 Hyperpolarizability coefficient, 123

I Imidazoline salt, 73 Indium nitride (InN), 148 Indium phosphide (InP), 148 Intercluster bonds extended networks, 328–329 of gold clusters, 322 of Zintl clusters, 323–328 Interconversion process, 376–378 Interstitial quasiatoms (ISQs), 276–277 Intrinsic bond strength index (IBSI), 241 Intrinsic interaction energy (DEint), 158, 162–165 Ionization energy (IE), 333–334 Ionization potentials (IPs), 88, 123–124, 142–143, 318–319 Ion-trap mass spectrometry, 253 IrB12¯ system, AdNDP analysis for, 4, 10f Iron(0) complex, 70–71 Isoelectronic borylene complexes, 63 Isoelectronic L!E L complexes, 63, 63f

J Jahn-Teller distortion, 29, 357–359 Jellium model, 30–33, 36, 247 atoms and atomic clusters within, 317–318 diphosphine-ligated gold clusters, 411–413 Jellium shell closure model, 336–337

427

K

M

Kinetic energy density (KED), 238–239 Kohn-Sham (KS) equation, 30–34, 42, 87, 314–315, 317 Koopman’s theorem, 238, 300

Magic clusters, 27, 253 Main group elements, 61 bond representation, 80–82 L!E L complexes borylenes, monovalent B(I) systems, 68–71 carbones, divalent C(0) systems, 63–68 isoelectronic L!E L complexes, 63, 63f nitreones, divalent N(I) systems, 71–80 low-valent state of donor!acceptor complexes, schematic representation of, 61, 62f L!E complexes, mesomeric representations and examples of, 61–62, 62f L!E-E L systems, examples of, 62, 62f multicentereddonor!acceptor complexes, examples of, 63, 63f Maximum hardness principle (MHP), 88, 300–301 M@C60endofullerenes, 175–177, 176–177f M(Bz)3 complex, 157, 167–169 M(N2)8 complex, 160, 162–165, 164t M(CO)8 complexes, 160–165, 164t M(pH3)8 complexes, 160–165, 160t, 163t Metal-carbon interaction, 1–2 Metal clusters alkali metal atoms, 124 carbon allotropes, 124 computations, 124–125 definition, 123 electrodynamic theory, 123 electron correlation, 345 electronic shell model, 123 extrinsic effects, 123 host-guest moieties atoms-in-a-molecule (AIM) analysis, 125, 130–132 electron decomposition analysis (EDA), 132–134 geometrical structures and thermodynamic feasibility, 125–126 guest@OA complexes, TDDFT analysis of, 134–135 optical and electronic properties, 127–130 hyperlithiated bonding, 345 hyperpolarizability values, 124 intrinsic effects, 123 ligand-protected gold clusters see Ligandprotected clusters Li3NM@GDY, optical response properties of, 124 lithium clusters, 345 multiple bonded second-row main group element beryllium-beryllium triple bond, 349–350 boron-boron quadruple bond, 352 photo-electron spectroscopy, 345 planarhypercoordinate main group elements, 345–348 planarpentacoordinate nitrogen (ppN), 348–349 quantum confinement effects, 123 small beryllium, 345

L Lagrangian multiplier, 88 Langmuir-Hinshelwood processes, 259–260 Laplacian electron density, 126, 127f, 130 Lawrencium (Lr)@C60endofullerene, 175–177, 176t Lennard-Jones (LJ) potential, 56 Lewis acids (LAs), 61–62, 387–388 Lewis bases (LBs), 387–388 Lewis chemical bonding model, 1 Li2B2H6 system, 4 Ligand-protected clusters, 420 diphosphine-ligated gold clusters electronic studies, 414–416 geometric and electronic structures, ligands effects on, 416–419 geometric studies, 413–414 jellium models and core shapes, 411–413, 413f thiolate-protected gold clusters, 411 Au38 and Au25 clusters, theoretically predicted structures of, 411, 412f Au25(SR)18 cluster, CO2 electroreduction on, 411 Au79 core, chiral structures and HCM values of, 411, 412f Au102 (p-MBA)44, density of states for, 411, 412f bare Au79 core (58e) without Au-thiolate staple layer, density of states for, 411, 412f Li3NM@GDY systems, 124 Lithium boron hydrides, 4 Lithium (Li)@C60endofullerenes PF6endofullerene complex, 177–179 vs. Lr@C60endofullerene, 175–177 vs. SA@C60endofullerene, 175 Local density approximation (LDA), 30–31, 107 Local electron energy density, 130 Local philicity, 89 Local potential energy density, 130 Lone pair occupancy (LPO), 75 Lowest unoccupied molecular orbital (LUMO), 35–36, 90, 92, 92f, 259–260, 387, 414–416, 417f Low ionization energy (IE) species doped endofullerenes, 174, 180–181 alkali atoms, 174–175 Li@C60—PF6endofullerene complex, 177–179 Li@C60vs. Lr@C60endofullerene, 175–177 Li@C60vs. SA@C60endofullerene, 175 SA@C60—BF4endofullerene complexes, 179–180

428

Index

Metal clusters (Continued) superalkali-superhalogen compounds, 124 Metalligand cooperation (MLC), 401 Metalloaromaticity, 91 Metal organic frameworks (MOFs), 297–298 Metal-organic polyhedra (MOPs) calculated BE and BE/H2 molecules, 226–227, 228t crystal structure of, 226–227, 227f Cub-Mg2+ complex, 228 H2 molecules, adsorption sites, 228 H2@MOP-9 and linker modified MOP-9, optimized geometries of, 226–227, 228f Metformin hydrochloride, 80 Mg(B3H8)2 system, reversible dehydrogenation of, 3 Milstein’s pyridine-based pincer complex, 401–402 Minimum electrophilicity principle (MEP), 88, 300–301 Minimum polarizability principle (MPP), 88 Mixed metallic clusters, 186 M3O+@GR/BGR/BNGR systems, 125–126, 128t Molecular electrides computational methodology alkali metal-doped electrides, 281–283 binuclear sandwich complexes, of alkaline earth metals, 287 Mg2@C60 and its electride characteristics, 286–287 [Mg4(HDippL)2]2– complex and its electride, bonding in, 285–286 Mg2EP and small molecule activation, 283–285 electron, confinement of, 275 inorganicelectrides, development of, 276–277 Li@calix[4]pyrrole, 277 norms and conditions of, 278–279 electron density, NNA, 278 ELF basin existence, NNA, 278 high NLO properties, 278 Laplacian of electron density negative value, NNA, 278 noncovalent interactions (NCI) index, 279 organicelectrides, development of, 275–276 Molecular electrostatic potential (MESP), 65–66, 279 Molecular orbitals (MOs), 1 of BLi6-BF4 compound, 124 correlation diagram, 157 (C6H6)3 in D3 symmetry, 161f, 168f tripletOh symmetric M(N2)8 complex, 161–162, 161f triplet symmetric Ca(PF3)8 complex, 161–162, 162f of valence (n)s(n)p(n–1)d AOs of M, 161f, 168f Molecular surgery, 173–174 Molecules theory, 89 Monte Carlo simulations, 43 Mulliken electronegativity, 88 Multicenter bond index (MCI), 90

Multiconfigurationalquasidegenerate perturbation (MCQDPT), 24 Multiconfigurational self-consistent field (MCSCF) level of theory, 24 Multi-metallic clusters, 186 Multiobjective (MO) optimization, 185–186 Multiple aromaticity, 21 Multiwfn software, 125

N Nanoclusters, small molecules catalytic activation Aln nanoclusters competitive H–X elimination, 260–262 C-X bond activation on, 260 selective C–H bond activation, 262–264 selectivity, during elimination, 262 CO2, activation and reduction of, 253–255 metal hydride, reduction of, 254–255 computational chemistry, 247 CO on Aun nanoclusters, oxidation of, 255–258 anionicaluminum nanoclusters, electron spin effect, 257–258 doping, effect of, 256–257 genetic algorithm (GA), 247 H2 activation, 251–252 H2O activation, 258–260 homogeneous electron gas model, 247 metallic nanoclusters, 247–248 N2 activation, 248–251 nanomaterials, 247 O2, activation of, 255–258 tight-binding based density functional theory (DFT), 247 Nanocluster without co-reagent, transformation in co-reactant-free transformations pH-induced transformation, 373–375 photo-induced size/structure transformation (PIST), 379–381 solvent-induced transformation, 375–379 temperature-induced transformation, 381–384 perspectives, 384 Nano functional materials, 186 Natural atomic orbitals (NAO), 2–3 Natural bond orbital (NBO), 1–2, 70–71, 279 Natural localized molecular orbitals (NLMOs), 333–334, 338 Natural population analysis (NPA), 71, 214 NCIPLOT program package, 280 Ng2M©B10– complexes, AdNDP analysis for, 12, 12f N-heterocyclic carbenes (NHCs), 61, 68–69 N-heterocyclic imine (NHI), 61–62 N-heterocyclic olefins (NHOs), 61–62 N-heterocyclic silylenes (NHSis), 61, 68–70 Nitreones, divalent N(I) systems applications in medicinally important molecules, 79–80 phase transfer catalysts (PTCs), 77–79 bonding analysis, 75, 76t

chemical reactivity and stability, 76–77 L2N+ systems protonatedbiguanides, quantum chemical studies of, 72, 72f representative structures, 72, 73f pentanidium salts, 72 synthesis of benzamidin-4-ylidene, 74, 74s of bis(cyclopropylidene) stabilized divalent N(I) compound, 73–74, 74s of bis(silylene) stabilized divalent N(I) compound, 74–75, 75s of chiral pentanidium type divalent N(I) compound, 73, 74s geometrical parameters and spectroscopic values, 75 tautomeric preference, 72 Nitride nanocages, 124 NLi4@C60, 175 Noble metal clusters roles and applications, 99 small silver/gold clusters (see Silver and gold clusters) Noncovalent interactions (NCIs), 237, 240, 279–280 Nonlinear optical (NLO) materials, 174 applications, 123 electric field, nonlinear function of, 123 inorganic NLO materials, 123 metal clusters, host-guest interactions (see Metal clusters) properties, 275, 280–283 second-order optical nonlinearity, 123 Nonnuclear attractor (NNA), 276–277 Nonnuclear maximum (NNM), 278 Nonstoichiometric clusters, 44 Novel cooperative co-evolutionary PSO (NCPSO), 185–186 Nucleus-independent chemical shift (NICS), 90–94, 237, 240, 242, 301, 347, 349, 361–363, 365

O Occupancy number (ON), 2–3 Oh symmetric M(N2)8 complex, 161–162, 161f OLi3@C60, 175 One-step fragmentation energy (Ef), 107–109, 108–109f Orbital interaction energy (DEorb), 158, 162–165 O symmetric Ca(PF3)8 complex deformation densities of, 165–166, 165f shape of occupied molecular orbitals, 161–162, 162f Oxazol-2-ylidene, 69, 69s

P Para-delocalization index (PDI), 89 Partial density of states (PDOS), 146–148 Particle swarm optimization (PSO) algorithm Aun and AunAgm clusters, 200 boron clusters B5 cluster, 192–200, 192f

Index

B6 cluster, 193–194, 193f potential energy surface (PES), 192 carbon clusters, 194–198 density functional theory-PSO approach, 186, 188, 188f equations, 187 experimental setup, 192 Gaussian 09 (G09) program, 187 global best (gbest) configuration, 187 global coefficient of acceleration, 188 hybrid PSO (HPSO) algorithm, 185–186 individual coefficient of acceleration, 188 inertia coefficient, 188 local optimum solution, pbest configuration, 187 N42- and N64- clusters, 199–200 novel cooperative co-evolutionary PSO (NCPSO), 185–186 randomly generated cluster configuration, 187 single point energy (SPE) of cluster, 187 Pauli exclusion principle, 239 Pauli repulsion, 158 Pentanidium salts, 72 Periodic boundary conditions (PBC), 279 Phase transfer catalysts (PTCs), 79 asymmetric PTC-catalyzed a-hydroxylation, 79, 79s chiralpentanidium chloride as, 77, 78f dihydrocoumarins, pentanidium-catalyzed alkylation of, 79, 80s enantioselectivea-hydroxylation, of 3substituted-2-oxindoles, 79, 79f Michael addition, of 3-substituted-2oxindoles, 79, 79s pentanidium-catalyzed Michael addition, 78, 78f silylamide as probase for, 79, 80f stereoselective PTC-catalyzed Michael addition of Schiff’s base, 77, 78s Phenomenological shell model (PSM), 99, 110–113, 118 Phosphine-protected gold clusters, 411 advancements in, 411 diphosphine-ligated gold clusters electronic studies, 414–416 geometric and electronic structures, ligands effects on, 416–419 geometric studies, 413–414 jellium models and core shapes, 411–413, 413f Photo-induced size/structure transformation (PIST), 379–381 Photoluminescence (PL) spectra, 373–374 Planar hexacoordinate boron (phB), 346, 348 Planar hexacoordinate carbon (phC), 365 Planar hypercoordinate carbon Ab initio computations, 357 higher coordinate carbon, 365–368 nonclassical molecules, 357 planarhexacoordinate carbon (phC), 365 planarpentacoordinate carbon (ppC), 361–364 planartetracoordinate carbon (ptC), 357–361 racemization process, 357 Planar pentacoordinate carbon (ppC), 361–364

Planar tetracoordinate carbon (ptC), 345–346, 357–361 p-mercaptobenzoic acid (p-MBA)-protected Au102 cluster, 411 Polarizability, atoms and atomic clusters, 313 chemical reactivity indices-based route, 318 DFT -based approach, 314–317 response properties, basics of, 314 spherically symmetric systems, 317–318 values of, 318–319 Polyhedral-only Au11 cluster model, 414–415 Polynitrogen clusters, 186 Potential energy surfaces (PESs), 1–2, 4–5, 43 Projected density of states (PDOS), 333–334, 337, 339 Projector-augmented wave method, 34 Proton affinity (PA), 66 “Pseudoatoms,”, 278 Python programing language, 186, 191

Q Quadratic configuration interaction including single and double excitation (QCISD), 280 Quantum confinement effect, 382–383 Quantum theory of atoms-in-molecule (QTAIM), 278

R Random sampling method, 185 Repulsive Coulomb barrier (RCB), 35 Reverse water gas synthesis (RWGS), 253–254 Reversible conformational isomerism, 382 RhB12¯ clusters, 4 Ruthenium clusters, 250–251

S Sanderson’s electronegativity equalization principle, 88 Second harmonic generation (SHG), 123 Second harmonic (SH) wave, 123 Second-order difference of energy (D2E), 107–109, 108–109f Shannon entropy, 90 Silver and gold clusters atomic aggregates, 99 atomic arrangements of, 99 electronic absorption spectra, 113–118 electron shell effects in, 99 equilibrium structures and growth mechanism, 102, 106–107 Ag7 and Ag8, stable isomers of, 102–104 Ag3 to Ag20 clusters, structural evolution from, 106, 107f Au3 to Au20 clusters, structural evolution from, 106, 106f C2v M3 clusters, a1 and b2 orbitals in, 100–101, 102f density functional theory (DFT) calculations, 100, 101t harmonic vibrational frequency computations, 100 lowest-energy isomers, 100–101, 102f

429

Mn clusters from n ¼ 12 to 16, lower-energy isomers of, 104, 104f Mn clusters with n from 17 to 20, lowerenergy isomers of, 105–106, 105f Mn from n ¼ 7 to 11 clusters, lower-energy isomers of, 101–102, 103f rhombic M4, 101, 103f square M4, degenerate HOMOs in, 101, 103f phenomenological shell model (PSM), 99, 110–113, 118 pure gold and binary silver-gold clusters, GM energy structures of, 186, 200, 200–201f relativistic effects, 99 thermodynamic stabilities, 117 atomic arrangement, 109 aurophilicity/aurophilic attraction, 110 binding energy per atom (BE), 107–108, 108–109f electron shell effect, 109 local density approximation (LDA), 107 one-step fragmentation energy (Ef), 107–109, 108–109f relativistic effect, 110 second-order difference of energy (D2E), 107–109, 108–109f transition metal series, 109 three-dimensional (3D) structures, 99, 117 Silylamide, 79, 80f Silylones, 63 Singly occupied molecular orbital (SOMO), 22, 70, 281 Si52– system, 4–5, 5–6f Size-focusing methods, 373 Slater-type orbitals, 42 Spherical aromaticity rules, 32 Stannylones, 63 Statistical geometry analysis, of Au13 clusters, 419 Sternheimer equation, 317 Stochastic search algorithms, 185 Stoichiometric clusters, 44 Strong covalent interaction (SCI), 239–240 Sum/difference frequency generation (SFG/ DFG), 123 Superalkalis (SAs)@C60endofullerene BF4endofullerene complexes, 179–180 vs. Li@C60endofullerene, 175 Superatoms, 321, 339–342 Superhalogens, 174, 335–339 Super valence bond (SVB) model, 321 Supervised learning, 186 Suzuki-Miyura cross coupling reaction, 67 Swarm intelligence (SI), 185

T Tail-and-humps patterns, 414–415 Tautomer analysis, 72 Tetraaminoallenes (TAA), 63–64, 64f Tetracyanoethene (TCNE), 283 Tetrafluoroborate salts, 76–77, 77s Tetrahydrofuran (THF), 387 Tetraphenylcarbodicyclopropenylidene, 65, 65s

430

Index

Thiolate-protected gold clusters, 411 Au38 and Au25 clusters, theoretically predicted structures of, 411, 412f Au25(SR)18 cluster, CO2 electroreduction on, 411 Au79 core, chiral structures and HCM values of, 411, 412f Au102 (p-MBA)44, density of states for, 411, 412f bare Au79 core (58e) without Au-thiolate staple layer, density of states for, 411, 412f Third harmonic generation (THG), 123 Thomas-Fermi formula, 238–239 Time-dependent density functional theory (TD-DFT), 125 BNF and guest@BNF moieties, results for, 127–130, 129t goldAun clusters, 113–115, 114f GR/BGR/BNGR and M3O+@GR/BGR/ BNGR moieties, results for, 130, 131–132t guest@OA complexes, 134–135 for silver Agn clusters, 115–117, 116f Tolman electronic parameter (TEP), 66–67

Total density of states (TDOS), 146–148 Transition metals (TMs), 91, 93–94, 248–250 Transition state (TS), 391–392 Transmutation electronic transmutation, 4–5 nuclear transmutation, 4 Two-center two-electron (2c-2e) bonds, 3–4

IrB12¯ system, AdNDP analysis for, 4, 10f Water endofullerenes (H2O@C60), 173–174 Weizs€acker KED formula, 238–239 Wiberg bond index (WBI), 279, 349–350, 397t Wigner-Witmer rules, 257–258

U

X

Ultrahigh vacuum molecular epitaxy technique, 29

XePtF6 complex, 174 X-ray diffraction, 185

V

Z

Variation-perturbation approach, 316 Vertical detachment energy (VDE), 336–337, 364 Vertical electron affinity (VEA), 334–335 VMD software, 280

W Wade-Mingos model, 336–337 Wankel motors B182– cluster, AdNDP analysis for, 7, 9f CB18 cluster, AdNDP analysis for, 7, 10f

Zeroth-order regular approximation (ZORA), 158 Zintl clusters, 333 computational methods, 333–334 intercluster bonds, 323–328 superalkali, 334–335 superatom, 339–342 superhalogens, 335–339 Zintlsuperalkali, 334–335 Zintlsuperatom, 339–342 Zintlsuperhalogens, 335–339