Polish Quantum Chemistry from Kolos to Now [1 ed.] 0443185948, 9780443185946

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EDITORIAL BOARD Remigio Cabrera-Trujillo (UNAM, Mexico) Hazel Cox (UK) Frank Jensen (Aarhus, Denmark) Mel Levy (Durham, NC, USA) Jan Linderberg (Aarhus, Denmark) Svetlana A. Malinovskaya (Hoboken, NJ, USA) William H. Miller (Berkeley, CA, USA) John W. Mintmire (Stillwater, OK, USA) Manoj K. Mishra (Mumbai, India) Jens Oddershede (Odense, Denmark) Josef Paldus (Waterloo, Canada) Pekka Pyykko (Helsinki, Finland) Mark Ratner (Evanston, IL, USA) Dennis R. Salahub (Calgary, Canada) Henry F. Schaefer III (Athens, GA, USA) John Stanton (Austin, TX, USA) Alia Tadjer (Sofia, Bulgaria) Harel Weinstein (New York, NY, USA)

Academic Press is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States 525 B Street, Suite 1650, San Diego, CA 92101, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 125 London Wall, London, EC2Y 5AS, United Kingdom First edition 2023 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: 978-0-443-18594-6 ISSN: 0065-3276 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Zoe Kruze Acquisitions Editor: Mariana L. Kuhl Developmental Editor: Akanksha Marwah Production Project Manager: Vijayaraj Purushothaman Cover Designer: Vicky Pearson Typeset by STRAIVE, India

Contributors Ludwik Adamowicz Department of Chemistry and Biochemistry, University of Arizona, Tucson, AZ, United States Wojciech Bartkowiak Department of Physical and Quantum Chemistry, Wrocław University of Science and Technology, Wrocław, Poland Maria Barysz Faculty of Chemistry, Nicolaus Copernicus University in Torun´, Torun´, Poland Nicholas P. Bauman Pacific Northwest National Laboratory, Richland, WA, United States Ewa Brocławik Polish Academy of Arts and Sciences, Krako´w, Poland Wojciech Cencek Department of Physics and Astronomy, University of Delaware, Newark, DE, United States Marta Chołuj Department of Physical and Quantum Chemistry, Wrocław University of Science and Technology, Wrocław, Poland Maciej Fidrysiak Institute of Theoretical Physics, Jagiellonian University, Krako´w, Poland Ireneusz Grabowski Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Torun´, Torun´, Poland Oleg V. Gritsenko Institute of Physics, Lodz University of Technology, Lodz, Poland; Section Theoretical Chemistry, VU University, Amsterdam, The Netherlands Michał Hapka Faculty of Chemistry, University of Warsaw, Warsaw, Poland Maciej Hendzel Institute of Theoretical Physics, Jagiellonian University, Krako´w, Poland Mohammad Reza Jangrouei Institute of Physics, Lodz University of Technology, Lodz, Poland Bogumił Jeziorski Faculty of Chemistry, University of Warsaw, Warsaw, Poland Jacek Komasa Faculty of Chemistry, Adam Mickiewicz University in Poznan´, Poznan´, Poland

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Contributors

Jacek Korchowiec Faculty of Chemistry, Jagiellonian University, Krako´w, Poland Tatiana Korona Faculty of Chemistry, University of Warsaw, Warsaw, Poland Karol Kowalski Pacific Northwest National Laboratory, Richland, WA, United States Stanisław A. Kucharski Institute of Chemistry, University of Silesia in Katowice, Katowice, Poland Vignesh Balaji Kumar Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Torun´, Torun´, Poland Maciej Lewenstein ICFO—Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona); ICREA, Barcelona, Spain Paweł Lipkowski Department of Physical and Quantum Chemistry, Wrocław University of Science and Technology, Wrocław, Poland Katarzyna Madajczyk Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Torun, Torun, Poland Artur Michalak Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Krakow, Poland Mariusz P. Mitoraj Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Krakow, Poland Robert Moszyn´ski Faculty of Chemistry, University of Warsaw, Warsaw, Poland Monika Musiał Institute of Chemistry, University of Silesia in Katowice, Katowice, Poland Roman F. Nalewajski Department of Theoretical Chemistry, Jagiellonian University, Cracow, Poland Konrad Patkowski Department of Chemistry and Biochemistry, Auburn University, Auburn, AL, United States Bo Peng Pacific Northwest National Laboratory, Richland, WA, United States Katarzyna Pernal Institute of Physics, Lodz University of Technology, Lodz, Poland Michał Przybytek Faculty of Chemistry, University of Warsaw, Warsaw, Poland

Contributors

xiii

Mariusz Puchalski Faculty of Chemistry, Adam Mickiewicz University in Poznan´, Poznan´, Poland Dorota Rutkowska-Zbik Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Krakow, Poland Aditi Singh Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Torun´, Torun´, Poland Szymon S´miga Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Torun´, Torun´, Poland Jo´zef Spałek Institute of Theoretical Physics, Jagiellonian University, Krako´w, Poland Anna Stachowicz-Kusnierz Faculty of Chemistry, Jagiellonian University, Krako´w, Poland Monika Stanke Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University, Torun´, Poland Krzysztof Szalewicz Department of Physics and Astronomy, University of Delaware, Newark, DE, United States Renata Tokarz-Sobieraj Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Krakow, Poland Małgorzata Witko Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Krakow, Poland Aleksander P. Woz´niak Faculty of Chemistry, University of Warsaw, Warsaw, Poland ˙ uchowski Piotr S. Z Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Torun, Torun, Poland ˙ urowska Olga Z Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Krakow, Poland

Preface Volume 87 provides a series of articles representing contributions to quantum chemistry by Polish scientists working in Poland and worldwide. The volume starts with reviews covering two leading fields of Polish activities in quantum chemistry: i. high-precision calculations for a few electrons systems (started with famous works of Kołos and Wolniewicz on hydrogen molecule) ii. development of the symmetry-adapted perturbation theory, an important and commonly used approach to treat intermolecular interactions at various levels of theory. A next review is devoted to the advancement of the theory of electronic correlation based on the coupled cluster approach, with its branches focused on excited states and multireference systems. The scientific range of other contributions to the current volume is broad: from works on information theory, quantum computing, electron dynamics in the hydrogen molecule, exciton transport, high-precision studies of the potential energy surfaces, explicitly correlated basis sets, relativistic calculations through studies on density functional theory and its applications to calculations on the catalyzed or confined reactions of organic compounds, and elementary considerations on the nature of the chemical bonding to the molecular dynamics simulations of water aggregates. We are confident that you will enjoy reading this collection of chapters as much as we have enjoyed preparing them for this special issue. MONIKA MUSIAŁ IRENEUSZ GRABOWSKI

xv

REVIEW

From the Kołos–Wolniewicz calculations to the quantum-electrodynamic treatment of the hydrogen molecule: Competition between theory and experiment
skib, and Jacek Komasaa,∗ Mariusz Puchalskia, Robert Moszyn a

Faculty of Chemistry, Adam Mickiewicz University in Pozna
n, Pozna
n, Poland Faculty of Chemistry, University of Warsaw, Warsaw, Poland Corresponding author: e-mail address: [email protected]

b ∗

Contents 1. Introduction 2. The nonrelativistic energy 2.1 The Born–Oppenheimer energy curves 2.2 Adiabatic and nonadiabatic corrections 2.3 Direct nonadiabatic calculations 3. Relativistic corrections 3.1 Kołos and Wolniewicz calculations 3.2 Relativistic correction in NAPT 3.3 Relativistic correction in DNA approach 4. Quantum electrodynamic corrections 4.1 The complete leading QED correction 4.2 Higher order QED corrections 5. Theory vs experiment 6. Summary Acknowledgments References

2 5 5 10 13 18 18 19 20 21 21 22 22 28 30 30

Abstract The hydrogen molecule—two and a half centuries after Cavendish’s discovery, a dozen decades after Lyman’s first spectroscopic observations and almost a centennial after Heitler and London’s quantum-mechanical explanation of its stability—is still the subject of intense research, both theoretical and experimental. Being the simplest neutral molecule, it serves as a benchmark for testing quantum-mechanical theories and Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.04.001

Copyright

#

2023 Elsevier Inc. All rights reserved.

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methods; being the most ubiquitous molecule in the Universe, plays an essential role in astronomy and the fundamentals of physics. Over decades, advances in increasing experimental precision have stimulated ever more accurate calculations. In this review, we summarize the progress made in the field of the quantum-mechanical description of the electronic structure of the hydrogen molecule, starting from the pioneering calculations of Kołos and Wolniewicz in the early 1960s and ending with the current state of knowledge, including applications of quantum electrodynamics. Bearing in mind the topic of this special issue, we will focus on the achievements of Polish scientists in this matter.

1. Introduction Molecular hydrogen is the simplest molecule that exhibits electron correlations and displays many aspects of chemical binding, to which fundamental studies can be addressed. Therefore it has been the ubiquitous reference system from the very beginning of quantum chemistry and molecular physics. It started in the pioneering work by Heitler and London in 19271 who approximately solved the Schr€ odinger equation and found that the system is stable against the dissociation into two hydrogen atoms. Consequently, the newly formulated quantum theory could satisfactorily explain the stability of the covalent bond between two neutral atoms. The milestone in the computational methods for the hydrogen molecule is the work of James and Coolidge from 19332 who employed the wave function expansion in terms of basis functions ϕ depending explicitly on the interelectron distance r12 ϕ ¼ eαðξ1 +ξ2 Þ ξk1 ξm2 ηl1 ηn2 r μ12 ,

ξi ¼

r Ai + r Bi , R

ηi ¼

r Ai  r Bi R

(1)

and achieved the millihartree accuracy (E ¼ 1.173 559 Eh) at equilibrium distance R ¼ 1.4 bohr using the 13-term expansion only. The calculated dissociation energy D0(H2) ¼ 4.454(13) eV was consistent with the available experimental value of 4.46(4) eV.3 It turned out that explicit inclusion of the r12-dependent term greatly improves the energy convergence compared to the approaches using the orbital basis sets. The Coulomb singularity in the Hamiltonian imposes a specific behavior on the wave function at the interparticle coalescence point. Indeed, the James–Coolidge (JC) wave function can reproduce Kato’s cusps. However, introducing the r μ12 factors to exponential functions involves the computation of extremely complicated integrals.

From the KW calculations to the QED treatment of the hydrogen molecule

3

Significant advances were achieved by Kołos and Wolniewicz in 1965,4 who extended the work of James and Coolidge and proposed a new form of the basis function which was also applicable to long internuclear distances ϕ ¼ eαξ1 αξ2 coshðβη1 + βη2 Þ ξn1 ηk1 ξm2 ηl2 r μ12 :

(2)

The developed computational methods, also accounting for the coupling of the electronic and nuclear motions and for relativistic effects, were implemented on electronic computers, which became available at that time. Since then, it has been many times demonstrated that the Schr€ odinger equation, when solved accurately with the explicitly correlated basis functions and corrected for small relativistic and quantum electrodynamic (QED) effects, can predict the molecular energy levels with very high precision. This, in turn, laid a foundation for the current confidence in the quantitative predictive power of quantum chemistry. Since 1970, the spectroscopic measurements have undergone significant progress.5–7 In contrast to atomic systems, the rotational and vibrational degrees of freedom rise to a multitude of states in the ground electronic manifold with long lifetimes. It enabled the contemporary measurements of transition frequencies between rovibrational levels of H2 and its deuterated and tritiated isotopologues to reach the relative accuracy of 109, see, e.g., Refs. 8–11. Advances in both theory and experiment have made molecular hydrogen an essential system in developing quantum chemistry. H2 has emerged as a benchmark for testing quantum electrodynamics in simple bound systems. Comparison between theory and experiment serves not only as a consistency check between them, but it can also potentially be exploited to determine physical constants such as the proton charge radius, the electron–proton mass ratio, or the Rydberg constant R∞. In this review, we want to present the most significant achievements of Polish scientists from the perspective of theoretical and computational developments of methods devoted to the hydrogen molecule, starting from the works of Kołos and Wolniewicz published 1960s. During the decades, progress has taken place in many directions of theory. Several research groups have devoted significant resources and participated in this progress. Nowadays, studies on H2 are also an area of intense research, touching on problems far beyond quantum chemistry. It shows that research on molecular hydrogen occupies a unique place, marking the frontier of research in highprecision applications and allowing a deeper insight into unexplored phenomena occurring in molecules.

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Mariusz Puchalski et al.

From today’s perspective, the theoretical research on molecular hydrogen can be systematized within the framework of nonrelativistic quantum electrodynamics (NRQED).12 Within this theory, all the corrections to the nonrelativistic energy are implemented on top of the nonrelativistic Hamiltonian using the standard perturbation theory with the nonrelativistic wave function. The principal assumption in NRQED is that the total energy can be expanded in powers of the fine-structure constant α ’ 1/137 EðαÞ ¼ Eð2Þ + Eð4Þ + E ð5Þ + E ð6Þ + E ð7Þ + Oðα8 Þ,

(3)

where E(n) are contributions of order αn. They are interpreted subsequently as the nonrelativistic energy E(2), the relativistic correction E(4), the leading QED correction E(5), and the higher order QED corrections E(n), n  6. All these contributions can be expressed as expectation values, with a Schr€ odinger’s wave function, of certain operators derived within NRQED theory.13 Based on the NRQED expansion (3), we can attempt to present in an orderly manner the development of theoretical methods for molecular hydrogen. The most general approach is to treat the hydrogen molecule as a fourbody system, where the starting point is the solution of the Schrodinger equation in which electrons and atomic nuclei are treated similarly. Such a direct nonadiabatic (DNA) approach, see Section 2.3, accounts strictly for the (nonadiabatic) effects of the finite mass of the nuclei and has been intensively developed in recent years. Such calculations are relatively costly considering a large number of rovibrational states of the diatomic molecule spectrum and the necessity of variational optimization for individual states. An alternative possibility is based on the nonadiabatic perturbation theory (NAPT) of the finite nuclear mass effects, in which the leading order is just the Born–Oppenheimer (BO) approximation commonly used in molecules. More details on this method and the numerical results obtained using different types of wave functions are given in Section 2.2.3. These two approaches, DNA and NAPT, often complemented in high-precision calculations and allowed for a nontrivial mutual verification. Equally important is the development of computational methods and their implementation on computing machines, the rapid advancement of which undoubtedly was and still is an impulse for further progress.

From the KW calculations to the QED treatment of the hydrogen molecule

5

2. The nonrelativistic energy The nonrelativistic energy of the hydrogen molecule can be obtained by solving the four-body Schr€ odinger equation directly or by separating nuclear and electronic variables. The error introduced by the separation can be later compensated by accounting for the finite nuclear mass effects in adiabatic and nonadiabatic corrections. In this section, we shall review first the results obtained within the BO approximation (Section 2.1), for both the ground and the electronically excited states, followed by reports on adiabatic and nonadiabatic corrections (Section 2.2) to BO energy. The next part of this section will be devoted to fully nonadiabatic (DNA) calculations. Selected numerical results will accompany a compact description of the methods.

2.1 The Born–Oppenheimer energy curves The BO separation of the electronic and nuclear motions in molecules is probably the most crucial approximation ever introduced in molecular quantum mechanics. It provides rigorous definitions of useful chemical concepts like the geometry of molecules, the molecular dipole moment, or the interaction potential. In this approximation, one assumes that electronic motions are much faster than nuclear ones. Therefore, one can postulate that slow changes in the nuclear positions do not strongly affect the physical description of the electronic motions, leading to an adiabatic separation of the electronic and nuclear problems. In the BO approximation, the electronic problem is solved for an (infinite) set of nuclear coordinates. The eigenvalue of the electronic Schr€ odinger equation, the electronic energy as a function of these coordinates, is nothing else than the potential energy surface. This energy surface is used as the potential operator in the second step of the BO approximation, the problem of nuclear motions in a given potential. 2.1.1 James–Coolidge and Kołos–Wolniewicz wave functions The basis set of JC was the first to be successfully applied in quantitative quantum-chemical calculations on a molecule2 and extended by Kołos and Roothaan in 1958.14,15 At that time, this was a veritable tour de force. In these works, a meticulous study of the optimal nonlinear parameter at different interatomic distances was reported, the convergence with the number

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Mariusz Puchalski et al.

of terms in the basis was checked, and finally, the accuracy of the result at the equilibrium distance was verified by using the virial theorem. In 1964 the work of Kołos and Roothaan was extended by Kołos and Wolniewicz.16 This seminal paper is known for calculating the adiabatic and relativistic corrections to the BO energy. Over time, it became clear that the JC basis set was not flexible enough to extend the calculations of the potential energy curve to the dissociation. Therefore, the JC basis was generalized by Kołos and Wolniewicz in 19654 and made compatible with the Heitler–London function at large distances. In 1966, the flexibility of the Kołos–Wolniewicz (KW) function was further increased in calculations for the excited B1 Σ+u state, and in 197417 and 197518 for the ground state. In the 1965 paper,4 the authors reported results for distances between 0.4 and 10 bohr, thus covering the full potential from the very repulsive region to the long-range attractive region with the different sizes of the basis sets for different interatomic distances R. At the beginning of the 1960s, Herzberg and Monfils performed a careful experimental investigation of the dissociation energy of the hydrogen molecule.19 Interestingly, the BO potential obtained by Kołos and Wolniewicz supplemented with the adiabatic and relativistic corrections predicted the dissociation energy D0(H2) ¼ 36 117.4 cm1 which was larger than the experimental value 36 113.0(3) cm1. This, in turn, implied that the total energy was lower than deduced from the experiment, in contradiction with the variational principle. Therefore, Kołos and Wolniewicz repeated the 1964/1965 calculations in the double precision arithmetic,20,21 confirming the existing discrepancy between theory and experiment. The controversy was resolved in 1970, when Herzberg,22 and independently Stwalley,23 repeated the measurement obtaining a value consistent with the theory. The BO potential for the ground state of H2 was revisited several times. In 1974 Kołos and Wolniewicz significantly improved the long-range tail of the potential17 using a much longer expansion (60 terms) for distances ranging from 6 to 10 bohr. Small and intermediate distances were carefully reinvestigated, and the most precise clamped nuclei energy has been computed for the equilibrium distance Re ¼ 1.4 bohr.18 All the theoretical and computational advances in the calculations on the ground state of the hydrogen molecule were briefly summarized by Kołos in 1978.24 Further improvements in the BO energies for the ground state of the hydrogen molecule had to wait for a new era of supercomputers. In 1986 Kołos et al.25 revisited the BO potential for H2 for distances ranging from 0.2 to 12 bohr. For all distances, improvements over the previous results

From the KW calculations to the QED treatment of the hydrogen molecule

7

were obtained. The size of the basis set varied depending on the interaction region (repulsive, minimum, or attractive). As many as 249 functions were used to improve the binding energy at the minimum by as little as 0.004 cm1. At other distances, the improvement was substantially better. After adding terms beyond the nonrelativistic BO approximation, perfect agreement between theory and experiment22,23 was obtained, while for the D2 isotopologue, a substantial discrepancy remained. Interestingly, a few years later, in 1993, Kołos and Rychlewski26 used a smaller basis set than in 1986, and still improved the accuracy of the BO potential energy curve, leading to a perfect agreement between theory and experiment for the dissociation energy of all stable isotopologues of the hydrogen molecule, but not for the ionization potentials. In the same year Wolniewicz27 revisited the 1986 paper by Kołos et al.,25 and constructed a basis of around 249 functions for all distances ranging from 0.2 to 12 bohr by eliminating only those that were responsible for the linear dependencies. With such a constructed basis and newly recomputed adiabatic and relativistic corrections, Wolniewicz reached a perfect agreement for both the dissociation energies and ionization potentials. The last calculation of the BO potential by Kołos was published in 1994. After a very careful reoptimization of the nonlinear parameters in the variational wavefunction, the well depth changed only by 0.009 cm1, still slightly above the best result at that time obtained with explicitly correlated Gaussian (ECG) function, showing that a large number of nonlinear parameters, long basis set expansions, and easy construction of the matrix elements of the Hamiltonian are often more efficient than basis sets carefully crafted to the analytic properties of the Hamiltonian. 2.1.2 Explicitly correlated Gaussian wave functions In 1960, Boys28 and Singer29 presented a set of compact analytic formulas of the integrals necessary to determine the Hamiltonian matrix elements of molecular systems in the basis of explicitly correlated Gaussians (ECGs). These works are considered the beginning of ECG computational methods, one of the most important tools for precise quantum chemistry. These formulas can be used in the BO approximation calculations for multicenter and multielectron molecules, which is their main advantage over the exponential-function methods. Moreover, the great simplicity of the ECG integrals enables the use of a large number of ECG functions in the expansion of wave functions. On the other hand, ECGs do not have the correct functional form near the coalescence point for Coulomb interactions, and also do

8

Mariusz Puchalski et al.

not have the correct asymptotics over long distances. For this reason, the potential of these methods for precise computation has not been evident for a long time. In 1979, Jeziorski and Szalewicz30 showed that this basis enables accurate calculations for the ground electronic state of H2. Even if the accuracy of the presented energy at the internuclear distance R ¼ 1.4 bohr was lower than the best results at that time obtained with KW functions, they demonstrated that the ECG basis could be used to calculate various properties accurately, which cannot be achieved by other methods.31–33 Finally, in the 1990s, ECG expansions were successfully employed by Rychlewski, Cencek, and Komasa34,35 thanks to the development of effective optimization algorithms allowing the use of much larger ECG bases than before. In the next years, the most accurate values of the H2 binding energy were those from ECG calculations, with estimated uncertainties below one nanohartree.36 In time, these results were improved by Sims and Hagstr€ om37 and by Nakatsuji and collaborators38,39 who employed large exponential basis sets and extended numerical precision packages. Little later, Cencek and Szalewicz40 increased the accuracy up to 8  1014 a.u. using 4800 ECG basis function and obtained E(R ¼ 1.4) ¼ 1.174 475 714 220 363 a.u., the most accurate energy obtained from the ECG functions until now. It required the implementation of algorithms going beyond the standard double-precision arithmetic, which drastically increased the costs of computer calculations. The ECG results in Ref. 40 confirm that the analytical deficiencies of ECG functions are not critical for nonrelativistic energies and can be compensated by thoroughly optimizing the nonlinear parameters to achieve a fast basis set convergence. It is a consequence of the completeness of the ECG basis set.41,42 2.1.3 Exponential wave functions Significant progress in the development of the methods based on exponential wave functions has been made by Pachucki in 2010.43,44 He constructed analytic formulas and recursion relations for two-center integrals that comprise matrix elements in exponential functions with arbitrary polynomials in all interparticle distances.45 For short- and middle-range internuclear distances, Pachucki employed the JC basis and, for long-range, the generalized Heitler–London basis. With the wave function expanded in about 22 thousand basis functions, he achieved the BO energy with the numerical precision of 1015, thus outweighing the competition between different methods in favor of the exponential functions benchmarking nonrelativistic energy BO calculations. His achievement remains the most accurate energy ever

From the KW calculations to the QED treatment of the hydrogen molecule

9

obtained for any neutral molecule. Later, this approach has successfully been applied also in BO calculations on excited states of H2 (see Section 2.1.4). These wave functions were applied to precisely determine adiabatic and nonadiabatic corrections to the BO energy of hydrogen molecule isotopologues (see Section 2.2.3). Fig. 1 summarizes the progress in reducing the error in the BO interaction energy observed over the years. The most important implication of the high accuracy offered by both the Gaussian and exponential wave functions was a strong impulse for their applications in competition with the precise spectroscopic measurements of the hydrogen molecule. This direction has dominated the development of theoretical and computational methods for many years and resulted in the accurate determination of adiabatic, nonadiabatic, relativistic, and QED corrections to the BO potential; see sections below. 2.1.4 Electronically excited states KW wave functions were applied to determine BO potential energy curves for many electronically excited states. An extensive review of such calculations performed by Kołos, Wolniewicz, Rychlewski, and their coworkers before 2003 can be found in Ref. 36. Pachucki and coworkers constructed a new method involving the exponential basis functions and again approached the problem of electronically excited BO energy curves. The culmination of these works was presented in recent articles by Pachucki, Siłkowski, et al.46–48 Relying on efficient

Fig. 1 Energy error of the Born–Oppenheimer interaction energy in H2 at the internuclear distance R ¼ 1.4 bohr.

10

Mariusz Puchalski et al.

evaluation of two-center integrals in KW basis, they obtained state-ofthe-art BO energy curves for the manifold of electronically excited states: 1-71,3Σ+g,u, 1-41,3Πg,u, 1-31,3Δg,u, 1-31,3Φg,u. The accuracy of new energy curves surpassed all the previous results49,50 by several orders of magnitude. What is worth emphasizing is the flexibility of the unrestricted KW basis set used in these calculations, which, combined with thorough optimization, allows the efficient application of this basis for all—very short, medium, and very long—internuclear distances, regardless of the symmetry and spin of the electronic state. The ECG wave function has also been successfully applied in evaluating the excited states of H2. To mention a few: calculations of the double-well potential of EF 1 Σ+g state51 and calculations of doubly excited states of 1,3 Σ g,u symmetry. The latter states are known to play an important role in the dissociative recombination of H+2 and have so far eluded experimental observation. Therefore, most information concerning such states’ location comes from quantum-mechanical computations.

2.2 Adiabatic and nonadiabatic corrections 2.2.1 James–Coolidge and Kołos–Wolniewicz wave functions Calculations of the adiabatic correction, also known as diagonal correction for the nuclear motions, in the basis set of JC or KW do not have such a long history as the calculations of the BO potential. The first calculation was performed by Kołos and Wolniewicz in 196416 using the same basis set expansion as in the calculations of the BO potential in terms of the JC functions. At the potential minimum, the adiabatic correction contributed 4.947 cm1 to the well depth. Surprisingly, these calculations’ results were considered sufficiently converged for the next twenty years. Only in 1983, Wolniewicz52 revisited these results. It turned out that around the minimum, they were very well converged but needed improvements for distances smaller than 0.55 bohr and larger than 2.20 bohr. Better converged calculations were reported in 1993 by Kołos and Rychewski with a carefully optimized basis set used in the BO calculations.26 The calculated contribution to the binding energy was 4.938 cm1, that is, 0.18% lower than the result from 1964. The same year Wolniewicz27 confirmed the result of Kołos and Rychlewski. 2.2.2 Born–Handy method In 1997 ECG geminals were employed by Cencek and Kutzelnigg53 to evaluate the adiabatic correction at R ¼ 1.4 bohr using the approach popularized by Handy.54 Within this approach, called Born–Handy method, the adiabatic

From the KW calculations to the QED treatment of the hydrogen molecule

11

correction is computed as the expectation value of the nuclear kinetic energy operator expressed in laboratory coordinates. The accuracy of such calculations has been analyzed for the ground state of H2 in Ref. 53. The optimum choice of ΔRI ¼ 0.0005 bohr allows obtaining 6–7 correct digits. For H2, this method was at least as accurate as that using KW wave functions. The Born–Handy method has also been applied to evaluate the adiabatic corrections of the EF 1 Σ+g state of H2.51 2.2.3 Nonadiabatic perturbation theory The classic calculations of adiabatic and nonadiabatic corrections by Kołos and Wolniewicz were based on the Born–Huang expansion in which the total wave function is represented as a sum of products of eigenfunctions of P the electronic Hamiltonian and suitable nuclear functions. ϕ(r, R) ¼ kϕel,k(r;R) χ k(R). In contrast, the NAPT, formulated in 2008 by Pachucki and Komasa,55,56 takes the total wave function as a sum of a single adiabatic product and the nonadiabatic correction obeying specific orthogonality conditions. The adiabatic correction can be considered as the first-order correction in the electron-to-nucleus mass ratio. The higher order corrections can be found by evaluation of the nonadiabatic version of the Schr€ odinger equation derived in the framework of NAPT " # J ð J + 1Þ 1 ∂ R2 ∂  2 + + YðRÞ χeJ ðRÞ ¼ E χeJ ðRÞ: (4) 2 μ? ðRÞR2 R ∂R 2 μk ðRÞ ∂R The finite nuclear mass effects appear in this equation as R-dependent corrections to the nuclear reduced mass, μk(R) and μ?(R), and to the interaction potential YðRÞ ¼ E el ðRÞ+E a ðRÞ+E na ðRÞ. The relativistic and QED potentials can further augment this internuclear potential. In such a case, the eigenvalue of the Schr€ odinger equation (4) accounts for all the leading corrections resulting from the finite nuclear mass and the finite speed of light, and should be directly comparable to the experimental value. This method, in connection with the ECG wave function, was applied to obtain all the bound energy levels of the hydrogen molecule.56–58 Several years later, NAPT was reformulated in the language of the matrix elements in the JC basis.59,60 Such matrix elements are expressible by combinations of the closed-form integrals derived by Pachucki in 2009.45 This method enabled the relative accuracy of 1012 to be reached for the adiabatic correction, which surpassed all the previous results by several orders of magnitude (see Table 1). The contribution of the adiabatic correction numerical

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Mariusz Puchalski et al.

Table 1 Illustration of the increase in accuracy of the adiabatic correction since the first estimates were published. Year Reference E a ð1:4Þ/a.u.

1936 1964 1993 1993

Van Vleck61

0.954

Kołos and Wolniewicz ( JC)

16

Kołos and Rychlewski (KW) Wolniewicz (KW)

0.958 926

26

0.958 691

27

0.958 683 53

1997

Cencek and Kutzelnigg (ECG)

2009

Pachucki and Komasa (ECG)56

2014

Pachucki and Komasa ( JC)

59

0.958 684 72 0.958 684 14 0.958 684 725 430

The adiabatic correction to the interaction BO potential is defined with respect to the separated atoms limit.

Table 2 The leading-order nonadiabatic corrections to the dissociation energy of the lowest rovibrational levels of the hydrogen molecule. (v, J) ΔD0,0/cm21

(0, 0)

0.433 961 8(1)

(0, 1)

0.440 635 4(1)

(0, 2)

0.453 975 4(1)

(0, 3)

0.473 967 5(1)

uncertainty to the overall error of rovibrational energy levels was estimated as less than 109 cm1.59 The radial nonadiabatic Schr€ odinger equation derived in the framework of NAPT (4) contains, apart from the clamped nuclei E el ðRÞ and adiabatic correction E a ðRÞpotentials, other five nonadiabatic, second-order potentials proportional to 1/μ2n. In 2015 these potentials were reevaluated to a much higher accuracy using JC wave functions.60 The convergence study performed at fixed internuclear distances revealed a relative accuracy better than 109 for all these potentials which resulted in 107 cm1 uncertainty contribution to the energy of rovibrational levels (see Table 2). The third-order nonadiabatic correction has been derived in Ref. 60 but has never been evaluated numerically. Its rough numerical estimation can be obtained by a simple scaling of the second-order correction by the mass factor 1/μn. The missing third- and higher order corrections remained the primary

From the KW calculations to the QED treatment of the hydrogen molecule

13

source of uncertainty in the energy of nonrelativistic rovibrational levels until this energy was evaluated in a direct manner (DNA), that is, without the expansion in mass factor as described in the next section. The DNA calculations also allowed the estimations mentioned above to be positively verified.

2.3 Direct nonadiabatic calculations This section reviews methods that do not assume the separation of electronic and nuclear degrees of freedom. In such an approach, the notion of the potential energy curve is no longer used, and both the Hamiltonian and the wave function treat all the four particles the hydrogen molecule is built of on equal rights. The four particles are entangled and described by a single state vector, a solution to the four-body Schr€ odinger equation. The eigenvalue corresponding to this state vector directly gives the nonadiabatic energy of a rovibrational level. We assigned such methods the acronym DNA for direct nonadiabatic. 2.3.1 Kołos and Wolniewicz calculations To the best of our knowledge, the only full nonadiabatic calculations based on the Born–Huang expansion were carried out in 196362 with the full nonrelativistic four-particle Hamiltonian and in a basis set of products of harmonic oscillator functions for the vibrational motions and JC functions for the electronic motions. The electronic part was taken from the work of Kołos and Roothaan.15 It consisted of 40 terms with an exponent that did not depend on the interatomic distance, and four consecutive harmonic oscillator functions with vibrational quantum numbers ranging from zero to three. The resulting dissociation energy was 27 cm1 above the experimental value. Given the fact that the relativistic effects contribute only ca. 0.6 cm1 to the dissociation energy, this relatively large discrepancy between theory and experiment is probably due to the short expansion of the nonadiabatic wave function and lack of flexibility of the electronic basis. 2.3.2 Nonadiabatic explicitly correlated Gaussian wave functions The hydrogen molecule is a special case of the four-body problem, where the two atomic nuclei have large masses. It requires the introduction of prefactors in a variational basis, which enable modeling of the oscillatory motion of the nuclei around the characteristic distance corresponding to the length of the bond. In 1999, Kinghorn and Adamowicz63 proposed a nonadiabatic,

14

Mariusz Puchalski et al.

explicitly correlated Gaussian (naECG) basis for rotationless states of small !0

!

“diatomic” systems. The basis functions are of the form, ϕ ¼ r m01 e r A r with the m-th power of the distance between the two heavy particles, hence the approach is restricted to diatomics with any number of electrons. The exponential factor provides a full correlation between all system particles. The challenge here was the need to develop methods for calculating fully correlated integrals that take into account the arbitrary power of the internuclear distance as well as finding the distribution of discrete m-parameters. An indicator of the high accuracy of the 2003 calculations was the upper bound for the H2 nonrelativistic nonadiabatic ground state energy equal to 1.164 025 030 0 hartree using 3000 basis functions and, analogously for excited rotationless states, that is, states represented by spherically symmetric wave functions.64 Another representation of the naECG basis functions ϕ has been used in DNA calculations since 201865 ϕ ¼ r n01 ea1 r 01 a2 r 02 a3 r 03 a4 r 12 a5 r 13 a6 r 23 , 2

2

2

2

2

2

(5)

where 0 and 1 denote nuclei and 2 and 3 electrons. For the rotational quan! tum number J ¼ 1 the functions ϕ were modified by the factor r , representing the angular momentum. The nonlinear ai parameters were optimized individually for each basis function ϕ. Also, the powers n of the internuclear coordinate r01 and the shares of different angular momen! tum factors r ab were subject to additional discrete optimization. The nonrelativistic wave function was constructed for several basis lengths to observe the numerical convergence of the energy. The obtained results reached the accuracy of about 1011 (see Table 3) and were in perfect agreement with the benchmark values from the nonadiabatic James–Coolidge (naJC) wave function66 described below. In comparison to former works based on a different representation of ECG functions,67 the efficient optimization algorithm enabled obtaining more accurate results despite using approximately five times smaller basis set. The naECG wave function was further employed in the evaluation of the relativistic correction (see Section 3.3). 2.3.3 Nonadiabatic James–Coolidge wave functions The direction set forward by Kołos and Wolniewicz in 196362,71 was based on the use of exponential functions in four-body calculations. Due to apparent technical difficulties, their idea was abandoned for half a century. New perspectives opened after Pachucki’s discovery in 200945 of a differential

15

From the KW calculations to the QED treatment of the hydrogen molecule

Table 3 Convergence of the nonrelativistic energy E (in a.u.) with the size of the naECG basis set. Basis size

E(H2)

E(HD)

E(D2)

128

1.164 023 669 155

1.165 470 991 485

1.167 167 911 358

256

1.164 024 987 878

1.165 471 628 967

1.167 168 756 439

512

1.164 025 027 334

1.165 471 916 621

1.167 168 805 491

1024

1.164 025 030 593

1.165 471 923 256

1.167 168 808 953

2048

1.164 025 030 843

1.165 471 923 906

1.167 168 809 193

naJC

1.164 025 030 883 1(3) 1.165 471 923 964 38(3) 1.167 168 809 284 0(1)

The “naJC” line contains benchmark values obtained with nonadiabatic James–Coolidge wave functions.68–70

equation satisfied by any integral of the Hamiltonian matrix elements in a four-body exponential basis. With the newly proposed exponential-type basis functions the Hamiltonian matrix elements became expressible in a closed form only in terms of logarithmic, dilogarithmic, and rational functions ensuring high accuracy and efficiency of calculations. The newly proposed basis function was of the form66 ϕfkg ¼ exp½α R  βðζ 1 + ζ2 Þ Rk0 r k121 ηk12 ηk23 ζ k14 ζ k25 ,

(6)

where the parameters α, and β are positive real numbers, whereas ki are nonnegative integers, and where the elliptic-type variables are defined as follows !

!

ζ 1 ¼ r 1A + r 1B , η1 ¼ r 1A  r 1B , ζ2 ¼ r 2A + r 2B , η2 ¼ r 2A  r 2B , R ¼ r AB :

This basis function was named the naJC function for its resemblance to the original JC basis function used in fixed-nuclei calculations. In contrast to JC function, the four-body nonadiabatic function contains an internuclear correlation factor, and the meaning of the ζ and η variables is different. Evaluation of matrix elements in the naJC basis presents a serious programming challenge for two reasons. First, this basis, although very efficient from a physical point of view, forms a nearly linearly dependent set of functions. This fact forces the use of multiple (octuple or higher) precision arithmetic software, which radically increases operating memory requirements and slows down the execution of the computer program. Second, large basis sets (of the order of 105) need to be employed to achieve the target precision, which calls for effective code parallelization. Despite the struggles with limitations in available computational resources, high-accuracy solutions to the Schr€ odinger equation for the hydrogen molecule and its isotopologues were obtained.68–70,72

16

Mariusz Puchalski et al.

The accuracy achievable using the naJC basis can be assessed by inspecting the energy convergence with the growing size of the wave function expansion, K. The so-called “shell parameter” Ω, which limits from P above the sum 5i¼1 ki of integer powers in Eq. (6), is a practical guide for such an inspection. As shown in Table 4, the relative numerical accuracy of an eigenvalue can reach the level of 1014.66,68 Further increase in accuracy is still possible but impractical. The nonrelativistic energies quoted in the table were evaluated with the proton-to-electron mass ratio μp recommended in CODATA 201873 compilation of physical constants. At the achieved level of accuracy, the significant digits of the eigenvalues are sensitive to changes in physical constants; therefore, it is crucial to accompany the reported energy with the actually used values of nuclear masses. Recomputing the energies every time the recommended nuclear masses change would be impractical. To avoid this, the published energy of each rovibrational level is accompanied by its derivatives with respect to nuclear mass parameters. It is demonstrated in Table 5 for HD molecule.69 Using these derivatives one can easily update the energy. The DNA method based on the naJC wave function outperforms the calculations with ECG wave function by several orders of magnitude in accuracy. It can be shown in the example of D2 molecule in Table 6. As mentioned in Section 2.2.3, the accuracy of the nonrelativistic energy offered by the NAPT is limited by the cutoff of the perturbation series. Currently, NAPT is implemented to the second order. The missing higher order terms are the source of the method’s uncertainty. The results obtained using the DNA method correspond to the infinite-order NAPT expansion, which gives the opportunity to determine the contribution Table 4 Convergence of the E0,0 (in a.u.) eigenvalue for H2 with the growing size of the basis set, K. Ω K E0,0

10

42588

1.164 025 030 873 423

11

61152

1.164 025 030 879 839

12

85904

1.164 025 030 881 369

13

117936

1.164 025 030 881 709

14

159120

1.164 025 030 881 803

15

210912

1.164 025 030 881 831

∞

∞

1.164 025 030 881 85(2)

The CODATA 2018 proton mass was used μp ¼ 1836.152 673 43(11).

17

From the KW calculations to the QED treatment of the hydrogen molecule

Table 5 Nonadiabatic eigenvalue (Ev,J) of the selected rovibrational energy levels of HD. (v, J)

Ev,J/hartree

∂E v,J ∂μp

(0, 0)

1.165 471 923 964 39(3)

1.69

0.424

(0, 1)

1.165 065 376 941 65(3)

1.84

0.459

(0, 2)

1.164 255 083 090 94(3)

2.13

0.533

(1, 0)

1.148 922 593 494 61(3)

4.54

1.14

(1, 1)

1.148 533 615 356 71(3)

4.68

1.17

(1, 2)

1.147 758 378 109 83(3)

4.95

1.24

(2, 0)

1.133 181 743 156 29(4)

7.11

1.78

(2, 1)

1.132 809 980 890 48(6)

7.23

1.81

(2, 2)

1.132 069 097 950 27(6)

7.49

1.87

(3, 0)

1.118 233 513 220 64(5)

9.39

2.35

 106

∂E v,J ∂μd

 106

The μn is the nucleus-to-electron mass ratio, with n ¼ p, d.

Table 6 Comparison of the best literature data for the nonrelativistic dissociation energy D0,0 of D2. Year Method and reference D0,0/cm21 Difference/cm21

2019

naJC70

36 749.090 989 81(1) 36 749.090 970

2.0  105

NAPT60

36 749.090 976

1.3  105

10000-term naECG74

36 749.090 974

1.6  105

2018

2048-term naECG

2015 2011

65

from the higher order NAPT terms. Such calculations were reported in Refs. 70 and 72 and revealed that the missing higher order remainder is of the order of 105 cm1 and, within given vibrational state, changes proportionally to J( J + 1). The nonadiabatic effects are the smallest in the heaviest isotopologues, that is, those containing triton: HT, DT, and T2. However, even in this favorable case, the DNA method in connection with the naJC wave function enabled reducing the error in the nonrelativistic energy by two to three orders of magnitude compared with NAPT results. Moreover, the error in the total energy, primarily dominated by the nonrelativistic component, was reduced by order of magnitude.72 It is well exemplified in Table 7 for the case of transition energies in T2.

18

Mariusz Puchalski et al.

Table 7 Comparison of theoretical data for the fundamental v ¼ 0 ! 1 vibrational splitting of selected transition lines in T2 (in cm1). Line NAPT DNA+NAPT

Q(0)

2 464.504 15(6)

2 464.504 142(8)

Q(1)

2 463.348 36(6)

2 463.348 350(8)

Q(2)

2 461.039 17(6)

2 461.039 163(8)

Q(3)

2 457.581 37(6)

2 457.581 366(8)

The results shown in the second column were obtained entirely from NAPT calculations,75 whereas those in the third column contain analogous results, but the nonrelativistic energy was replaced by that from the DNA calculations.

Table 8 Comparison of the accuracy of the nonadiabatic dissociation energy achieved over the decades of the theoretical studies on H2. H2 D0,0/cm21

Reference

Pachucki and Komasa (2018), DNA/naJC68 Pachucki and Komasa (2015), NAPT/JC

60

Pachucki and Komasa (2009), NAPT/ECG56 Kinghorn and Adamowicz (1999), ECG Wolniewicz (1995), KW

63

76

36 118.797 675 36 118.797 8 36 118.797 74 36 118.795

Kołos, Szalewicz, and Monkhorst (1986), KW25 Wolniewicz (1983) KW

36 118.797 746 10(3)

27

36 118.088 38 118.01

77

Bishop and Cheung (1978), JC

36 118.60

Kołos and Wolniewicz (1964), JC71 Kołos and Wolniewicz (1963), JC

62

36 114.7 36 091

To summarize the struggles toward accurate determination of the nonrelativistic dissociation energy of the hydrogen molecule, we have assembled selected representative results in Table 8.

3. Relativistic corrections 3.1 Kołos and Wolniewicz calculations The investigations of the relativistic effects in the hydrogen molecule started in 1961. Kołos and Wolniewicz78,79 derived the expressions for the relativistic integrals in the JC basis contributing to the matrix elements of the

From the KW calculations to the QED treatment of the hydrogen molecule

19

Breit-Pauli Hamiltonian.80 These expressions were coded and used in the calculations of the relativistic corrections to the BO potential in 196416 obtaining 0.526 cm1 as the contribution to the well depth. In the same year, Kołos and Wolniewicz71 performed nonadiabatic calculations with the full four-particle nonrelativistic Hamiltonian and including the electronic part of the relativistic Breit-Pauli Hamiltonian. The basis set used in the calculations consisted of products of the harmonic oscillator functions for the vibrational motions and JC functions for the electronic motions leading to dissociation energies larger by 0.6 cm1, again in contradiction with the variational principle. Surprisingly, the relativistic calculations for H2 in the JC or KW basis functions were not attempted till 1993 when Wolniewicz27 derived the relativistic integrals for the KW basis with a completely new algorithm. It turned out that despite very carefully optimized basis functions and much longer basis set expansions, the result did not dramatically change compared to the 1964 paper,16 and the contribution to the binding energy amounts to 0.517 cm1, that is, changed by less than 2%.

3.2 Relativistic correction in NAPT In 2009 there was a new opening in this matter when ECG functions were employed to evaluate the relativistic matrix elements.57,58 In the BO approximation, the leading relativistic correction is that to the nonrelativistic potential EðRÞ. This correction can be expressed in terms of the expectation value E rel ðRÞ ¼ hϕel jH rel jϕel i of the Breit-Pauli Hamiltonian80 ! j i ij  r r 1 4 1 δ j H rel ¼  p1 + p42 + π δ3 ðr 12 Þ  pi1 + 123 12 p2 r 12 8 2 r 12 (7)  π 3 + δ ðr 1A Þ + δ3 ðr 2A Þ + δ3 ðr 1B Þ + δ3 ðr 2B Þ , 2 where the spin-dependent terms vanish for the electronic states of 1Σ symmetry. Highly singular operators, Dirac-δ and p4, present in the Hamiltonian (7) require special attention if relativistic correction of high accuracy is to be obtained. The expectation value of these operators converges very slowly with the increasing expansion size of a wave function. This effect is especially noticeable when ECG-type functions are used. For that reason, regularization schemes have been developed, which speed up the convergence significantly.81,82 Apart from the standard regularization, applied already by Wolniewicz,27 an original regularization, dedicated to wave functions obeying the Kato’s cusp condition, has been proposed.83 Such ECG functions,

20

Mariusz Puchalski et al.

with the 1 + r12/2 prefactor, were introduced in 2016 and nicknamed rECG. A novel algorithm for numerical quadrature of nonstandard ECG two-center integrals has been worked out.83 Apart from the fast convergence, the new regularization scheme has additional advantages. First, it enables complete elimination of the π δ3(r12) term from the relativistic Hamiltonian; second, a whole class the time-consuming integrals with three odd powers of interparticle distances do not appear in the matrix elements. This nontrivial cancellation has a remarkable impact on the efficiency of the calculations.84 The leading finite-nuclear-mass relativistic correction has been derived and evaluated numerically using NAPT. The so-called relativistic recoil potential E ð4,1Þ ðRÞ has been reported in Ref. 85. The error contributed by the potential is estimated to be 2  104  h χjE ð4,1Þ j χi. Thus far, no higher order finitenuclear-mass relativistic corrections are known, the effect of their omission is approximated by E(4, 1)  me/μn and included in the total E(4) error estimate.

3.3 Relativistic correction in DNA approach In DNA, the relativistic correction to the energy of a rovibrational level can be expressed in terms of the expectation value of the four-particle BreitPauli Hamiltonian    πX 1 4 δs 3 4 H rel ¼  p2 + p3 + 1 + 2 δ ðr xa Þ + π δ3 ðr 23 Þ 8 2 x, a mx !   j i j ij r 23 r 23 j r ixa r xa 1 i δ 1 X 1 i δij p3 + + 3 p + 3 pja  p2 r 23 2 2 x, a mx x r xa r xa r 23 ! j r i01 r 01 j 1 1 i δij p1  p + 3 (8) 2 m0 m1 0 r 01 r 01 where index x goes over 0,1 and a over 2,3. The coefficient δs ¼ 0 for the nuclear spin s ¼ 0 or 1, and δs ¼ 1 for s ¼ 1/2. In the above formulas, we have omitted all the electron spin-dependent terms because they vanish for the ground electronic state of 1 Σ+g symmetry. We have also omitted the p4x =ð8m3x Þ terms because their numerical values are smaller than the uncertainty of the whole relativistic correction. As in NAPT, regularization techniques and the cusp obeying rECG wave functions are mandatory to achieve high-accuracy results.65,86 Table 9 shows how the total relativistic correction E(4) changed over time when evaluated using two different approaches, NAPT and DNA.

21

From the KW calculations to the QED treatment of the hydrogen molecule

Table 9 Representative results showing the accuracy of the relativistic correction to the dissociation energy of H2 achieved using various wave functions and methods. Reference Erel/cm21

NAPT Kołos and Wolniewicz (1964), JC71 Wolniewicz (1993), KW

Piszczatowski et al. (2009), ECG Komasa et al. (2011), ECG

0.5 0.533 0

27 57

0.531 9(2) 0.531 8(2)

58

Czachorowski et al., (2018), ECG85

0.531 213(2)

DNA 0.569 1

Stanke et al. (2013), naECG87 Wang and Yan (2018), naHy88 Puchalski et al. (2018), naECG

86

0.531 255(63) 0.531 215 6(5)

4. Quantum electrodynamic corrections 4.1 The complete leading QED correction In 1993, Wolniewicz computed the QED correction of H2 in an approximate way.27 The first complete, leading QED correction valid in BO approximation was derived and applied with ECG functions by Piszczatowski et al. in 2009.57 Subsequent studies aimed at improving the quality of such calculations.58 The component limiting the accuracy of this correction is the so-called Bethe logarithm—a quantity for which exact values were known for a long time only for the hydrogen atom. The Bethe logarithm and the whole leading QED correction potential have recently been computed to a relative precision of 108 for an extensive range of internuclear distances by Siłkowski et al.89 This potential has been implemented in the publicly available H2Spectre program75,90 and can be used in the evaluation of the QED correction for all the rovibrational levels. Within the fully nonadiabatic approach, the complete formula for the leading QED correction E(5) was presented in Refs. 86, 91. So far, however, numerical calculations have been reported only for the ground rovibrational levels of H2 and its isotopologues.

22

Mariusz Puchalski et al.

4.2 Higher order QED corrections The derivation of the E(6) term of the α-expansion (3) for H2 was first reported in 2016.83 Calculations were performed only in the framework of the BO approximation. Because this term is represented by a sum of a first-order expectation value and a second-order quantity, both being divergent, a new technique of dimensional regularization to eliminate these divergences from the matrix elements had to be applied. Despite difficulties in its evaluation, the relative numerical accuracy of 3  103 for this term has been achieved, and its contribution to D0,0(H2) amounts to 0.002 065(6) cm1. E(7) is the first term of the expansion of energy in powers of α whose explicit form for H2 is not entirely known yet. Currently, it is approximated using dominating contributions known from the hydrogen atom theory. The first estimation of its magnitude appeared in 2016.83 For D0,0(H2) the E(7)  0.000 118(59) cm1. Both the higher order corrections, in the form of a function of internuclear distance, have been implemented in the H2Spectre program.75,90

5. Theory vs experiment Since the advent of the first theoretical predictions concerning physical observables of the hydrogen molecule—dissociation energy of the ground or excited levels or transitions between them—the reference to experimental data was treated as a mandatory check of the correctness and accuracy of such results. And vice versa—accurate theoretical results served as a verification tool for experimentalists. For over a century, a very fruitful competition between theory and experiment has been observed, resulting in continuous progress in both. The beginning of this competition was not very optimistic—in 1913, Niels Bohr theoretically estimated the dissociation energy of H2 as 21,000 cm1,92 whereas Irving Langmuir’s measurements gave over two times larger value of 46,000 cm1.93 Since then, the theory–experiment gap has only narrowed, as exemplified by the famous story of the contradiction KW vs Herzberg–Monfils, mentioned in Section 2.1, which ended with the reinterpretation of the experiment. Over the last century, the predictive power of theoretical methods and the precision of measurements have increased by several orders of magnitude. Only in the last decade, the advances in spectroscopic techniques enabled the experimental accuracy to be increased by three orders of magnitude. It is shown in Fig. 2, which collects a selection of measurements performed for the hydrogen molecule and its isotopologues. It is a great

From the KW calculations to the QED treatment of the hydrogen molecule

23

Fig. 2 The relative uncertainty of spectroscopic experiments performed recently. The measurements concern a variety of observables—dissociation energy of rovibrational levels, transition energies, ionization potential—of all isotopologues of molecular hydrogen.

challenge for the theory to keep up with this progress. The theory is expected not only to increase the accuracy of the leading components of the total energy but also to account for the unknown higher order terms in the α-expansion (3). Although the contribution from such terms is relatively small and could have been neglected for some time, the progress on the experimental side and the consequent expectations placed upon the theory compel the study of these ever-smaller components. An important physical quantity studied theoretically and experimentally is the dissociation energy of the rovibrational ground level, D0,0. A survey of advances in determination of D0,0 for H2 covering period of 1926–2009 can be found in Ref. 94. Here we shall quote a small sample of existing theoretical data contributed by Polish scientists, which illustrates the progress made in this field in juxtaposition with the recent experimental result (Table 10). The agreement achieved between calculations and experimental results is not given once and for all. An excellent example is the dissociation energy of the lightest isotopologues of the hydrogen molecule, for which full compliance was achieved in 2010. However, after a dozen years, tightening the uncertainties on both sides of the competition has caused discrepancies to reappear (see Fig. 3).

24

Mariusz Puchalski et al.

Table 10 Total dissociation energy D0,0 of H2 including nonrelativistic (NR), relativistic (REL), quantum-electrodynamic (QED), and higher order QED (HQED) components. Year Reference, components, wave function D0,0/cm21

1964 Kołos and Wolniewicz,71 NR+REL, JC 1983 Wolniewicz, 1986 Kołos et al.,

52

25

NR+REL+QED, KW

NR+REL, KW

36 114.2 36 118.01(1) 36 118.088

1993 Wolniewicz,

27

NR+REL, KW

36 118.060

1995 Wolniewicz,

76

NR+REL, KW

36 118.069

2009 Piszczatowski et al.,57 NR+REL+QED, ECG

36 118.069 5(10)

58

2011 Komasa et al., NR+REL+QED+HQED, JC+ECG 36 118.069 6(1) 2019 Puchalski et al.,86 NR+REL+QED+HQED, naJC+naECG+ECG

36 118.069 632(26)

2018 Beyer et al.,95 EXPERIMENT

36 118.069 605(31)

The meaning of the wave function symbols is as follows: JC, James–Coolidge; KW, Kołos–Wolniewicz; ECG, explicitly correlated Gaussian; naECG, nonadiabatic ECG; naJC, nonadiabatic JC. For comparison, the most accurate experimental D0,0 is shown at the bottom of the table.

Fig. 3 Changes in accuracy and agreement between theory86,91,96 and experiment10,97–99 of the dissociation energy D0,0 of H2, HD, and D2 observed over the last decade. The zero level corresponds to the mean of the theoretical and experimental values.

From the KW calculations to the QED treatment of the hydrogen molecule

25

In the further part of this section, we will present, in a tabular or graphical form, a comparison of theoretical and experimental results for the energy of rovibrational excitations in H2, starting from the energetically lowest and ending with excitations to states close to the dissociation threshold. We will also show similar comparisons for other isotopic variations of the hydrogen molecule. Table 11 shows the transition energy between the ground and the first rotationally excited level for three isotopologues H2, HD, and D2. Theoretical results were obtained using state-of-the-art methods and wave functions.86 The experiments come from three different experimental research groups.10,97,100 As inferred from the table, the differences fit in the combined uncertainties. Fig. 4, in turn, displays just the differences between theoretical90 and experimental101 results for the growing rotational quantum number, J, in the ground vibrational level of H2. As can be seen, the theoretical uncertainty band fits within the experimental error bars. The only exception is the J ¼ 4 dot, which lies outside the expected region for unknown reasons. There is an agreement between theory and experiment at the level of 103 cm1 also for higher energy regions. It is demonstrated in Fig. 5 and Table 12 referring to higher vibrational bands. There is a growing number of experimental data on transition energies in heavier isotopologues of the hydrogen molecule, which enables simultaneous verification of calculations and measurements. Generally, the heavier the nuclei, the smaller the finite nuclear mass effect present in all the terms of α-expansion (3). Hence, the corresponding theoretical predictions usually have tighter uncertainties related to this effect. Table 13 exemplifies the level of agreement for selected rovibrational transitions in D2 molecule. Advances in handling radioactive samples enabled accurate spectroscopic measurements of tritiated isotopologues of hydrogen molecule.108 The current state of agreement between theory and experiment can be illustrated by Fig. 6, which presents a comparison of the accuracy obtained in recent Table 11 Fundamental rotational splitting (0, 1) ! (0, 0) in cm1. H2 HD D2

Theory

118.486 812 7(11) 89.227 930 9(8)

59.780 615(3)

Experiment 118.486 770(50)

89.227 931 6(8)

Difference

0.000 000 7(11) 0.000 68(95)

0.000 042(50)

A comparison of calculated86 and experimental10,97,100 data.

59.781 30(95)

26

Mariusz Puchalski et al.

Fig. 4 Difference between theory and experiment for the lowest rotational energy levels of H2 (in units of 103 cm1). The blue band shows the range of theoretical uncertainties obtained in calculations using the H2Spectre program.90 The red dots with error bars come from the experiment.101

Fig. 5 Difference between theory and experiment for the rotational energy levels (J ¼ 1  7) of v ¼ 3  0 vibrational band of H2 (in units of 103 cm1). The blue band shows the range of theoretical uncertainties.102 The red dots with error bars come from the experiment.103

27

From the KW calculations to the QED treatment of the hydrogen molecule

Table 12 Highly excited vibrationally (v ¼ 11, 12) rotational energy levels (in cm1, relative to the ground level) of H2. Level (v, J)

Theory

(11, 1)

58

Experiment

(11, 3)

(11, 4)

(11, 5)

32 937.7494(53) 33 186.4802(52) 33 380.1019(52) 33 615.5293(51) 104

32 937.7554(16) 33 186.4791(16) 33 380.1025(33) 33 615.5371(18)

Difference

0.0060(55)

0.0011(54)

0.0006(62)

0.0078(54)

Level (v, J)

(12, 0)

(12, 1)

(12, 2)

(12, 3)

Theory58 Experiment Difference

34 302.1741(47) 34 343.8483(46) 34 426.2179(46) 34 547.3332(45) 105

34 302.1823(35) 34 343.8531(35) 34 426.2216(35) 34 547.3362(35) 0.008(6)

0.005(6)

0.004(6)

0.003(6)

Table 13 Comparison of selected rovibrational transitions (v0 , J0 ) ! (v00 , J00 ) in D2. (1, 0) ! (0, 0)

Theory

(2, 2) ! (0, 2)

(2, 2) ! (0, 0)

(2, 4) ! (0, 2)

2 993.617 048(9) 5 855.583 348(22) 6 034.650 446(24) 6 241.127 614(25)

Experiment 2 993.617 06(15) 5 855.583 358(14) 6 034.650 463(10) 6 241.127 637(17) Difference

0.000 01(15)

0.000 010(26)

0.000 017(26)

0.000 023(30)

Theoretical line positions were obtained using H2Spectre program90 whereas the experimental data come from Refs. 106, 107

Fig. 6 Differences between theory and experiment in the Q branch of the v ¼ 1  0 band in HT, DT, and T2. Theoretical results (blue dots with error bars)72 superimposed on the experimental uncertainty band (in red).110

measurements11,109,110 for the line positions in Q branch of the v ¼ 1  0 band with the corresponding theoretical predictions.72 The increasing accuracy of both calculations and measurements can reveal discrepancies at a higher level. For instance, there is a systematic

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Fig. 7 The discrepancy between theory and experiment in the R2(1) line position of HD. Theoretical result85 (open blue square) is compared to the spectroscopic data (filled red circles) of several independent measurements.111–115

difference of ca. 6  105 cm1(1.9 MHz) between several observed and calculated line positions in HD molecule. A sample of such a discrepancy is depicted in Fig. 7 for the R2(1) line, for which independent measurements agree with each other but differ with the theoretical value. Currently, the source of this discrepancy is unknown. The presumed cause may lie in the as-yet-unknown finite mass effects in the QED correction.

6. Summary The total theoretical energy of a bound rovibrational level of the hydrogen molecule is composed of terms defined in the α-power series (Eq. 3). The least accurate or cutoff terms of this series determine the accuracy of the total energy. Therefore, increasing the accuracy of the dominating terms has to be accompanied by accounting for subsequent smaller and smaller components of this series. For example, the accuracy of the theoretical dissociation energy of H286 is currently limited by the estimated uncertainty of the unknown correction of the order α7 (see Fig. 8A). Another instance concerns the most accurately measured energy of the R(1) transition of (2-0) vibrational band in HD. The line position is calculated as a difference between two rovibrational energy levels, and its uncertainty accounts

From the KW calculations to the QED treatment of the hydrogen molecule

29

Fig. 8 Examples of state-of-the-art uncertainties of the components of selected observables. (A) Uncertainties of the components of the dissociation energy D0,0 of H2. (B) Uncertainties of the components of the R(1) transition energy of (2–0) vibrational band in HD.

for the cancellation of errors of individual contributions. In this case, the total uncertainty is dominated by the uncertainty of the QED term; see Fig. 8B. These two examples show where the current studies should concentrate on to make further progress. In this article, we have presented an overview of research conducted by Polish scientists aimed at constructing methods for a more and more accurate description of the electronic structure of the hydrogen molecule. Studies initiated by Włodzimierz Kołos over 60 years ago were continued by the next generations of researchers at the Universities of Warsaw, Toru
n, and Pozna
n. The timeline of the most significant events related to the development of such methods is summarized in Fig. 9.

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Fig. 9 The chronological overview of key events in the development of theory and methods of description of the hydrogen molecule involving Polish researchers.

Acknowledgments This research was supported by National Science Center (Poland) Grants No. 2019/34/E/ ST4/00451, 2017/25/B/ST4/02698 and 2021/41/B/ST4/00089.

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REVIEW

How to make symmetry-adapted perturbation theory more accurate? Tatiana Koronaa, Michał Hapkaa, Katarzyna Pernalb, and Konrad Patkowskic,∗ a

Faculty of Chemistry, University of Warsaw, Warsaw, Poland Institute of Physics, Lodz University of Technology, Lodz, Poland Department of Chemistry and Biochemistry, Auburn University, Auburn, AL, United States ∗ Corresponding author: e-mail address: [email protected] b c

Contents 1. Introduction 2. Theoretical foundations of SAPT 3. Making SAPT more accurate for typical systems 3.1 Formulation through monomer properties 3.2 SAPT(CC) 3.3 Improvements to SAPT(DFT) 3.4 Exchange energies beyond the S2 approximation 3.5 Explicitly correlated SAPT 4. Enabling accurate SAPT data for new systems 4.1 Multireference SAPT 4.2 Spin-flip SAPT for multiplet splittings 5. Summary Acknowledgments References

38 39 44 44 49 53 55 59 61 61 63 66 67 67

Abstract Symmetry-adapted perturbation theory (SAPT), one of the crowning achievements of the Polish quantum chemistry school with contributions from many other countries, provides accurate intermolecular interaction energies together with their decomposition into physically meaningful terms. Over the past 15 years, SAPT has evolved in several directions intended to extend both its accuracy for typical intermolecular complexes and its applicability to a broader group of interacting systems. In this contribution, we review some of these recent developments, including SAPT(CC), i.e., SAPT with monomers described at the coupled-cluster level, multireference SAPT and spinflip SAPT, where the last two allow to treat molecules and complexes that cannot be

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.04.002

Copyright

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2023 Elsevier Inc. All rights reserved.

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Tatiana Korona et al.

described well by a single determinant, explicitly correlated SAPT for an enhanced basis set convergence, further developments enhancing the accuracy and performance of SAPT(DFT), and the elimination of the commonly used S2 approximation from several SAPT exchange corrections.

1. Introduction Weak intermolecular interactions are ubiquitous, but their strength is substantially lower than the covalent and ionic interactions underlying chemical bonding. Therefore, a reliable description of these interactions calls for both high accuracy and meaningful insights into the nature of interaction. Such insights can be distilled from an interaction energy decomposition, a technique that partitions the entire energetic effect of an intermolecular interaction into physically meaningful terms. Many computational methods can provide accurate interaction energies but no decomposition; the supermolecular coupled-cluster approach with singles, doubles, and noniterative triples (CCSD(T)), commonly used to produce benchmark “gold standard” noncovalent data, is a prime example. There are also many approaches that provide meaningful energy decompositions but no accurate overall interaction energy, such as the broad class of energy decomposition analysis (EDA) methods which identify physical components of the supermolecular Hartree–Fock (HF) or density functional theory (DFT) interaction energy.1,2 Symmetry-adapted perturbation theory (SAPT) is a unique approach that provides both reliable intermolecular interaction energies and their decomposition into physically meaningful components. The SAPT expansion starts from a noninteracting wavefunction (a product of, exact or approximate, wavefunctions of isolated fragments) and computes interaction energy terms in different orders with respect to the intermolecular interaction. While the first attempts at a SAPT expansion date back to the very beginning of quantum chemistry,3 the foundations of the modern SAPT methodology were laid out in the 1970s and 1980s by Polish quantum chemists: Jeziorski, Kołos, Chałasi
nski, Szalewicz, and coworkers.4–7 Since then, many further developments, which collectively transformed SAPT into a practical and successful general-purpose method, have been made by these original authors, their Polish students and coworkers (including the authors of this review), and many other researchers all over the world. It is therefore fitting that a volume highlighting the accomplishments of Polish quantum chemistry includes a detailed account of SAPT.

39

How to make symmetry-adapted perturbation theory more accurate?

The most comprehensive review covering the first two decades of SAPT development and use cases was written in 1994 by Jeziorski, Moszy
nski, and Szalewicz.8 Since then, the number of available SAPT variants, and the breadth of applications, has grown so much that no single work has matched the completeness of Ref. 8, but various specific aspects of SAPT have been thoroughly described in a number of reviews.9–11 This review, while highlighting the accomplishments of Polish quantum chemists in the development of SAPT, is focused on the accuracy of both the total interaction energies and their SAPT components, and on some recent efforts by multiple groups, including ours, to raise this accuracy to a higher level for a wide variety of intermolecular complexes.

2. Theoretical foundations of SAPT In its core, SAPT is a theory where the unperturbed wavefunction ψ 0 ¼ ψ Aψ B is a product of wavefunctions for noninteracting molecules (monomers) A and B, and the perturbation operator V contains all Coulomb interactions of the electrons and nuclei of A with those of B. In this way, the first-order correction describes the electrostatic interaction ð1Þ

E elst between the unperturbed charge distributions of molecules A and B. At long range, this interaction can be expressed by the multipole expansion and interpreted through its dipole–dipole, dipole–quadrupole, … contribuð1Þ

tions, but E elst remains valid at short range when the multipole approximation breaks down and charge penetration effects appear. The second-order correction naturally decomposes into three parts: ð2Þ

E ð2Þ ¼ E ind ðA

ð2Þ

BÞ + E ind ðB

ð2Þ

AÞ + E disp ,

(1)

where the first two terms quantify the induction energy, where the electric field generated by the (unperturbed) monomer B polarizes monomer A and vice versa. The last term in Eq. (1) denotes the dispersion energy, an effect of intermolecular electron correlation coupling the charge density fluctuations ð1Þ

on A and B. Just like for Eelst, the second-order SAPT terms are not limited to their multipole approximations. The simple perturbation theory outlined above, usually referred to as the polarization approximation,12 has two important limitations. First, while the exact wavefunction ψ is antisymmetric with respect to an exchange of any two electrons, the zeroth-order wavefunction ψ 0 is only antisymmetric

40

Tatiana Korona et al.

when the electrons are exchanged within the same monomer (A or B). As a result, ψ and ψ 0 are not close in a Hilbert space sense, and the perturbation results in a large change in the wavefunction even at very large intermolecular distances. A practical consequence of the incomplete antisymmetry of ψ 0 is that the polarization approximation misses the effects resulting from exchange tunneling of electrons between the interacting monomers. To account for the exchange terms, one needs to insert appropriate symmetry projections into the perturbation equations, that is, perform a symmetry adaptation or symmetry forcing. The specific form of symmetry forcing is not unique, and several different variants have been proposed— see Ref. 8 for a review. However, except for convergence studies in model few-electron complexes, all practical SAPT calculations utilize the simplest (and weakest) symmetry forcing, the symmetrized Rayleigh–Schr€ odinger (SRS) method.7 In the SRS approach, all perturbation corrections to the wavefunction are obtained straight from the polarization approximation, and only afterward the exchange terms are computed using an antisymmetrized energy expression. As a result, each of the SAPT corrections ð1Þ

ð2Þ

described so far acquires an exchange counterpart: E exch , E exchind , ð2Þ

E exchdisp , …These exchange corrections are the primary source of shortrange repulsion between molecules, quenching the respective polarization terms at small intermolecular distances but not affecting their long-range asymptotic behavior. The other limitation of the single perturbation theory outlined so far is that it formally requires exact wavefunctions ψ A, ψ B for the noninteracting monomers. Getting such wavefunctions requires full configuration interaction (FCI) calculations for A and B, a task that is unfeasible for all but the smallest monomers. As a simplistic (often too simplistic) alternative, one could start the perturbation expansion from the product of monomer HF HF determinants, ϕHF A ϕB . The resulting SAPT corrections are entirely missing intramolecular electron correlation while including some intermolecular correlation, e.g., via dispersion. The simplest approach of this kind, SAPT0, is obtained when the expansion in V is truncated at the second order: ð10Þ

ð10Þ

ð20Þ

ð20Þ

ð20Þ

ESAPT0 ¼ Eelst + E exch + E ind;resp + E exchind;resp + E disp int ð20Þ

ð2Þ

+ Eexchdisp + δE HF ,

(2)

41

How to make symmetry-adapted perturbation theory more accurate?

where the second, zero superscripts indicate the neglect of intramolecular correlation, the additional subscript “resp” denotes response (relaxation) of monomer HF orbitals in the electric field of the interacting partner, and the last “HF delta” term accounts for third- and higher-order induction and exchange-induction effects within the supermolecular HF approximation.8 The SAPT0 approach defined by Eq. (2) is quite inaccurate at the ð20Þ

complete basis set (CBS) limit, mainly due to Edisp overestimating the true dispersion energy. However, it exhibits quite a consistent error cancellation in smaller basis sets, leading to qualitatively correct total interaction energies and a meaningful and useful energy decomposition.13 In this way, an important application of SAPT0 is as a qualitative tool to classify different intermolecular complexes according to the type of interaction present. When the SAPT0 approximation is too crude, the first step to HF improvement is noting that the choice of ϕHF A ϕB for the zeroth-order wavefunction implies the choice of FA + FB, the sum of monomer Fock operators, for the zeroth-order Hamiltonian. This operator differs from the true Hamiltonian of the complex not just by the lack of the intermolecular interaction operator V, but also by the lack of the Møller– Plesset fluctuation potential W ¼ WA + WB, where WX ¼ HX  FX distinguishes the exact molecular Hamiltonian HX, X ¼ A,B, from its HF approximation. Thus, the effects of both V and W can be obtained from a truncated double perturbation expansion, in the formalism that is comðmnÞ

monly called the many-body SAPT.14 The resulting corrections E SAPT are labeled by the order with respect to each perturbation, m for V and n for W (note that this notation has already been used in Eq. (2)). A number of leading-order many-body SAPT corrections incorporating intramolecular ð21Þ

ð22Þ

correlation, for example, E disp and E disp , have been developed and implemented in several quantum chemistry codes. One can also go beyond second order in V: the third-order polarization correction separates into pure induction, pure dispersion, and mixed induction–dispersion terms, and each of these terms has a corresponding exchange counterpart resulting from the symmetry adaptation.15 Adding a specific set of intramolecular correlation terms (and possibly also some third-order terms) on top of SAPT0 leads to one of the predefined levels of many-body SAPT interaction energies, from the most approximate SAPT2, through SAPT2+ and SAPT2+(3), to the formally least approximate SAPT2+3 level.13

42

Tatiana Korona et al.

An exhaustive set of SAPT level and basis set combinations has been benchmarked against high-accuracy CCSD(T)/CBS interaction energies to identify the best strategies for computing many-body SAPT data.13 Another possibility of going beyond SAPT0 is based on the coupledcluster treatment of the wavefunctions for monomers, giving rise to the SAPT(CC) methods described in detail in Section 3.2. Overall, higherlevel SAPT variants provide significantly more accurate interaction energies than SAPT0, and their accuracy, unlike for SAPT0, generally increases with the basis set size. However, these variants are also significantly more expensive: the most demanding currently available SAPT corrections ð22Þ

ð30Þ

E disp and E exchdisp scale like N 7 with the system size, just like CCSD(T). A more economical alternative for including intramolecular correlation effects in SAPT is provided by DFT. The original idea, dating back to Williams and Chabalowski,16 was to replace the HF orbitals and orbital energies in many-body SAPT expressions by their Kohn–Sham (KS) counterparts, hoping that corrections beyond the zeroth order in “W” (the post-KS fluctuation potential) will not be needed as DFT already accounts for intramolecular correlation. This is a very appealing idea for multiple reasons: DFT is only asked to do things it does well (recover short-range correlation), while the treatment of long-range correlation (dispersion), a notoriously difficult problem for standard DFT,17 is delegated to SAPT, which is also tasked with providing an energy decomposition. However, the simple replacement of HF quantities by KS ones led to a very poor interaction energy accuracy, and several important improvements needed to be made to transform Williams and Chabalowski’s idea into a practical approach. First, standard density functionals yield exchange-correlation (XC) potentials with an incorrect long-range asymptotic behavior, which causes an artificial “overflow” of charge density into the asymptotic region.18,19 While this issue has no bearing on supermolecular DFT calculations, the KS orbital-based SAPT corrections are highly inaccurate unless the XC potential is asymptotically correct, that is, decays with distance like 1/r + (I + EHOMO), where I is the molecular ionization potential and EHOMO is the energy of the highest occupied molecular orbital.20 This issue can be alleviated by asymptotically correcting the XC potential, which can be accomplished by splicing the short-range potential from the underlying XC functional with a properly shifted long-range form.21 Alternatively,

How to make symmetry-adapted perturbation theory more accurate?

43

the shifting and splicing can be avoided by the use of an optimally tuned long-range-corrected (LRC) density functional.22,23 In either case, the use of SAPT0 expressions employing KS orbitals and orbital energies coming from an asymptotically correct XC functional (a method usually denoted as SAPT(KS)) leads to accurate electrostatic, exchange, and induction corrections, but the dispersion terms remain poor. The latter issue is resolved when the SAPT0-like dispersion expression, corresponding to the uncoupled KS theory, is replaced by a generalized Casimir– Polder integral involving monomer frequency-dependent density susceptibilities (FDDSs) evaluated at imaginary frequencies.24,25 The FDDSs, just like other linear response functions, can be evaluated with standard algorithms of time-dependent DFT (TD-DFT), leading to the coupled Kohn–Sham (CKS) value for dispersion energy. The replacement of the uncoupled KS dispersion and exchange-dispersion by the CKS ones in the asymptotically corrected SAPT(KS) approach leads to a method called SAPT(DFT) or DFT-SAPT.26,27 Multiple benchmark studies have shown that SAPT(DFT) provides high and consistent accuracy, vastly surpassing SAPT0 and rivaling high-level many-body SAPT. At the same time, the computational complexity of SAPT(DFT) is only somewhat higher than for SAPT0: the most expensive part of the calculation, the CKS dispersion energy, scales like N 6 with the system size, brought down to N 5 when density fitting (DF) is introduced.26,27 The many-body SAPT and SAPT(DFT) methods form the core of the modern toolkit for computing and decomposing intermolecular interactions. Both approaches have highly efficient computer implementations, enabling large calculations through DF of all four-electron tensors and many other improvements.28–32 SAPT0 and SAPT(DFT), both initially developed for closed-shell monomers, have been extended to interactions between high-spin open-shell molecules, described by either spin-restricted open-shell determinants (ROHF/ROKS) or unrestricted determinants (UHF/UKS).33–35 Finally, both many-body SAPT and SAPT(DFT) have been applied to compute and decompose interaction energies in numerous complexes of experimental interest, leading to many groundbreaking insights. Enumerating the successful SAPT applications is beyond the scope of this article; two recent reviews11,36 point to a broad range of examples. Despite their popularity and numerous successes, the established manybody SAPT and SAPT(DFT) approaches have some limitations. First, the

44

Tatiana Korona et al.

range of open-shell complexes that can be studied is severely limited: not only each monomer has to be described by a single high-spin determinant, but the dimer has to be in a high-spin state as well. This means that the core SAPT variants cannot describe interactions of molecules with a multireference character and cannot be used to compute splittings between different spin states of the complex. As far as the accuracy of the existing SAPT variants is concerned, one could naively think that the best way to further increase the accuracy would be to derive and implement more higher-order SAPT corrections. However, such a strategy is not only very difficult (as the terms get more and more complex), but unlikely to succeed as (1) at the currently available level of many-body SAPT, there is no single missing correction that can be expected to dominate the residual errors, and (2) the underlying perturbation expansions in both V and W are actually divergent.37,38 Instead, one could strive to reduce residual SAPT errors arising from an imperfect treatment of intramolecular electron correlation, an incomplete basis set, or approximations used to compute most SAPT exchange terms. The next sections of this review showcase recent efforts, by us and by others, to combat all those limitations.

3. Making SAPT more accurate for typical systems 3.1 Formulation through monomer properties An elegant and systematic way of introducing various electronic structure levels of monomers is based on expressing individual SAPT components through properties of monomers, such as densities, density matrices, FDDSs, etc. This allows for a development of SAPT based on any electronic theory for which these properties are properly defined. For a concise notation, a generalized interaction potential v(i, j)39 will be introduced as (in atomic units), 1 1 X Zα 1 X Zβ 1 X Z αZ β vði, jÞ ¼   + : (3) Rij N A α Rαj N B β Rβi N A N B αβ Rαβ In Eq. (3), i (j) label electrons of monomer A (B), α (β) label nuclei of A (B) (with atomic numbers Zα, Zβ), Rxy denotes the distance between

45

How to make symmetry-adapted perturbation theory more accurate?

particles x and y, and NA, NB denote the number of electrons for the appropriate monomer. With this notation, the first-order electrostatic contribution can be rigorously calculated from the formula, ð1Þ Eelst

Z ¼

γ A ð1Þvð1, 10 Þγ B ð10 Þdτ1 dτ01 ,

(4)

where γ A(1) and γ B(1) are unperturbed one-electron densities of A and B, and the number(s) in parentheses symbolize the coordinates of electron 1. These densities represent a diagonal part of the one-electron reduced density matrix (1-RDM), defined from the wavefunction of N electrons as γð1j10 Þ ¼ N

Z

Ψð1, 2, …, N Þ? Ψð10 , 2, …, N Þdτ2 …dτN ,

(5)

with γ(1) ≡ γ(1j1). The second-order induction and dispersion contributions require monomer FDDSs, which are defined as, αðr, r’jωÞ ¼ hh^ ρðrÞ; ρ^ðr’Þiiω

ρ^ðrÞ ¼

N X

δðr  ri Þ,

(6)

i¼1

^ Y^ iiω is the polarization propagator, which describes the and where hhX; linear response of a molecule to perturbing operators X^ and Y^ oscillating with a frequency ω, 

 ^ Q ^ Y^ iiω ¼  Ψ0 jY^ ^ 0 hhX; XΨ ^  E0 + ω H   ^ Q  Ψ0 jX^ Y^ Ψ0 : ^  E0  ω H

(7)

^ , Ψ0, and E0 are the electronic Hamiltonian, ground-state In Eq. (7) H wavefunction, and energy for the unperturbed molecule, respectively, ^ is a projector on the space orthogonal to Ψ0. With these definitions, and Q the second-order dispersion energy can be calculated as the integral40 ð2Þ E disp

1 ¼ 2π

Z 0

∞

Z

αA ð1, 10 jiωÞαB ð2, 20 jiωÞ

1 1 dτ dτ0 dτ dτ0 dω, (8) r 12 r 10 20 1 1 2 2

46

Tatiana Korona et al.

and the second-order induction energy as Z 1 ð2Þ E ind ¼  ωB ð1ÞαA ð1, 10 j0ÞωB ð10 Þdτ1 dτ01 + A , B, 2

(9)

where ωB(1) is the complete electrostatic potential of monomer B, Z γ B ð10 Þ 0 ωB ð1Þ ¼ vB ð1Þ + dτ1 , r 110

(10)

and A , B denotes that the second half of the formula is obtained by replacing A and B indices. The induction energy can also be computed from a simpler formula based on the first-order density matrix, which is defined as, Z ð1Þ 0 γ ð1j1 Þ ¼ N Ψð1, 2, …, NÞ? Ψð1Þ ð10 , 2, …, N Þdτ2 …dτN , (11) where Ψ(1) is the first-order wavefunction of the molecule in response to some perturbation, which in the case of monomer A is the ωB operator. With this definition, a more practical formula for the second-order induction is, Z ð2Þ ð1Þ Eind ¼ γ A B ð1j1ÞωB ð1Þdτ1 + A , B, (12) where one can note that Eq. (11) can be applied to the definition of a transition density matrix (TDM) γ K!L after replacing the ket and bra wavefunctions by two eigenfunctions ΨK and ΨL of the Hamiltonian. Exchange contributions result from electron exchanges between monomers. Within the S2 approximation discussed in detail in Section 3.4, the first- and second-order exchange contributions can be expressed by two-electron reduced density matrices (2-RDMs), Γð12j10 20 Þ ¼ N ðN  1Þ

Z Z

Ψð1, 2, 3, …, N Þ? Ψð10 , 20 , 3, …, N Þdτ3 …dτN , (13) ð1Þ

and some generalizations of FDDSs. The first such expression for E exch was proposed by Moszy
nski et al.41 Subsequently, Korona42 utilized the partitioning of the 2-RDM into the antisymmetrized product of 1-RDMs and the remaining true 2-body part (denoted as a cumulant Λ),

47

How to make symmetry-adapted perturbation theory more accurate?

Γð12j10 20 Þ ¼ γð1j10 Þγð2j20 Þ  γð1j20 Þγð2j10 Þ+Λð12j10 20 Þ,

(14)

to express the first-order exchange energy as a sum of four contributions: the first expressed through 1-RDMs, the second and the third being mixed terms utilizing 1-RDMs and cumulants, and the last—defined through cumulants only.42 None of the terms in the resulting expressions can be factorized as products of two integrals; therefore, these formulae are sizeextensive as long as the applied property method is size-extensive. The second-order exchange-dispersion energy in the S2 approximation can be expressed through generalizations of FDDSs, which can be defined43 either through the polarization propagator formalism, or by using a sum-over-states expression and one- and two-electron transition density matrices. The second way gives the following formulas for the so-called density-matrix susceptibilities α and αe, 0

αð1j1 ; 2jiωÞ ¼ 2 αeð12j10 20 ; 3jiωÞ ¼ 2

∞ X K¼1 ∞ X

γ 0!K ð1j10 Þγ K!0 ð2j2Þ

ωK ; ðωK Þ2 + ω2

Γ0!K ð12j10 20 Þγ K!0 ð3j3Þ

K¼1

ωK , ðωK Þ2 + ω2

(15) (16)

where ωK is the excitation energy of state K and Γ0!K is the two-electron TDM, which can be expressed as an antisymmetrized product of 1-RDMs and 1-TDMs and a cumulant in a spirit of Eq. (14), Γ0!K ð12j10 20 Þ ¼ γ 0!K ð1j10 Þγð2j20 Þ  γ 0!K ð1j20 Þγð2j10 Þ γ 0!K ð2j10 Þγð1j20 Þ + γ 0!K ð2j20 Þγð1j10 Þ + Λ0!K ð12j10 20 Þ: (17)

This, in turn, allows for the representation of the αe susceptibility through products of α’s and 1-RDMs and an additional cumulant susceptibility component. The α(1j10 ;2jiω) quantity for a purely imaginary frequency iω can be defined analogously to the FDDS, with the left density operator replaced by its density-matrix analog.43 We can now express the second-order exchange-dispersion energy in ð1Þ

terms of monomer properties. Similarly to E exch , four terms can be distinguished depending on the presence or absence of the cumulant contribution. The noncumulant (n-n) term is given by the expression,

48

ð2Þ

Eexchdisp,nn ¼

Tatiana Korona et al.

Z Z 1 ∞ αA ð1j10 ; 3jiωÞαB ð10 j1; 30 jiωÞvð1, 10 Þ 2π 0 1 dτ dτ0 dτ dτ0 dω r 330 1 1 3 3 Z Z 1 ∞ B 0 0 0 0 vð1, 10 ÞαA ð1j20 ; 3jiωÞAB + 10 20 A10 1 αB ð1 j1 ; 3 jiωÞγ B ð2 j1Þ 2π 0 1 dτ dτ0 dτ0 dτ dτ0 dω r 330 1 1 2 3 3

Z Z 1 ∞ A 0 0 0 + vð1, 10 ÞAA 12 A110 αA ð1j1; 3jiωÞγ A ð2j1 ÞαB ð1 j2; 3 jiωÞ 2π 0 1 dτ dτ0 dτ dτ dτ0 dω r 330 1 1 2 3 3

+

Z Z 1 ∞ A B B A A B B vð1, 10 ÞðAA 12 A120 A10 20 A10 2  1  π 12 π 120 π 10 20 π 10 2 Þ 2π 0 αA ð1j1; 3jiωÞγ A ð2j20 Þ

 αB ð10 j10 ; 30 jiωÞγ B ð20 j2Þ

1 dτ dτ0 dτ dτ0 dτ dτ0 dω, r 330 1 1 2 2 3 3

(18)

where the following auxiliary operations are defined on a product of two functions F(ijj)G(kjl): π jl (π *ik ) permutes coordinates which are placed on the right (unstarred version) or on the left (the starred version) of the vertical bar, Ajl ¼ 1  π jl and A*ik ¼ 1  π *ik , and finally the superscript A or B denotes that the operations are performed on quantities for the monomer A or B only. Similar expressions containing cumulants will not be listed here (see Eq. (23) of Ref. 43). Similarly to the second-order induction energy, which can be expressed through FDDSs for ω ¼ 0 (see Eq. (9)), also the second-order exchangeinduction energy in the S2 approximation can be described through static density-matrix susceptibilities (see Eq. (49) in Ref. 43). However, it is more practical to use an expression with just first-order density matrices and related quantities. With this in mind, the second-order exchange-induction energy in the S2 approximation resembles closely the structure of the ð1Þ

E exch ðS2 Þ expression with some 1-RDMs or cumulants replaced by their first-order counterparts (the order is counted with respect to the oneelectron electrostatic potential of the other monomer). Additionally, each product of 1-RDMs transforms into a sum of products with the left

How to make symmetry-adapted perturbation theory more accurate?

49

1-RDM replaced by the first-order 1-RDM and the right 1-RDM unchanged and vice versa. This can be understood by looking at the representation of the first-order 2-RDM (defined by replacing Ψ by Ψ(1) for the ket function in Eq. (13)), which can be described in terms of 1-RDMs, first-order 1-RDMs, and first-order cumulants (Λ(1)) as,   Γð1Þ ð12j10 20 Þ ¼ A*12 γ ð1Þ ð1j10 Þγð2j20 Þ  γ ð1Þ ð1j20 Þγð2j10 Þ + Λð1Þ ð12j10 20 Þ:

(19)

This naturally leads to more terms in the equations for the n-n, noncumulant-cumulant, and cumulant-noncumulant parts of the secondorder exchange-induction contribution relative to Eq. (14) (see Eq. (18) of Ref. 44).

3.2 SAPT(CC) The formalism described in the previous section can be used for all electronic structure theories for which the required monomer properties have been derived. In many cases, this task is neither straightforward nor unique because the approximate theory does not fulfill the Hellmann–Feynman theorem, connecting the derivative of the energy over the perturbational parameter with the average value of the perturbing operator. A prominent example is the CC theory, for which one can either define first-order properties by utilizing the derivative and applying the energy Lagrangian, or use the average value of the perturbing operator as a starting point. For second- and higher-order properties, the Lagrangian approach has been further developed to either the so-called response theories45 or time-independent theories.46 It turns out (see the previous section) that the second approach possesses the properties necessary to define a proper partitioning of 2-RDMs and 2-TDMs into 1-RDMs, 1-TDMs, and cumulants obtained at the same level of theory, while the response approach cannot be used for this purpose.47 Therefore, for the development of SAPT(CC), the so-called expectation-value CC theory (XCC)48 and its generalizations to second-order properties46 are utilized. In the XCC method, the first-order property for the operator X^ is calculated from the following expression, { ^ T eS{ Φi, X ¼ hΦjeS eT Xe

(20)

50

Tatiana Korona et al.

which has been derived from the expectation-value expression with the normalized CC ansatz inserted in place of the exact wavefunction. In this formula, a new size-extensive excitation operator S has been introduced,48 which in the first approximation is just equal to T and which differs from T by terms of the third power of the fluctuation potential W (i.e., OðW 3 Þ). The S operator is specifically approximated as ^1 S¼T +P

h i hh i i 1^ T {1 , T 2 + P T {2 , T 2 , T 2 : 2 2

(21)

^ n is used, which projects the operator In this equation, the superoperator P ^ on the space spanned by n-tuple excitation operators, e.g., for the operator X, X X an a1 ^ n ðXÞ ^ ¼ 1 ^ P hΦjeia11 …eiann XΦie i1 …ein , ðn!Þ2 a1 …an i1 …in

(22)

where eqp ¼ aq ap for ar and ar being a creation and annihilation operator, respectively, for the rth spinorbital. The form of Eq. (20) ensures that the first-order property is size-extensive for a connected operator X^ , since it can be expanded in the form of nested commutators. In practical implementations of Eq. (20), systematic size-extensive truncations can be proposed, ignoring small terms of high T or W orders. Eq. (20) serves as the starting point for deriving both 1-RDMs and 2-RDMs. The exponential similarity-transformed form allows to represent a 2-RDM as an antisymmetrized product of 1-RDMs calculated on the same XCC level, while the XCC cumulant obtained as a difference is connected by construction and as such size-extensive.42,47 In contrast, CC properties obtained from the CC response theory do not possess this property and formally cannot be used to define a proper cumulant. The explicit form of the cumulant elements derived from Eqs. (20) and (14) reads D  {  E { { { Λpq11qp22 ¼ ΦjeS eT epq11 eT eS P^ eS eT epq22 eT eS Φ D ED E { { { {  ΦjeS eT aq1 ap2 eT eS Φ ΦjeS eT epq21 eT eS Φ ,

(23)

^ stands for the superoperator P^1 + P^2 + P^3 + … . In practice, where P truncated forms of Eq. (23) are utilized, with the use of the nested commutator expansion in order to preserve size-extensivity. To this end, the cumulant within the CCSD(3) approximation is given by the formula,49

51

How to make symmetry-adapted perturbation theory more accurate?

 h i h i  h i Λpq11qp22 ¼ hΦj epq11 P^1 epq22 , T 2 + S{2 , epq11 P^1 epq22 + epq22 , T 2 nh +

i h io nh h ii  o  S{2 , epq11 P^2 epq22 , T 2 + S{2 , epq11 , T 2 P^1 epq22 Φi C

  + O W3 ,

C

(24)

where the notation XC means the connected part of the operator X. The FDDSs necessary for second-order SAPT terms are derived from a general formula for the time-independent CC polarization propagator,46 which for imaginary frequencies and real functions is written as, D   E ^ Y^ iiiω ¼ 2 eS eT { Y eT { eS ΦjP^ eS{ ΩX ðiωÞ eS{ Φ , hhX;

(25)

where ΩX(iω) is the first-order response of the CC amplitudes to the perturbation X^ oscillating with frequency iω (the same as in response CC theory). Analogously to the first-order property, this expression could be evaluated by using the nested commutator expansions with truncations made in a similar manner as those leading to Eq. (24). For the CCSD theory, the CCSD(3) approximation has been proposed49 where the propagator is truncated after all terms which are of at most third order in terms of the fluctuation operator W, ^ Y^ iiiω ¼ 2ðhΦjY ΩX1 ðiωÞΦi + hSΦj½Y , ΩX ðiωÞΦi hhX; + h½Y , S2 ΦjΩX1 ðiωÞΦi + h½½T { , Y , SΦjΩX ðiωÞΦiÞ,

(26)

with the S operator approximated as in Eq. (21). The algebraic expressions for the first- and second-order properties necessary for the evaluation of the SAPT(CCSD) components are presented in Refs. 42–44,49,50. Here, we want to make some remarks on the computational cost of the SAPT(CCSD) method for the case of the second-order dispersion and exchange-dispersion components. For other SAPT terms, i.e., electrostatics, induction, and their exchange counterparts, the standard CCSD equations are solved for the T amplitudes for both monomers. In the case of induction, one additional operator (Ω) per monomer should be obtained, which describes the response of one monomer to the complete electrostatic poten^X of the other monomer, tial ω ^ T , ΩΦ ¼ eT ω ^X eT Φ: ½eT He

(27)

52

Tatiana Korona et al.

Since the calculation of such a response scales with the same power of 6 molecular size N as the calculation of the T operator, i.e., OðN ), the calculation of second-order induction and exchange-induction energies in the SAPT(CC) formalism is only about two times more expensive than the calculation of the first-order SAPT(CC) components. In all cases, the cost is determined by the calculations of CC operators T and Ω. On the other hand, for dispersion (and exchange-dispersion) the whole polarization propagator matrix is needed, which in the algebraic picture is represented as, X αðr1 , r2 jωÞ ¼ Πprqs ðωÞ ϕp ðr1 Þϕq ðr1 Þϕr ðr2 Þϕs ðr2 Þ, (28) pqrs

where ϕm is the mth orbital, Πprqs ðωÞ is the four-index orbital representation of the polarization propagator (Πprqs ðωÞ ¼ hhE pq ; E rs iiω with E pq being the orbital replacement operator), and where the summation in Eq. (28) goes over all N orbitals (occupied and virtual). In practice, the integration over ω is performed via a numerical quadrature, so the polarization propagator needs to be evaluated at a number nfreq of frequencies. The value nfreq depends on the required accuracy and the selected quadrature, but usually about 10 frequencies is a safe choice. The most problematic part arises from the fact that for a given frequency, the responses for all orbital replacement operators E sr oscillating with this frequency should be obtained in order to construct the full four-index propagator matrix in Eq. (28). The cost of one such calculation is of the same magnitude as the cost of solving of Eq. (27), 8 so the total cost of these calculations is of the order of nfreq N . Luckily, this cost can be reduced by one order of magnitude by applying density fitting (DF) to Eq. (28) (for details, see Refs. 43,50). The main idea of this approximation is expressing the density operator in a basis of auxiliary functions instead of products of orbitals, X X ρ^ðrÞ ¼ ϕ*p ðrÞϕq ðrÞE pq  χ K ðrÞ^ χ K , with pq

χ^K ¼

X pq

K

DKpq E pq

and

ϕ*p ðrÞϕq ðrÞ

¼

X

DKpq χ K ðrÞ,

(29)

K

which reduces the number of calculated first-order responses (a dimension of auxiliary basis is usually 2–3 times larger than N). However, even with the DF formalism, the SAPT(CCSD) remains to be expensive and is usually used as benchmark for more approximate SAPT approaches.51,52

How to make symmetry-adapted perturbation theory more accurate?

53

3.3 Improvements to SAPT(DFT) A prerequisite for SAPT(KS) or SAPT(DFT) calculations is ensuring the correct asymptotic behavior of the XC potential. As mentioned, this may be achieved with LRC functionals provided that the range-separation parameter ω is adjusted so that the orbital energies become aligned with ionization potentials.53 Currently, finding system-specific ω values is possible either via IP tuning or using the global density-dependent scheme (GDD).54 A serious limitation of the former is that for large, small-gap systems, it leads to parameters which are too small—with 1/ω exceeding the size of the molecule the LRC functional effectively becomes semilocal. The GDD scheme avoids this problem.22,54,55 Additionally, it provides the optimal ω value in a single calculation, while optimal tuning requires several additional computations for each of the monomers. The potential benefit of GDD tuning in SAPT(DFT) was pointed out by one of us on the example of static polarizabilities and long-range van der Waals coefficients of polyacetylene chains.22 The first comparison of optimal tuning and GDD-based SAPT calculations was included in the work of Lao and Herbert56 who verified the accuracy of the “extended” XSAPT model based on Kohn–Sham orbitals for popular large complex datasets. The recommendation of Ref. 56 to combine SAPT with the GDD scheme has later been corroborated in a systematic study of Gray and Herbert.55 The authors reported that interaction energies obtained with the GDD tuning are either on par with or superior to the optimal IP tuning both for XSAPT and SAPT(KS) methods. Several years ago, Holzer and Klopper57 proposed avoiding asymptotic correction altogether by combining SAPT(DFT) with Green’s functionbased methods. The key idea behind the introduced GW-SAPT approach is to evaluate second-order induction and dispersion energies based on response properties from solutions of the Bethe–Salpeter equations which take quasiparticle energies as input. A different path to circumvent the limitations of XC potentials originating from DFT functionals was taken by Boese and Jansen.58 Following Ref. 59, the authors derived a ZMPSAPT method which uses potentials constructed from ab initio densities via the Zhao–Morrison–Parr scheme.60 Although promising results were obtained with both GW- and ZMP-SAPT, a broader assessment of their performance is still lacking. Perhaps the most straightforward way of improving the SAPT(DFT) accuracy, at least when polarization components are concerned, is to employ

54

Tatiana Korona et al.

better density functional approximations for the underlying Kohn–Sham description of the monomers. Hapka et al.61 examined whether metaGGA functionals offer any advantage over the (hybrid-)GGAs routinely used in SAPT(DFT) calculations. Their work revealed that pure nonempirical meta-GGAs agreed better with reference SAPT(CCSD) contributions than nonhybrid GGA models. SCAN062 was identified as the only functional which gave first-order energy terms of good quality without the asymptotic correction. However, none of the tested meta-GGAs outperformed asymptotically corrected SAPT(PBE0) both in terms of individual energy components and complete interaction energies. The impressive accuracy of SAPT(DFT) notwithstanding, it is worthwhile to recall two generic problems of the method.10,11,36 First, by its nature KS-DFT would not predict exact reduced density matrices even if the exact XC functional were available. RDMs entering the exchange components of SAPT(DFT) pertain to a KS noninteracting system so that they can only approximate RDMs corresponding to the interacting Hamiltonians of the monomers. Second, the FDDS computation is based on the frequency-independent XC kernels, i.e., the adiabatic approximation for the kernel is adopted. Going beyond this approximation is still an open problem in TD-DFT. Recent progress in the SAPT(DFT) codes should also be mentioned. The SAPT2020 program of Szalewicz and coworkers36 includes a new implementation of the method with improved disk space requirements and better handling of density-fitted integrals. In particular, the computation ð2Þ

of the E exchdisp (CKS) term is now possible for systems with hundreds of atoms using nonhybrid functionals.31 SAPT(DFT) has recently been introduced in the PSI4 code.63 The implementation by Xie et al.32 features a modified algorithm of Ref. 64 for the FDDS computation with hybrid exchange-correlation kernels. The new algorithm retains the N 5 scaling but, unlike the original method, it is noniterative. By adopting a density-fitted two-index representation of the kernel combined with a grid-free algorithm of Pitonˇa´k and Hesselmann65 and using QR factorizations for numerical stability, the code is capable of computing coupled second-order dispersion energy with hybrid functionals for systems ð2Þ

up to 3000 basis functions. The coupled E exchdisp contribution is approximated by a scaling of the uncoupled term with a fixed factor optimized for hybrids.32

55

How to make symmetry-adapted perturbation theory more accurate?

3.4 Exchange energies beyond the S2 approximation The exchange energies in SAPT result from the appearance of the (NA + NB)electron antisymmetrizer A in the SRS energy expressions. For example, in the first order in V, the complete SRS correction is given by ð1Þ

ð1Þ

ð1Þ

E SRS ¼ Eelst + E exch ¼

hψ 0 jV Aψ 0 i : hψ 0 jAψ 0 i

(30)

The first step to evaluating this expression for many-body SAPT or SAPT(DFT), where the zeroth-order wavefunction is taken as the product of monomer HF (or KS) determinants, is noticing that the denominator can be rewritten as4,66 

HF HF HF ¼ ϕHF A ϕB jAϕA ϕB

N A !N B ! det S, ðN A + N B Þ!

(31)

where the (NA + NB)  (NA + NB) matrix S collects the overlap integrals between all occupied spinorbitals on A and B. In a similar manner, the  HF HF HF numerator ϕHF can be expressed by both the deterA ϕB jV AϕA ϕB minant and first and second cofactors of S, and the latter can in turn be related to the elements of matrix D ¼ S1. Once the derivation is continued to comð10Þ

pletion, the expression for Eexch is obtained by simply subtracting the electroð10Þ

static contribution E elst . This construction, free from any approximations ð10Þ

beyond the assumed form of ψ 0, was carried out for Eexch at the very outset of the many-body SAPT theory,4 but generalizing it to higher-order exchange corrections turned out to be quite cumbersome. A simpler alternative is using an approximate expression for A. The full form for this operator can be written as8 A¼

N A !N B ! A A ð1 + P + P 0 Þ, ðN A + N B Þ! A B

(32)

where the single-exchange operator P¼

XX

P ij ,

(33)

iA jB

collects all transpositions of a single pair of electrons between A and B, and the remainder P 0 contains all higher exchanges, that is, transpositions of 2, 3, …, minðN A , N B Þ pairs of electrons. A useful simplification is to neglect

56

Tatiana Korona et al.

P 0 completely, which is called the single-exchange approximation or, alternatively, the S2 approximation as it neglects terms higher than quadratic in j the intermolecular overlap integrals Si . This interpretation in terms of the power expansion in S explains why the S2 approximation should be very reasonable as long as the intermolecular overlap is not very large, but it might break down at very short intermolecular distances. ð10Þ

ð10Þ

In a typical SAPT calculation, both full E exch and Eexch ðS2 Þ are obtained. However, until recently, all other SAPT exchange corrections could only be computed within the S2 approximation. Finally, in 2012, Sch€affer and Jansen derived the nonapproximate formula for the complete second-order induction energy, including both the polarization and exchange part66: ð20Þ

ð20Þ

ð20Þ

E Ind;resp ¼ Eind;resp + Eexchind;resp D E   ð10Þ HF ð10Þ AjΨ ϕHF ϕ j V  E A B ind;resp  HF HF ¼ , HF HF ϕA ϕB jAjϕA ϕB

(34)

ð10Þ

where the induction wavefunction Ψind;resp is a linear combination of products of the ground-state HF determinant for one monomer and a singly excited HF determinant for the other one, X X ð10Þ B Ψind;resp ¼ C ia ϕAi!a ϕHF C jb ϕHF (35) B + A ϕj!b , ia

jb

with the occupied (virtual) spinorbitals of monomer A denoted by i (a), the occupied (virtual) spinorbitals of B by j (b), and the coefficients Cia, Cjb obtained by solving the standard coupled-perturbed HF equations. ð20Þ

Note that the capitalized correction name in E Ind;resp denotes the entire SRS term, that is, the sum of the polarization and exchange contributions. The numerator of Eq. (34), including the full antisymmetrizer A, can still be expressed through first and second cofactors, but the underlying matrices Si!a/Sj!b differ from S by a single excitation (a replacement i ! a or j ! b in one column). Sch€affer and Jansen evaluated these excited cofactors66 to ð20Þ

derive a nonapproximated spinorbital expression for E Ind;resp , Eq. (34). A year later, the same authors extended their derivation to cofactors of a doubly excited matrix Si!a, j!b and arrived at a nonapproximated spinorbital ð20Þ

expression for E Disp , that is, the combined second-order dispersion and

57

How to make symmetry-adapted perturbation theory more accurate?

exchange-dispersion energy67 (note that the level of complication of the resulting expressions quickly grows with both the excitation order and the cofactor order). The formalism is equally applicable to HF-based SAPT and SAPT(DFT)—the only difference is the set of dispersion amplitudes, evaluated from the monomer FDDSs in the latter case. The performance of the S2 approximation is also similar between SAPT0 and SAPT(DFT).66–68 The approximation is very adequate at the van der Waals minimum distance and at larger separations, but the residual errors ð10Þ

increase as one moves to shorter distances. Briefly, Eexch ðS2 Þ somewhat ð10Þ

ð20Þ

underestimates the true E exch , E exchind;resp ðS2 Þ underestimates the true ð20Þ

E exchind;resp to a larger extent (with typical relative errors 2–3 times larger), ð20Þ

ð20Þ

and E exchdisp ðS2 Þ slightly overestimates the true Eexchdisp. The inaccuracies ð10Þ

ð20Þ

of short-range Eexch ðS2 Þ and E exchind;resp ðS2 Þ become particularly proð20Þ

nounced for complexes involving anions; however, E exchdisp ðS2 Þ remains accurate in this case.68 In practical SAPT calculations, the leading errors due to the S2 approximation are suppressed by the hybrid SAPT approach, ð2Þ

that is, the addition of the δE HF correction of Eq. (2), canceling the errors in ð10Þ

ð20Þ

ð30Þ

E exch ðS2 Þ, E exchind;resp ðS2 Þ, and, if needed, Eexchind;resp ðS2 Þ. However, the ð2Þ

primary role of the δEHF correction (accounting for high-order induction and exchange-induction terms) is unrelated to the S2 approximation, and it contains some unphysical contributions in addition to physical effects.69 ð2Þ

In addition, δE HF is the only part of a SAPT/SAPT(DFT) workflow that requires a calculation for the entire complex (making the computation more expensive), and it is inherently nonlocal and cannot be decomposed in terms of interactions between molecular fragments.70 Therefore, it is worthwhile to eliminate at least some of the reasons that require the use of hybrid SAPT, such as the S2 approximation in further exchange corrections. Existing applications of SAPT and SAPT(DFT) including nonð20Þ

ð20Þ

approximate E exchind;resp and E exchdisp corrections are quite limited as no public code could compute these terms until about 2019 when they were added to MOLPRO71 and PSI4.63 As already mentioned, the addition ð2Þ

of δEHF eliminates the (sometimes sizable) difference between using ð20Þ

ð20Þ

E exchind;resp and E exchind;resp ðS2 Þ , and the only difference remaining is

58

Tatiana Korona et al.

ð20Þ

the comparably small effect of forgoing the S2 approximation in E exchdisp. If ð2Þ

δEHF is not added, it is important to account for some of the high-order inducð30Þ

tion effects directly by computing the third-order terms E ind;resp and ð30Þ

E exchind;resp 72 (the latter term is approximated by scaling the nonrelaxed ð30Þ

ð30Þ

ð30Þ

contribution E exchind by the ratio of E ind;resp to Eind ). However, the sum ð30Þ

ð30Þ

E ind;resp + E exchind;resp is much smaller than its two parts, and the S2 approximation might adversely affect the cancellation between the polarization and exchange terms. To investigate this issue, Waldrop and one of us recently ð30Þ

derived and implemented73 the complete, nonapproximated E exchind correcð20Þ

tion. Analogous to E exchdisp, first and second cofactors of up to doubly excited matrices Si!a, j!b, Si!a,i0 !a0 , and Sj!b, j0 !b0 are sufficient to evaluate this term, so the techniques of Sch€affer and Jansen67 can be directly adapted (other nonð30Þ

ð30Þ

approximate third-order terms, E exchinddisp and Eexchdisp , would require triply and quadruply excited determinants, respectively). Numerical tests ð30Þ

of Ref. 73 indicated that E exchind;resp ðS2 Þ ð30Þ

underestimates the true ð20Þ

E exchind;resp value by a similar percentage as for E exchind;resp ðS2 Þ. However, the larger absolute values of the third-order exchange-induction effects, and their near complete cancellation with the corresponding induction energies, make the third-order errors more severe in practice. At the extreme case of some short-range ionic interactions, the S2 errors lead to an unphysical ð2Þ

short-range attraction predicted by SAPT2+3 without δE HF ,74 an issue that ð30Þ

is completely resolved when E exchind;resp ðS2 Þ is replaced by the full ð30Þ

E exchind;resp .73 Numerical investigations involving S2-based corrections15 indicated ð2Þ

that while the addition of δE HF may be somewhat counterproductive for nonpolar complexes, it is absolutely essential for polar ones. The recent incorporation of second- and higher-order beyond-S2 exchange corrections into popular quantum-chemistry codes makes it worthwhile to investigate on a large and diverse dataset if the balanced, nonapproximate treatment of induction and exchange-induction makes the accuracy of ð2Þ

pure SAPT (without δE HF ) competitive, or possibly even superior, to hybrid SAPT. Work in this direction is in progress in the group of one of us.

59

How to make symmetry-adapted perturbation theory more accurate?

3.5 Explicitly correlated SAPT It is well known that molecular correlation energies converge very slowly to their complete basis set (CBS) limits, and the same is true for supermolecular interaction energies computed using correlated methods such as MP2 and CCSD(T). Several remedies to this issue have been proposed, including CBS extrapolation75 and adding basis functions centered in the space between the molecules (midbond functions).76 Perhaps the most impressive CBS convergence speedups are afforded by the explicitly correlated F12 approach—methods such as MP2-F12 and CCSD(T)-F12 improve the basis set convergence of conventional MP2 and CCSD(T) by allowing an explicit dependence on the interelectronic distance r12.77 In the SAPT context, the terms that form a part of the supermolecular ð10Þ

ð10Þ

ð20Þ

ð20Þ

HF interaction energy such as E elst , Eexch, E ind;resp, and Eexchind;resp converge quickly to their CBS limits. However, correlated SAPT terms such as dispersion and exchange-dispersion (as well as intramolecular correlation contributions to electrostatic, exchange, and induction energies) exhibit slow CBS convergence just like supermolecular MP2 or CCSD(T). Therefore, it is worth applying the F12 strategy to speed up basis set convergence of difficult SAPT corrections, and the primary target for improvement, with largest potential performance gains, are the dispersion and exchange-dispersion energies. The calculation of explicitly correlated dispersion and exchangedispersion corrections for few-electron systems dates back to a seminal study by Szalewicz and Jeziorski.78 These authors proposed a dispersion Hylleraas functional that can be minimized using a chosen explicitly correlated Ansatz to produce the dispersion pair function. This pair function can ð20Þ

be used directly to compute explicitly correlated estimates of Edisp and ð20Þ

E exchdisp as well as combined with an MP2-like pair function for intramoð21Þ

lecular correlation to obtain Edisp . The theory of Ref. 78, formulated specifically for the interaction between two helium atoms, enabled constructing a series of ever-improving SAPT helium pair potentials with the leading ð20Þ

ð21Þ

ð20Þ

E disp + E disp + E exchdisp dispersion part (as well as the entire FCI-based ð1Þ

ð1Þ

first-order energy E elst + Eexch) virtually converged to the CBS limit using explicitly correlated Gaussian functions.79,80 The formalism of Ref. 78 can be generalized to arbitrary many-electron complexes, but a full nonlinear optimization of dispersion pair functions is

60

Tatiana Korona et al.

not feasible beyond few-electron systems.81 An obvious alternative is the F12 approach, where a significant CBS convergence improvement can be afforded by a fairly simple Ansatz that does not require full optimization.82,83 The first attempt at adapting the F12 approach to dispersion energy was made in 2016 by Klopper and coworkers,84 where a dispersion-like correction was carved out of the MP2-F12 correlation energy for the complex by including only pairs of occupied orbitals localized on different fragments. ð20Þ

The first true E disp -F12 correction was developed by Przybytek85 who generalized the Hylleraas functional of Ref. 78 to many-electron systems and obtained the resulting conventional and F12 dispersion amplitudes via full optimization. Ref. 85 tested a number of correlation factors and confirmed that the Slater-type factor, the standard in F12 methods, provides ð20Þ

the best performance, leading to E disp -F12 values essentially converged to the CBS limit at the aug-cc-pVTZ level. Unfortunately, the full optimization of F12 dispersion amplitudes is 8 expensive (scaling like N ) and sometimes numerically unstable. Therefore, in 2019 Kodrycka et al.86 proposed several approximate algorithms for determining the dispersion amplitudes, in the spirit of methods where the F12 amplitudes are fixed using cusp conditions.83 Specifically, it is assumed that only the diagonal F12 amplitudes are nonzero. These diagonal amplitudes can be either freely optimized (the resulting optimized diagonal Ansatz (ODA) is an N 5 algorithm) or fixed to a common value which is a single optimized parameter. In addition to introducing new approximations to ð20Þ

E disp -F12, Ref. 86 also derived and implemented the corresponding ð20Þ

exchange correction E exchdisp -F12. Numerical tests indicated86 that F12 calculations in an X-tuple zeta basis are about as accurate as conventional ð20Þ

calculations in (X + 2)-tuple zeta basis for E disp -F12 and either (X + 1)ð20Þ

or (X + 2)-tuple zeta one for Eexchdisp-F12. The full amplitude optimization led to the most accurate SAPT-F12 corrections, but the more affordable ODA approach came very close. To make SAPT-F12 calculations competitive to the conventional ones, all two-electron quantities need to be efficiently approximated by robust ð20Þ

density fitting.87 This task was recently accomplished for E disp -F12 and ð20Þ

E exchdisp -F12 in the ODA variant,88 and the DF errors turned out to be negligible with standard auxiliary bases. The computational efficiency of

61

How to make symmetry-adapted perturbation theory more accurate?

density-fitted SAPT-F12 allowed for an extensive benchmarking study88 of ð20Þ

ð20Þ

both the CBS convergence of Edisp -F12 and E exchdisp-F12 (confirming the ð20Þ

ð20Þ

substantial improvement over conventional E disp and Eexchdisp ) and the ð20Þ

ð20Þ

influence of replacing Edisp and Eexchdisp by their better-converged F12 counterparts on total SAPT interaction energies. As expected, the effect of F12 on SAPT0 is detrimental as it disturbs error cancellation between basis set incompleteness effects and the errors of the uncoupled dispersion.13 However, at the highest available SAPT levels, including the virtually converged F12 values of leading-order dispersion and exchange-dispersion terms eliminates one of the main sources of error and in most cases substantially improves the accuracy. Still, a consistent improvement to overall SAPT requires a development of F12 corrections beyond the SAPT0 level, in particular, for the intramonomer correlation terms in dispersion such as ð21Þ

ð22Þ

E disp and E disp . Work in this direction is in progress in one of our groups.

4. Enabling accurate SAPT data for new systems 4.1 Multireference SAPT Most of the development and applications of the modern SAPT toolkit have so far been based on a single-reference description of the interacting monomers. Transition-metal complexes, interactions involving excitedstate molecules or open-shell systems (see also the next section), are all examples of situations which demand either a multireference or a multiconfigurational treatment, which calls for development of a new SAPT method based on a multireference description of monomers. To address this challenge, two of us have introduced a multiconfigurational (MC) SAPT variant, SAPT(MC).89,90 The method is based on the SRS approach including energy contributions through second order in the intermolecular interaction operator V, with all exchange energy terms limited to the S2 approximation. Second-order energies are expressed via response properties obtained from the extended random phase approximation (ERPA).91 In the latter, the ^ {ν comprises only single excitation (particle–hole) excitation operator O operators i Xh ^ {ν ¼ X pq ^a{p ^aq + Y pq ^a{q ^ap (36) O p>q

62

Tatiana Korona et al.

but, as opposed to the RPA approach, the operator acts on a multiconfigurational state. The other difference with respect to RPA is that the applicability of ERPA is not limited to single-determinantal states. The appealing feature of ERPA is that it uses reduced density matrices of order not higher than second. First-order terms in SAPT(MC) are computed from spin-free 1- and 2-RDMs γ Ψ and ΓΨ, respectively, pertaining to the zeroth-order monomer wavefunctions Ψ. The accuracy of E(1) is restricted by the level of correlation accounted for by these wavefunctions. The second-order induction and dispersion energies in SAPT(MC) are given in terms of spin-free 1-RDMs and 1-TDMs, the latter connecting a given state Ψ (ground or excited) with the νth state. 1-TDMs are obtained as ERPA eigenvectors. The accuracy of the second-order polarization terms increases with the amount of correlation recovered by the wavefunction model, but is constrained by the underlying ERPA approximation. The computation of the second-order exchange energies involves 2-RDMs and 2-TDMs. The expressions for 2-TDMs in terms of the ERPA eigenvectors and 1-RDMs were derived in Ref. 89. While discussing SAPT(MC), it is worth mentioning that recently two of us have proposed a CAS+DISP method, constructed as a supermolecular complete active space self-consistent field (CASSCF) calculation followed by the computation of dispersion energy from the CASSCF wavefunctions of the monomers.92 Importantly, CAS+DISP is designed to avoid dispersion double counting in a rigorous manner. The method offers a similar or better accuracy than SAPT(MC) and its scope of applicability is just as vast. In contrast to SAPT, CAS+DISP requires the calculation of the dimer energy and lacks SAPT’s interpretive power. The combination of SAPT with ERPA gives the possibility to apply SAPT(MC) with a broad range of multiconfigurational (MC) methods. So far, SAPT(MC) has been applied with the CASSCF and generalized valence bond perfect-pairing (GVB-PP) wavefunctions. Comparison with SAPT(CCSD) for small single-reference systems52 showed that SAPT(MC) systematically improves upon the SAPT0 energy components, but is less accurate than SAPT2 or DFT-based SAPT.89,90 This is hardly surprising—the affordable size of the GVB/CAS active space is not enough to account for the intramonomer correlation even in molecules limited to several atoms. While SAPT(GVB) and SAPT(CAS) give first-order energies of a similar quality, the latter is clearly superior in the second order. The nominal scaling of SAPT(MC) is with the sixth power of the system size. As in single-reference SAPT variants, the bottleneck steps are the

How to make symmetry-adapted perturbation theory more accurate?

63

solution of the full ERPA eigenproblem and the computation of secondorder dispersion terms. Recently, the SAPT(MC) efficiency has been improved by employing the Cholesky decomposition of two-electron integrals together with the N 5 -scaling algorithm for solving the ERPA equations. The novel algorithm employs a group product function partitioning of the monomer Hamiltonian and a consecutive expansion of response functions in the coupling parameter.93 This has brought the scal6 5 ing of the dispersion energy calculation from N to N for the CASSCF description of monomers. Apart from model multireference systems, SAPT(MC) has been applied to study interactions involving low-lying exciton-localized states.94 A SAPT analysis confirmed that the breaking of hydrogen bonds by n ! π* excitations is a purely electrostatic effect, i.e., the excitation is followed by an acute drop in the electrostatic attraction. For π ! π* excitons, the picture is typically more nuanced: changes in E(1) elst are accompanied by contributions (2) from both E(1) exch and Edisp. Note that in excited-state interactions the dispersion energy can no longer be expressed only via the FDDS functions, since transitions from the excited state to the ground state have to be taken into account.94 These extra terms can be easily computed in the ERPA framework95 and may constitute a significant fraction of the dispersion energy, as demonstrated on the example of the Ar⋯ C2H*4 complex.90 The key challenge facing further development of SAPT(MC) is to improve the accuracy of the method. An apparent target is a better representation of the short-range electron correlation effects, which are practically missing in both GVB-PP and CASSCF descriptions of the monomers. An extension to the range-separated multiconfigurational DFT96 could ameliorate this problem. Moving beyond single-perturbation theory in the spirit of many-body SAPT is also worth exploring. Another facet of the enhanced accuracy are higher-order corrections. In particular, a multiconfigurational analog to the δE(2) HF term cannot be uniquely defined. For excited-state interactions, satisfactory results were obtained using a simple scaling of the δE(2) HF contribution via the ratio of the induction energy computed for ground and excited states.94

4.2 Spin-flip SAPT for multiplet splittings Both the original, single-reference formulation of open-shell SAPT (using ROHF/ROKS33 or UHF/UKS34,35 determinants to describe the monomers) and the multireference SAPT approach of the previous section are

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restricted to the high-spin state of the interacting complex. The fundamental reason for this is the form of the zeroth-order wavefunction ψ 0 ¼ ψ Aψ B, with all unpaired spins in both ψ A and ψ B pointing the same way. Consequently, ψ 0 and all further wavefunction corrections are eigen2 functions of the total spin operator S^ , and the high-spin state of the complex is obtained automatically, without a need for any spin projection. However, the other dimer spin states resulting from the coupling of the same monomer spin states are not accessible, so spin splittings due to intermolecular interaction cannot be computed by standard open-shell SAPT. A fairly obvious alternative is starting the SAPT expansion from a product function ψ 0 ¼ ψ Aψ B in which the unpaired spins on A and B point in the opposite directions (say, spin-up on A and spin-down on B). Assuming that each monomer has at least one unpaired electron, such 2 a ψ 0 is not an eigenfunction of S^ , but rather a linear combination of spin states corresponding to all possible couplings of monomer spin angular momenta, jSA  SB j, ⋯ , SA + SB . Thus, each of these spin states can be extracted from the SAPT wavefunction by a suitable projection, and splittings between different spin states can now be computed. The spin projection should be applied together with the symmetry projection to lead to a fully antisymmetric state of pure spin. As a result, the polarization expansion does not yet contain a spin projection, and all nonexchange corrections are the same for all possible couplings of monomer spin states, that is, for an entire bundle of asymptotically degenerate states of the complex. The only reason for the spin splittings are the SAPT exchange corrections, that is, the splittings arise from the quantum tunneling of electrons between A and B. Thus, to compute all spin states of an interacting complex, one has to evaluate exchange corrections with the antisymmetry projector applied together with a suitable spin projector (note that these two operators commute with each other). So far, the formalism for computing SAPT exchange corrections for an arbitrary spin state has been proposed for a zeroth-order wavefunction that is a product of spin-restricted (ROHF or ROKS) determinants for A and B.97 When the S2 approximation is applied, the resulting exchange terms are linear combinations of two matrix elements: a diagonal (spin-averaged) contribution and a spin-flip contribution that determines the splittings. The diagonal exchange matrix elements involve only the unprojected ψ 0, while the spin-flip ones involve both ψ 0 and a function of the form ψ #Aψ "B in which one of the unpaired spins in ψ A (pointing up in ψ 0) has been lowered and

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one of the unpaired spins in ψ B (pointing down in ψ 0) has been raised. The resulting approach was named spin-flip SAPT (SF-SAPT) in analogy to the spin-flip electronic structure theories of Krylov and coworkers98: while the goals and the equations of the two methods are entirely different, both algorithms are able to access multireference, low-spin states in an entirely single-reference manner. One can show that, within the singleexchange approximation, double and higher spin flips do not contribute97 so that the entire bundle of asymptotically degenerate spin states can be described by two parameters, the diagonal exchange term and the single spin-flip term. The linear combination coefficients leading from these two terms to SAPT exchange energies for each spin state result exclusively from the angular momentum algebra. Thus, SF-SAPT effectively describes the bundle of multiplets within the popular Heisenberg model with a single spin-coupling parameter. The initial SF-SAPT study97 derived and implemented the first-order ð10Þ

ROHF-based correction E exch ðS2 Þ for an arbitrary spin state of the complex. The first-order estimate of the splittings is a very inexpensive one and the results follow the expected trends, but one cannot expect quantitative accuracy from such a simple approximation. Moreover, a problematic short-range behavior was sometimes observed: for example, the singlet and ð10Þ

triplet E exch ðS2 Þ curves for the interaction of two (doublet) lithium atoms intersect each other at a short distance. The obvious culprit is the S2 approximation, and one may adapt the techniques of avoiding this approximation described in Section 3.4 to the SF-SAPT case. To this end, the second ð10Þ

SF-SAPT paper99 improved the ROHF-based arbitrary-spin Eexch calculation by replacing the S2 approximation by a much milder single spin-flip (1-flip) approximation, that is, neglecting the matrix elements involving double and higher spin flips. While the 1-flip approximation is exact if either monomer has only one unpaired electron and remains highly accurate otherwise, the single exchange approximation is not exact even in the closed-shell case with no unpaired electrons (for technical reasons, the single-exchange treatment is not even exact when one of the monomers has only one electron so that mulð10Þ

tiple exchanges are not possible99). The SF-SAPT E exch values within the 1-flip approximation show reasonable short-range behavior without unphysical crossings. However, to turn SF-SAPT into an efficient practical approach to compute exchange splittings, it needs to be generalized to higher orders of perturbation theory. The development of the (ROHF-based)

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SF-SAPT version of the second-order exchange-induction and exchangedispersion corrections is in progress in one of our groups. In the future, the spin-flip formalism could also be combined with the multireference SAPT development of the previous section to provide spin splitting estimates in challenging multireference complexes.

5. Summary Symmetry-adapted perturbation theory is a mature and popular approach to compute intermolecular interaction energies as sums of distinct, physically meaningful terms. The theoretical foundations of SAPT were laid out from the 1970s until the early 2000s in a multiteam effort led by Professors Bogumił Jeziorski, Włodzimierz Kołos, Krzysztof Szalewicz, Grzegorz Chałasi
nski, Robert Moszy
nski, and other members of the Polish quantum chemistry school. As a result, multiple flavors of intermolecular SAPT have been developed, with the primary difference between them being the treatment of intramolecular electron correlation. The working equations for SAPT, at various levels of theory, are programmed in several general-purpose and specialized quantum chemistry codes, and the numerous enhancements to the SAPT computational efficiency (again, carried out by many groups) were indispensable for turning SAPT into a highly successful practical approach with a multitude of chemical applications. The SAPT methodology remains an active research area, and significant progress in the SAPT accuracy, efficiency, and range of applicability has been achieved in the last 15 years. The four of us have been involved in many of the latest enhancements; many others were proposed by other researchers. In this review, we describe an incomplete list of recent improvements to SAPT, focusing on the developments that either make the results more accurate or enable applications to new classes of complexes. For the first group, we described the SAPT(CC) method as the source of high accuracy, reference energy contributions, several notable enhancements to SAPT(DFT), the removal of the customary S2 approximation from the exchange corrections, and an F12-based approach to improve basis set convergence of dispersion terms. The latter group is represented by the newly developed SAPT(MC) approach for interactions between multireference systems and the SF-SAPT method to access low-spin states of the interacting complex.

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Acknowledgments M.H. was supported by the National Science Centre, Poland under Grant No. 2021/ 43/D/ST4/02762. K.P. was supported by the US National Science Foundation award CHE-1955328.

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REVIEW

Advanced models of coupled-cluster theory for the ground, excited, and ionized states Monika Musiał∗ and Stanisław A. Kucharski∗ Institute of Chemistry, University of Silesia in Katowice, Katowice, Poland *Corresponding authors: e-mail address: [email protected]; [email protected]

Contents 1. 2. 3. 4.

Introduction Elementary definitions Single reference coupled-cluster (SRCC) approach Equation-of-motion coupled-cluster (EOM-CC) approach 4.1 Electronic excited states: EE-EOM-CC 4.2 Ionized and electron-attached states 5. Multireference coupled-cluster (MRCC) approach 5.1 General considerations 5.2 Hilbert-space formulation of multireference coupled-cluster approach 5.3 Fock-space multireference coupled-cluster approach 5.4 Intermediate Hamiltonian: IH-FS-MRCC 6. Nonstandard realizations of the coupled-cluster theory 7. Final remarks Acknowledgments References

74 74 77 82 84 85 90 90 92 94 97 101 103 104 105

Abstract Main directions of the development of the coupled-cluster (CC) theory are presented: the single reference (SR) CC approach applied to the ground state; equation-of-motion (EOM) CC scheme applied to the excited, ionized, and electron-attached states; and multireference (MR) CC theories. A common issue in all of them is a focus on the contributions from Polish quantum chemists. The most pronounced of the latter are: the Jeziorski–Monkhorst ansatz for the Hilbert-space MRCC theory, the intermediate Hamiltonian formulations of the Fock-space MRCC theory, and an incorporation of the higher cluster operators into the SR, EOM, and MR variants.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.03.003

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2023 Elsevier Inc. All rights reserved.

73

74

Monika Musiał and Stanisław A. Kucharski

1. Introduction Coupled-cluster (CC) theory has an established position as a most powerful computational tool for the description of the electronic correlation in atoms and molecules. It has been introduced to quantum chemistry over 50 years ago, and during this period it evolved into a gold standard within the quantum chemistry methods. Although formally it is a simple theory, due to its rich internal structure it offers a lot of formal paths, threads, and alleys, attractive from the mathematical viewpoint and also useful for computational chemistry. One may always hope that among numerous theoretical variants, there is a shortcut that needs to be discovered which will offer a breakthrough solution for chemical problems. These features are the reason that the CC theory is very much appreciated and valued by quantum chemists and became a method of choice for those focusing on the ab initio approaches. In this review, we intend to discuss some advanced realizations of the CC theory with the emphasis on Polish contributions to the subject in accordance with a profile of the present volume. In the main part of the review, we are going to focus on three principal directions of the CC development: (i) a single reference (SR) formulation used in the ground-state calculations but incorporating higher cluster operators, (ii) an extension of the CC method to the studies of excited, ionized and electron-attached states via an equation-of-motion (EOM) approach, and (iii) generalization of the CC theory toward multireference formulations. In later sections, we will refer also to the—as we call it—nonstandard realizations of the CC theory which in many cases offer an attractive computational scheme.

2. Elementary definitions The Schr€ odinger equation for the kth electronic state takes the form: ^ k i ¼ Ek jΨk i HjΨ

(1)

The ground-state solution corresponds to the lowest energy state which will ^ is expressed in the secondbe denoted as E0 and jΨ0i. The Hamiltonian H quantized formalism as: X 1 X rs { { ^ ¼ F^ + V^ ¼ f rs ^ r {^s + v ^r ^s u^^t (2) H 4 rstu tu rs

Advanced models of coupled-cluster theory for the ground, excited and ionized states

75

where ^ r { …ð^t …Þ are second-quantized creation (annihilation) operators, r and f s and vrstu are one- and two-electron integrals. We can rewrite the Schr€ odinger equation using the Hamiltonian operator in a normal-order ^N : form, H ^ N jΨ0 i ¼ ΔE0 jΨ0 i H

(3)

^N ¼ H ^  hΦ0 jHjΦ ^ 0 i , ΔE0 ¼ E 0  hΦ0 jHjΦ ^ 0 i is a correlation where H energy, and jΦ0i is a single Slater determinant, usually chosen to be the Hartree–Fock (HF) function. A ground-state wave function within the CC formalism is expressed as: ^

jΨ0 i ¼ eT jΦ0 i

(4)

where T^ is a cluster operator creating singly (T^ 1), doubly (T^ 2), ⋯ , excited configurations: T^ ¼ T^ 1 + T^ 2 + ⋯ + T^ N X 1 X ab { ^{^^ 1 X abc { ^{ { ^^^ ¼ tai ^a{^i + tij ^a b ji + t ^a b ^c kji + ⋯ 4 ab,ij 36 abc,ijk ijk a, i

(5)

{ N is a number of electrons in the system. The ^a{ , b^ , ⋯ (^i, ^j, ⋯ ) are second-quantized creation (annihilation) operators and the coefficients ^^ tab… ij… are antisymmetrized cluster amplitudes. Note that the i, j, ⋯ operators, which annihilate electrons on occupied levels, in the particle–hole formalism become creation operators (responsible for creation of holes). As a consequence, the second-quantized operators present in the definition of the T^ operators are all creation operators. Inserting the CC wave function, Eq. (4), into the Schr€ odinger equation, ^ Eq. (3), multiplying from the left with eT and projecting against jΦ0i, we get the correlation energy, ΔE0: ^ ^ N jΦ0 i ^ N eT^ jΦ0 i ¼ hΦ0 jH ΔE 0 ¼ hΦ0 jeT H

(6)

Projecting against configurations jΦab… ij… i, we obtain CC equations: ^ hΦab… ij… j H N jΦ0 i ¼ 0

(7)

76

Monika Musiał and Stanisław A. Kucharski

^ N symbol representing a In the equations above, we introduced the H similarity transformed Hamiltonian: ^ N ¼ eT^ H ^ N eT^ H

(8)

The following index convention is adopted throughout the paper: symbols i, j, … (a, b, …) indicate occupied (virtual) one-particle levels and r, s, … general indices. According to this, the configuration Φab… ij… represents a Slater determinant in which occupied orbitals i, j, … are replaced with the virtuals a, b, …; Φab… (Φij…) represents determinant with additional electrons placed (removed) on the unoccupied levels a, b, … (from the occupied levels i, j, …). ^ N , we may decompose Emphasizing the many-body structure of the H n it into individual n-body contributions I^ as follows: ^ N ¼ I^1 + I^2 + I^3 + I^4 + ⋯ H

(9)

n n We may partition a particular I^ term into I^k components where k indicates n n the number of annihilation lines in the I^ . The I^k operator can be further n split into I^hp components (k ¼ h + p) with h annihilation hole lines and p annihilation particle lines (terminology used in the diagrammatic formulation) or—equivalently—with h hole and p particle annihilation operators ^  N without annihilation lines since those appear (we leave out the forms of H in the expansion of the cluster operators): 1 1 1 1 1 1 I^ ¼ I^1 + I^2 ¼ I^10 + I^01 + I^11 2 2 2 2 2 2 2 2 2 2 2 2 2 I^ ¼ I^1 + I^2 + I^3 + I^4 ¼ I^01 + I^10 + I^02 + I^11 + I^20 + I^12 + I^21 + I^22 3 3 3 3 3 3 3 3 3 3 3 I^ ¼ I^1 + I^2 + I^3 ¼ I^01 + I^10 + I^02 + I^11 + I^20 + I^12 + I^21

⋮ It is obvious that the total number of annihilation lines (k) in all n-body ^  N does not exceed 3 due to the two-body character (n > 2) elements of H ^  N elements with n greater than 3 will of the V^ operator. Hence, all n-body H have the same structure as the I^ component. 3

Advanced models of coupled-cluster theory for the ground, excited and ionized states

77

3. Single reference coupled-cluster (SRCC) approach  ˇ ek The CC theory has been introduced to quantum chemistry by Ciz 1 in his classic paper followed by first applications reported by Paldus et al.,2 Bartlett and Purvis,3 and Pople et al.4 A real step forward came with the seminal paper by Purvis and Bartlett5 who developed the first general purpose CC program incorporating into the cluster expansion all singles and doubles and creating a scheme known until today as CCSD (the T^ operator in Eq. (5) is limited to T^ 1 and T^ 2 ). The next steps aimed at the inclusion of connected triples, T^ 3 operator in Eq. (5), were made mainly in Bartlett’s group and resulted in the development of the first iterative method incorporating connected triples in a partial way, CCSDT-1.6 This paper was soon followed by an attempt to include triples in a computationally efficient, noniterative way, which resulted in the scheme known as CCSD[T].7 At the same time, the iterative schemes CCSDT-n (n ¼ 2, 3, 4) including the triple contributions in a more advanced way were reported.8 Soon the full CCSDT method emerged, developed by Noga and Bartlett9 and independently by Scuseria and Schaefer.10 A few years later Watts and Bartlett11 reported a new CCSDT code aimed at the general reference function (RHF—Restricted HF and UHF—Unrestricted HF) and made it available for routine calculations. A remarkable summary of all efforts aiming at the efficient incorporation of T^ 3 operator into the CC theory was the scheme introduced by Raghavachari et al.12 known as CCSD(T), now considered as a gold standard within the plethora of quantum chemical methods based on the electronic wave function. An important advantage of all iterative and noniterative approximations to the full CCSDT relies on a lower n-scaling, i.e., a lower dependence of the cost of calculations on the size of the studied system: full model is characterized by n8 scaling while the approximate ones—by n7 (note that for the CCSD this factor is equal to n6). A pursuit for the next cluster operator, T^ 4, had perturbational origins. It was well known that the simplest inclusion of the T^ 3 operator into the CCSD scheme (like CCSD[T] or CCSD(T)) creates the method correct through fourth order of the many-body perturbation theory (MBPT) and the same MBPT level is ensured by the full CCSDT method. In order to gain correctness through the next, i.e., fifth MBPT order, one has to reach

78

Monika Musiał and Stanisław A. Kucharski

for the simplest quadruples. These terms were identified and reported in Bartlett’s group13 and—later on—implemented to create the method correct through fifth order.14 The analogous implementation was done by Pople and coworkers.15 The first partial incorporation of the T^ 4 operator into the CC framework was reported soon,16 and it was done both in the iterative (CCSDTQ-1) and noniterative (Q(CCSDT)) manner. The full inclusion of the connected quadruples into the CC scheme, the CCSDTQ method, was done in Bartlett’s group17 and at the same time reported by Oliphant and Adamowicz.18 When focusing on the implementation of higher rank clusters into the CC scheme, it is worth to consider a reduction of the CC equations to a quasilinear form as described in Ref. 19. It relies on the fact that nonlinear contributions to the cluster equations can be ^ N operator with the T^ k operators factorized, i.e., contractions of the H occurring in a nonlinear term can be done subsequently, one by one, producing intermediates which can be further contracted with the remaining T^ operators. This subsequent series of contractions is a reason that the n-scaling attributed to the given CC model depends on the rank of the cluster equation and not on the excitation level of the particular expansion term. The factorization procedure has been developed in Ref. 19 in such a way that in each nonlinear term, containing—as it is well known—at most four T^ operators, all but one of the latter are “hidden” in the intermediate and the T equations contain only linear terms. Since the final linearity of the CC equations is a consequence of the reorganization of the original nonlinear terms, the name quasilinear is justified. Expressing the CC equations in the quasilinear form:

01

02

02

02

hΦai jðI^00 + I^01 T 1 + I^10 T 1 + I^11 T 2 + I^11 T 1 + I^12 T 2 + I^21 T 2 + I^22 T 3 Þc jΦo i ¼ 0 1

1

1

5

1

3

2

1

1

2

1

sum:

1

(10)

15

^2 ^1 ^1 ^1 ^02 ^02 ^02 ^2 ^2 ^2 ^2 ^2 hΦab ij jðI 00 + I 01 T 2 + I 10 T 2 + I 11 T 3 + I 01 T 1 + I 10 T 1 + I 02 T 2 + I 11 T 2 + I 20 T 2 + I 12 T 3 + I 21 T 3 + I 22 T 4 Þc jΦo i ¼ 0 1

5

5

2

6

2

3

5

4

2

2

sum:

1

38

  1 1 1 2 002 2 2 2 2 2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ hΦabc j I T + I T + I T + I T + I T + I T + I T + I T + I T + I T ijk 01 3 10 3 11 4 01 2 10 2 02 3 11 3 20 3 12 4 21 4 jΦo i ¼ 0

(12)

c

5

5

2

13

11

4

5

4

2

sum:

2

53

^1 ^1 ^2 ^2 ^2 ^2 ^2 ^03 ^03 ^03 ^03 hΦabcd ijkl jðI 01 T 4 + I 10 T 4 + I 01 T 3 + I 10 T 3 + I 02 T 4 + I 11 T 4 + I 20 T 4 + I 01 T 2 + I 10 T 2 + I 20 T 3 + I 11 T 3 Þc jΦo i ¼ 0 5

5

13

13

4

5

4

9

14

1

1

sum:

(11)

74

(13)

80

Monika Musiał and Stanisław A. Kucharski

relies on the replacement of the F^ and V^ integrals, see definition of the ^  N element. The unprimed Hamiltonian in Eq. (2), with the appropriate H n I^hp operator denotes an intermediate representing a complete form of the

^  N element, whereas the primed one refers to the incomplete one. H For a proper bookkeeping, we may use the general formula for the numn ber of antisymmetrized terms, K, contributing to the I^k elements (cf. the formula reported in Ref. 20): n2 7n n n KðI^10 Þ ¼ KðI^01 Þ ¼ +  2δ + δ2n  kodd + 4 4 2   1n n n n KðI^20 Þ ¼ KðI^02 Þ ¼ INT  2δ1n + δ2n + 2 2 n KðI^11 Þ ¼ n  δ1n + δ2n + 2 n n KðI^21 Þ ¼ KðI^12 Þ ¼ 1  δ1n + δ2n

(14) (15) (16) (17)

where δij denotes a Kronecker symbol (equal 1 for i ¼ j and zero otherwise), kodd is zero for n even and 34 for n odd, and INT(x) is an integer-valued floor function. Below each CC equation, Eqs. (10)–(13), we give a number of terms occurring in the original CC equations (cf. Ref. 17) replaced by the term in the quasilinear form. The numbers referring to the complete ^  N elements correspond to KðI nhp ) given in Eqs. (14)–(17). We see that H the number of terms in the quasilinear form of the CCSDTQ method is significantly reduced. From the numbers placed under above equations (indicated by sum) equal to 15, 38, 53, and 74, it is reduced to 8, 12, 10, and 11 for the T1, T2, T3, and T4 equations, respectively. Using this strategy, the full CCSDTQ method was implemented.17 Testing the range of the feasible realizations of the CC method, the full CC scheme through T^ 5 operator has been implemented in Bartlett’s group by Musial et al.21 An incorporation of the T^ 5 operator into the CC equations requires small modification of the T3 equation: one new term representing a contribution from T^ 5 , and similar one in the T4 equation: two new terms originating also from the T^ 5 operator. The most difficult step is connected with the construction of the new equation for the T5 amplitudes comprising 99 antisymmetrized diagrammatic terms. Quasilinearization of this equation reduced the number of terms to 14, see the bottom part of Table 2 in Ref. 21. The linearized T5 equation contains terms obtained by a contraction of the com N elements and the T^ 5 operator. The first seven terms in plete or incomplete H

Advanced models of coupled-cluster theory for the ground, excited and ionized states

81

N the T5 equation (replacing 49 out of 99 diagrammatic terms) engages the H elements defined for lower sectors. Next four contributions (32 original terms) use the intermediates already defined at the T4 level but modified for the T5 step, and only last three terms in the T5 equation (Table 2 in Ref. 21) require 03 04 04 ^ N elements can be comnew intermediates: I^02, I^10, and I^01. Note that, the H puted in a recursive way, i.e., higher rank terms can be expressed through the lower rank intermediates, see, e.g., Table IV in Ref. 21, reducing in this way both the programmer’s task and the cost of calculations. The performance of the schemes engaging high cluster is usually tested by comparison with the FCI (full Configuration Interaction) values. As it can be seen in Table 1 of Ref. 22, the full inclusion of the T^ 4 operator reduces the error by an order of magnitude compared to the CCSDT method. The same level of improvement (with respect to the CCSDTQ) can be observed upon the inclusion of the T^ 5 operator. For example, for the FH and H2O (DZP basis set) the CCSDTQP deviations from FCI for stretched geometries (R ¼ 2Re) are of the order of 1 and 26 μH, respectively. Similarly, the deviations at the CCSDTQP level for the N2 and C2 molecules go down from 192 to 16 μH and from 622 to 103 μH, respectively, see Ref. 22, compared to the CCSDTQ scheme. One should note that the high n-scaling for the methods engaging T^ 4 and T^ 5 operators: n10 and n12 makes their wider application prohibitive. Impressive results in the quest for inclusion of the high cluster operators into the CC theory were obtained by Hirata and Bartlett.23 Using the full CI program, properly modified, they managed to solve the CC equations through the octuple excitations. A similar achievement was reported by Kallay and Surjan24 who by combining the diagrammatic formalism and string-based FCI algorithm were able to go up to the T^ 10 operator reaching a full CC limit in the calculations for H2O. Considering a computationally effective inclusion of the T^ 4 operator into the CC method, one can exploit the fact that the lowest order quadruples are factorizable. This means that the fifth-order contribution to the energy (originating from the third-order T^ 4 via T2 equation) can be replaced with the n7 scaling. This opens the way to formulate the CCSDT(Qf) method25 which on top of the CCSDT model includes the factorized T^ 4 contribution. Bartlett and his collaborators combined the noniterative triples and quadruples contributions to obtain the efficient method based on the CCSD iterative scheme and correct through fifth order, known as CCSD(TQ).26

82

Monika Musiał and Stanisław A. Kucharski

4. Equation-of-motion coupled-cluster (EOM-CC) approach Application of the CC theory to the studies of excited states of atoms and molecules has long history and is connected with merging the CC approach with the EOM27 formalism. The first successful attempts to use the EOM approach within the CC theory have been reported in Refs. 28–31. Foundations for the routine use of the EOM-CC approach in the description of the molecular excited states were laid by paper of Stanton and Bartlett32 where working equations for the EOM-CCSD scheme were derived. Later on, the papers extending the EOM formalism to the ionized potential33 and electron-attached34 states came from the same group. In the current review, we will consider the development of the EOM-CC approach to investigate electronically excited (EE) states, ionization potentials (IP), double ionization potentials (DIP), electron-attachment (EA), and double electron-attachment (DEA) processes. We should mention also that an attempt to carry out the EOM-CC calculations for the triple electron-attached states has been done.35 We are aware that the alternative approach to study excited states via CC method is offered by the linear response theory (LRT).29,36–38 The eigenvalues obtained with the latter approach are the same as those of EOM on condition that the full CC models are considered. In the EOM formalism, the wave function jΨki arises from the action of ^ the linear operator RðkÞ on the ground-state wave function jΨ0i: ^ jΨk i ¼ RðkÞjΨ 0i

(18)

At the start, we do not need to specify the type of the process involved so the ^ RðkÞ operator may represent any of the above processes. The most general ^ form of the RðkÞ operator can be written out as: ^ ^1 ðkÞ + R ^2 ðkÞ + ⋯ + R ^N ðkÞ RðkÞ ¼ r 0 ðkÞ + R

(19)

^ Some of the components of RðkÞ disappear for specific processes, e.g., r0 ^1 does vanishes for the processes of ionization and electron attachment, R not occur for the double ionization and double electron attachment, etc. Combining the ground-state Schr€ odinger equation for CC wave function

Advanced models of coupled-cluster theory for the ground, excited and ionized states

83

and that for the excited, ionized, or electron-attached states, we obtain the EOM equation in the form32: ^ ^  N , RðkÞjΦ ^ ½H 0 i ¼ ωk RðkÞjΦ 0i

(20)

where ωk denotes the energetic effect of the studied process. The common element of these realization of the EOM-CC approaches is a construction of ^  N operator based on the CC wave function obtained for the reference the H ^  N operator is diagonalized within the particular system. Thus, the same H sector, i.e., a subspace of the Fock space: the subspace of the excited configurations (EE-EOM), the subspaces of singly, doubly ionized configurations (IP-EOM, DIP-EOM), and the subspaces of single-, double-, and triple-electron attached configurations (EA-EOM, DEA-EOM, TEA-EOM). The level of accuracy of the computed eigenvalues depends ^ N operator on the quality of the CC reference function defining the H (i.e., obtained at the CCSD, CCSDT, or with inclusion of higher-rank cluster operators) and on the size of the subspace corresponding to the given ^2 , R ^3 , or higher operators. sector, i.e., engaging R ^ N On the other hand, we may underline the CI-like structure of the H  N takes the operator rewriting it in the matrix form. The EOM matrix H ^  N jXi elements since they are irrelevant to form (we leave out the hΦ0 jH the diagonalization process): 3 2 ^ ^ ^ N jT i  N jSi hSjH  N jDi hSjH hSjH 7 6 ^ ^ N jT i 7 ^  N ¼ 6 hDjH  N jDi hDjH  N jSi hDjH (21) H 5 4 ^ ^ ^ N jT i  N jSi hTjH  N jDi hT jH hT jH ^  N jY i (X, Y ¼ S, D, T) represent the subwhere the matrix elements hXjH matrices corresponding to the configurations engaged in the one- (S), two(D), and three- (T) electron processes. The diagonalization of the above matrix is carried out using the Davidson procedure39 modified by Hirao and Nakatsuji40 for non-Hermitian matrices. In some cases, where the size of the EOM matrix is small, the more general QR scheme developed for the diagonalization of non-Hermitian matrices41 can be used. Remaining in the spirit of the current issue, in the following sections we will discuss an incorporation of the connected triples operators into the various sectors of the Fock space developed by Polish quantum chemists working with the Bartlett’s group.

84

Monika Musiał and Stanisław A. Kucharski

4.1 Electronic excited states: EE-EOM-CC Routine calculations of the excitation energies with the EOM-CC scheme have been initiated by the mentioned above work of Stanton and Bartlett32 who introduced the EOM-CCSD scheme and implemented it into the ACES2 program package for general, i.e., RHF/UHF reference function. Within the EOM-CCSD approximation, the EOM matrix, Eq. (21), ^ contains only Sð≡ Φai Þ and Dð≡ Φab ij Þ configurations, while H N is constructed on the basis of the T1 and T2 amplitudes, obtained with the CCSD method. The paper32 provides a thorough analysis of the EOM N matrix CCSD approach presenting algebraic formulas for the H elements, diagrammatic expressions for the density matrix elements, and results of the preliminary calculations in comparison to the FCI values or to the results of other theoretical approaches. A natural extension of the EE-EOM-CCSD is an inclusion of the triple excitations. First works reporting a partial incorporation of the triple excitations into computationally feasible schemes were published by Watts and Bartlett.42–44 They proposed—in analogy to the existing variants of the standard ground-state method—various approximate schemes, e.g., noniterative versions of EOM-CCSD(T) as well as the EOM counterpart to the iterative formulations of the ground-state CC method, like EOMCCSDT-1 and EOM-CCSDT-3.44 The results, confronted in most cases with the FCI values and—in several cases—with experiment, were encouraging. As it may be expected, even a crude evaluation of the triple-excitation contribution is crucial for the states of a double-excitation character. An extension of the EOM-CCSD model to the full inclusion of con^ N operator has nected triple excitations45 requires two steps. First, the H to be constructed on the basis of the CCSDT solution, i.e., it must engage also the T3 amplitudes. Second, the EOM matrix, Eq. (21), must include also triply excited configurations Tð≡ Φabc ijk ). As one can expect, the inclusion of T configurations is connected with much more complicated formulas for the ^ ^ elements and with much larger computational effort due to higher NR H scaling, n8 vs n6 (in the case of EOM-CCSD). Formally, the elements ^ N jSi and hT jH ^ ^ N and their  N jDi engage the four-body terms of H hTjH 9 constructions require n scaling. This can be avoided by decomposition ^  N element and reducing the scaling to n7. A similar situation takes of the H ^ N contribute to the place in the case when the three-body components of H ^  N jT i element of the EOM matrix. The extra computational effort hTjH

Advanced models of coupled-cluster theory for the ground, excited and ionized states

85

Table 1 MAEs (mean absolute errors) (eV) related to the FCI and experimental values for excitation energies (from Refs. 45–48). FCI Experimental N2(4) C2(2) CH2(5) H2O(3) Ne(4)

N2(4) CO(4) O3(4)

0.13 0.14 0.30

0.08

0.25

0.26 0.25

1.93

EE-EOM-CCSDT 0.01 0.07 0.01

0.03

0.03

0.03 0.07

0.39

EE-EOM-CCSD

connected with the triples pays off very well when we compare the improvement in accuracy of the results.45–48 In Table 1, we show the mean absolute errors (MAEs) of the excitation energy values related to the FCI reference for several small molecules and Neon atom. The selection of molecules and a number of considered excited states (in parentheses) were limited to those with available FCI data. To obtain the results comparable with experimental values, the larger basis sets were used. In the last three columns in Table 1, we compare performance of the EOM-CCSD and EOM-CCSDT methods with respect to the experimental values for the N2 and CO molecules (aug-cc-pVQZ basis set) and for ozone molecule (POL1 basis set). In all cases, the inclusion of triple excitations in a complete manner reduces the error from a few tenth to a few hundredth of eV. It should be mentioned also that the effect of triples and of higher excitations has been computed by Hirata using modified FCI program.49 The full EOM-CCSDT was reported also by Piecuch et al.,50 Kallay and Surjan,51 and Hirata.52 A comprehensive review of the wide range of applications of various realizations of the EOM-CC theory can be found in Ref. 53.

4.2 Ionized and electron-attached states ^ operIn this section we discuss briefly the EOM approaches in which the R ator is responsible for removing or adding electrons to the reference system. As before, we focus on the role of triple excitations included into the EOM ^3 operator. An implementation of the latter operator was formalism via R reported by Musial et al. in several papers35,46,54–58 done in collaboration with Bartlett’s group. We consider four, mentioned before, situations: IP (removing one electron from the reference), DIP (removing two electrons from the reference), EA (adding one electron to the reference), and DEA (adding two electrons to the reference).

86

Monika Musiał and Stanisław A. Kucharski

4.2.1 IP-EOM-CC and EA-EOM-CC approaches The IP and EA approaches are structurally very similar (compare general diagrammatic equations reported in Refs. 54, 55) and can be obtained from each other by exchanging hole and particle lines; hence, we will discuss both ^ N cases in the same section. Within the IP-EOM-CCSD33 model, the H operator is diagonalized within the subspace spanned by S(≡ Φi) and Dð≡ Φaij Þ configurations. Extending the approach to the CCSDT model, 54 the T ð≡ Φab ijk Þ configurations are needed. In the IP-EOM scheme, a step determining effectiveness of the calculations is a solution of the CCSD (CCSDT) equations for the reference state which scales as n2o n4v (n3o n5v), while the EOM part scales as n3o n2v (n3o n4v ). Note that, similarly as in the EE case, the standard expressions for some of the EOM matrix elements require ^  N . For the IP-EOM-CCSD model, higher than two-body components of H ^ ^ N compo N jDi to which the three-body H this happens only for the hDjH ^ N jSi, hTjH ^ N jDi, ^3 level those occur in hT jH nents contribute, while at the R ^ ^ N element con N jTi elements. In addition, the four-body H and in hT jH ^ N jDi , see Fig. 1 in Ref. 54. In order to adhere to  tributes to the hT jH ^ N is required. the declared n-scaling, the decomposition of the higher rank H

^3 operators provide significant improveWe see in Table 2 that the T^ 3 and R ment in the MAE values when related to the FCI reference or, when large basis sets can be used, also to the experimental data, cf. molecules N2, CO, and F2. For larger molecules: C2H2, C2H4, and H2CO the triples effect when compared with experiment is also noticeable, but the jump in accuracy is not so spectacular mainly due to the inadequacies in the basis set. The EA realization of the EOM-CC method is fully analogous to the IP counterpart when the removal of the electron is replaced by the electron ^  N is diagonalized in the space of attachment. At the CCSD level, the H

Table 2 MAEs (eV) related to the FCI54 and experimental46,59 values for vertical ionization potentials with the EOM-CC methods. FCI Experimental C2

BH

H2O

N2

CO

F2

C2H2 C2H4 H2CO

No. of states

2

2

3

3

3

3

3

IP-EOM-CCSD

0.50 0.19 0.14

0.24 0.23 0.17 0.40 0.13 0.39

IP-EOM-CCSDT 0.04 0.01 0.02

0.03 0.08 0.09 0.27 0.07 0.26

5

5

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87

configurations S with single attached electron (≡ Φa) and D (≡ Φab i , electron attachment accompanied by an excitation).34 At the CCSDT level, the 55 T ð≡ Φabc ij Þ configurations must be included. Computationally, however, EA-EOM is more demanding than its IP analog due to the fact that the number of EA amplitudes is of the order of n2o n3v (n2o n5v for CCSDT) (no—number of occupied levels, nv—number of virtual ones) compared to the n3o n2v (n3o n4v ) and usually the number of virtual levels is much greater than that of occupied ones. Hence, the factorization introduced in the EA-EOM-CCSDT approach is much deeper than in the IP scheme. The EA-EOM-CC offers an attractive field of applications for these open-shell systems which upon elimination of one electron assume a closed-shell structure. In such case, we may adopt as a reference a singly ionized structure with RHF reference function, and by performing the EA-EOM-CC calculations, we get the results for open-shell system without going through inconvenient UHF methodology. An example of the EA-EOM calculations carried out in this manner are the energies of the atomic levels of the Na atom reported in Ref. 60. Assuming as a reference Na+ ion and adopting a relatively large basis set (128 functions), the EA-EOMCCSDT provided quite accurate term values with the error for the first four terms lower than 0.01 eV. We may observe also a noticeable improvement upon incorporation of the triple excitations into the EOM scheme (reduction of the error by half ).60 Even more convincing example of the usefulness of the EA-EOM approach can be demonstrated with calculations of the potential energy curves (PEC) for electronic states of cations of alkali metal dimers, Me+2 . By removing one electron from the Me+2 , we obtain a double cation which is a closed-shell system at the equilibrium and also dissociates into closed+ + shell fragments: Me2+ 2 ! 2Me (Me is isoelectronic with the noble gas as a reference, we may apply the RHF atom). By adopting the Me2+ 2 solution for the whole range of interatomic distances. Performing in each point the EA-EOM-CC calculations, we recover the original cationic structure and generate the smooth potential energy curve for the Me+2 .61,62 The same approach has been applied to alkali metal heterodiatomics, e.g., see Ref. 60 for LiK+ and NaK+ results and alkali metal hydrides: NaH+63 and KH+.64 For each system, potential energy curves as well as selected molecular constants were computed for the lowest lying electronic states. In most cases, those were first rigorous ab initio results to confront with the values obtained

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on the basis of effective core potential. Majority of the theoretical data reported in Refs. 60–64 were obtained at the CCSD level. The EOMCCSDT values for the alkali metal diatomics do not affect the accuracy in the critical manner although in all cases improve the results. 4.2.2 DIP-EOM-CC and DEA-EOM-CC approaches For the DIP and DEA approaches, the EOM matrix, Eq. (21), takes much simpler form: h ^ i ^  N ¼ hDjH^ N jDi hDjH^ N jT i H (22)  jDi hTjH  jT i hT jH N

N

^ N jDi block, In the CCSD model, the EOM matrix is reduced to the hDjH 57 where in the case of DIP the D indicates doubly ionized configurations Φij, whereas in the DEA case58 it represents the configurations with two attached electrons Φab. At the CCSDT level, the T configurations enter the EOM matrix: for the DIP case these are Φaijk , while for the DEA— 2 Φabc i . In the former case, the EOM matrix at the CCSD level is of the no 41 size and can be easily diagonalized with the standard QR scheme. For the small- and medium-size molecules, the QR algorithm can be applied also at the CCSDT level (size of matrix: n3o nv ). For the DEA case, the EOM matrices are relatively large: n2v and no n3v for SD and SDT case, respectively, and their diagonalization requires more elaborated technique.39,40 In Table 3, we compare performance of both CC models in the evaluation of the energies57 of the doubly ionized states of several molecules. The MAE values were obtained for 3 (2 in case of H2O) lowest lying states. In all cases, the errors introduced at the CCSDT level are 5 to 10 times lower than those for the CCSD model. Note that the deviations from experiment are large even for more accurate scheme. Keeping in mind, however, that the DIP values are of the order of 30 to 50 eV, the relative error—particularly for the SDT model—is small.

Table 3 MAEs (eV) of the energies of the doubly ionized states related to experiment (from Ref. 57). H2O CO C2H2 C2H4

No. of states

2

3

3

3

DIP-EOM-CCSD

9.4

5.2

5.9

5.6

DIP-EOM-CCSDT

0.8

0.3

1.5

1.1

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The results collected in Table 3 represent an example of a direct application of the DIP-EOM scheme, i.e., starting from neutral closed-shell structure we obtain results for double positive cation. We may consider also an indirect approach in which we assume as a reference the double anion for which the DIP-EOM will recover original neutral molecule. The examples of such indirect approach are DIP-EOM calculations of the automerization barrier of the cyclobutadiene.57 Starting with the C4H2 4 reference and performing the DIP-EOM-CC calculations, we go smoothly through the critical point corresponding to square configuration. Similar problem can be met in the calculation of the potential energy curves for twisted ethylene, i.e., the curve depending on the dihedral angle. Here, the singularity occurs at the dihedral angle equal to 90 degrees due to HOMO-LUMO degeneracy, and to avoid the latter we assume as a reference the double anion form of C2H4. Due to this, we generate smooth PEC in the whole range of values of dihedral angle.57 A good example of indirect applications of the DEA-EOM approach are calculations of the potential energy curves for alkali metal diatomics. A direct approach to the dissociation of the Me2 dimer would be to use the EE-EOM scheme to compute the energy values for the interatomic distances from equilibrium to the dissociation limit. We observe, however, that the closed-shell structure, occurring at the equilibrium bond length, turns to the open shell during homolytic dissociation. We may apply to this problem the same solution as in the mentioned above case of the dissociation of the cationic dimer Me+2, i.e., to adopt as a reference system the double positive ion Me2+ 2 which + 2+ dissociates into closed-shell fragments Me . By applying to the Me2 reference the DEA-EOM approach, we recover a neutral structure obtaining at the same time a smooth curve in the whole range of interatomic distances. The usefulness of this strategy has been confirmed in Ref. 58 where the accurate PECs were obtained for the ground state of Na2 dimer. Similarly by doing DEA-EOM-CC calculation for double cations of hydrazine and ethane molecules, the PECs describing correctly the dissociation of the N–N and C–C bonds, respectively, were obtained. They were confronted with the results obtained with the standard CCSD, CCSDT, and CCSD(T) methods, and the latter either exhibit unphysical behavior (CCSD(T)) or it cannot be converged for the stretched bond (example: CCSDT for the N2H4). A satisfactory performance is obtained also in the calculations of PECs for excited states of Na265,66 with the DEA-EOM-CC at the SD and SDT levels. Both potential energy curves as well as excitation energies remain in satisfactory agreement with experiment.

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In the next section, we will apply the same strategy to calculate the PECs for alkali metal diatomics using the multireference formulation of the CC theory. The latter scheme offers in addition to the competitive accuracy of results also a size-extensivity property, essential in the studies of interatomic potentials.

5. Multireference coupled-cluster (MRCC) approach 5.1 General considerations One of the possibilities mentioned above to go beyond the SR formulation of the CC theory is to recall to its multireference extension.67 The latter scheme offers a way to treat the radicals and ionized species or, in general, the systems which require a multideterminantal description of the reference state. Obviously, there is a certain overlap of the fields covered by the EOM and MR approaches. The principal idea of the MR approach is to solve the Schr€ odinger equation, Eq. (3), in a specific manner: to obtain only a few (hopefully most important) eigenvalues out of the whole spectrum of the Hamiltonian. The solutions sought span the M-dimensional model space P  with the relevant projection operator P^ ¼ M I jΦI ihΦI j. The remaining part of the configurational space is a subspace known as the orthogonal one, ^ projector, P^ + Q ^ ¼ ^1. The action of P^ on the exact wave ?, with the Q o ^ k ¼ Ψk , generates the model function Ψok , defined within the function, PΨ P model space, Ψok ¼ M I c Ik ΦI . The latter function can be converted back to ^ o. ^ Ψk ¼ ΩΨ the exact one by the action of the wave operator Ω, k ^ and P^ operators, we obtain from Eq. (3) a With the help of the Ω well-known Bloch equation written out as: ^ P^ ¼ Ω ^H ^N Ω ^ ef f P^ H

(23)

^ ef f is defined as where the effective Hamiltonian operator H ^ P^ ^ ef f ¼ P^H ^NΩ H

(24)

^ ef f Ψok ¼ ΔEk Ψok and pro^ ef f operates on the model space functions H The H vides exact eigenvalues ΔEk. Within the CC formalism, the wave operator ^ is defined in the same way as in the single reference situation, i.e., via Ω ^  eS^, where the operator S^ is, generally, responexponential expansion, Ω sible for the transitions from the model space to the orthogonal one.

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An annoying problem of the multireference formulations of the CC theory are intruder states. The latter arise when the low-lying excited determinants from the orthogonal space are close in energy to those in the model space. This is a reason for the numerical instabilities which make the MRCC equations difficult to converge or even divergent. A detailed form of the wave operator depends on the assumed variant of the MRCC theory. Basically, we can distinguish two principal definitions of the wave operator which lead to two different MRCC methods: the Hilbert-space (HS) MRCC and the Fock-space (FS) MRCC. Within the Hilbert-space approach, all engaged configurations, belonging either to the model or to the orthogonal space, as well as the resulting electronic states, refer to the constant number of electrons. The Fock-space formulation admits electronic configurations with variable number of electrons. In both formulations, we can identify significant contributions by Polish quantum chemists. The paper by Jeziorski and Monkhorst68 introduced an ansatz (known as JM ansatz) which is a basis for a so-called state-universal (SU) or state-specific (SS) formulations of the HS-MRCC theory. Within the Fock-space approach, also known as a valence-universal (VU) method, we can point to the essential contribution by Meissner and Bartlett69 introducing the intermediate Hamiltonian formulation of the FS-MRCC approach, crucial when thinking about wide molecular applications of the method. In both approaches, an important initial step is a selection of the active space involving a number of occupied (indicated by μ, ν, …) and unoccupied one-electron states (indicated by α, β, …). The excitations among active levels create the configurations Φαβ,… μν,… spanning the model space. To make both versions easier to define, we introduce unique index notation with the following relations between subsets of indices: fag ¼ fag + fμg ¼ fag + fαg + fμg fi g ¼ fig + fαg ¼ fi g + fμg + fαg

(25) (26)

Indices a, b, … (i, j, …) run over active and inactive particles (holes), i.e.,  … (i, j, …)—over inactive particles levels unoccupied (occupied) in Φ0; a, b, (holes); α, β, … (μ, ν, …)—over active particles (holes), and a, b, … (i, j, …) run over all active levels and inactive particles (holes). Below we give a brief description of both formulations of the MRCC theory.

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Monika Musiał and Stanisław A. Kucharski

5.2 Hilbert-space formulation of multireference coupled-cluster approach The wave operator for the state-universal approach is based on the JM ansatz68 and is expressed as: ^¼ Ω

M X I

^I ¼ Ω

M X

^

eSðIÞ P^I ¼

I

M X

^

eSðIÞ jΦI ihΦI j

(27)

I

and consequently we may define the JM MRCC wave function as: Ψk ¼

M X

^

c Ik eSðIÞ jΦI i

(28)

I

where the cIk coefficients are obtained by diagonalization of the Heff matrix, ^ is defined as Eq. (24). The cluster operator SðIÞ ^ ¼ S^1 ðIÞ + S^2 ðIÞ + ⋯ + S^n ðIÞ SðIÞ X { 0 ab⋯ sij⋯ ðIÞ^a{ b^ ⋯^j^i S^n ðIÞ ¼

(29) (30)

ab⋯ij⋯

where the summation over a, b, …, (i, j, …) runs over levels unoccupied (occupied) in the model determinant ΦI and 0 indicates that the excitations within the model space are excluded. Note that the definition of the hole and particle levels is modified for each reference determinant (holes—levels occupied in ΦI, particles—levels unoccupied in ΦI). Substituting the oper^ Eq. (27), into the Bloch equation, Eq. (23), and projecting from the ator Ω, right against model determinant ΦJ and from the left against the orthogonalspace determinant Φab⋯ ij⋯ (J), we obtain a set of SU equations determining ab… amplitudes sij… ð JÞ: X ^ eff Sð ^ JÞ SðLÞ ^ ^ eff jΦJ i hΦab… jΦJ i ¼ hΦab… jΦL ihΦL jH ij… ð JÞje ij… ð JÞjðH N  HJJ Þe L6¼J

(31) Analysis of the MRCC equations formulated within the SU formalism, like spin adaption, relation to the multireference CI as well as initial applications to the model system were reported in the papers by Jeziorski, Paldus, and collaborators.70–72 The detailed SU-MRCC equations at the CCSD level were reported in Ref. 73. Some simple applications for small molecules were

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done by Balkova et al.74,75 They indicate that the SU-CC method is capable to provide reliable results on condition that the CC solution is not damaged by intruder states. To reduce this risk (i.e., to avoid the intruder state problem), the novel formulations of the HS-CC theory emerged, known as state-specific (SS) approaches. A characteristic feature of the latter is a focus on the single electronic state, corresponding to one of the states belonging to the model space. The first attempt to construct the MR solution based on the SR wave function was made by Piecuch et al.76 employing the CCSDTQ solution for the single reference case (denoted as state-selective approach). The other formulation, introduced by Hubac and Masic,77,78 known as the Brillouin–Wigner (BW) state-specific MRCC method has been extensively exploited in the Pittner group.79,80 The BW-MRCC equations ^ JÞ ^ JÞ Sð ^ Sð jΦJ i ¼ EhΦab⋯ jΦJ iare obtained in a straightforhΦab⋯ ij⋯ ð JÞjH N e ij⋯ ð JÞje ward manner by inserting the JM wave function, Eq. (28), into the Schr€ odinger equation, Eq. (1), and projecting against excited determinants. The BW formulation suffers from the size-extensivity error and in order to minimize the latter a posteriori corrections are worked out.78,80 Inclusion of the latter on top of the full triples version of the Brillouin–Wigner scheme (BW-MRCCSDT) provided satisfactory results in particular in the evaluation of the singlet–triplet separation in organic biradicals.81,82 In recent years, the HS-MRCC calculations were dominated by another successful application of the JM ansatz proposed by Mukherjee and known under acronym MkMRCC.83–85 The MkMRCC equations were obtained by introducing the JM wave function, Eq. (28), into Schr€ odinger equation, Eq. (3), and after somewhat more complicated algebra based on the resolution of identity one can arrive at the following results: ^

^

Sð JÞ ^ c J hΦab⋯ H N eSð JÞ jΦJ i ¼ ij⋯ ð JÞje

X L6¼J

^

^

Sð JÞ SðLÞ ^ ef f jΦL i c L hΦab⋯ e jΦJ ihΦJ jH ij⋯ ð JÞje

(32)

This equation is slightly different from the SU approach, Eq. (31), since it depends on the coefficients cJ, which requires diagonalization of the Heff matrix in each iteration. The other difference is that the bra and ket vectors in the right-hand side of Eq. (32) refer to the same vacuum, which makes the working equations simpler. The MkMRCC approach has been proved to be

94

Monika Musiał and Stanisław A. Kucharski

rigorously size extensive.85 A construction of the production-level code has been reported by Evangelista, Allen, and Schaefer.86 An extension to the full inclusion of the triple excitations has been also done87 followed by the attempt to construct the MkMRCC analog of the CCSD(T).88 A competent and inspiring review of the MRCC theories has been published by Evangelista.89 The third realization of the JM ansatz, denoted as MRexpT, was proposed by Hanrath.90,91 The originality of the Hanrath’s approach lies in the definition of the orthogonal space ?. For example, for the MRCCSD model the latter contains unique set of excited determinants related to the model determinants by single and double substitutions in opposition to the previous definition of the ? space constructed by the action of the S1(I ) + S2(I ) on the model function ΦI (I ¼ 1, M ) which admits certain redundancy within the excited determinants. In summary, the JM-based realizations of the MRCC theory, particularly MkMRCC approach, became recently an important tool in solving the real chemical problems. The potential and prospects of the MkMRCC approach may be illustrated with the spectacular calculations of the singlet–triplet gaps for the system with 105 atoms and 2158 basis functions by Brabec et al.92

5.3 Fock-space multireference coupled-cluster approach An alternative formulation of the MRCC theory, known as a Fock-space approach, was shaped up in the Mukherjee’s group (see the thorough review ^ is by Mukherjee and Pal93). Within this formulation, the wave operator Ω defined in a valence universal manner as: ðk,lÞ

^ ¼ feS^~ gP^ Ω

(33)

where k, l denote the number of valence particles and valence holes, respectively. The distinction between hole and particle levels in this case is unique and is specified by the determinant Φ0 which serves as a Fermi vacuum and is usually selected as a Hartree–Fock solution for the reference state. The model space, defined by the projector P^ , is of a Fock-space type, i.e., it includes manifold of configurations with variable number of electrons assigned to different sectors: the Fermi vacuum Φ0—sector (0,0), determinants with one valence hole Φμ—sector (0,1), and those with one valence particle Φα—sector (1,0). Two-valence sectors are: (0,2) (Φμν), (2,0) (Φαβ), ^~ðk,lÞ is the excitation and (1,1) (Φα ) and similarly for higher sectors. The S μ

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operator which can be written out as a sum of operators responsible for single, double, triple, …excitations ^ ~ S

ðk,lÞ

ðk,lÞ

^ ~1 ¼S

ðk,lÞ

^ ~2 +S

ðk,lÞ

^~ +⋯+S N

(34)

ðk,lÞ ^ ~n can be defined as follows: In general, S ðk,lÞ

^ ~n S

¼

{ X  1 0 a b⋯  ^a{^b ⋯ ^j^i g s f   i j⋯ ðn!Þ2 a b⋯  i j⋯

(35)

where 0 indicates that the excitations within a model space are excluded. The n index in the last expression indicates the number of creation (annihilation) operators. Note that the summation ranges in Eq. (35) for the creation (superscripts a ⋯ ) and annihilation (subscripts i ⋯ ) operators are no longer exclusive, see Eqs. (25) and (26); hence, the contraction within the particular ^ expansion is possible. To prevent this, the normal product term of the Ω ^ operator, Eq. (33), is assumed.94 form, indicated by braces {}, of the Ω ðk,lÞ ~^ It follows from the definition in Eq. (35) that the S operator includes ^ðk,lÞ ¼ Pk Pl S^ði,jÞ . ~ all lower rank components: S i¼0 j¼0

ðk,lÞ ^ ~n , Eq. (35), introduces one additional implication. The definition of S ^~jΦα i, we ^ ~ operator, e.g., hΦaμ jS Namely, taking the matrix element of the S μ

obtain as the results a sum of amplitudes saα + saαμμ and we cannot separate the saα amplitude (sector (1,0)) and saαμμ amplitude (sector (1,1)) (the latter amplitude contains a so-called spectator line μ). In order to make the FS equations well determined, we have to find the saα from the lower, i.e., (1,0) sector. This observation, made by Haque and Mukherjee,95 imposes a hierarchical manner of solving the FS equations. For example, if we want to calculate the EE’s using the (1,1) sector of the model space, it implies also the calculations for the sectors of the lower valence rank, in this case, (0,1), (1,0), and (0,0) ones. ^ ~ expression an operator corresponding We may extract from the S ð0,0Þ ^ ¼ S^ ^ ¼ T^ + S ^ , i.e., replacing tilde with bar ~ to the (0,0) sector, S +S ^~ operator, i.e., indicates that we excluded the (0,0) sector from the S ^ ¼ eT^ feS^gP^ with T^ being a standard single reference cluster operator. Ω ^ into Eq. (23), multiplying with the Introducing the latter form of Ω

96

Monika Musiał and Stanisław A. Kucharski

^ from the left and operating with the orthogonal space projector expðTÞ ^ ðk,lÞ , we obtain FS amplitude equations: Q   ðk,lÞ   ðk,lÞ   ðk,lÞ ^ ðk,lÞ ^ ðk,lÞ H ^ ðk,lÞ ^ ef f P^ðk,lÞ  N eS^ H P^ Q ¼Q eS (36) c

c

whereas the model space projection produces the effective Hamiltonian: ðk,lÞ ^ ef f ¼ H

i¼k , j¼l X

i, j ¼ 0 i+j>0

^ ði,jÞ H ef f ¼

i¼k , j¼l X

i, j ¼ 0 i+j>0

ði,jÞ ^ P^ði,jÞ ^N Ω P^ H

(37)

which upon diagonalization gives energies of the considered process. Selecting the active space composed of Nah occupied and Nap unoccupied one-particle levels, we include into the model space all determinants obtained by the distribution of 0 to Na(¼ Nah + Nap) electrons into the Na active levels. In such a way, the model space contains all possible (k, l) determinants where k refers to the number of valence particles and l—to the number of valence holes, created with respect to the Fermi vacuum. The FS equations, Eq. (36), are solved in the iterative manner and a number of applications have been reported both in Mukherjee’s group (e.g., Refs. 96–98) as well as by Bartlett and coworkers (e.g., Refs. 99–102). A detailed formal analysis of the relationship between the cluster amplitudes of the valence universal and the linear coefficients of multireference CI wave function were carried out by Jeziorski and Paldus.103 A dominant model of these early applications is the CCSD scheme with some attempts to reach for approximate triples.98,100 Jankowski et al.104 tested various approximations introduced at the CCSD level for the H4 model system followed by the evaluation of the importance of approximate triples contributions.105 A general characteristic of the valence universal wave operator based on the Fock-space similarity transformations was given by Stolarczyk and Monkhorst.106,107 The full inclusion of the connected triples at the (0,0), (1,0), (0,1), and (1,1) sectors was reported by Musial and Bartlett108; however, the applications of the new scheme were limited (N2 molecule and Ne atom only) due to mentioned earlier intruder state problem which becomes even more severe upon incluð1,1Þ sion of the S^3 operator. Another factor which affects the attractivity of the Fock-space approach in the IP and EA calculations ((0,1) and (1,0) sectors, respectively) is an observation, made by Mukherjee and coworkers109 and

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demonstrated by Meissner and Bartlett,110 indicating an equivalence of the IP-EOM-CC results and those obtained with the FS-MRCC in the (0,1) sector (an analogous correspondence occurs in the case of the EA-EOMCC and (1,0) sector). Solving the IP(EA)-EOM-CC equations (i.e., diagonalizing the IP(EA)-EOM matrix) is computationally much more efficient than the iterative procedure used in the FS one-valence sector. An analogous correspondence between the EOM and FS solutions obviously does not hold for higher sectors, e.g., two-valence ones, but also in these cases the remedy has been found to cure the intruder state problem or, generally, to eliminate inconveniences resulting from the iterative solving of the FS equations. This scheme, known as an intermediate Hamiltonian approach, has been introduced to the FS-MRCC theory, as it was mentioned before, by Meissner and Bartlett.69

5.4 Intermediate Hamiltonian: IH-FS-MRCC An intermediate Hamiltonian (IH)69,111,112 approach to the FS theory relies on the replacement of the iterative technique with diagonalization of the intermediate Hamiltonian matrix. The IH matrix is defined within the intermediate space and the latter is that part of the orthogonal space which can be reached from the model space by the action of the current-sector cluster ði,jÞ  N matrix, i.e., EOM operator S^ . Thus, the full IH matrix consists of H  N matrix matrix, constructed within the intermediate space. Some of the H elements are further modified by the connected and disconnected contributions (called dressing in the original paper69) originating from the term ^ N feS^ g where the feS^ g expansion is limited to the terms that produce exciH

^ operator is always built from tations outside the intermediate space and the S the amplitudes belonging to lower sectors. The type of the EOM submatrix and the form of dressing elements depend on the studied sector and the adopted CC model. 5.4.1 Sector (1,1) The intermediate Hamiltonian matrix within the (1,1) sector is built on the basis of the submatrix of the EE-EOM matrix, Eq. (21), corresponding to the S configurations for the CCSD model (size of the IH matrix: nonv) or S and D configurations, for the CCSDT model (size: n2o n2v)). The connected ^ N eS^ expression enter the IH matrix through and disconnected terms of the H 2 ^ N jSi or through the I^3 element of the I^ elements of submatrix hSjH 11

11

98

Monika Musiał and Stanisław A. Kucharski

^ N jDi. The IH-FS within the (1,1) sector at the CCSD level submatrix hDjH was applied in Ref. 113 to study excitation energies of selected small molecules with relatively large basis sets and adequate model spaces generating the satisfactory results. The next step was an extension of the CCSD model to include connected triples by Musial and Bartlett.114–117 The IH-FS-CCSDT model was a more challenging task both at the implementation and at the application level. It requires the full CCSDT solution at the lower sectors (zero-valence and one-valence) and the constructed IH matrix needs more sophisticated tools for diagonalization.39,40 Nevertheless, the results—as it can be seen in Table 4—are worth the effort. In Table 4, we quote the vertical excitation energies for the four lowest lying singlet and triplet states obtained with the aug-cc-pV5Z basis set. The active space includes 12 virtual and 4 occupied levels which creates the model space of size equal to 48. An inclusion of the S^3 operator improves the accuracy of the results by more than an order of magnitude, bringing the errors to ca. 0.01 eV nearly for all considered excited states. Up to our knowledge, these are the best theoretical results obtained with the CC method.118 5.4.2 Sector (2,0) An intermediate Hamiltonian approach applied successfully to the (1,1) sector114 can be equally easily used in the other two-valence sectors.65,121 The most noteworthy applications were reported within the (2,0) sector. In this case, the IH matrix at the CCSD level is of the size n2v . The FS(2,0)MRCC (FS(2,0) for brevity) approach turned out to be particularly advantageous in the evaluation of the excited states of alkali metal diatomics. Adopting the same strategy as in the mentioned earlier case of DEA-EOM-CC calculations, i.e., by assuming the doubly ionized diatomic molecule as a reference, the FS approach in the (2,0) sector recovers the electronic structure of the neutral species. Since the double ionized structure dissociates into the closed-shell fragments, we may use the RHF reference in the whole range of interatomic distances and obtain smooth and continuous potential energy curves from the equilibrium to the dissociation limit. The advantage of the FS(2,0) scheme over the DEA-EOM approach relies on the fact that the former one belongs to the method rigorously size extensive which is a desired feature when studying dissociation processes. The FS(2,0) has been successfully applied to alkali metal dimers: Li2,122 Na2,123 alkali metal hydrides: NaH,63 KH,64 and alkali heterodiatomics: NaLi,124 KLi,125 and NaK.126

Table 4 Vertical excitation energies (in eV) of N2 as obtained with the IH-FS-CC at the CCSD and CCSDT levels with the aug-cc-pV5Z basis set.118 IH-FS IH-FS Sym. 1

Πg

9.354

Exp.a

Sym.

CCSDT

Exp.b

9.340

3 + Σu

7.852

7.760

7.750

Πg

8.167

8.040

8.040

CCSD

1  Σu

10.097

9.890

9.880

1

Δu

10.576

10.294

10.270

3

Δu

9.039

8.901

8.880

1

Πu

13.332

13.063

13.050

3  Σu

10.026

9.702

9.670

0.214

0.015

0.186

0.016

Ref. 119. Ref. 120. Active space (12,4). b

9.392

CCSDT

3

MAE a

CCSD

MAE

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Monika Musiał and Stanisław A. Kucharski

Calculations of the potential energy curves for alkali metal diatomics are dominated by the approaches treating explicitly only valence electrons and replacing the inner shells by the effective core potentials (ECPs). The latter admit some adjustable parameters which improve the accuracy compared to the rigorous ab initio methods. However, the results obtained in series of calculations63,64,122–126 demonstrate that the FS(2,0) method outperforms the ECP-based approaches. The potential energy curves obtained in the cited works fully coincide with the experimental (RKR or IPA) potentials, reproducing correctly all multiple minima and asymptotic dissociation limits. The excellent performance of the FS(2,0) scheme can be appreciated when the calculated results are compared to experiment, see Table 5. In this table, we list the mean absolute errors for selected molecular parameters: equilibrium bond distance, well depth, and adiabatic excitation energy for three molecules: Li2, Na2, and LiNa. The reported errors were obtained by comparison of the computed values with experiment. The number of considered electronic states was depending on the availability of the experimental data. Table 5 shows that the average error for De (well depth) values for all electronic states for which experimental values are available (15 for Li2, 13 for Na2, and 8 for LiNa) is about 30 cm1 which is much below 0.01 eV. The same order of accuracy is observed for the adiabatic excitation energies, Te, for the Li2 and LiNa molecules. Only for the Na2 molecule, the average (13 states) error amounts to 120 cm1 which corresponds to 0.015 eV. In the summary of two principal formulations of the MR theory, we would like to mention about the method which do not fall into these two categories. This is an internally contracted (ic) MRCC approach. The ic-MRCC wave function is obtained when SR-type wave operator operates on the multideterminantal reference function Ψok.127 The advantage of the ic-MRCC relies on the fact that contrary to the state-universal and valence-universal approaches, the number of amplitudes does not grow Table 5 MAEs of selected spectroscopic constants related to the experiment—IH-FS-CCSD (2,0) (the number of considered electronic states in parenthesis)—from Refs: Li2,122 Na2,123 and LiNa.124 Molecule Re(Å) De(cm21) Te(cm21)

Li2

0.011(17)

24(15)

48(15)

Na2

0.024(14)

33(13)

120(13)

LiNa

0.026(7)

37(8)

61(7)

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linearly with the size of model space. The ic-MRCC approach seems to be an attractive alternative to the other MR methods; however, its weak point is an enormous complexity. Nevertheless, the initial results obtained in the K€ ohn group128 are very encouraging.

6. Nonstandard realizations of the coupled-cluster theory In this part, we will briefly discuss those variants of the CC theory which stay aside the main development lines considered in the previous sections. From the computational viewpoint, the most appealing approaches are those based on the SRCC theory due to its formal simplicity and easy implementation. To this category, we may count the tailored coupled-cluster (TCC) method introduced within the CCSD scheme by Bartlett and coworkers.129–131 An essence of the TCC relies on using a small subset of the cluster amplitudes from external sources, e.g., from complete active space (CAS) CI solution, which are kept fixed during the CC iterations. Similar idea has been introduced by Jankowski and coworkers132–134 and referred to as a split amplitude strategy. In this approach, the cluster amplitude is treated as a sum of two parts: one is fixed a priori and the other, a small correction, is determined from the CC equations. The fixed part is taken from the independent sources, e.g., various MBPT solutions or single and multireference CI. This scheme, referred to by the authors as almost linear CC, has been tested for two model systems (H4 and H8) and a several small molecules (H2O, BH, FH). The results demonstrate that even a small number of fixed amplitudes, taken from the CI calculations, significantly improve the results for the strong degeneracy regions (stretched bonds). In similar spirit may be seen an approach proposed by Stolarczyk,135 although in this case the amplitudes taken from the external sources (e.g., CASSCF) belong to the higher cluster operators T3 and T4, generating a constant contribution to the T1 and T2 amplitudes. A correlation energy and properties can be related to the exponentially parameterized wave function also via expectation value expression (XCC) or the unitary CC approach. The XCC energy is defined as ΔE ¼ { ^ N eT Þc jΦ0 i where c subscript indicates that only connected terms hΦ0 jðeT H enter the energy expansion. The first formulation of the XCC goes back to the paper by Bartlett and Noga136 where the amplitude equations and energy

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expression were derived for the XCC(n) expansion truncated at order n. A different formulation has been proposed by Jeziorski and Moszy nski137 who observed that the expectation value for the operator X^ formulated T{

^

within the XCC framework as X ¼ hΦ0 je T {XeT hΦ0 je

jΦ0 i e jΦ0 i T

can be expressed in terms

of the finite commutator series of T^ and an appropriately defined auxiliary ^ This idea has been later exploited by Korona and Jeziorski138 to operator S. calculate one-electron density matrix and a set of one-electron properties for a number of small- and medium-size molecules (e.g., benzene dimer) with quite satisfactory results. The unitary coupled-cluster (UCC) approach goes back to Kutzelnigg139 resumed later by Bartlett et al.140 It is based on the definition of the unitary { cluster operator τ^ ¼ T^  T^ . The UCC approach gains recently a lot of interest due to its potential usefulness for quantum computing. A number of papers focused on that subject which were published recently are coauthored by Kowalski.141–143 The approaches worked out by Piecuch and coworkers were aimed at the capture of substantial amount of correlation effects due to higher cluster operators. One route leading to this purpose is an active space SRCC approach144 which relies on the inclusion of higher clusters (e.g., T3 and T4) into the CC expansion while limiting the summation over their indices to the smaller subset of the one-particle levels called an active space. The clusters restricted fully or partially to the active space are indicated by small letters in the respective acronyms, e.g., CCSDt, CCSDtq, and CCSDTq, e.g., the latter denotes the CC model with the amplitudes of the T4 operators summed within the limited range of indices.144 The same scheme was later on introduced to the EOM-CC methods: with respect to the excitation energy calculations50,145 and ionized (IP-EOMSDt) and electron-attached (EA-EOM-CCSDt) states.146,147 In the case of DIP and DEA calculations, the adopted model at the EOM level goes as far as to 4 particle–2 holes (DEA) or 2-particle–4-holes (DIP) excitations148–150 which corresponds to the inclusion of R4(k) operator in Eq. (19). In the DEA (DIP) calculations, the reference state function is obtained at the CCSD level and ^ N DEA(DIP)-EOM-CC matrix is constructed with the CCSD-based H operator. In order to make this scheme computationally tractable, the deep restriction on T and Q configurations is imposed. The thorough analysis of the performance of the active space approaches can be found in the review.151

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Other approaches introduced by the Piecuch’s group are focused on the development of the robust noniterative corrections based on the CCSD wave function. The relevant expressions are modified with respect to the standard CCSD(T) or CCSD[T] schemes by the renormalization of the noniterative contribution, i.e., by introduction of the denominator resulting from the overlap of the exact wavefunction with the CCSD one.152 The modified noniterative approach named by the authors CR (completely renormalized)–CCSD(T) scheme (there are also more advanced variants: CR-CCSD(TQ) or CR-CCSDT(Q)) has some advantage over regular CCSD(T) scheme in particular when applied to the potential energy curves at the cost of losing rigorous size extensivity.153 The modified treatment of the noniterative terms has been applied also to the excited states resulting in the CR-EOM-CC schemes.154–156 Recent developments are focused on combining both above approaches, i.e., renormalized noniterative contributions and active space treatment.157 One of the original novel developments is a combination of the EOM-CC approach with Monte Carlo sampling.158 Another important contribution to the CC theory made by Korona is connected with the implementation of the local CC method applied to excited states. This relied on the extension of the local CCSD scheme to the local EOM-CCSD done in collaboration with Werner159 and analogous implementation for the CC2 variant (with Sch€ utz).160 This extension combined with the density fitting technique made it possible to run calculations for the large systems exceeding 100 atoms. The test calculations were successfully done for the complex containing 127 atoms and 370 correlated electrons.160 Much attention has been paid recently to the geminal-based wave function obtained via pair couple-cluster doubles (pCCD) introduced by Ayers.161 The pCCD scheme is surprisingly efficient in the situations when nondynamic correlation is important. The results obtained recently by Boguslawski and her team162,163 confirm the usefulness of the geminal-based approaches both in the proper reproduction of potential energy curves for several diatomics162 and in the studies of excited states of selected actinide compounds.163

7. Final remarks In this short review, we focused on these realizations of the CC theory to which Polish quantum chemists made some contributions. Many of them were done in cooperation with American/Canadian research centers. The

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researchers working in Poland on the development of various aspects of CC theory are connected with two universities: Quantum Chemistry Laboratory at the University of Warsaw—group established by Włodzimierz Kołos and currently led by Bogumił Jeziorski—and Nicolaus Copernicus University (UMK) in Torun—connected with the name of Lutosław Wolniewicz where the main research groups were established by Karol Jankowski (UMK, Department of Physics) and Andrzej Sadlej (UMK, Department of Chemistry). Many of the members of the Quantum Chemistry Laboratory at the University of Warsaw were cited in the current review: Bogumił Jeziorski, Robert Moszynski, Tatiana Korona, and Leszek Stolarczyk. At the Nicolaus Copernicus University in Torun, the research relevant to the topic of this review is carried out mainly by Leszek Meissner who has been many times cited in the multireference section. A quantum chemistry at this University is represented by Ireneusz Grabowski, Maria Barysz, ˙ uchowski, Mirosław Katharina Boguslawski, Paweł Tecmer, Piotr Z Jabło nski, and Piotr Jankowski. To make things complete, we should mention also the quantum chemistry group established at the University of Silesia in Katowice to which the authors of the current review belong. It should be emphasized how beneficial for the development of quantum chemistry in Poland was and is a collaboration with American quantum chemistry groups. For the research focused on the electronic correlation, the most often visited research center is a Quantum Theory Project and a fruitful collaboration with the research teams of Rodney J. Bartlett and Hendrik Monkhorst. More than 15 long-term visitors have been invited to the Bartlett’s group from Poland both as postdoctoral associates and visiting professors (also to pursue a Ph.D. program) which resulted in over 130 common publications (R.J. Bartlett + Polish coauthors). Another research center open for collaboration with Polish scientists is located at the University of Waterloo and was established by Josef Paldus. There are many papers resulting from the cooperation with Josef Paldus by Karol Jankowski group, Bogumił Jeziorski, and Piotr Piecuch. Nearly all Polish collaborators presently have research groups in Poland or in the United States/Canada.

Acknowledgments The research activities are cofinanced by the funds granted under the Research Excellence Initiative of the University of Silesia in Katowice.

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162. 163.

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CHAPTER ONE

Electronic convection in resultant information-theoretic description of molecular states and communications Roman F. Nalewajski* Department of Theoretical Chemistry, Jagiellonian University, Cracow, Poland *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Continuities of wavefunction components 3. Phase supplements of classical entropic descriptors 4. Probability and current networks 5. Internal ensembles of charge-transfer states 6. Continuity of chemical potential descriptors 7. Conclusion References

116 117 120 126 128 133 136 137

Abstract Time-evolutions of the modulus and phase components of electronic states are examined, their continuity relations are discussed, and additive separation of components in wavefunction logarithm is emphasized. The classical (probability) density exhibits a vanishing source, while the nonclassical (phase/current) quantities exhibit nonvanishing productions. Probability current establishes the “convection” velocity, which determines flows of all electronic properties. Its source is related to probability acceleration reflecting the gradient of phase-production. The local energy concept implies the phaseequalization in stationary equilibria. The resultant descriptors of the entropy/information content in complex electronic states are summarized, their integral sources are examined, and the phase-dynamics in a mutual opening of reactants is tackled. The charge transfer (CT) phenomena in donor-acceptor systems are approached using the internal ensemble approach, with one reactant providing a microscopic electron reservoir for the complementary subsystem. A statistical mixture of the integral CT states generates the parabolic energy function of Mulliken’s interpolation. It implies a continuous electronegativity difference driving CT, while the macroscopic reservoir of thermodynamic grand ensemble predicts the chemical potential discontinuity.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.010

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2023 Elsevier Inc. All rights reserved.

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1. Introduction The Quantum Mechanics (QM) and Information Theory (IT) provide a solid basis for both determining and understanding the electronic structure of molecules, and for explaining general trends in their chemical behavior. The former establishes the system electronic wavefunction and its physical attributes, while the latter provides additional interpretative tools for an entropic, chemical interpretation of its structural organization and dynamics. The IT treatment has been successfully used to probe and interpret in chemical terms the electronic structure of molecules, e.g.1–5 To paraphrase Prigogine,6 the modulus and phase components of electronic wavefunctions, respectively, determine the (static) structure “of being” and the (dynamic) structure “of becoming.” The latter indeed generates the velocity of electronic current, the divergence of which shapes the time evolution of probability density. This “convection” flux, measured by the current-per-particle, also defines the flow descriptors of other physical and IT entities in their respective continuity relations. It is of interest in theory of electronic structure and chemical reactivity to reexamine implications of the Schr€ odinger eq. (SE) for the dynamics of the wavefunction components and related physical distributions of electronic probability and its current. In particular, it is crucial to extract local sources of the generalized IT descriptors combining the probability and phase/current contributions, explore a role of probability convection in electronic communications, and determine the nonclassical (phase-related) entropic descriptors of the chemical bond. In the resultant IT description4,5,7–20 both these structural organizations contribute to the overall content of the entropy or information in quantum states. The classical, probability terms of the Fisher21–25 gradient information or Shannon26 global entropy predict the vanishing sources in their local continuity balances, while the nonclassical, phase/current-related supplements in resultant measures give rise to finite productions of IT properties.3–5,27,28 Probability flows should also affect electronic communications in molecules. In complex electronic wavefunctions the “static” probability channel of classical communication theory26,29,30 is supplemented by the associated “flow” network reflecting a convection of electronic probability.4,31 In Orbital Communication Theory (OCT) of the chemical bond1–5,31–41 these information systems reflect the static and dynamic patterns of the probability scattering between the adopted “input” and “output” electronic events via

Electronic convection in resultant information-theoretic description

117

the system occupied Molecular Orbitals (MO) defining the configuration Slater determinant. In the resolution defined by Atomic Orbitals (AO) the channel communication “noise” (conditional entropy) descriptor reflects the overall “covalency” in such a molecular state, while the information “flow” (mutual information) indexes the network bond “ionicity.” Compared to probability distribution alone, a presence of the local current pattern introduces an additional predictability (information) about the system, and hence signifies less uncertainty (entropy) content. The classical and nonclassical contributions exactly cancel each other in phase-transformed (thermodynamic) states, the phase-equilibria.4,5,7–13 The statistical mixtures of “equilibrium” states generate the ensemble entropy of von Neumann,42 reflecting electronic uncertainty due to external state-probabilities in applied thermodynamic conditions. The internal phase-equilibria then correspond to local phases generating nonvanishing electronic flows. In molecular subsystems, e.g., reactants, these fluxes reflect the current activation of the polarized, mutually-closed substrates, which constitutes a vital information for predicting reactivity trends in the equilibrium reactive system consisting of mutually-open fragments.5,18–20 While an external openness of a molecule, relative to macroscopic (infinitely “soft”) electron reservoir, predicts chemicalpotential discontinuity,43,44 in the ensemble approach45 to the chemical potential (negative electronegativity) concept,46–48 little is known about a behavior of electronegativity difference driving the internal ChargeTransfer (CT) between externally-closed reactants. It would be interesting to connect such an internal-ensemble analysis to the classical approach by Mulliken.46

2. Continuities of wavefunction components Consider the simplest case of a quantum state at time t, jψ(t)i, of a single electron moving in external potential v(r) due to the “frozen” nuclear framework. The two components of its (complex) wavefunction ψ ðr, tÞ ¼ hrjψ ðtÞi ¼ Rðr, tÞ exp½iϕðr, tÞ

(1)

generate the state physical descriptors of the probability density p(r, t) ¼ R(r, t)2 and its current: j ðr, t Þ ¼ ½ħ=ð2miÞ ½ψ ∗ ðr, tÞ rψ ðr, tÞ  ψ ðr, tÞ rψ ∗ ðr, tÞ ¼ pðr, tÞ V ðr, tÞ ≡ J p ðr, tÞ:

(2)

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The current-per-electron defines a “convection” velocity of the probability fluid reflecting the phase gradient: V ðr, tÞ ¼ j ðr, t Þ=pðr, tÞ ¼ ðħ=mÞ rϕðr, tÞ:

(3)

The Schr€ odinger eq. (SE) generates the state dynamics, iħ ½∂ψ ðr, tÞ=∂t ¼ HðrÞ ψ ðr, tÞ,

(4)

where the particle Hamiltonian
 HðrÞ ¼  ħ2=ð2 mÞ Δ + vðr Þ ≡ TðrÞ + vðr Þ:

(5)

It predicts the sourceless continuity of the particle probability density: σ p ðr, tÞ  dpðr, t Þ=dt ¼ ∂pðr, t Þ=∂t + ½drðtÞ=dt½∂pðr, tÞ=∂r  ¼ ∂pðr, t Þ=∂t +V ðr, t Þ∇pðr, tÞ ¼ ∂pðr, tÞ=∂t + ∇ j ðr, t Þ

(6)

¼ ∂pðr, t Þ=∂t + ½V ðr, t Þ∇pðr, tÞ + pðr, t Þ∇V ðr, t Þ ¼ 0, and hence also rV ðr, tÞ ¼ ðħ=mÞ Δϕðr, tÞ ¼ 0

or

Δϕðr, t Þ ¼ 0:

(7)

Above, ∂p(r, t)/∂t denotes the probability derivative at the fixed point in space, while the total derivative dp(r, t)/dt is measured at a point moving with the “convection” velocity V(r, t). The state components are additively separated in the wavefunction logarithm: ln ψ ðr Þ ¼ lnRðrÞ + iϕðr Þ ≡ θðrÞ + iϕðrÞ:

(8)

Expressing SE in terms of lnψ then gives:   i½∂ ln ψ ðr Þ=∂t  ¼ i½∂θðr Þ=∂t   ∂ϕðr Þ=∂t ¼ ½ħ=ð2mÞ ½∇ ln ψ ðrÞ2 + Δlnψ ðr Þ + vðrÞ=ħ   ¼ ½ħ=ð2mÞ ½∇θðrÞ2  ½∇ϕðrÞ2 + ΔθðrÞ + 2i∇θðr Þ∇ϕðrÞ + vðr Þ=ħ: (9)

Comparing the real parts in this complex relation generates Schr€ odinger’s phase-dynamics,   ∂ϕðrÞ=∂t ¼ ½ħ=ð2mÞ ½rθðrÞ2  ½rϕðrÞ2 + Δθðr Þ  vðr Þ=ħ, (10) while its imaginary components give rise to the modulus or θ-dynamics: ∂θðrÞ=∂t ¼ ðħ=mÞ rθðr ÞrϕðrÞ ¼ V ðrÞrθðrÞ ≡ r J θ ðrÞ:

(11)

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These equations can be conveniently interpreted as the associated continuity equations: ∂ϕðrÞ=∂t ¼ r J ϕ ðrÞ + σ ϕ ðrÞ and ∂θðrÞ=∂t ¼ r J θ ðr Þ,

(12)

where component sources, σ ϕ ðrÞ ¼ dϕðrÞ=dt ¼ ∂ϕðrÞ=∂t + r J ϕ ðrÞ and σ θ ðr Þ ¼ σ R ðrÞ=RðrÞ ¼ dθðr Þ=dt ¼ ∂θðrÞ=∂t + r J θ ðr Þ ¼ 0,

(13)

reflect rates of changes in moving volume element. In ascribing fluxes to general distributions one realizes that all physical properties are carried by the probability convection of Eqs. (2) and (3), as already recognized in Eqs. (11) and (12). Therefore, the phase flow is characterized by the phase current Jϕ(r) ¼ ϕ(r)V(r) ≡ J(r), the modulus flux reads JR(r) ¼ R(r)V(r), Jθ(r) ¼ θ(r)V(r), etc. Analogous fluxes can be introduced to characterize flows of information descriptors, with the probability convection providing the velocity factor in their currents. The vanishing production of the wavefunction modulus is in accordance with the sourceless continuity relation for the particle probability distribution in QM (Eq. 6): σ p ðrÞ ¼ dpðrÞ=dt ¼ ∂pðr Þ=∂t + rjðrÞ ¼ 2RðrÞσ R ðr Þ ¼ 2pðrÞσ θ ðrÞ ¼ 0:

(14)

A comparison between Eq. (10) and first continuity balance of Eq. (13) gives the following expression for the local phase production:   σ ϕ ðrÞ ¼ ½ħ=ð2mÞ ½rθðrÞ2 + Δθðr Þ  vðrÞ=ħ (15)   ¼ ½ħ=ð2mÞ ½rϕðrÞ2 + Rðr Þ1 ΔRðrÞ  vðrÞ=ħ 6¼ 0: Eqs. (10) and (15) are also in accord with the first Hohenberg-Kohn theorem of Density Functional Theory (DFT).44,49–53 Indeed, since the ground-state electron density p0 or its logarithm θ0 both correctly identify the system external potential, v ¼ v[p0] ¼ v[θ0], the phase dynamics and its source are also uniquely determined by these distributions. Therefore, they both uniquely recognize the quantum state itself, ψ ¼ ψ[ p0] ¼ ψ[θ0], and the time-evolution of its phase component: ϕ ¼ ϕ[ p0] ¼ ϕ[θ0]. The local production of electronic current similarly reads: σ j ðr, tÞ ¼ dj ðr, tÞ=dt ¼ pðr, tÞ½dV ðr, tÞ=dt ≡ pðr, tÞaðr, tÞ ¼ ðħ=mÞpðr, tÞrσ ϕ ðr, tÞ:

(16)

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It involves the local “acceleration” of probability “fluid”: aðr, tÞ ¼ dV ðr, t Þ=dt ≡ σ V ðr, t Þ ¼ ðħ=mÞ rσ ϕ ðr, tÞ:

(17)

which generates the electronic probability “force” F ðr, tÞ ¼ m aðr, tÞ ≡ rW ðr, tÞ,

(18)

negative gradient of the underlying probability “potential” W ðr, tÞ ¼ ħ σ ϕ ðr, t Þ:

(19)

To summarize, the net production of the classical probability-variable of electronic states identically vanishes, while that of their nonclassical, phasecomponent remains finite. It ultimately results in non-vanishing sources of the resultant entropy/information descriptors of electronic states, which combine the classical (modulus) contributions and the relevant nonclassical (phase/current) terms due to probability convection.

3. Phase supplements of classical entropic descriptors In molecular applications of classical IT of Fisher and Shannon the entropic descriptors quantify the information content in the electronic probability distribution or the state modulus. It has been argued, however, that in order to fully account for the whole structure information in the given quantum state one has to include in the resultant measure also the nonclassical convection (coherence) contribution due to the state phase ϕðr, t0 Þ ≡ ϕðrÞ  0 ðphase conventionÞ,

(20)

or its gradient determining the convection velocity. Since a presence of finite local currents introduces more predictability (information) in such a combined IT description, it diminishes the system overall uncertainty (entropy) content. The alternative instantaneous measures of the resultant entropy in a quantum state of Eq. (1), ψ(r, t0) ≡ ψ(r), include both the real (scalar) and complex (vector) descriptors3–5,7–17: ð S½ψ  ¼ hψ j ln p + 2ϕjψ i ¼ 2 pðrÞ½ln RðrÞdr + ϕðrÞdr ð (21) ¼ pðrÞsðr Þdr ≡ S½ p + S½ϕ ≡ S½ p, ϕ,

Electronic convection in resultant information-theoretic description

S½ψ  ¼ hψj2lnψjψ i ¼ S½ p + i S½ϕ ≡ S½p, ϕ,

121

(22)

where s(r) ¼ 2[lnR(r) + ϕ(r)] stands for the density-per-electron of the scalar functional. The former represents the quantum expectation of the Hermitian operator S(r) ¼ [lnp(r) + 2ϕ(r)] and combines the (positive) Shannon global entropy S[p], the quantum average value of the multiplicative operator lnp(r) ¼ 2lnR(r), ð h lnpi ¼ hψjlnpjψ i ¼ 2 pðrÞ lnRðrÞ dr ≡ 2h lnRi, (23) andÐ its (negative) phase complement S[ϕ] ¼ 2hψjϕjψi ¼ 2 p(r)ϕ(r)dr ≡ 2hϕi, which reflects the state average phase hϕi. These two contributions are seen to constitute the real and imaginary parts of the related complex mesure S[ψ], expectation value of the state non-Hermitian operator S(r) ¼ 2lnψ(r). Consider now the resultant analog of the classical Fisher (gradient) information for locality events, given by the familiar probability/modulus measure ð ð ð I ½ p ¼ pðr Þ ½rlnpðr Þ2 dr ¼ pðr Þ1 ½rpðr Þ2 dr ¼ 4 ½rRðr Þ2 dr ¼ I ½R: (24)

Its generalization I[ψ] for a complex quantum state ψ represents the expectation value of the electronic information operator23   IðrÞ ¼ 4Δ ¼ 8m=ħ2 Tðr Þ ≡ κ Tðr Þ, (25) proportional to the particle average kinetic-energy T [ψ] ≡ hT i ¼  [ħ2/(2 m)] hψjΔjψi ≡ hψjTjψi, ð ð   I ½ψ  ¼ κ hT i ¼ hψ j4Δjψ i ¼ 4 j∇ψ ðr Þj2 dr ¼ 4 pðrÞ ½∇ ln Rðr Þ2 + ½∇ϕðrÞ2 dr ð  I ½R + I ½ϕ ¼ I ½ p + ð2m=ħÞ2 pðrÞ1 j ðrÞ2 dr  I ½ p + I ½ j  ð ð ¼ I ½ p + ð2m=ħÞ2 pðr ÞV ðr Þ2 dr  I ½ p + I ½V   pðr ÞI ðrÞdr: (26)

Here, I(r) stands for the density-per-electron of this overall measure. This descriptor is thus proportional to the system average kinetic energy of

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an electron. It supplements the classical measure of Eq. (24) with the (positive) nonclassical (convection) complement due to the wavefunction phase or the associated current/velocity: I[ϕ] ¼ I[ j] ¼ I[V ]. Due to the vanishing production of electronic probability density [see Eq. (14)] the sources of classical entropy/information descriptors also identically vanish: ð dS½ p=dt ¼ σ p ðr Þ ½δS=δpðrÞ dr ¼ 0 and ð dI ½ p=dt ¼ σ p ðrÞ ½δI=δpðrÞ dr ¼ 0: (27) The nonclassical contributions of resultant measures, however, give rise to finite sources, due to a nonvanishing phase production of Eq. (15)4,5,17,27: ð ð σ S ðt0 Þ ¼ σ ϕ ðrÞ ½δS½ϕ=δϕðrÞ dr ¼ 2 pðrÞ σ ϕ ðrÞ dr, (28) ð ð σ I ðt0 Þ ¼ σ ϕ ðrÞ ½δS½ϕ=δϕðrÞ dr ¼ κħ j ðrÞrσ ϕ ðrÞ dr ð ¼ κ pðrÞ V ðrÞrvðrÞ dr: (29) The nonclassical information I[ϕ] thus generates a nonvanishing integral source in state ψ: ð ð σ I ½ψ  ¼ dI ½ϕ=dt ≡ pðr, t Þ σ I ðr, tÞ dr ¼ κ j ðr, tÞrvðrÞ dr ð ¼ κ ħ j ðr, tÞrσ ϕ ðr, t Þ dr: (30) Its density-per-electron σ I(r, t) ¼ dI(r, t)/dt is seen to be determined by the product of a local probability “flux” j(r, t) and “affinity” factor proportional to gradient of the external potential, related to the gradient of a local phase-production σ ϕ(r, t). As an illustrative example consider the stationary states of an electron, corresponding to the sharply specified particle energy Est. and timeindependent probability distribution: ψ st: ðr, t Þ ¼ Rst: ðrÞ exp½iðE st=ħÞt  Rst: ðr Þ expði ωst: t Þ  Rst: ðr Þ exp½i ϕst: ðt Þ:

(31)

123

Electronic convection in resultant information-theoretic description

In such states the phase is thus exclusively time-dependent, ϕst.(r, t) ¼ ϕst.(t), and the phase-source amounts to a constant, σ ϕ st: ðr, tÞ ¼ ωst:, thus generating a vanishing local probability force of Eq. (18), since then W st.(r, t) ¼ ħωst. ¼ const. For this “equilibrium” density pst.(r, t) ¼ Rst.(r)2 ¼ pst.(r) the convection current also identically vanishes, j st.(r) ¼ (ħ/m) pst.(r) rϕst.(t) ¼ 0, thus conserving in time this probability distribution. These eigenstates of electronic Hamiltonian, Hðr Þψ st: ðr, t Þ ¼ E st: ψ st: ðr, tÞ

or

Hðr Þ Rst: ðr Þ ¼ E st: Rst: ðrÞ,

(32)

thus correspond to the spatially equalized local energy E ðr, tÞ ≡ ψ ðr, t Þ1 HðrÞψ ðr, t Þ, 1

E ðr Þ ≡ R ðr Þ st:

st:

(33)

HðrÞ R ðrÞ ¼ E : st:

st:

(34)

This principle can be also interpreted as an equalization rule for the spatial phase. Indeed, the local wave-number/phase concepts for a general state of Eq. (1), ωðr, tÞ ≡ E ðr, tÞ=ħ

and

ϕðr, t Þ ¼ ωðr, tÞ t,

(35)

directly imply their local equalization in ψ st.(r, t): ωst: ðr, tÞ ¼ Est:=ħ ¼ ωst: ¼ const:

and

ϕst: ðr, t Þ ¼ ðE st:=ħÞ t ¼ ωst: t ¼ ϕst: ðtÞ:

(36)

Therefore, the stationary equilibrium in quantum mechanics is marked by the local phase equalization throughout the whole physical space. It should be realized that, due to the complex character of the wavefunction, the local energy of Eq. (33) is also complex: E(r, t) 6¼ E(r, t)⁎. This also implies the complex nature of the local phase/wave number concepts. The two components of the latter, ωðr, tÞ ¼ c ðr, tÞ + i bðr, tÞ,
 c ðr, t Þ ¼ Re ωðr, tÞ ¼ ωðr, t Þ + ωðr, tÞ∗ =2,
 bðr, tÞ ¼ Im ωðr, t Þ ¼ ωðr, tÞ  ωðr, tÞ∗ =ð2iÞ,

(37)

ultimately shape the dynamics of the additive components of Eq. (8). Rewriting SE in terms of these two components of the (complex) wave-number gives:

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ψ ðr, tÞ1 ½∂ψ ðr, t Þ=∂t ¼ ∂ ln ψ ðr, tÞ=∂t ¼ ∂θðr, tÞ=∂t + i∂ϕðr, tÞ=∂t ¼ iωðr, t Þ ¼ ic ðr, tÞ + bðr, t Þ:

(38)

The real terms in this equation determine the familiar modulus-dynamics, ∂θðr, t Þ=∂t ¼ bðr, tÞ,

(39)

while its imaginary terms generate the time-evolution of wavefunction phase component: ∂ϕðr, tÞ=∂t ¼ c ðr, t Þ:

(40)

This local energy perspective thus generates a transparent description of a (nonequilibrium) time-evolution of wavefunction components: the real contribution in Eq. (37) shapes the phase dynamics, while the modulus evolution is governed by the imaginary components of ω(r, t). The spatial equalizations of these complex wave-number or local phase concepts mark the stationary equilibrium in molecular QM. Finally, by comparing Eqs. (39) and (40) with Eqs. (11) and (10), respectively, one arrives at the following expressions for the real and imaginary parts of the local wave-number of Eq. (37) in terms of the additive components of the wavefunction logarithm in Eq. (8):   c ðr, t Þ ¼ ∂ϕðr, tÞ=∂t ¼ ½ħ=ð2mÞ Rðr, t Þ1 ΔRðr, tÞ  ½∇ϕðr, tÞ2 + vðrÞ=ħ   ¼ ½ħ=ð2mÞ Δθðr, tÞ + ½∇θðr, t Þ2  ½∇ϕðr, tÞ2 + vðrÞ=ħ, bðr, t Þ ¼ ∂θðr, t Þ=∂t ¼ ðħ=mÞ rϕðr, t Þrθðr, tÞ ¼ V ðr, tÞrθðr, tÞ:

(41) (42)

It follows from these dynamic equations for the additive wavefunction components that the phase gradient enters the time evolutions of both the modulus and phase components of the system wavefunction. Thus, the presence of a nonvanishing local phase in the (nonstationary) quantum state affects the time evolutions of both the probability and current distributions. In the phase-equalization problem one also examines a transition from the initial state of the mutually-closed subsystems in the polarized composite system Rc ¼ (A+jB+), defined by the product of wavefunctions describing the two (distinguished) groups of electrons,  
 
 Ψα + N α 0 ; v ¼ Rα N α 0 exp iϕα N α 0 , α ¼ A, B, (43)

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Electronic convection in resultant information-theoretic description

 
0  +
0  Ψc N A 0 , N B 0 ¼ Ψ+ A N A ; v ΨB N B ; v     
      Ψc ðN Þ ¼ RA N A 0 RB N B 0 exp i ϕA N A 0 + ϕB N B 0  RðN Þ exp½iϕðN Þ,

(44)

to the final, isoelectronic state Ψ(N) in the whole Ro ¼ (A⁎¦B⁎), N 0 ≡ N A 0 + N B 0 ¼ N A ∗ + N B ∗ ≡ N ∗ ¼ N,

(45)

Ψo ðN Þ ¼ RðN Þ expfi½ϕðN Þ + ΦAB ðN Þg ≡ RðN Þ exp½iΦðN Þ ≡ ΨðN Þ, (46) exhibiting the same electron distribution P(N) ¼ R(N)2 and density ρo ðr Þ ¼ ρA ∗ ðrÞ + ρB ∗ ðrÞ ¼ ρA + ðrÞ + ρB + ðrÞ ≡ ρc ðr Þ ¼ ρðrÞ:

(47)

In this (internal) “opening” issue both fragments ultimately acquire the “molecular” probability distribution of all electrons in the whole composite system: po ðrÞ ≡ ρðrÞ=N ¼ ρα ∗ ðr Þ=N α ∗ ≡ pα ∗ ðrÞ ρα ∗ ðrÞ ¼ ðN α ∗=N Þ ρðr Þ ≡ P α ∗ ρðrÞ:

or (48)

The phase-shift ΦAB(N) thus represents the very effect of such a mutual opening of both fragments, for the “frozen” overall electron density of the whole system. Therefore, for the conserved spatial distribution of electrons, the wavefunctions of the mutually-closed parts {α+} exhibit separate phases {ϕαðN α 0 Þ}, while their open analogs {α⁎} are described by the common “molecular” phase Φ(N) in the wavefunction of the whole system, including the phase-shift ΦAB(N). Let us again assume, for reasons of simplicity, a stationary character of the initial states {ψ α + ½Nα 0 ; v ¼ Rα + ½Nα 0 ; v exp[iϕα(t)]} in the system polarized parts,  

 Hα N α 0 ; v R α + N α 0 ; v ¼ E α 0 R α + N α 0 ; v , (49) with time-independent moduli {Rα + ðNα 0 Þ} and purely time-dependent phases.     ϕα ðtÞ ¼  Eα 0=ħ t ≡  ωα 0 t , α ¼ A+ , B+ : (50)

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Roman F. Nalewajski

The subsystem Hamiltonians {Hα(Nα 0; v)} for the molecular external potential v(r) ¼ vA(r) + vB(r), determine the additive part H0(N, v) ¼ P 0 α Hα(Nα ; v) of the energy operator of the whole composite system,   (51) HðN, vÞ ¼ H0 ðN , vÞ + V AB N A 0 , N B 0 , with the embedding term VAB(NA0, NB0) grouping all nonadditive interactions. The dynamics of the global equilibrium in the open system is determined by SE applied to the phase-transformed state of Eq. (46),
   iħ½ð∂Ψ=∂t=ΨÞ ¼ iħð∂ ln Ψ=∂t Þ ¼ iħ i ωA 0 + ωB 0 + ið∂ΦAB=∂t Þ ¼ E A 0 + E B 0  ħð∂ΦAB=∂t Þ ¼ ðHΨÞ=Ψ ¼ EA 0 + EB 0 + V AB or ∂ΦAB=∂t ¼ V AB=ħ:

(52)

Therefore, in this interaction picture of QM the fragment embedding energy of Eq. (51) shapes the dynamics of the phase-shift describing the mutual opening of both subsystems.

4. Probability and current networks In this section, we would like to comment on entropic descriptors of chemical bonds in resultant communication theory. For simplicity, let us assume the AO channel, with the input and output “events” corresponding to AO basis χ ¼ ( χ 1, χ 2, …, χ m) used to expand N (singly) occupied Spin Molecular Orbitals (SMO), ψ ¼ (ψ 1, ψ 2, …, ψ N) ¼ χ C, C ¼ {Ck,s ¼ h χ kjψ si}, in the molecular state of an electron configuration defined by the Slater determinant. ΨðN Þ ¼ det ψ ≡ |ψ 1 , ψ 2 , …, ψ N |: These occupied SMO delineate the bonding subspace ofPthe molecule, corresponding to the projection operator Pψ ¼ jψihψj ¼ s jψ sihψ sj and the density operator of underlying MO ensemble: Dψ ¼ N1Pψ . The AO representation of Pψ represents the (idempotent) Charge-andBond-Order (CBO) matrix of the familiar SCF LCAO MO theory, n o X  ∗ , γ ¼ χ jPψ jχ ¼ CC{ ¼ γ k,l ¼ C C (53) γ2 ¼ γ, k,s l,s s the elements of which define the conditional probabilities P( χ 0 j χ ) ¼ {P( χ lj χ k)} in classical (probability) network in state Ψ(N). This channel

Electronic convection in resultant information-theoretic description

127

Fig. 1 Probability (Panel a) and current (Panel b) flow charts of the classical and nonclassical AO networks. For greater transparency only the χ k ! χ l connection is shown. One observes that the second, current panel represents the V-scaled probability panel. Both probability and current scatterings are effectively described by the same set of the joint AO probabilities P( χ , χ 0 ) ¼ {P(k, l)}.

consists of the AO input events χ ¼ {χ k} in its “source” S, and AO outputs χ 0 ¼ {χ l}, in its “receiver” R. The condensed AO probabilities in the configuration bond system then read p( χ ) ¼ {pk ¼ γ k,k/N} and simultaneous input-output orbital events are characterized by the joint probabilities P( χ , χ 0 ) ¼ {P(k,l) ¼ γ k,l γ l,k/N} (see Fig. 1), which satisfy the required normalizations: i X X hX ð Þ ¼ P k, l p ¼ 1: k l k k The communication links between basis functions are then reflected by conditional probabilities of the output AO events, given the specified AO inputs:   X Pð χ 0 jχ Þ ¼ P ðljkÞ ¼ P ðk, l Þ=pk ¼ γ k,l γ l,k=γ k,k , P ðljkÞ ¼ 1: (54) l This probability channel also generates the associated current system for the convectional probability flows between AO.4,31 It corresponds to the local MO phase “intensity” ϕ(r), common to all occupied SMO in the molecular phase-equilibrium: ψs ðrÞ ¼ Rs ðrÞ exp½iϕðrÞ,

s ¼ 1, 2, …, N :

(55)

This equalized phase descriptor ultimately determines the configuration current of Eq. (2) and the MO convection velocity of Eq. (3), for molecular probability distribution the “shape” function of the state elecP p(r) ¼ ρ(r)/N, P tron density ρ(r) ¼ s ρs(r) ¼ s Rs(r)2. The classical scattering of AO probabilities generates the probability network shown in Fig. 1a. This channel exhibits the molecular probabilities p( χ ) ¼ {pk} in its input and output, p( χ ) ¼ p( χ 0 ), and generates the following IT bond descriptors (in natural units):

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Roman F. Nalewajski

conditional entropy of “outputs-given-inputs,” the channel covalency (noise) measure, XX SðRjSÞ ¼ Sð χ 0 jχ Þ ¼  k l P ðk, l Þ lnP ðljkÞ; (56) mutual information in inputs and outputs, the network ionicity (information “flow”) index, XX I ðS:RÞ ¼ Sðχ :χ 0 Þ ¼ P ðk, lÞ ln½P ðljkÞ=pl  k l ¼ SðRjSÞ + S½pðχ 0 Þ;

(57)

overall (entropic) bond-multiplicity (order): M ðS;RÞ ¼ SðRjSÞ + I ðS:RÞ ¼ S½ pð χ 0 Þ:

(58)

Here, the Shannon entropy in the effective (condensed) AO probabilities reads: X S½pðχ Þ ¼  k pk lnpk : (59) The classical channel of Fig. 1a also determines the associated network of convectional probability flows between AO in the molecular bond system, which is shown in Fig. 1b. It corresponds to the configuration velocity V(r) ¼ (ħ/m) rϕ(r) ¼ j(r)/p(r) reflecting the system quantum state Ψ(N). The current flow chart is seen to represent the V-multiplied probability network, with the same input and output AO flows, j(S) ≡ {jP ≡ j(R), k} ¼ {jl} P giving rise to identical overall (molecular) currents: j(S) ≡ k jk ¼ l jl ≡ j(R). Therefore, entropic bond descriptors of Eqs. (56)–(58) apply to both the classical and nonclassical networks of Fig. 1. Indeed, it has been amply demonstrated that these IT measures of “bond orders” (in bits) already provide quite adequate representation of an accepted chemical “intuition,”41 and the bond-multiplicity descriptor of quantum chemistry,54 which represents an electron density/probability concept.

5. Internal ensembles of charge-transfer states Let us consider next the Charge-Transfer (CT) system R0 ¼ A0----B0, consisting of the initially separated acidic (A0, net electron-acceptor) and basic (B0, net electron-donor) reactants. The optimum amount N CT ∗ of a spontaneous, internal CT between these externally-closed but mutually-open subsystems,

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Electronic convection in resultant information-theoretic description

N CT ∗ ¼ N A ∗  N A 0 ¼ N B 0  N B ∗ ,

(60)

ultimately generates the equilibrium reactive system R⁎ ¼ (A⁎¦B⁎), where the broken vertical line symbolizes the freedom for electron exchanges between both fragments. At equilibrium these substrates exhibit fractional electron populations {N α ∗}, which conserve the system overall number of electrons: N ∗ ≡ N A∗ + N B∗ ¼ N A0 + N B0 ≡ N 0:

(61)

They result from the Electronegativity Equalization (EE) principle, μA  ¼ μA + + ηA + N CT  ¼ μB  ¼ μB +  ηB + N CT   μR 

or



μCT ¼ ∂E R=∂N CT j ¼ μCT + ηCT N CT ¼ 0, +

+

(62)

formulated above in terms of the system in situ characteristics, derivatives of the CT-energy function ECT ðΔN Þ ¼ E R ðN A 0 + ΔN , N B 0  ΔN Þ ≡ E R ðΔN Þ: the gradient, determining chemical potential difference43–48 μCT + , and the Hessian, defining the CT chemical hardness55 ηCT + , in the initial (polarized) system R+ ¼ (A+jB+) composed of the mutually closed subsystems, when {N α + ¼ N α 0 }, μCT + ¼ ∂E CT=∂N CT j+ ¼ μA +  μB + < 0 and ηCT + ¼ ∂2 E CT=∂N CT 2 j+ ¼ ∂μCT +=∂N CT     ¼ ηA,A  ηB,A + ηB,B  ηA,B  ηA + + ηB + > 0:

(63)

Here, the hardness tensor in reactant resolution, n

o ηR ¼ ηβ,α ¼ ∂2 E R ðN Þ=∂N β ∂N α + ¼ ∂μα + ðN Þ=∂N β + , is defined by the populational derivatives of fragment chemical potentials in the polarized system R+. The corresponding change, to 2nd-order, in the system electronic energy due to the specified amount NCT of electron transfer, ΔER (2)(NA, NB) ≡ ΔER (2)(N) ≡ ER(NCT), then reads: E R ðN CT Þ ¼ μCT + N CT + ½ ηCT + N CT 2 :

(64)

It follows from Eq. (62) that the optimum amount of CT equalizes chemical potentials of the mutually-open reactants, μA⁎(NCT⁎) ¼ μB⁎(NCT⁎) ¼ μR or μCT⁎(NCT⁎) ¼ 0, and hence: N CT ∗ ¼ μCT +=ηCT + > 0 + 2

and

E R ðN CT ∗ Þ ¼ ½ ðμCT Þ =ηCT ≡ E CT ∗ < 0: +

(65)

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Roman F. Nalewajski

Each equilibrium subsystem is then described by the same level of molecular chemical potential and exhibits a fractional electron population, CT modified relative to R+. As argued in the preceding section, the open reactants are also described by a common phase component, describing the whole reactive system. Clearly, the same equilibrium in R⁎( μR) results from coupling both reactants to their separate external electron reservoirs {Rα(μR)} exhibiting the molecular level μR of the chemical potential, in the macroscopic composite system: MðμR Þ ¼ ½R A ðμR Þ¦A∗ jB∗ ¦R B ðμR Þ ≡ ½R ðμR Þ¦R∗ ðμR Þ:

(66)

The internal CT is best described by the accessible states of reactants in R+ corresponding to their different (integer) numbers of electrons, e.g., the stationary states of Eq. (49), for   (67) N α  N α 0  1, N α 0 , N α 0 + 1 , α ¼ A, B, Hα ðN α ; vÞ ψ j α ðN α ; vÞ ¼ Ej α ψ j α ðN α ; vÞ,

j ¼ 0, 1, 2…

(68)

They include the neutral fragments (for Nα ¼ Nα0) and ionic species: cations (for Nα ¼ Nα0  1) and anions (for Nα ¼ Nα0 + 1). The products of the corresponding ground ( j ¼ 0) states then allows one to construct the N-conserving, integer population states for NCT  (1, 0, +1) in        RCT  R1 ¼ A+1 jB1 , R0 ¼ A0 jB0 , R+1 ¼ A1 jB+1 : (69) In the zero temperature limit they determine the internal ensemble describing a fractional CT in the range 1 < N CT ∗ < 1. These integer-CT basis states, {jCTi, CT ¼ (1, 0, +1)}, thus combine the lowest electronic states of the neutral fragments in. |R0 i ¼ |A0 B0 i ¼ |N CT ¼ 0i ≡ |0i,

(70)

or represent the ion-pairs: |R1 i ¼ |A+1 B1 i ¼ |N CT ¼ 1i ≡ |  1i 1

|R+1 i ¼ |A B i ¼ |N CT ¼ +1i ≡ |+1i: +1

and (71)

Setting the reference energy E0 ¼ h0jHRj0i ≡ 0, and denoting E1 ¼ h-1jHRj-1i, E+1 ¼ h+1jHRj + 1i, where the system Hamiltonian.    
    HR N A 0 , N B 0 ; v ¼ HA N A 0 ; v + HB N B 0 ; v + V R N A 0 , N B 0     (72) ≡ Hadd: N X 0 , N B 0 + Hnadd: N X 0 , N B 0 ,

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Electronic convection in resultant information-theoretic description

Energy IA – AB EA®B

EB®A

IB – AA NCT

0 ECT* –1 円A+1 ᎏ B–1典

NCT*

0 円A0ᎏ B0典

+1 円A–1 ᎏ B+1典

Fig. 2 Integer-CT states in DA reactive system R ≡ A—B, the parabolic (ensemble) interpolation of their energies, the optimum amount NCT ⁎ of CT, and the associated stabilization energy E CT ⁎ .

with VR grouping the “embedding” (nonadditive) interactions Hnadd., gives the CT energy as quadratic function of the current amount of CT in Mulliken’s interpolation: ER ðN CT Þ ≡ a N CT 2 + b N CT + c:

(73)

This energy function of the continuous variable NCT is shown in Fig. 2. It is defined by the following parameters: a ¼ ½ ðE 1 + E +1 Þ,

b ¼ ½ ðE +1  E 1 Þ

and

c ¼ 0:

(74)

The minimum of this energy function is found for the optimum amount of CT, N CT ∗ ¼ b=ð2aÞ ¼ ½ ðE1  E +1 Þ=ðE1 + E+1 Þ,

(75)

which reproduces the stabilization energy E⁎CT of Eq. (65) resulting from this spontaneous flow of electrons: E CT ∗ ≡ ER ðN CT ∗ Þ ¼ b2=ð4aÞ ¼ ðE 1  E+1 Þ=½8ðE 1 + E +1 Þ: (76) The chemical potentials {μα} and hardnesses {ηα} of separate reactants α ¼ A0, B0, as well as the energy changes corresponding to integer electron transfers, can be all expressed in terms of the subsystem electron affinities {Aα} and their ionization potentials {Iα}: μα ¼ ½ ðI α + Aα Þ < 0,

μA < μB ,

ηα ¼ I α  Aα > 0,

(77)

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Roman F. Nalewajski

E 1 ¼ I A  AB > 0,

E+1 ¼ I B  AA > 0,

(78)

The associated expressions for the finite-difference CT criteria, the in situ chemical potential and the process resultant chemical hardness, which ultimately determine the optimum (fractional) amount of the CT displacement, then read: μCT + ¼ μA +  μB +  ½ ðI B + AB  I A  AA Þ < 0 ηCT + ¼ ηA + + ηB +  I A  AA + I B  AB > 0:

and (79)

This internal CT can be also given the ensemble representation in the T ! 0 limit. The state of R is then characterized by the statistical mixture of the above integer CT states {jCTi}, defined by their probabilities. P CT ðN CT Þ ¼ fP 1 ðN CT Þ, P 0 ðN CT Þ, P +1 ðN CT Þg ¼ fP CT ðN CT Þg, (80) and the associated density operator: X |CTi P CT ðN CT Þ hCT|: DðN CT Þ ¼ CT

(81)

It determines the statistical average of any physical observable O: X P ðN CT Þ hCTjOjCTi: (82) hOðN CT Þiens: ¼ tr½DðN CT Þ O ¼ CT CT For example, the equilibrium electron populations of both reactants corresponding to the specified value of NCT read,    N A ∗ ðN CT Þ ¼ N A ∗ ðN CT Þ ens: ¼ P 1 ðN CT Þ N A 0  1 + P 0 ðN CT ÞN A 0   + P +1 ðN CT Þ N A 0 + 1 ¼ N A 0 + N CT ,    N B ∗ ðN CT Þ ¼ N B ∗ ðN CT Þ ens: ¼ P 1 ðN CT Þ N B 0 + 1 + P 0 ðN CT ÞN B 0   + P +1 ðN CT Þ N B 0  1 ¼ N B 0  N CT ,

(83)

and the average CT energy in this internal ensemble. hE R ðN CT Þiens: ¼ P 1 ðN CT Þ E1 + P +1 ðN CT Þ E +1 ¼ ER ðN CT Þ:

(84)

It follows from Eq. (83) that for the specified amount of CT, N CT ¼ P 1 ð1Þ + P 0 ð0Þ + P +1 ð + 1Þ ¼ P +1  P 1 ,

(85)

the normalized probabilities in such an internal ensemble, P 1 ðN CT Þ + P 0 ðN CT Þ + P +1 ðN CT Þ ¼ 1,

(86)

Electronic convection in resultant information-theoretic description

133

read: P 1 ðN CT Þ ¼ ½ N CT ðN CT  1Þ, P +1 ðN CT Þ ¼ ½ N CT ðN CT + 1Þ:

P 0 ðN CT Þ ¼ 1  N CT 2 , (87)

They indeed confirm the quantum pure-state character of the integer-CT basis, P 1 ð1Þ ¼ 1, fP 1 ðN CT 6¼ 1Þ ¼ 0g;

P 0 ð0Þ ¼ 1, fP 0 ðN CT 6¼ 0Þ ¼ 0g;

P +1 ð +1Þ ¼ 1, fP 1 ðN CT 6¼ 1Þ ¼ 0g: (88)

These probabilities also reproduce three energies of these states: hER ðN CT Þiens: ¼ E1 for P 1 ð1Þ ¼ 1; hER ðN CT Þiens: ¼ E+1 for P +1 ð1Þ ¼ 1; hER ðN CT Þiens: ¼ 0

(89)

for P 0 ð0Þ ¼ 1;

and recover Mulliken’s parabolic energy function of Fig. 2 defined in Eqs. (73) and (74).

6. Continuity of chemical potential descriptors Thermodynamic parameters of the grand ensemble, the chemical potential μ of an external electron reservoir R(μ), and temperature T of the heat bath B(T), which are imposed on the molecular system M[N, v] in the composite (macroscopic) system Mðμ, T Þ ¼ ½R ðμÞ¦Mðμ, T Þ¦BðT Þ,

(90)

establish the equilibrium probabilities {Pj i(μ, T; v)} of the system stationary states {ψ j i ≡ ψ j[Ni, v]}. The statistical mixture of these eigenstates of Hamiltonians {H(Ni, v) ≡ Hi}, Hi ψ j i ¼ E j i ψ j i , is then defined by the equilibrium ensemble probabilities,
  P j i ðμ, T; vÞ ¼ Ξ1 exp β μN i  E j i ,

(91)

(92)

where β ¼ (kBT)1, kB is the Boltzmann constant and Ξ stands for the grand partition function:

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Ξ¼

XX i

j


  exp β μN i  E j i :

In the limit T ! 0 this mixed quantum state involves only the system lowest (ground) states {ψ0 i ≡ ψ i} for different (integer) numbers of electrons {Ni ≡ i}, appearing with the ensemble probabilities {P0i(T ! 0) ≡ Pi(T ! 0)} and corresponding to energies {E0i ¼ E0[Ni, v] ≡ Ei}. This mixture represents the lowest equilibrium state of an externally-open system in the applied thermodynamic conditions. It has been demonstrated43 that in this limit only two ground states, jψ ii and jψ i+1i, corresponding to the neighboring integers “bracketing” the given fractional value of the average electron number hN(T ! 0)iens. ≡ N, i N i + 1, appear in this thermodynamic equilibrium: NðT ! 0Þ ¼

X

i

P i ðT ! 0Þ N i ¼ i P i ðT ! 0Þ + ði + 1Þ ½1  P i ðT ! 0Þ, (93)

P i ðT ! 0Þ ¼ 1 + i  N ≡ 1  ω

and

P i+1 ðT ! 0Þ ¼ N  i ≡ ω:

(94)

The continuous average energy function hEM(T ! 0)iens. ≡ EM(N, T ! 0) of the (average) electron number N in the externally-open molecule M thus consists of the straight-line segments between the neighboring integer values. This implies constant values of the chemical potential μM (negative electronegativity χ M) in all such admissible (partial) ranges of the system fractional number of electrons, and the μ-discontinuity at integer values {N ¼ i} (see Fig. 2): μM ðN, T ! 0Þ ¼ ∂EM ðN, T ! 0Þ=∂N ¼ χ M ðN, T ! 0Þ:

(95)

This external-ensemble analysis thus predicts a discontinuity of the global chemical potential of molecular/reactive systems in M( μM, T ! 0) or M( μR, T ! 0): μM ðT ! 0Þ ¼  ∂EM ðN, T ! 0Þ=∂N ¼ I M , for N < N M 0

and

  AM , for N > N M 0 : (96)

Consider such external exchanges of dN R ¼ dN A⁎ + dN B⁎ ¼ jΔN Rj > 0 electrons, between the internally-open reactive system R⁎ ¼ (A⁎¦B⁎) and its (macroscopic) electron reservoir R. In the process of an electron removal from R⁎, producing an effective cation R(+) for ΔNR ¼ -dNR < 0, it is the average population NB⁎ of its basic part, which is affected most strongly, dNB ∗ ¼ F B ð+Þ dNR < dNA ∗ ¼ F A ð+Þ dNR < 0 or

F B ð+Þ > F A ð+Þ , (97)

135

Electronic convection in resultant information-theoretic description

where the fragment response indices in cationic reactive system FR (+) ¼ {FA (+), FB (+)} reflect their populational polarizabilities in the global outflow () of electrons:

FB ð+Þ ¼ ∂N B ∗=∂N R  > FA ð+Þ ¼ ∂N A ∗=∂N R j , FA ð+Þ + FB ð+Þ ¼ ∂N R=∂N R j ¼ 1:

(98)

⁎

()

In an electron inflow to R , generating an effective anion R for ΔNR ¼ dNR > 0, the incoming extra electrons end up mainly on the acidic fragment A⁎, due to its relatively less shielded nuclei: dNB ∗ ¼ F B ðÞ dNR < dNA ∗ ¼ F A ðÞ dNR > 0

or

F B ðÞ < F A ðÞ : (99)

Here, the response properties in anionic reactive system, for the global inflow (+) of electrons. n o F R ðÞ ¼ F A ðÞ ¼ ∂NA =∂NR j+ , F B ðÞ ¼ ∂NB =∂NR j+ , , F A ðÞ + F B ðÞ ¼ ∂NR=∂NR j+ ¼ 1:

(100)

As a consequence of the discontinuity in the global chemical potential, the populational responses of reactants accompanying an external inflow/outflow of electrons to/from the equilibrium reactive system R⁎ are thus characterized by different sets of the Fukui function56 responses: FR(+) ¼ 6 FR(). In Mulliken’s interpolation between energies of the integer electron transfer states, and in the internal ensemble representation the electronic energy function appears as the parabolic function of the CT amount. The internalensemble analysis fully validates the Mulliken interpolation scheme and indicates that the in situ chemical-potential difference for the internal CT between reactants, μCT ðN CT Þ ¼ ∂E R ðN CT Þ=∂N CT ¼ ∂hE R ðN CT Þiens:=∂N CT ,

(101)

when one molecular subsystem serves as a microscopic reservoir for the complementary fragment, represents a continuous function of the CT populational variable: μCT ðN CT Þ ¼ E 1 ½∂P 1 ðN CT Þ=∂N CT  + E +1 ½∂P +1 ðN CT Þ=∂N CT  ¼ 2a N CT + b: (102) This populational “force” descriptor indeed vanishes for the optimum CT amount of Eq. (65), μCT(NCT⁎) ¼ 0, when reactants equalize their chemical potentials at the global level in R⁎.

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Therefore, this result of the internal continuity of the in situ chemical potential differs from the external discontinuity of the molecular chemical potential of R⁎ in M⁎ ¼ (R⁎¦R), when R is coupled to the macroscopic (infinitely soft) electron reservoir R. Indeed, in Fig. 2 the “bracketing” state perspective applied to R⁎ would predict different (constant) values of μCT in the A ! B (left part) and B ! A (right part) electron flows, for the negative and positive CT displacements, respectively. They are measured by slopes of the straight-line segments of the linear energy changes EA!B and EB!A in the figure: μCT ð+Þ ¼ ðI A  AB Þ ¼ AB  I A < 0 and

μCT ðÞ ¼ I B  AA > 0: (103)

These different slopes explain integer numbers of electrons in fragment dissociation43,44: AB ! A0 + B0 :

(104)

7. Conclusion We have emphasized throughout this work the nonclassical concepts in IT of electronic structure, explored continuities of the wavefunction components and resultant-information densities, and examined the internal ensemble description of CT phenomena in the externally-closed acid-base complexes. It has been argued, that due to the sourceless character of the state classical “variables” of the electron density or associated probability distribution, only the nonclassical, phase-related state “parameters” contribute to the local net production of the information density in molecular systems. Using Schr€ odinger’s wavefunction dynamics the phase and current sources have been extracted, and their role in determining the resultant IT densities has been explored. The nonclassical channel of electronic flows between AO in the molecular bond system has been shown to represent the convection-scaled classical channel of electron communications. Therefore, IT descriptors of the bond multiplicity and composition, which result from the probability scattering network, also apply to the current/phase channel. Indeed, these classical IT indices have been shown to already reflect the accepted chemical intuition fairly well. It thus appears that the chemical bond pattern indeed constitutes a classical IT concept, in accordance with the basic premise of DFT.

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In the context of the known discontinuity of the chemical potential in an externally-open molecular system, in contact with the macroscopic electron reservoir, the question arises of a behavior of the electronegativity difference driving CT in the externally-closed DA systems. In the latter case each reactant constitutes a microscopic reservoir for the reaction partner, and the equilibrium fractional CT is described by the internal ensemble of the integer CT states. We have explicitly demonstrated that this populational CT gradient is continuous and the resulting (ensemble-average) energy function reconstructs the parabolic interpolation of Mulliken.

References 1. Nalewajski, R. F. Information Theory of Molecular Systems; Elsevier: Amsterdam, 2006. 2. Nalewajski, R. F. Information Origins of the Chemical Bond; Nova Science Publishers: New York, 2010. 3. Nalewajski, R. F. Perspectives in Electronic Structure Theory; Springer: Heidelberg, 2012. 4. Nalewajski, R. F. Quantum Information Theory of Molecular States; Nova Science Publishers: New York, 2016. 5. Nalewajski, R. F. Chemical Reactivity in Quantum Mechanics and Information Theory; Elsevier: Amsterdam, 2022 (in press). 6. Prigogine, I. From Being to Becoming: Time and Complexity in the Physical Sciences; Freeman, W. H. & Co.: San Francisco, 1980. 7. Nalewajski, R. F. Exploring Molecular Equilibria Using Quantum Information Measures. Ann. Phys. (Leipzig) 2013, 525, 256–268. 8. Nalewajski, R. F. On Phase/Current Components of Entropy/Information Descriptors of Molecular States. Mol. Phys. 2014, 112, 2587–2601. 9. Nalewajski, R. F. On Phase Equilibria in Molecules. J. Math. Chem. 2014, 52, 588–612. 10. Nalewajski, R. F. Information Theory, Molecular Equilibria and Patterns of Chemical Bonds. In Advances in Quantum Systems Research; Ezziane, Z., Ed.; Nova Science Publishers: New York, 2014; pp. 119–167. 11. Nalewajski, R. F. Quantum Information Approach to Electronic Equilibria: Molecular Fragments and Elements of Non-Equilibrium Thermodynamic Description. J. Math. Chem. 2014, 52, 1921–1948. 12. Nalewajski, R. F. Phase/Current Information Descriptors and Equilibrium States in Molecules. Int. J. Quantum Chem. 2015, 115, 1274–1288. 13. Nalewajski, R. F. Quantum Information Measures and Molecular Phase Equilibria. In Advances in Mathematics Research; Baswell, A. R., Ed.; 19; Nova Science Publishers: New York, 2015; pp. 53–86. 14. Nalewajski, R. F. Complex Entropy and Resultant Information Measures. J. Math. Chem. 2016, 54, 1777–1782. 15. Nalewajski, R. F. Quantum Information Measures and Their Use in Chemistry. Current. Phys. Chem. 2017, 7, 94–117. 16. Nalewajski, R. F. Entropy Continuity, Electron Diffusion and Fragment Entanglement in Equilibrium States. In Advances in Mathematics Research; Baswell, A. R., Ed.; 22; Nova Science Publishers: New York, 2017; pp. 1–42. 17. Nalewajski, R. F. Information Equilibria, Subsystem Entanglement and Dynamics of Overall Entropic Descriptors of Molecular Electronic Structure. J. Mol. Model. 2018, 24, 212–227.

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18. Nalewajski, R. F. On Entropy/Information Description of Reactivity Phenomena. In Advances in Mathematics Research; Baswell, A. R., Ed.; 26; Nova Science Publishers: New York, 2019; pp. 97–157. 19. Nalewajski, R. F. Understanding Electronic Structure and Chemical Reactivity: Quantum-Information Perspective. Appl. Sci. 2019, 9, 1262–1292. 20. Nalewajski, R. F. Phase Equalization, Charge Transfer, Information Flows and Electron Communications in Donor-Acceptor Systems. Appl. Sci. 2020, 10, 3615–3646. 21. Fisher, R. A. Theory of Statistical Estimation. Proc. Cambridge. Phil. Soc. 1925, 22, 700–725. 22. Frieden, B. R. Physics From the Fisher Information—A Unification; Cambridge University Press: Cambridge, U. K., 2004. 23. Nalewajski, R. F. Use of Fisher Information in Quantum Chemistry. Int. J. Quantum Chem. 2008, 108, 2230–2252. 24. Nalewajski, R. F.; K€ oster, A. M.; Escalante, S. Electron Localization Function as Information Measure. J. Phys. Chem. A 2005, 109, 10038–10043. 25. Nalewajski, R. F.; de Silva, P.; Mrozek, J. Use of Nonadditive Fisher Information in Probing the Chemical Bonds. J. Mol. Struct.: THEOCHEM 2010, 954, 57–74. 26. Shannon, C. E. The Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–493, 623–656. 27. Nalewajski, R. F. Continuity Relations, Probability Acceleration, Current Sources and Internal Communications in Interacting Fragments. Acad. J. Chem. 2020, 5, 58–68. 28. Nalewajski, R. F. Independent Sources of Information-Theoretic Descriptors of Electronic States. Acad. J. Chem. 2020, 5, 106–115. 29. Abramson, N. Information Theory and Coding; McGraw-Hill: New York, 1963. 30. Pfeifer, P. E. Concepts of Probability Theory; Dover: New York, 1978. 31. Nalewajski, R. F. Entropic Descriptors of Quantum Communications in Chemistry. J. Math. Chem. 2015, 53, 1–28. 32. Nalewajski, R. F. Entropic Measures of Bond Multiplicity From the Information Theory. J. Phys. Chem. A 2000, 104, 11940–11951. 33. Nalewajski, R. F. Entropy Descriptors of the Chemical Bond in Information Theory: I. Basic Concepts and Relations. Mol. Phys. 2004, 102, 531–546. 34. Nalewajski, R. F. Entropy Descriptors of the Chemical Bond in Information Theory: II. Application to Simple Orbital Models. Mol. Phys. 2004, 102, 547–566. 35. Nalewajski, R. F. Partial Communication Channels of Molecular Fragments and Their Entropy/Information Indices. Mol. Phys. 2005, 103, 451–470. 36. Nalewajski, R. F. Reduced Communication Channels of Molecular Fragments and Their Entropy/Information Bond Indices. Theoret. Chem. Acc. 2005, 114, 4–18. 37. Nalewajski, R. F. Entropy/Information Bond Indices of Molecular Fragments. J. Math. Chem. 2005, 38, 43–66. 38. Nalewajski, R. F. Entropic Bond Indices from Molecular Information Channels in Orbital Resolution: Ground-State Systems. J. Math. Chem. 2008, 43, 265–284. 39. Nalewajski, R. F. Chemical Bond Descriptors From Molecular Information Channels in Orbital Resolution. Int. J. Quantum Chem. 2009, 109, 425–440. 40. Nalewajski, R. F. Multiple, Localized and Delocalized/Conjugated Bonds in the Orbital-Communication Theory of Molecular Systems. Adv. Quantum Chem. 2009, 56, 217–250. 41. Nalewajski, R. F.; Szczepanik, D.; Mrozek, J. Bond Differentiation and Orbital Decoupling in the Orbital Communication Theory of the Chemical Bond. Adv. Quantum Chem. 2011, 61, 1–48. 42. Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton: Princeton University Press, 1955.

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43. Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Density Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy. Phys. Rev. Lett. 1982, 49, 1691–1694. 44. Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. 45. Gyftopoulos, E. P.; Hatsopoulos, G. N. Quantum-Thermodynamic Definition of Electronegativity. Proc. Natl. Acad. Sci. U. S. A. 1965, 60, 786–793. 46. Mulliken, R. S. A New Electronegativity Scale: Together With Data on Valence States and on Ionization Potentials and Electron Affinities. J. Chem. Phys. 1934, 2, 782–793. 47. Iczkowski, R. P.; Margrave, J. L. Electronegativity. J. Am. Chem. Soc. 1961, 83, 3547–3551. 48. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. Electronegativity: The Density Functional Viewpoint. J. Chem. Phys. 1978, 69, 4431–4439. 49. Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. B 1964, 136, 864–971. 50. Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. A 1965, 140, 1133–1138. 51. Levy, M. Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 6062–6065. 52. Dreizler, R. M.; Gross, E. K. U. Density Functional Theory: An Approach to the Quantum Many-Body Problem; Springer: Berlin, 1990. 53. Nalewajski, R. F., Ed., Vols. 180-183; Density Functional Theory I-IV: Topics in Current Chemistry; Springer: Berlin, 1996. 54. Wiberg, K. B. Application of the Pople-Santry-Segal CNDO Method to the Cyclopropylcarbinyl and Cyclobutyl Cation and Bicyclobutane. Tetrahedron 1968, 24, 1083–1096. 55. Parr, R. G.; Pearson, R. G. Absolute Hardness: Companion Parameter to Absolute Electronegativity. J. Am. Chem. Soc. 1983, 105, 7512–7516. 56. Parr, R. G.; Yang, W. Density Functional Approach to the Frontier-Electron Theory of Chemical Reactivity. J. Am. Chem. Soc. 1984, 106, 4049–4050.

CHAPTER TWO

Coupled-cluster downfolding techniques: A review of existing applications in classical and quantum computing for chemical systems Nicholas P. Bauman∗, Bo Peng∗, and Karol Kowalski∗ Pacific Northwest National Laboratory, Richland, WA, United States *Corresponding authors: e-mail address: [email protected]; [email protected]; [email protected]

Contents 1. Introduction 2. Theory 2.1 Non-Hermitian CC downfolding 2.2 Hermitian CC downfolding 3. Quantum flows 3.1 Non-Hermitian CC flows 3.2 Hermitian CC flows 4. Time-dependent CC extensions 5. Green’s function applications 6. Review of applications 6.1 Numerical validation of the SES-CC theorem 6.2 Approximations based on quantum flows 6.3 Quantum computing 7. Conclusions Acknowledgments References

142 143 144 146 148 148 149 153 155 157 157 157 159 160 161 162

Abstract In this chapter, we provide an overview of the recent developments of the coupledcluster (CC) downfolding methods, where the ground-state problem of a quantum system is represented through effective/downfolded Hamiltonians defined using active spaces. All CC downfolding techniques discussed here are derived from a singlereference exponential ansatz for the ground-state problem. We discuss several

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.03.006

Copyright

#

2023 Elsevier Inc. All rights reserved.

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extensions of the non-Hermitian and Hermitian downfolding approaches to the time domain and the so-called quantum flows. We emphasize the important role of downfolding formalisms in transitioning chemical applications from noisy quantum devices to scalable and error-corrected quantum computers.

1. Introduction The coupled-cluster (CC) theory1–11 has evolved into one of the most accurate formulations to describe the correlation effects in chemistry,9–11 material sciences, and physics.6,12–22 Although the CC formalism originates in the Linked Cluster Theorem,23,24 it has been successfully extended to describe excited states, properties, and time evolution of the system.6,8,25–35 Over the last few decades, a significant effort has been exerted to address the steep scaling of canonical CC formulations and apply them to realistic chemical processes. Parallel computing, especially with recently developed exascale computing architectures, has extended the applicability of conventional CC methods, but only modestly before encountering prohibitive costs once more. As a result, there has been much development in recent years on new reduced-scaling approaches for classical and quantum computing paradigms to push the envelope of the system sizes tractable by CC formalisms. Mathematically rigorous formulations for reducing the dimensionality/ cost of quantum formulations are urgently needed to shift the envelope of system-size tractable by accurate many-body formulations in chemistry, material sciences, and physics. Among the most successful formulations, one should mention local CC formulations, various partitioning and incremental schemes, and embedding methods.36–45 These approaches are driven by various design principles from the locality of correlation effects in the wave function approaches to properties of self-energy in correlated systems. Thanks to these formulations, significant progress has been achieved in describing correlation effects in large molecular systems allowing for simulations based on the utilization of modest computational resources. The dimensionality reduction techniques also play a crucial role in enabling the early stages of quantum computing driven by noisy intermediate-scale quantum devices (NISQ). This is associated with the reduction of the qubits required to represent the quantum problem of interest. As an illustration, one should mention several techniques developed to take full advantage of the ubiquitous variational quantum eigensolvers (VQE) approach46–63 in addressing problems beyond the situation where few electrons are correlated.

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In the context of the development of quantum algorithms for quantum chemistry, the main goal of dimensionality reduction methods is to provide a mathematically rigorous way of representing interdependencies between static and dynamical correlation effects. However, while the inclusion of static effects can be achieved for small-size systems on currently available quantum hardware, much needed dynamical correlation effects, usually manifesting in a large number of fermionic degrees of freedom (amplitudes) characterized by small values, are beyond the reach of current quantum technologies. The recently introduced downfolding techniques based on the double unitary coupled-cluster Ansatz (DUCC)64–71 provide one of the solutions to the above-mentioned problem. The DUCC formalism offers a special representation of the ground-state wave function that, in analogy to singlereference subsystem embedding subalgebras (SES-CC),72,73 allows one to construct effective Hamiltonians that integrate out all out-of-active-space degrees of freedom usually identified with dynamical amplitudes. Although the effective Hamiltonian formulations have a long history in quantum history and physics, especially in dealing with strong correlation effects, there are notable distinct features of the DUCC and SES-CC formalisms: (1) both formulations are embedded in the single-reference language employing a straightforward definition of the excitation domain (i.e., wave function parameters) in the vain of single-reference formulations, and (2) the possibility of describing a quantum problem in the form of quantum flows, i.e., coupled small-dimensionality eigenvalue problems. In this way, one can probe large subspaces of the Hilbert space without unrealistic quantum resource demands. Since the eigenvalue problems involved in the quantum flow represent physically well-defined problems (defined by the corresponding effective Hamiltonians and density matrices), the quantum flow formulation naturally lent itself to capture possible sparsity characterizing the quantum system. This chapter provides a compact overview of the main development threads originating in the single-reference SES-CC formulation (Section 2.1) and its unitary extension (Section 2.2). Section 3 introduces and discusses the basic tenets of quantum flows. The extension of the CC downfolding methods to the time domain and Green’s function formalism is discussed in Sections 4 and 5. Finally, Section 6 discusses applications of the downfolding formalisms.

2. Theory The SES-CC and DUCC formulations have been amply discussed in recent papers (see Refs. 64, 70, 72). Here, we overview only the salient

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features of these approaches. While the SES-CC technique forms the basis for non-Hermitian downfolding, the DUCC expansions provide its Hermitian formulations. In both cases, the ensuing downfolding procedures are encoded in the properties of exponential ansatzes for the ground-state wave functions jΨi: jΨi ¼ eT jΦi,

(1)

for non-Hermitian formalism given by standard SR-CC expansion, and jΨi ¼ eσ ext eσ int jΦi,

(2)

for the Hermitian downfolding defined by the DUCC Ansatz. In these equations, jΦi is the so-called reference function usually identified with the Hartree–Fock determinant, T is the SR-CC cluster operator, and σ ext and σ int are the anti-Hermitian external and internal cluster operators (vide infra). Both types of downfolding lead to many-body forms of effective or downfolded Hamiltonians acting in the appropriate active spaces. Although effective Hamiltonian formulations have a long history in electronic structure theory, especially in treating strong correlation effects, the present methods have several unique features compared to the multireference effective Hamiltonian approaches. Among the most distinct, one should mention: (1) the possibility of developing effective Hamiltonian formalisms using a very simple single-reference language to define the manifold of excitations used to construct downfolded Hamiltonians and (2) the concept of the quantum flows (QF), which boils down to coupling downfolding procedures corresponding to various active spaces. The former formalism allows for sampling large subspaces of the Hilbert space using reduced-dimensionality eigenvalue problems. The QF formalism is not only a convenient representation of appropriate many-body formulations in the form of numerically feasible computational blocks, which plays a crucial role in the early stages of quantum computing development, but also provides a natural environment for capturing the sparsity characterizing correlated systems.

2.1 Non-Hermitian CC downfolding Active spaces play a central role in the development of CC downfolding techniques and are defined by the subset R of occupied active orbitals (R ¼ fRi , i ¼ 1, …, xR gÞ and a subset S of active virtual orbitals (S ¼ fSi , i ¼ 1, …, ys g ). Using many-body language, the excited Slater

Coupled-Cluster downfolding techniques

145

determinants spanning the active space (along with the reference function jΦi) can be generated by generators E aill ¼ a{al ail (il  R and al  S) acting on the reference function jΦi. These generators define the so-called gðN Þ ðR, SÞ subalgebra. Due to the utilization the particle–hole formalism all generators E aill commute and the gðN Þ ðR, SÞ is commutative. For the sake of the following analysis, it is convenient to characterize various types of subalgebras gðN Þ ðR, SÞ by specifying the numbers of active occupied (xR) and active virtual (yS) orbitals, namely, gðN Þ ðxR , yS Þ. As shown in Refs. 65, 72, 74, each subalgebra h ¼ gðN Þ ðR, SÞ induces partitioning of the cluster operator T: T ¼ T int ðhÞ + T ext ðhÞ,

(3)

where T int ðhÞ belongs to h, while T ext ðhÞ does not belong to h. If the expansion eT int ðhÞ jΦi produces all Slater determinants (of the same symmetry as the jΦi state) in the active space, we call h the subsystem embedding subalgebra for the CC formulation defined by the T operator. In Ref. 72, we showed that each standard CC approximation has its own class of SESs. The existence of the SESs for standard CC approximations provides alternative ways for calculating CC energies, which can be obtained in contrast to the standard CC energy E expression E ¼ hΦjeT HeT jΦi,

(4)

as an eigenvalue of the active-space non-Hermitian eigenproblem H eff ðhÞeT int ðhÞ jΦi ¼ EeT int ðhÞ jΦi,

(5)


ext ðhÞðP + Qint ðhÞÞ H eff ðhÞ ¼ ðP + Qint ðhÞÞH

(6)


ext ðhÞ ¼ eT ext ðhÞ HeT ext ðhÞ : H

(7)

where

and

The above result is known as the SES-CC Theorem. In Eq. (6), P stands for the projection operator onto reference function and Qint ðhÞ is a projection operator onto all excited Slater determinants with respect to the reference function jΦi that corresponds to h . One should also mention that the standard energy expression given by Eq. (4) can be reproduced from Eq. (5) when h represents the simplest case when there are no generators

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(both sets R and S are empty). When SES-CC Theorem is applied to the exact cluster operator corresponding to the full CC approach, the lowest eigenvalues of effective Hamiltonians correspond to the exact ground-state full configuration interaction (FCI) energy. Standard CC approximations (such as CCSD, CCSDT, CCSDTQ, etc. methods) are characterized by specific classes of SESs. For example, typical CCSD SESs are gðN Þ ð1R , yS Þ or gðN Þ ðxR , 1S Þ subalgebras. For the CCSDTQ approach, corresponding SESs are of the gðN Þ ð2R , yS Þ or gðN Þ ðxR , 2S Þ form. From these definitions, it is easy to see that lower-rank CC approximations are SESs for high-rank approaches. For example, the CCSD SESs gðN Þ ð1R , yS Þ are SESs for the CCSDTQ formalism. The SES-CC Theorem is flexible in the choice of active spaces, providing a number of alternative ways for calculating CC energies using effective Hamiltonians corresponding to various SESs. For example, for the CCSD approximation with a fixed molecular orbital basis, where the SES gðN Þ ðR, SÞ defined at the orbital level73 contains either one occupied active orbital or one virtual active orbital, the number of different SESs SCCSD and corresponding effective Hamiltonians that upon diagonalization reproduce the standard CCSD energy is SCCSD ¼ no ð2nv  1Þ + nv ð2no  1Þ  no nv :

(8)

This formula is a consequence of binomial expansion, in which k active

 virtual/occupied orbitals can be chosen in nkv = nko different ways, where no and nv stand for the numbers of correlated occupied and virtual orbitals, respectively. The validity of the SES-CC Theorem has recently been confirmed numerically on the example of several benchmark systems. In addition to the standard spatial orbital-based definition of SES-generated active spaces, it was shown that the SES-CC Theorem also holds for the active spaces defined by a nontrivial number of active spin orbitals. In the extreme case, we demonstrated that the SES-CC Theorem is also satisfied for the active space describing one active electron “correlated” in two active α-type spin-orbitals.73

2.2 Hermitian CC downfolding The Hermitian form of the downfolded Hamiltonian is obtained as a consequence of utilizing DUCC representation of the wave function64,65

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Coupled-Cluster downfolding techniques

jΨi ¼ eσ ext ðhÞ eσ int ðhÞ jΦi,

(9)

where σ ext ðhÞ and σ int ðhÞ are general-type anti-Hermitian operators σ {int ðhÞ ¼ σ int ðhÞ,

(10)

σ {ext ðhÞ

(11)

¼ σ ext ðhÞ:

In analogy to the non-Hermitian case, the σ int ðhÞ operator is defined by parameters carrying only active spin-orbital labels and σ ext ðhÞ operators are defined by parameters with at least one in-active spin-orbital label. The use of the DUCC Ansatz (9), in analogy to the SES-CC case, leads to an alternative way of determining energy, which can be obtained by solving active-space Hermitian eigenvalue problem: H eff ðhÞeσ int ðhÞ jΦi ¼ Eeσ int ðhÞ jΦi,

(12)


ext ðhÞðP + Qint ðhÞÞ H eff ðhÞ ¼ ðP + Qint ðhÞÞH

(13)


ext ðhÞ ¼ eσ ext ðhÞ Heσext ðhÞ : H

(14)

where

and

When the external cluster amplitudes are known (or can be effectively approximated), the energy (or its approximation) can be calculated by diagonalizing the Hermitian effective/downfolded Hamiltonian (13) in the active space using various quantum or classical diagonalizers. For quantum computing applications, second-quantized representation of H eff ðhÞ is required. In the light of the noncommuting character of components defining the σ ext ðhÞ operator, one has to rely on the finite-rank commutator expansions, i.e., H eff ðhÞ ’ ðP + Qint ðhÞÞ l X 1 ðH + ½…½H, σ ext ðhÞ, …, σ ext ðhÞi ðP + Qint ðhÞÞ: i! i¼1

(15)

Due to the numerical costs associated with the contractions of multidimensional tensors and the rapidly expanding number of terms in this expansion, only approximations based on the inclusion of low-rank commutators are feasible. In recent studies, approximations based on single, double, and part of triple commutators were explored where one- and

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two-body interactions in the second quantized form of H eff ðhÞ were retained. In practical applications, one also has to determine the approximate form of σ ext ðhÞ. For practical reasons, we used the following approximation σ ext ðhÞ ’ T ext ðhÞ  T ext ðhÞ{ ,

(16)

where Text were defined through the external parts of the T1 and T2 operators obtained in CCSD calculations.

3. Quantum flows 3.1 Non-Hermitian CC flows In the case of non-Hermitian downfolding, the SES-CC Theorem can be used to form computational frameworks (quantum flow) that integrate eigenvalue problems70,74 H eff ðhi ÞeT int ðhi Þ jΦi ¼ EeT int ðhi Þ jΦi ði ¼ 1, …, MÞ,

(17)

where M is the total number of SESs or active space problems included in the flow. In Refs. 70,74, we demonstrated that a problem defined in this way is equivalent (at the solution) to the standard CC equations with cluster operator defined as a combination of all unique excitations included in T int ðhi Þði ¼ 1, …, MÞ operators, i.e., T¼

M [

T int ðhi Þ

(18)

i¼1

and QeT HeT jΦi ¼ 0, T

hΦje

He jΦi ¼ E, T

(19) (20)

where the Q operator is a projection operator onto a subspace of excited Slater determinants generated by the action of T operator of Eq. (18) onto the reference function. The discussed equivalence is known as the Equivalence Theorem. Initially, as discussed in Ref. 72, the quantum flows were introduced as a form of the invariance of the SES-CC Theorem upon separate rotations of occupied and virtual orbitals. Although the form given by Eqs. (19) and (20) is generally better suited in canonical calculations to take advantage of parallel computing architectures, the representation given by Eq. (17) is well-poised to capture a

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Coupled-Cluster downfolding techniques

general type of the sparsity characterizing quantum systems. This is because Eq. (17) represents reduced-dimensionality computational blocks representing quantum problems defined by non-Hermitian Hamiltonians H eff ðhi Þ. In analogy to the bivariational CC formulations,6 one can introduce the left eigenvectors of active-space Hamiltonians H eff ðhi Þ, using either CC-Λ hΦjð1 + Λint ðhi ÞÞeT int ðhi Þ

ði ¼ 1, …, MÞ,

(21)

or the extended CC formalism hΦjeSint ðhi Þ eT int ðhi Þ

ði ¼ 1, …, MÞ,

(22)

where Λint ðhi Þ and Sint ðhi Þ are deexcitation operators acting in the corresponding active spaces, to form one-particle reduced density matrices γðhi Þ. For the Λ-CC formalism, the matrix elements of the γðhi Þ are given by the formula γ pq ðhi Þ ¼ hΦjð1 + Λint ðhi ÞÞ eT int ðhi Þ a{p aq eT int ðhi Þ jΦi, 8i¼1,…,M :

(23) (24)

The above construct, in contrast to the existing local CC formulations where the one-particle reduced density matrices are postulated, allows one to introduce them in a natural way. A more detailed analysis of the local formulations stemming from the Equivalence Theorem is discussed in Refs. 70,74. This procedure can also be extended to systems driven by different types of interactions, such as in nuclear structure theory or quantum lattice models, where the extension of the standard local CC formulations as used in quantum chemistry may be less obvious.

3.2 Hermitian CC flows Using non-Hermitian formulation as a guide, the idea of quantum flow can be generalized to the DUCC case. We start our analysis by assuming that we would like to perform DUCC effective simulations for SES h problem expressed in Eq. (12) for an active space that is too big to be handled either in classical or quantum computing. We will assume that external amplitudes σ ext ðhÞ can be effectively approximated. For simplicity, we will introduce a new DUCC Hermitian Hamiltonian AðhÞ which is defined as H eff ðhÞ or its approximation in the ðP + Qint ðhÞÞ space (in the simplest case it can be just the ðP + Qint ðhÞÞHðP + Qint ðhÞÞ operator). We will denote

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AðhÞ simply by A. Next, we assume that excitations in h that are relevant to the state of interest can be captured by excitation subalgebras: h1, h2, …, hM , where, in analogy to the SR-CC case, we admit the possibility of “sharing” excitations/deexcitations between these subalgebras. We also assume that the number of excitations belonging to each hiði ¼ 1, …, MÞ is significantly smaller than the number of excitations in h and therefore numerically tractable in simulations. The AðhÞ Hamiltonian and the ðP + Qint ðhÞÞ space can be treated as a starting point for the secondary DUCC decompositions generated by subsystem algebras hi ði ¼ 1, …, MÞ defined above, i.e., Aeff ðhi Þeσint ðhi Þ jΦi ¼ Eeσint ðhi Þ jΦi ði ¼ 1, …, MÞ

(25)

or in the VQE-type variational representation as min hΨðθðhi ÞÞjAeff ðhi ÞjΨðθðhi ÞÞi ði ¼ 1, …, MÞ, θðhi Þ

(26)

where jΨðθðhi ÞÞi approximates eσ int ðhi Þ jΦi. Each Aeff ðhi Þ is defined as
ext ðhi ÞðP + Qint ðhi ÞÞ Aeff ðhi Þ ¼ ðP + Qint ðhi ÞÞA

(27)


ext ðhi Þ ¼ eσ ext ðhi Þ Aeσ ext ðhi Þ , A

(28)

and

where we defined external σ ext ðhi Þ operator with respect to h or ðP + Qint ðhÞÞ space (i.e., cluster amplitudes defining σ ext ðhi Þ must carry at least one index belonging to active spin orbitals defining h and not belonging to the set of active spin orbitals defining hi). In other words, subalgebras hi generate active subspaces in the larger active space h, i.e., ðP + Qint ðhi ÞÞ  ðP + Qint ðhÞÞ. Due to the noncommutativity of components defining σ-operators, connecting DUCC computational blocks given by Eqs. (25) or (26) directly into a flow is a rather challenging task. To address these issues (see Ref. 74) and define practical DUCC flow, we will discuss the algorithm that combines secondary downfolding steps with Trotterization of the unitary CC operators. Let us assume that the σ int ðhÞ operator can be approximated by σ int ðhi Þði ¼ 1, …,MÞ, i.e., σ int ðhÞ ’

M X i¼1

σ int ðhi Þ + Xðh, h1 , …, hM Þ,

(29)

151

Coupled-Cluster downfolding techniques

where the Xðh, h1 , …, hM Þ operator (or X for short) eliminates possible overcounting of the “shared” amplitudes. It enables to reexpress σ int ðhÞ as σ int ðhÞ ¼ σ int ðhi Þ + Rðhi Þ ði ¼ 1, …, MÞ,

(30)

where Rðhi Þ¼ðiÞ

M X

σ int ðhj Þ + X

(31)

j¼1

P and ðiÞ M j¼1 designates the sum where the ith element is neglected. Consequently, we get eσint ðhÞ jΦi ¼ eσ int ðhi Þ + Rðhi Þ jΦi ði ¼ 1, …, MÞ:

(32)

Using the Trotter formula, we can approximate the right-hand side of Eq. (32) for a given j as N

eσ int ðhÞ jΦi ’ ðeRðhi Þ=N eσ int ðhi Þ=N Þ jΦi:

(33)

ðN Þ

Introducing auxiliary operator Gi  N 1 ðN Þ Gi ¼ eRðhi Þ=N eσint ðhi Þ=N eRðhi Þ=N ði ¼ 1, …, MÞ,

(34)

the “internal” wave function (32) can be expressed as ðN Þ σ int ðhi Þ=N

eσ int ðhÞ jΦi ’ Gi ðN Þ

where Gi

e

jΦi ði ¼ 1, …, MÞ,

(35)

is a complicated function of all σ int ðhj Þ ðj ¼ 1, …, MÞ and ðN Þ

term. the above expression does not decouple σ int ðhi Þ from the Gi However, using this expression, one can define the practical way of determining computational blocks for flow equations. To this end, let us introduce the expansion in Eq. (35) to Eq. (12) (with H eff ðhÞ replaced by the A operh i ðN Þ 1 ator), premultiply both sides by Gi , and project onto ðP + Qint ðhi ÞÞ subspace, which leads to nonlinear eigenvalue problems h i1 ðN Þ ðN Þ AGi eσ int ðhi Þ=N jΦi ðP + Qint ðhi ÞÞ Gi ’ Eeσ int ðhÞi Þ=N jΦi ði ¼ 1, …, MÞ:

(36)

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Nicholas P. Bauman et al.

These equations define computational blocks for the DUCC flow. To make practical use of Eq. (36), let us linearize them by defining the downfolded h i ðN Þ ðN Þ ðN Þ 1 ðN Þ Hamiltonian Γi , Γi ¼ ðP + Qint ðhi ÞÞ Gi AGi ðP + Qint ðhi ÞÞ as a function of all σ int ðhj Þ ð j ¼ 1, …, MÞ from the previous flow cycle(s) (pc). We will symbolically designate this fact by using special notation for ðN Þ

ðN Þ

Γi effective Hamiltonian, i.e., Γi ðpcÞ Hamiltonian. Now, we replace eigenvalue problems in Eq. (36) by optimization procedures described by Eq. (26) which also offer an easy way to deal with “shared” amplitudes. Namely, if, in analogy to SR-CC flow, we establish an ordering of hi subalgebras, with h1 corresponding to the importance of active spaces with respect to the wave function of interest. Then in the hi problem we partition set of parameters θN ðhi Þ into subset θCP N ðhi Þ that refers to common pool of amplitudes determined in preceding steps (say, for hj ð j ¼ 1, …, i  1Þ ) and subset θX N ðhi Þ that is uniquely determined in the hi minimization step, i.e., ðN Þ

CP min hΨðθX N ðhi Þ, θN ðhi ÞÞjΓi X

θN ðhi Þ

CP ðpcÞjΨðθX N ðhi Þ, θN ðhi ÞÞi ði ¼ 1, …, MÞ,

(37)

CP σ int ðhi Þ=N where jΨðθX jΦi. In this way, each N ðhi Þ, θN ðhi ÞÞi approximates e computational block coupled into a flow corresponds to a minimization procedure that optimizes parameters θX N ðhi Þ using quantum algorithms such as the VQE approach. At the end of the iterative cycle, once all amplitudes are converged, in contrast to the SR-CC flows, the energy is calculated ðN Þ

using h1 problem as an expectation value of the Γ1 operator. The discussed formalism introduces a broad class of control parameters defining each computational step’s dimensionality. These are the numbers of occupied/ unoccupied active orbitals defining hi subalgebras xRi =ySi , respectively. An essential feature of the DUCC flow equation is associated with the fact that each computational block (37) can be encoded using a much smaller number of qubits compared to the qubits requirement associated with the original problem. This observation significantly simplifies the qubit encoding of the effective Hamiltonians included in quantum DUCC flows, especially in formulations based on the utilization of localized molecular basis set (for quantum algorithms exploiting locality of interactions, see Refs. 75, 76).

Coupled-Cluster downfolding techniques

153

4. Time-dependent CC extensions The SES-CC-based downfolding techniques could also be extended to the time-dependent domain.65,70 As in the stationary case, we will assume a general partitioning of the time-dependent cluster operator T(t) into its internal (T int ðh, tÞ) and external (T ext ðh, tÞ) parts (we also assume that the employed molecular orbitals are time independent), i.e., jΨðtÞi ¼ eT ext ðh,tÞ eT int ðh,tÞ jΦi, 8h  SES:

(38)

For generality, we also include phase factor T 0 ðh, tÞ in the definition of the T int ðh, tÞ operator. After substituting (38) into time-dependent Schr€ odinger equation and utilizing properties of SES algebras, we demonstrated that the ket dynamics of the subsystem wave function eT int ðh,tÞ jΦi corresponding to arbitrary SES h iħ

∂ T int ðh,tÞ jΦi ¼ H eff ðh, tÞeT int ðh,tÞ jΦi, e ∂t

(39)

where
ext ðh, tÞðP + Qint ðhÞÞ H eff ðh, tÞ ¼ ðP + Qint ðhÞÞH

(40)


ext ðh, tÞ ¼ eT ext ðh,tÞ HeT ext ðh,tÞ : H

(41)

and

If T ext ðh, tÞ operator is known or can be efficiently approximated, then the dynamics of the entire system can be described by effective Hamiltonian H eff ðh, tÞ. In analogy to the stationary cases, various subsystem’s computational blocks can be integrated into a flow enabling sampling of large subspaces of Hilbert space through a number of coupled reduced-dimensionality problems (time-dependent quantum flows), i.e., iħ

∂ T int ðhi ,tÞ jΦi ¼ H eff ðhi , tÞeT int ðhi ,tÞ jΦi, ði ¼ 1, …, M SES Þ: e ∂t

(42)

Given analogies between stationary CC flow equations based on the localized orbitals and local CC formulations developed in the last few decades in quantum chemistry, time-dependent flow equations given by Eq. (42) can be utilized to design reduced-scaling variants of the time-dependent CC formulations.

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Nicholas P. Bauman et al.

The time-dependent variant of the DUCC Ansatz is represented by the normalized time-dependent wave function jΨDUCC ðh, tÞi, jΨDUCC ðh, tÞi ¼ eσ ext ðh,tÞ eσ int ðh,tÞ jΦi, 8h  SES,

(43)

where σ int ðh, tÞ and σ ext ðh, tÞ are general-type time-dependent anti-Hermitian operators σ int ðh, tÞ{ ¼ σ int ðh, tÞ,

(44)

{

σ ext ðh, tÞ ¼ σ ext ðh, tÞ:

(45)

Again, as in the SES-CC case, the dynamics of the entire system are given by the active-space time-dependent effective Hamiltonian H eff ðσ ext ðh, tÞ, ∂σ ext∂tðh, tÞÞ 

 ∂σ ext ðh, tÞ H σ ext ðh, tÞ  ∂t   ∂σ ð h, t Þ ext
ext ðh, t Þ  iħA σ ext ðh, t Þ ðP + Qint Þ, (46) ¼ ðP + Qint Þ H ∂t eff

where anti-Hermitian operator Aðσ ext ðh, tÞ,

∂σ ext ðh, tÞ Þ ∂t

is expressed as



 X   ∞ ∂σ ext ðh, tÞ ð1Þk ∂σ ext ðh, tÞ ¼ (47) I σ ðh, tÞ, A σ ext ðh, tÞ, ∂t ∂t ðk + 1Þ! k ext k¼0 and  Ik

∂X ðhtÞ X ðht Þ ∂t









  ∂X ðht Þ ¼ X ðht Þ X ðht Þ… X ðhtÞ X ðhtÞ … : (48) ∂t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} k times

If the fast-varying in time part of the wave function (or σ ext ðh, tÞ-dependent part of the wave function) is known or can be efficiently approximated, then the slow-varying dynamic (captured by the proper choice of the active space and σ int ðh, tÞ operator) of the entire system can be described as a subsystem   dynamics generated by the Hermitian H eff σ ðh, tÞ, ∂σ ext ðh, tÞ operator. ext

∂t

This decoupling of various time regimes (slow- vs fast-varying components) is analogous to decoupling high- and low-energy Fermionic degrees of freedom in stationary formulations of the SES-CC and DUCC formalisms.

155

Coupled-Cluster downfolding techniques

5. Green’s function applications The CC Green’s function formulations have recently evolved into important formulations to describe spectral functions in various energy regimes77–83 and as high-accuracy solvers for quantum embedding formulations. Following original formulations based on the CC bivariational approach, the corresponding frequency-dependent Green’s function for an N-particle system can be expressed as Gpq ðωÞ ¼ hΦjð1 + ΛÞeT a{q ðω + ðH  EÞ  iηÞ1 ap eT jΦi + hΦjð1 + ΛÞeT ap ðω  ðH  EÞ + iηÞ1 a{q eT jΦi,

(49)

where ω denotes the frequency parameter, and the imaginary part η is often called a broadening factor. The cluster operator T and deexcitation operator Λ define correlated ket (jΨi) and bra (hΨj) ground-state wave functions for N-electron system jΨi ¼ eT jΦi,

(50) T

hΨj ¼ hΦjð1 + ΛÞe

:

(51)

The ground-state energy E0 and the amplitudes defining T and Λ operators are obtained from the following sequence of CC equations. To combine GFCC and DUCC formalisms, we replace T, Λ, and H oper~ int), and Hermitian H eff ðhÞ ~ int), deexcitation (Λ ators in Eq. (49) by cluster (T [Eq. (13), which is further denoted as Γ] operators acting in the some active space generated by h(for the notational simplicity we also skip the hsymbol).69 We will also consider the case when the set of active orbitals consists of all occupied orbitals and a small subset of active virtual orbitals (containing nact v active virtual orbitals), where, in general, nact v ≪nv , where nv designates the total number of virtual orbitals. The standard CC equations for T, CC energy E, and Λ are replaced by their “active” counterparts ~

~

~ int T

~ int T

Qint eT int ΓeT int jΦi ¼ 0 , hΦje

~ int T

~ int Þe hΦjð1 + Λ

Γe

~ int T

Γe

jΦi ¼

Qint ¼

(52)

Eint 0 ,

Eint 0 hΦjð1

(53) ~ int ÞQint : +Λ

(54)

The CC Green’s function employing the DUCC Hamiltonian Γ can be expressed for active orbitals as follows

156

Nicholas P. Bauman et al.

~ int ÞeT~ int a{Q ðω + ðΓ  E int Þ  iηÞ1 aP eT~ int jΦi GDUCC ðωÞ ¼ hΦjð1 + Λ 0 PQ 1

~ int ÞeT int aP ðω  ðΓ  E int Þ+iηÞ a{Q eT int jΦi, + hΦjð1 + Λ 0 ~

~

(55) where indices P, Q, … designate active spin orbitals. Again, applying the ~ ~ resolution of identity eT int eT int in the above equation, one gets the following expressions for DUCC Green’s function matrix elements int

~ int Þa{Q ðω + Γ
N  iηÞ1 aP int jΦi GDUCC ðωÞ ¼ hΦjð1 + Λ PQ 1

int

~ int ÞaP int ðω  Γ
N + iηÞ a{Q jΦi, + hΦjð1 + Λ

(56)

where we used the following definitions:
¼ eT~ int Γ eT~ int , Γ
N ¼ Γ
 Eint , Γ

(57) (58)

0

~ int T

~ int T

aP e , a
P int ¼ e int ~ ~
a{Q ¼ eT int a{Q eT int :

(59) (60)

In the active-space-driven DUCC-GFCC approach, the Xp(ω) and Yq(ω) int operators are replaced by X int P ðωÞ and Y Q ðωÞ , respectively, which are given by the following expressions: X X IJ xI ðP, ωÞint aI + xA ðP, ωÞint a{A aJ aI + ⋯ , (61) X int P ðωÞ ¼ I I 12 bohr since the asymptotic expansion, which is a part of the fits, is sufficiently accurate in this region). These initial fits revealed that, in contrast to the fits obtained in Refs. 43, 61, and 62, the density of data points was insufficient due to the much higher accuracy achieved in the present calculations. Therefore, we performed calculations at additional 30 points located at Ri + (1/3)δi and Ri + (2/3)δi, where Ri , i ¼ 1, …, 15, are the first 15 initial points and δi ¼ Ri+1  Ri (all distances were rounded to 0.01 bohr). Since the full nonlinear optimization of four-electron ECG functions is very expensive, only the linear parameters ck in Eq. (1) were calculated at these new points, while the nonlinear parameters were taken from the wave

How competitive are expansions in orbital products with explicitly correlated ones

237

function of the nearest neighbor in the original set. Note that this procedure requires a scaling of the Gaussian centers Aki to account for the changed nuclear positions, which can be easily done by expressing the bond-centered Gaussian functions as products of Gaussian functions centered on nuclei.66 To check if the new set of points is sufficient to obtain an accurate fit in the whole range of R, we performed the following test. For each of the 48 data points Ri (except for the two external ones, Ri ¼ 1.0 bohr and Ri ¼ 12 bohr), we generated a separate fit using the data set with Ri removed. The search for the optimal fit parameters was started from random values in each case, i.e., did not use the parameters of the full fit. Then, for each of these new 46 fits, we compared the energy predicted at the removed point with the CBS energy. The only instances where the predicted and CBS energies deviated by significantly more than the uncertainty estimate σ (defined under Eq. (6)) were at the three shortest distances (by 4.9 σ, 2.7 σ, and 4.8 σ at 1.17, 1.33, and 1.50 bohr, respectively). Since these distances correspond to the very high-energy region on the repulsive wall (above 105 K), this behavior is inconsequential. Moreover, it should be stressed that fitting to the data set with a removed point is more stringent a test than necessary, because what is only important is how the total fit behaves between the existing 48 points. Finally, let us note that this test was first performed for the initial set of 16 data points, where it revealed that this number of points is insufficient and had to be enlarged. In Ref. 62, the same test confirmed that 18 points used there were sufficient because of the lower level of accuracy than achieved in the present work (see section VI.F of Ref. 62). The convergence of the upper bounds to the interaction energies (obtained by subtracting the exact monomer energies from the calculated dimer energies) is presented in Table 1 for six selected distances from the original set and for the leftmost point of the new set (R ¼ 1.17 bohr). Let us denote by ΔK the energy change brought about by increasing the expansion pffiffiffi size from K= 2 to K. The values of Δ6788 that can be calculated from the energies listed in Table 1 decrease with increasing R, from 0.2 K at R ¼ 1.0 bohr to just 0.02 mK at R ¼ 9.0 bohr. There is one exception, namely, the value at R ¼ 1.17 bohr, Δ6788 ¼ 0.7 K, is larger than at 1.0 bohr. This is due to the fact that the nonlinear parameters of the wave function at R ¼ 1.17 bohr were not optimized for this distance but rather for R ¼ 1.0 bohr and then scaled to 1.17 bohr, as described above. This effect is larger at shorter distances, where the wave function changes more rapidly with R, and the point shown in Table 1 represents the worst-case scenario: we observed here the slowest convergence out of the whole set of 46 points.

Table 1 The convergence of helium dimer interaction energies (in kelvin) at selected interatomic distances R (in bohr). R K

1.0

1.17

2.0

4.0

5.6

7.0

9.0

2400

286415.031

204059.019

36142.3675

292.585060

11.000152

4.622326

0.989578

3394

286413.769

204055.966

36142.3110

292.583645

11.000290

4.622384

0.989593

4800

286413.134

204054.243

36142.2798

292.582918

11.000389

4.622427

0.989615

6788

286412.917

204053.551

36142.2694

292.582592

11.000464

4.622466

0.989637

η4800

1.99

1.77

1.81

1.95

1.39

1.35

0.68

η6788

2.93

2.49

3.00

2.23

1.32

1.10

1.00

E1.32

286412.239

204051.389

36142.2369

292.581573

11.000698

4.622588

0.989706

E3.00

286412.809

204053.205

36142.2642

292.582429

11.000501

4.622485

0.989648

K denotes the number of terms in the expansion of Eq. (1). The product of two 337-term helium atom functions is used as ϕ0 in all calculations. ηK is defined by Eq. (5) and Eη is the CBS-extrapolated value obtained using the value of η in Eq. (6).

How competitive are expansions in orbital products with explicitly correlated ones

239

To extrapolate our interaction energies to the CBS limits, we use the well-established empirical observation that the ratio pffiffiffi ΔK=pffiffi2 EðK= 2Þ  EðK=2Þ pffiffiffi , ηK ¼ ¼ (5) ΔK EðKÞ  EðK= 2Þ calculated from ECG energies for a given system, is approximately independent of K. Note that in earlier papers ηK (and ΔK) were usually defined for pffiffiffi doubling of the basis set sizes, whereas we now use 2 as the ratio of two consecutive sizes. The values of η4800 and η6788 calculated at the selected distances are listed in Table 1. At the largest distances, the behavior of ηK is counter-intuitive, especially at R ¼ 9.0 bohr. The values of ηK equal to one show constant increments of E, whereas the values smaller than one mean that the increments get larger so that the energies apparently diverge. We believe that this phenomenon is at least partially caused by the fact that, due to the almost complete saturation of the interaction energy, at such distances the nonlinear optimization is to a significant extent directed toward removing the residual inaccuracies of the monomer contraction. As the result, the convergence pattern is a superposition of various effects and no longer uniform. Therefore, we disregarded the values of ηK for R > 5.6 bohr for the purpose of the extrapolation. By analyzing other distances and rejecting several outliers (such as around R ¼ 5.0 bohr, where the potential crosses zero), we found that most of the values of ηK are between 1.32 and 3.00. Assuming either of these limiting values as η for all K > 6788 and adding to E(6788) the sum of geometric series with the first term equal to Δ6788/η and the ratio equal to 1/η, i.e., using the formula Eð∞Þ ¼ Eð6788Þ +

Δ6788 , η1

(6)

we obtain the two sets of CBS interaction energies, denoted in Table 1 as E1.32 and E3.00. One possibility is to choose the recommended value in the middle between these two energies and their difference (possibly multiplied by two) as the estimated uncertainty.40 In the present case, we chose E1.32, which is always below E3.00, as the recommended interaction energy and the difference as the uncertainty σ(E). The reason is that the nonlinear optimization of the ECG parameters for a given K does not end up in an exact minimum and there is a residual optimization error, which is always negative due to the variational nature of our interaction energies. Whereas the incompleteness due to the truncation at a given K is taken care of by the CBS extrapolation procedure, this residual error can be partly compensated for

240

Krzysztof Szalewicz et al.

by selecting E1.32. Note also that our “upper limit” of the uncertainty interval, E3.00, is rather well established in the sense that it would be virtually unchanged if any other value of η close to 3.00 were used. In contrast, the “lower limit,” E1.32, is quite sensitive to η. The choice of η ¼ 1.32 seems conservative to us (cf. Table 1). The recommended values of the interaction energies E for all 46 distances and their uncertainties, σ(E), calculated as described above, are listed in Table 2. We also list there the best upper bounds to the exact interaction Table 2 Born–Oppenheimer interaction energy of the helium dimer, E (in kelvin), and its uncertainty, σ(E), as a function of the internuclear separation, R (in bohr). R Eub E σ(E) E (Ref. 43) σ(E) (Ref. 43)

1.00

286412.92

286412.24

0.57

1.17

204053.6

204051.4

1.8

1.33

147823.14

147822.29

0.72

1.50

104313.28

104313.13

0.12

1.67

73149.22

73148.92

0.25

1.83

52101.65

52101.51

0.11

2.00

36142.269

36142.237

0.027

2.17

24941.891

24941.829

0.053

2.33

17509.271

17509.238

0.028

2.50

11960.869

11960.857

0.010

2.67

8125.008

8124.992

0.014

2.83

5615.2239

5615.2134

0.0088

3.00

3767.7329

3767.7284

0.0038

3.17

2509.2831

2509.2776

0.0047

3.33

1698.2341

1698.2303

0.0033

3.50

1110.6570

1110.6551

0.0016

3.67

717.4936

717.4916

0.0017

3.83

468.9001

468.8982

0.0017

4.00

292.58259

292.58157

0.00086

4.17

177.52897

177.52788

0.00091

4.33

106.84205

106.84102

0.00086

286417

5

104314

1

36142.3

0.5

11960.9

0.2

3767.68

0.071

1110.649

0.028

292.570

0.015

241

How competitive are expansions in orbital products with explicitly correlated ones

Table 2 Born–Oppenheimer interaction energy of the helium dimer, E (in kelvin), and its uncertainty, σ(E), as a function of the internuclear separation, R (in bohr).—cont’d R Eub E σ(E) E (Ref. 43) σ(E) (Ref. 43)

4.50

58.40515

58.40459

0.00048

4.67

28.18122

28.18063

0.00050

4.83

10.64336

10.64268

0.00057

5.00

0.47072

0.47114

0.00036

5.10

4.55159

4.55198

0.00032

5.20

7.34317

7.34349

0.00027

5.30

9.16445

9.16470

0.00021

5.40

10.26175

10.26197

0.00019

5.50

10.82505

10.82535

0.00025

5.60

11.00046

11.00070

0.00020

5.73

10.82804

10.82825

0.00018

5.87

10.32053

10.32079

0.00022

6.00

9.68059

9.68078

0.00016

6.17

8.73446

8.73463

0.00014

6.33

7.82260

7.82282

0.00018

6.50

6.89298

6.89313

0.00013

6.67

6.03609

6.03621

0.00010

6.83

5.30862

5.30879

0.00014

7.00

4.62247

4.62259

0.00010

7.33

3.52880

3.52891

0.00009

7.67

2.68068

2.68084

0.00013

8.00

2.06670

2.06678

0.00007

8.33

1.60639

1.60647

0.00006

8.67

1.25053

1.25063

0.00008

9.00

0.98964

0.98971

0.00006

58.397

0.010

0.4754

0.0065

9.1681

0.0048

11.0037

0.0031

9.6819

0.0023

6.8936

0.0012

4.6225

0.0006

2.0669

0.0003

0.98984

0.00015

Eub are strict upper bounds to the interaction energy, obtained by subtracting the exact helium monomer energies from the variational dimer energies in the largest basis set. The available ab initio values of the best previous potential of Ref. 43 are given for comparison.

242

Krzysztof Szalewicz et al.

energies, Eub, i.e., the energies calculated from K ¼ 6788 by subtracting twice the exact energy of the helium atom. Note that in four-electron ECG calculations it is not possible to compensate for a counterpart of the basis set superposition error (BSSE) of orbital bases, i.e., to assure that the basis set completeness of monomer energies within the dimer is exactly the same as in the isolated monomer calculations. Removal of BSSE is possible in perturbational calculations with ECG functions.67 The BO energies calculated in Ref. 43 are presented in Table 2 for comparison. At all 16 distances where energies from both sets are available, the respective error bars overlap so that both sets of results are mutually consistent. However, the present error bars are tighter by about an order of magnitude (from 8 to 23 times for R < 7 bohr and from 2.5 to 6 times for the remaining distances).

4. Calculations in orbital bases The calculations of He2 interaction energies in orbital bases were similar to those of Refs. 44 and 43, but used larger basis sets at both the CCSD (T) and FCI levels. In Ref. 44, the Δ(T) ¼ CCSD(T)CCSD and ΔFCI ¼ FCICCSD(T) contributions to interaction energy were obtained in a series of extrapolations from two basis set sequences, d-aug-cc-pVXZ and aug-cc-pVXZ68,69 plus a (6s6p6d3f1g1h) set of 95 midbond functions70 (these bases will be denoted by dXZ and aXZb95, respectively). In the present work, we do not need to use Δ(T), we just compute interaction energies at the CCSD(T) level. In Refs. 44 and 43, the basis set cardinal number X up to 7 for CCSD(T) and up to 6 for ΔFCI was used. At present, we were able to extend both calculations one cardinal number further, that is, to the d8Z and a8Zb95 bases at the CCSD(T) level, and d7Z and a7Zb95 at the FCI level. Both of these extensions were nontrivial. First, octuple-zeta basis sets for helium involve k functions. Such functions were not supported in the efficient CCSD(T) implementation in MOLPRO71 at the time our calculations were performed. Among the few quantum-chemistry packages that support k functions, we found the PSI4 code72 to be acceptably efficient and free from any numerical accuracy problems, and we used it for all d8Z and a8Zb95 calculations. The d8Z basis set was constructed in Ref. 73 and the same work also proposed a new d7Z set, but we employed here the original d7Z basis of Ref. 74, consistent with Refs. 44 and 43. For the FCI calculations, we used the LUCIA program75 with the integral and SCF interface redesigned in Ref. 44 to combine the data from DALTON2.076 and 77 ATMOL, allowing for more than 255 basis functions. Because of numerical convergence issues, this dual interface could not be made to work for the

How competitive are expansions in orbital products with explicitly correlated ones

243

d7Z and a7Zb95 bases at some interatomic distances (both very short and very long). If that was the case, we employed an alternative sequence of bases, optimized to provide better asymptotic constants than the original aXZ and dXZ sequences. We will denote these alternative bases by aXZMP and dXZMP, respectively. These bases were optimized by Przybytek78 and the dXZMP ones were published in Ref. 62, but the aXZMP ones remain unpublished. The dXZMP bases are of the same size as the corresponding dXZ ones, while the aXZMP sets have one more s function then the corresponding aXZ ones. We used the aXZMPb95 sequence in place of aXZb95, and dXZMP in place of dXZ, whenever the original bases could not be employed in the X ¼ 7 FCI calculations. While the “MP” basis sets appear to provide somewhat less converged interaction energies at most distances, they are free from numerical issues plaguing the d7Z and a7Zb95 calculations and provide FCI interaction energies for all interatomic distances down to R ¼ 3.5 bohr (R ¼ 3.0 bohr for d7ZMP only). The complete set of CCSD(T) and FCI interaction energies obtained in the four basis set sequences for X  6 is given in Table 3. Our orbital-basis estimates of the total FCI interaction energy and of the ΔFCI term were obtained using two basis sequences (with X ¼ 6, 7) for each R  3.5 bohr: aXZb95 whenever convergent (otherwise aXZMPb95) and dXZ whenever convergent (otherwise dXZMP). At R ¼ 3.0 bohr, only the dXZMP sequence could be used. The CCSD(T) and ΔFCI CBS limits of Ref. 44 were obtained using four kinds of extrapolations. In addition to the conventional X3 extrapolation of any correlated energy contribution,79 Ref. 44 employed the so-called “versus” (EH vs E) approach of Ref. 80, assuming that a hard-tocalculate contribution EH exhibits the same basis set convergence pattern as a different contribution E for which the CBS limit is known from GTG calculations. More specifically, the underlying assumption is that EH is a linear function of E in the vicinity of the CBS limit, and this assumption is sufficient to calculate the CBS limit of EH knowing the values of EH and E at two consecutive X and the GTG-level CBS limit of E.80 Three choices of E were used: the helium-atom correlation energy at the second-order level of many-body perturbation theory based on the Møller–Plesset partition of the Hamiltonian (MP2), denoted as EMP2,cr , the correlation part of the MP2 He-He interaction A cr energy Eint,MP2 , and the correlation part of the factorizable CCD (FCCD)49 cr He–He interaction energy Eint,FCCD . Converged GTG values of these three terms at all interatomic distances were taken from Ref. 44, while the needed orbital calculations were performed in the present work. We will employ the same four extrapolations to estimate the FCI interaction energy and

Table 3 Interaction energies (in kelvin) computed using the aXZb95, dXZ, aXZb95MP, and dXZMP orbital bases using X ¼ 6, 7, 8 for CCSD(T) and X ¼ 6, 7 for FCI. CCSD(T) ΔFCI FCI R

Basis

X56

X57

X58

X56

X57

X56

X57

3.0

aXZb95

3777.67345

3776.17787

3775.19983

5.37213

3772.30132

3.0

dXZ

3780.20725

3777.30246

3775.65716

5.42732

3774.77993

3.0

aXZb95MP

3777.93879

3776.47235

5.40634

3772.53245

3.0

dXZMP

3786.45638

3780.85382

5.53272

3.5

aXZb95

1115.76500

1115.19048

1114.78548

3.28878

1112.47622

3.5

dXZ

1117.28312

1115.96171

1115.18135

3.32487

1113.95825

3.5

aXZb95MP

1115.90117

1115.31504

3.30751

3.28233

1112.59367

1112.03271

3.5

dXZMP

1121.51635

1118.86942

3.39579

3.34644

1118.12057

1115.52298

4.0

aXZb95

295.17058

294.95461

294.81429

1.91269

4.0

dXZ

296.04776

295.42470

295.05419

1.93392

1.91049

294.11385

293.51421

4.0

aXZb95MP

295.24091

295.01885

1.92360

1.90986

293.31732

293.10900

4.0

dXZMP

297.67910

296.84853

1.97191

1.94724

295.70719

294.90129

4.5

aXZb95

59.72756

59.65144

59.60180

1.08791

1.08071

58.63965

58.57073

4.5

dXZ

60.19166

59.90649

59.73605

1.09896

1.08664

59.09270

58.81985

4.5

aXZb95MP

59.76401

59.68312

1.09423

1.08669

58.66978

58.59643

4.5

dXZMP

60.63300

60.36769

1.11675

1.10446

59.51625

59.26323

5.0

aXZb95

0.22388

0.19884

0.61928

0.61546

0.39540

0.41662

0.18219

5.44387

3780.92366

3775.40995

293.25789

0.62436

0.61801

0.17887

0.29615

0.21450

0.62309

0.61907

0.37957

0.40457

0.53349

0.44454

0.63272

0.62648

0.09923

0.18194

aXZb95

8.68318

8.69591

8.70424

0.44480

0.44224

9.12798

9.13815

5.3

dXZ

8.54644

8.61762

8.65855

0.44778

0.44356

8.99422

9.06118

5.3

aXZb95MP

8.66966

8.68542

0.44761

0.44455

9.11727

9.12997

5.3

dXZMP

8.52095

8.56884

0.45331

0.44906

8.97426

9.01790

5.6

aXZb95

10.66130

10.66800

10.67209

0.32209

0.32027

10.98339

10.98827

5.6

dXZ

10.57808

10.61765

10.64133

0.32375

0.32094

10.90184

10.93858

5.6

aXZb95MP

10.65217

10.66077

0.32426

0.32205

10.97643

10.98282

5.6

dXZMP

10.57543

10.60232

0.32756

0.32461

10.90300

10.92693

6.0

aXZb95

9.46221

9.46532

9.46688

0.21261

0.21136

9.67482

9.67668

6.0

dXZ

9.41863

9.43641

9.44828

0.21331

0.21166

9.63194

9.64807

6.0

aXZb95MP

9.45695

9.46087

0.21416

0.21269

9.67111

9.67356

6.0

dXZMP

9.42437

9.43763

0.21576

0.21385

9.64013

9.65147

6.5

aXZb95

6.76183

6.76319

6.76369

0.12993

6.5

dXZ

6.74060

6.74769

6.75377

0.13012

0.12923

6.87073

6.87692

6.5

aXZb95MP

6.75930

6.76084

0.13097

0.13007

6.89027

6.89091

6.5

dXZMP

6.74736

6.75348

0.13162

0.13055

6.87898

6.88403

5.0

dXZ

0.44549

0.32186

5.0

aXZb95MP

0.24351

5.0

dXZMP

5.3

0.25162

6.89175

Continued

Table 3 Interaction energies (in kelvin) computed using the aXZb95, dXZ, aXZb95MP, and dXZMP orbital bases using X ¼ 6, 7, 8 for CCSD(T) and X ¼ 6, 7 for FCI.—cont’d CCSD(T) ΔFCI FCI R

Basis

X56

X57

X58

X56

X57

X56

X57

7.0

aXZb95

4.54037

4.54113

4.54131

0.08185

7.0

dXZ

4.52928

4.53230

4.53571

0.08190

0.08136

4.61118

4.61366

7.0

aXZb95MP

4.53937

4.54001

0.08256

0.08199

4.62193

4.62200

7.0

dXZMP

4.53454

4.53402

0.08283

0.07863

4.61738

4.61265

8.0

aXZb95

2.03094

2.03140

2.03142

0.03543

8.0

dXZ

2.02802

2.02842

2.02927

0.03546

0.03521

2.06348

2.06363

8.0

aXZb95MP

2.03112

2.03130

0.03577

0.03551

2.06689

2.06681

8.0

dXZMP

2.03025

2.03101

0.03583

0.03556

2.06608

2.06656

9.0

aXZb95

0.97224

0.97260

0.97259

0.01696

9.0

dXZ

0.97165

0.97167

0.97184

0.01701

0.01686

0.98866

0.98853

9.0

aXZb95MP

0.97262

0.97273

0.01713

0.01701

0.98975

0.98973

9.0

dXZMP

0.97256

0.97266

0.01715

0.01693

0.98971

0.98959

The internuclear separation R is given in bohr.

4.62222

2.06638

0.98919

How competitive are expansions in orbital products with explicitly correlated ones

247

the ΔFCI term using X ¼ 6, 7. As we were not able to calculate the FCCD energies in the X ¼ 8 bases, the CCSD(T) interaction energy estimate will be obtained using the three remaining X ¼ 7, 8 extrapolations: X3, vs cr EMP2,cr , and vs Eint,MP2 . Thus, the FCI and CCSD(T) contributions and their A uncertainties are computed (similar to Ref. 44) as the midpoints and half-widths, respectively, of the intervals encompassing all eight (FCI) and six (CCSD (T)) results extrapolated from two basis set sequences. Note that only the correlation energy contributions are extrapolated (the SCF interaction energy is taken from the larger basis set of the two used in the extrapolation and assumed not to contribute to the overall uncertainty). For R  4.5 bohr, some FCCD calculations in bases aXZb95 and aXZMPb95 did not converge (in such a case, cr we had to exclude the extrapolation vs Eint,FCCD using those sequences from consideration). Table 4 presents the CBS limit values of the CCSD(T) interaction energies and of the ΔFCI contributions obtained in this way for the same set of interatomic distances as in Ref. 44. The corresponding complete set of extrapolated results is given in the Supplementary Information available online at https://doi.org/10.1016/bs.aiq.2023.03.007.81. Two different ways of evaluating the total nonrelativistic BO interaction energy are possible: a direct extrapolation of the X ¼ (6, 7) FCI interaction energies and a hybrid method using the sum of the CCSD(T) interaction energy extrapolated using X ¼ (7, 8) and of the ΔFCI correction extrapolated using X ¼ (6, 7). Both types of values are given in Table 4.

5. Comparison of ECG and orbital calculations The comparisons of the ECG and orbital calculations are presented in Fig. 1 and on the logarithmic scale in Fig. 2. In Fig. 1, the zero line are the ECG upper bounds to the interaction energies. The extrapolated ECG results are below the upper bounds, as expected taking into account our extrapolation method. The absolute deviations between the two approaches are the largest at small R. However, in relative terms, the differences between the two methods range from 0.0001% at R ¼ 3 bohr to 0.007% at R ¼ 9 bohr, not counting the value at R ¼ 5 bohr, close to the point where the potential crosses zero. The estimated error bars of the extrapolated results are in all cases smaller (up to 20%) than the difference between these results and the upper bounds. The fact that the upper bounds are almost within these error bars is a conformation of soundness of our uncertainty estimates. The results of orbital calculations are quite far from the ECG results, with the CCSD(T) + ΔFCI results in a better agreement than the FCI ones. This is one more example that hybrid methods of this type should be preferred.

Table 4 Interaction energies (in kelvin) obtained in this work from extrapolated dXZ and aXZb95 orbital-basis calculations with X ¼ 7, 8 for CCSD(T) and X ¼ 6, 7 for FCI. R CCSD(T)(7,8) ΔFCI(6,7) CCSD(T)+ΔFCI FCI(6,7)

3.0

3772.77990  0.44656

 5.29872  0.01976

3767.48118  0.46632

3766.48733  1.21480

3.5

1113.78788  0.18521

 3.23928  0.02323

1110.54860  0.20844

1110.58835  0.55276

4.0

294.41916  0.11550

 1.87690  0.00959

292.54226  0.12509

292.59699  0.15853

4.5

59.44681  0.05637

 1.06498  0.00348

58.38183  0.05985

58.39168  0.06460

5.0

0.12922  0.02004

 0.60693  0.00202

 0.47771  0.02206

 0.48158  0.02923

5.3

 8.73110  0.01043

 0.43651  0.00139

 9.16761  0.01182

 9.16978  0.01444

5.6

 10.68471  0.00460

 0.31621  0.00097

 11.00092  0.00556

 11.00122  0.00489

6.0

 9.47221  0.00226

 0.20864  0.00059

 9.68085  0.00286

 9.67819  0.00289

6.5

 6.76597  0.00130

 0.12782  0.00071

 6.89379  0.00202

 6.88999  0.00273

7.0

 4.54220  0.00054

 0.08040  0.00061

 4.62261  0.00115

 4.61994  0.00225

8.0

 2.03122  0.00029

 0.03421  0.00088

 2.06543  0.00117

 2.06526  0.00143

9.0

 0.97237  0.00020

 0.01251  0.00429

 0.98488  0.00449

 0.98547  0.00423

The internuclear separation R is given in bohr.

0.02

0.2

0.004

0.015

0.15

0.003 0.01 0.1 0.005

0.002

Eint [K]

0.05 0 0 3

4

5

6

7

R [bohr]

8

3

4

5

6

7

8

R [bohr]

9

9

0.001

–0.005

–0.05

0

–0.01

5

–0.1 –0.015

–0.001

–0.15 –0.02 –0.2

–0.002

ECG

Fig. 1 Interaction energies relative to the ECG upper bounds.

CCSD(T)/FCI

FCI

6

7

8

R [bohr]

9

250

Krzysztof Szalewicz et al.

Absolute deviation from ECG & uncertainties of ECG [K]

10

1

0.1

0.01

3

4

5

R [bohr]

6

7

8

9

|CC/FCl-ECG| |FCl-ECG| sigma-ECG sigma-CC/FCI sigma-FCI

0.001

0.0001

0.00001

Fig. 2 Interaction energies relative to the ECG extrapolated results. Legend abbreviations indicate the following types of interaction energy differences: jCC/FCI-ECGj: CCSD(T) + ΔFCI-ECG; jFCI-ECGj: FCI-ECG; and the following estimated uncertainties: sigma(ECG): σ(ECG); sigma(CC/FCI): σ(CCSD(T)+ΔFCI); sigma(FCI): σ(FCI).

We will therefore discuss only the former results. The actual absolute errors of CCSD(T)+ΔFCI energies are up to 80 times larger than the uncertainty estimates of the extrapolated ECG energies, with the average value of 30 times. Thus, one may say that the uncertainties of the former method are nearly two orders of magnitude larger than of the latter one. The CCSD(T)+ΔFCI values lie significantly below the ECG ones up to R ¼ 5.3 bohr and above ECG ones beyond R > 7 bohr. In the intermediate region, 5.6–7 bohr, they are only slightly below, and in two points, 5.5 and 7 bohr, the agreement is amazingly good, but we have to assume it is fortuitous. Thus, literature calculations with the CCSD(T) + ΔFCI method, performed mostly for these two points, might have resulted in a deceivingly positive opinion about the performance of this method. The estimates of uncertainties of CCSD(T)+ΔFCI results are generally too conservative: they are from 0.9 to 58 times larger than the actual errors of these calculations, with the average value of 12. Yet, for the two largest distances the errors were underestimated. The advantage of these conservative bounds is that the ECG result is always within the bars, except at R ¼ 8 and 9 bohr. Clearly, the performance of the CCSD(T)+ΔFCI orbital method is nowhere close to that of the ECG method. Could it be improved? For several reasons this is difficult. The largest basis sets used by us are at the edge of numerical accuracy, i.e., with the double-precision (DP) arithmetics, linear

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dependencies in the basis sets lead to errors on digits that are relevant for the investigated problem. This is probably the main reason for the jagged dependence on R in Fig. 1. Thus, to cure this problem, one would have to move to quadruple-precision (QP) arithmetics. This would require not only changing DP to QP arithemetics, but also reanalyzing integral subroutines and other fragments of the codes that use convergence thresholds. Furthermore, to improve accuracy, one would have to optimize bases with X > 8, while already optimizations with X ¼ 8 are quite tricky. Thus, it appears that the ECG approach is at present the only available one for establishing benchmark results for few-electron systems, such as needed for metrology.

6. Comparison of the ECG potential with BO potentials from literature The interest in the helium dimer potential is nearly as old as quantum mechanics. In 1928, Slater81 developed the first potential for this system which gave the interaction energy of 8.8 K at the internuclear distance R ¼ 5.6 bohr. There is a wide range of helium dimer potentials available in the literature (see Ref. 82 for a comparison). Fig. 3 illustrates the remarkable progress in accuracy of predictions achieved since 1979. Empirical potentials dominated the field until the end of 1980s, the two most widely used ones, HFDHE283 and HFD-B,84 developed by Aziz et al. The first really successful ab initio potential was LM-2 by Liu and McLean.85 Those authors performed CI calculations and, by analyzing the configuration space and basis set convergence, obtained extrapolated interaction energies with estimated uncertainties (several versions of the LM-2 potential were proposed). Aziz and Slaman86 used the HFD-B functional form with refitted parameters to “mimic” the behavior of LM-2, of unpublished ab initio data computed by Vos et al.,87 and of the small-R Green-function Monte Carlo (GFMC) data88 to obtain the potentials denoted as LM2M1 and LM2M2, differing by assuming the smallest and largest LM-2 potentials well depths, respectively. The LM2M2 was considered the best helium potential until the mid-1990s, when purely ab initio calculations took the lead. TTY is a remarkably simple potential, developed by Tang et al.,89 based on perturbation theory. HFD-B3-FCI1 was obtained by Aziz et al.,90 who used the HFD-B functional form with parameters adjusted to mimic the GFMC data of Ref. 88 and the FCI results of van Mourik and van Lenthe.91 The SAPT96 potential58,59 opened an era of helium potentials based on ECG functions. The complete first-order and most of the second-order

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Fig. 3 Comparison of ECG BO interaction energies E (in kelvin) at R ¼ 4.0, 5.6, and 7.0 bohr with those from selected earlier potentials. For empirical potentials (HFDHE2 and HFD-B), the sum of post-BO corrections was subtracted in each case. The energies are plotted as error bars from E  σ to E + σ (with dots at E) whenever uncertainty estimates σ are available and as crosses otherwise (for three cases at R ¼ 7.0 bohr the energies are not available). The horizontal dotted lines denote the positions of the BO energies calculated in Ref. 42. For acronyms, see the text.

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(in the intermolecular interaction operator) contributions to interaction energies were calculated using GTGs. All the remaining contributions were computed using SAPT in orbital bases at two levels: using the standard SAPT codes92 and using high-order He2-specific SAPT codes or, equivalently, FCI. It is interesting to note that the errors of SAPT96 turned out to be completely dominated by the residual orbital (rather than GTG) contributions. For instance, at R ¼ 5.6 bohr, the orbital part constitutes only 1.81 K out of 11.00 K, but its error was 0.05 K out of the total SAPT96 error of 0.06 K. The underestimation of the uncertainties seen in Fig. 3 for R ¼ 5.6 bohr was entirely due to this issue. With an added retardation correction, SAPT96 was used (under the name of SAPT2) by Janzen and Aziz93 to calculate properties of helium and found to be the most accurate helium potential at that time. Van Mourik and Dunning94 calculated CCSD(T) energies in basis sets up to d6Z, CCSDTCCSD(T) differences in the dQZ basis set, and FCICCSDT differences in the dTZ basis set. The CCSD(T) energies were CBS-extrapolated and then refined by adding a correction equal to the Rinterpolated differences between the accurate CCSD(T)-R122,95 results (available at a few distances in Ref. 96) and the obtained CBS limits. Supermolecular ECG-based potentials started to appear in the late 1990s97,98 and were initially aimed at providing upper bounds to the interaction energies (by subtracting exact monomer energies), as the authors did not attempt to extrapolate their results to the CBS limits. Another application of explicitly correlated functions to the helium interactions was a series of papers by Gdanitz,74,99,100 who used the multireference averaged coupled-pair functional method with linear r12 factors, r12-MR-ACPF. The extrapolated results from the last paper of the series, Ref. 100 (denoted “Gdanitz01” in Fig. 3), were among the most accurate results available at that time. However, the reported uncertainties were strongly underestimated at shorter distances (as much as 5 times at 5.6 bohr and 17 times at 4.0 bohr). Another important series of papers was published by Anderson et al.,101–103 who reported quantum Monte Carlo energies with progressively reduced statistical uncertainties. Although these results were obtained only for a few internuclear distances, they represented very valuable benchmarks for mainstream quantum chemistry methods. In fact, until the publication of the CCSAPT07 potential, the result from Ref. 103, 10.998(5) K (see “Anderson04” in Figure 3), was the most accurate value available for 5.6 bohr.

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In Refs. 80 and 104, a hybrid supermolecular ECG plus orbital method was applied to the helium dimer. The bulk of the correlation effect on the interaction energy, at the CCSD level, was evaluated using GTG functions with nonlinear parameters optimized at the MP2 level. The effects of noniterative triple excitations (the “(T)” contribution), i.e., the differences between the CCSD(T) and CCSD energies, were calculated using large orbital basis sets (up to a6Zb95 and d6Z) and extrapolated to the CBS limit. Finally, the ΔFCI corrections were obtained in basis sets up to a5Zb95 and d5Z and also extrapolated. Results for three distances were reported in Ref. 104 (see “Cencek04” in Fig. 3). Hurly and Mehl (HM) analyzed the best existing ab initio data for the helium dimer and created a new potential105 representing a compromise based on uncertainties of existing data and their mutual agreement (for instance, as can be seen in Fig. 3, the result from Ref. 104 was used at R ¼ 7.0 bohr). The diagonal adiabatic corrections of Ref. 106 were added to the final potential, which was then used to calculate the second virial coefficient, viscosity, and thermal conductivity of helium. The CCSAPT07 potential43 combined three different computational techniques, according to the criterion of the lowest uncertainty available for each internuclear distance. Variational four-electron ECG calculations were used for distances smaller than 3.0 bohr and SAPT+ΔFCI was employed for R > 6.5 bohr. At intermediate distances, the hybrid supermolecular method described above, i.e., CCSD(GTG)+ΔCCSD(T)+ΔFCI, provided the highest accuracy. Compared to Refs. 80 and 104, several computational improvements were introduced,44 which resulted in significantly reduced uncertainties. The SAPT calculations43 of CCSAPT07 followed the SAPT96 recipe, but again with larger basis sets and some computational improvements. Another highly accurate potential, by Hellmann, Bich, and Vogel (HBV),73 appeared almost at the same time as CCSAPT07. Those authors used very large basis sets (up to d8Z with bond functions at the CCSD level, progressively smaller for higher levels of theory up to FCI) and CBS extrapolations. After augmenting the HBV potential with adiabatic, approximate relativistic, and retardation corrections, the authors used it to calculate thermophysical properties of helium.107 However, the uncertainties of the HBV potential have not been estimated, which restricts its usefulness. A direct accuracy comparison between the pure BO component of HBV and CCSAPT07 is now possible because of the much higher accuracy of the present benchmark energies, and we performed such analysis, using the values reported in the last column of table 3 in Ref. 73. Out of 11

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distances for which all three energies are available, the largest relative error (with respect to the ECG result), equal to 0.90%, occurs for the CCSAPT07 interaction energy at 5.0 bohr, while the error of the HBV interaction energy at this distance is only 0.48%. If one excludes this distance, which is close to where the helium potential crosses zero, and calculates the average relative error at the remaining distances, one obtains 0.007% for CCSAPT07 and 0.011% for HBV. Therefore, both potentials exhibit a similar level of accuracy. Note, however, that the full (post-BO) helium properties calculations using the CCSAPT07 potential in Refs. 61 and 62 are more accurate than the analogous calculations in Ref. 107, because the former include the complete α2 relativistic and α3 QED corrections, while the latter omit two-electron terms in these corrections, which are nonnegligible.

7. Conclusions The BO interaction energies computed in the ECG basis sets established a new accuracy benchmark for the helium dimer. This was achieved by a combination of three factors. First, a pure ECG approach was used, i.e., with all four electrons explicitly correlated and no contributions were calculated with orbital methods. Indeed, the residual errors of older hybrid ECG-orbital potentials, SAPT9658,59 and CCSAPT07,43 were dominated by insufficient basis set saturation of the relatively small orbital contributions. Second, the use of the monomer contraction method dramatically improved the energy convergence with respect to the ECG expansion size and an optimized monomer contraction (rather than a simple product of monomer wave functions) reduced the computational cost. Third, the complete optimization of nonlinear parameters in large basis set expansions was possible because of the use of optimized contractions and other algorithmic improvements. We have also computed the interaction energies in orbital bases at the hybrid CCSD(T) plus FCI level. The largest available basis sets were applied. For most points, the CCSD(T)+ΔFCI approach gives errors nearly two orders of magnitude larger than ECG-estimated uncertainties. For a couple points, the CCSD(T)+ΔFCI results are fairly close in accuracy to ECG ones, but this is mainly due to the former method overestimating the magnitude of the interaction energy at small R and underestimating at large R. Since these points are near the van der Waals minimum, some previous evaluations of the performance of orbital methods restricted to this region might have been

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overoptimistic. When the whole range of R is considerd, the CCSD(T) +ΔFCI orbital approach is no match for the ECG approach. We also point out that improvements of accuracy of the CCSD(T)+ΔFCI approach would require a huge effort. Finally, we compare the ECG potential with literature BO potentials for He2. Enormous progress has been achieved over the past 40 or so years, with uncertainties reduced by three orders of magnitude. This progress was needed to assist metrology research.

Acknowledgments This research was supported by the NSF grant CHE-2154908, by the Real-K project 18SIB02 that has received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme, and by the National Science Center of Poland within Project No. 2017/27/B/ST4/02739.

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63. Czachorowski, P.; Przybytek, M.; Lesiuk, M.; Puchalski, M.; Jeziorski, B. Second Virial Coefficients for 4He and 3He From an Accurate Relativistic Interaction Potential. Phys. Rev. A 2020, 102, 042810. 64. Cencek, W.; Komasa, J.; Rychlewski, J. Benchmark Calculations for 2-Electron Systems Using Explicitly Correlated Gaussian Functions. Chem. Phys. Lett. 1995, 246, 417–420. 65. Patkowski, K.; Cencek, W.; Jankowski, P.; Szalewicz, K.; Mehl, J. B.; Garberoglio, G.; Harvey, A. H. Potential Energy Surface for Interactions Between Two Hydrogen Molecules. J. Chem. Phys. 2008, 129, 094304. 66. Cencek, W.; Kutzelnigg, W. Accurate Adiabatic Correction for the Hydrogen Molecule Using the Born-Handy Formula. Chem. Phys. Lett. 1997, 266, 383. 67. Bukowski, R.; Jeziorski, B.; Szalewicz, K. Basis Set Superposition Problem in Interaction Energy Calculations With Explicitly Correlated Bases. Saturated Secondand Third-Order Energies for He2. J. Chem. Phys. 1996, 104, 3306. 68. Woon, D. E.; Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. IV. Calculation of Static Electrical Response Properties. J. Chem. Phys. 1994, 100, 2975–2988. 69. van Mourik, T.; Wilson, A. K.; Dunning, T. H., Jr. Benchmark Calculations With Correlated Molecular Wavefunctions. XIII. Potential energy curves for He2, Ne2 and Ar2 Using Correlation Consistent Basis Sets Through Augmented Sextuple Zeta. Mol. Phys. 1999, 96, 529–547. 70. Partridge, H.; Bauschlicher, C. W. The Dissociation Energies of He2, HeH, and ArH: A Bond Function Study. Mol. Phys. 1999, 96, 705. 71. Werner, H. J.; Knowles, P. J.; Lindh, R.; Sch€ utz, M.; Celani, P.; Korona, T.; Manby, F. R.; Rauhut, G.; Amos, R. D.; Bernhardsson, A.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Hampel, C.; Hetzer, G.; Lloyd, A. W.; McNicholas, S. J.; Meyer, W.; Mura, M. E.; Nicklass, A.; Palmieri, P.; Pitzer, R.; Schumann, U.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T. MOLPRO, Version # 2010.1, A Package of Ab Initio Programs, Birmingham, UK; 2010. See # http://www.molpro.net. 72. Parrish, R. M.; Burns, L. A.; Smith, D. G. A.; Simmonett, A. C.; DePrince, A. E.; Hohenstein, E. G.; Bozkaya, U.; Sokolov, A. Y.; Di Remigio, R.; Richard, R. M.; Gonthier, J. F.; James, A. M.; McAlexander, H. R.; Kumar, A.; Saitow, M.; Wang, X.; Pritchard, B. P.; Verma, P.; Schaefer, H. F.; Patkowski, K.; King, R. A.; Valeev, E. F.; Evangelista, F. A.; Turney, J. M.; Crawford, T. D.; Sherrill, C. D. PSI4 1.1: An Open-Source Electronic Structure Program Emphasizing Automation, Advanced Libraries, and Interoperability. J. Chem. Theory Comput. 2017, 13, 3185–3197. 73. Hellmann, R.; Bich, E.; Vogel, E. Ab Initio Potential Energy Curve for the Helium Atom Pair and Thermophysical Properties of Dilute Helium Gas. I. Helium-Helium Interatomic Potential. Mol. Phys. 2007, 105, 3013–3023. 74. Gdanitz, R. Accurately Solving The Electronic Schrodinger Equation of Atoms and Molecules By Extrapolating to the Basis Set Limit. I. The Helium Dimer (He-2). J. Chem. Phys. 2000, 113, 5145–5153. 75. J. Olsen and with contributions from H. Larsen and M. Fulscher. LUCIA, A Full CI, General Active Space Program. 76. Dalton. A Molecular Electronic Structure Program, Release 2.0; 2005. http://www.kjemi. uio.no/software/dalton/dalton.html. 77. Saunders, V. R.; Guest, M. F. ATMOL Program Package. SERC Daresbury Laboratory: Daresbury, Great Britain. 78. Przybytek, M. 2003 (unpublished results). 79. Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-Set Convergence In Correlated Calculations on Ne, N2, and H2O. Chem. Phys. Lett. 1998, 286, 243.

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80. Jeziorska, M.; Bukowski, R.; Cencek, W.; Jaszu nski, M.; Jeziorski, B.; Szalewicz, K. On the Performance of Bond Functions and Basis Set Extrapolation Techniques in High-Accuracy Calculations of Interatomic Potentials. A Helium Dimer Study. Coll. Czech. Chem. Commun. 2003, 68, 463–488. 81. Slater, J. C. The Normal State of Helium. Phys. Rev. 1928, 32, 349–360. 82. Spirko, V.; Sauer, S. P. A.; Szalewicz, K. Relation Between Properties of Long-Range Diatomic Bound States. Phys. Rev. A 2013, 87, 012510. 83. Aziz, R. A.; Nain, V. P. S.; Carley, J. S.; Taylor, W. L.; McConville, G. T. An Accurate Intermolecular Potential for Helium. J. Chem. Phys. 1979, 70, 4330–4342. 84. Aziz, R. A.; McCourt, F. R. W.; Wong, C. C. K. A New Determination of the Ground-State Interatomic Potential for He2. Mol. Phys. 1987, 61, 1487–1511. 85. Liu, B.; McLean, A. D. The Interacting Correlated Fragments Model for Weak Interactions, Basis Set Superposition Error, and the Helium Dimer Potential. J. Chem. Phys. 1989, 91, 2348–2359. 86. Aziz, R. A.; Slaman, M. J. An Examination of Ab Initio Results From the Helium Potential Energy Curve. J. Chem. Phys. 1991, 94, 8047–8053. 87. Vos, R. J.; van Mourik, T.; van Lenthe, J. H.; van Duijneveldt, F. B. (unpublished). 88. Ceperley, D. M.; Partridge, H. J. The He-2 Potential at Small Distances. J. Chem. Phys. 1986, 84, 820. 89. Tang, K. T.; Toennies, J. P.; Yiu, C. L. Accurate Analytical He-He van der Waals Potential Based on Perturbation Theory. Phys. Rev. Lett. 1995, 74, 1546–1549. 90. Aziz, R. A.; Janzen, A. R.; Moldover, M. R. Ab Initio Calculations for Helium: A Standard for Transport Property Measurements. Phys. Rev. Lett. 1995, 74, 1586–1589. 91. van Mourik, T.; van Lenthe, J. H. Benchmark Full Configuration-Interaction Calculations on the Helium Dimer. J. Chem. Phys. 1995, 102, 7479–7483. 92. Jeziorski, B.; Moszy nski, R.; Ratkiewicz, A.; Rybak, S.; Szalewicz, K.; Williams, H. L. SAPT: A Program for Many-Body Symmetry-Adapted Perturbation Theory Calculations of Intermolecular Interaction Energies. In Methods and Techniques in Computational Chemistry: METECC-94; Clementi, E., Ed.; Vol. B; STEF: Cagliari, 1993; p. 79. 93. Janzen, A. R.; Aziz, R. A. An Accurate Potential Energy Curve for Helium Based on Ab Initio Calculations. J. Chem. Phys. 1997, 107, 914. 94. van Mourik, T.; Dunning, T. H., Jr. A New Ab Initio Potential Energy Curve for the Helium Dimer. J. Chem. Phys. 1999, 111, 9248–9258. 95. Klopper, W.; Manby, F. R.; Ten-No, S.; Valeev, E. F. R12 Methods in Explicitly Correlated Molecular Electronic Structure Theory. Int. Rev. Phys. Chem. 2006, 25, 427–468. 96. Noga, W.; Klopper, W. Kutzelnigg CC-R12: An Explicitly Correlated CoupledCluster Theory. In Recent Advances in Coupled-Cluster Methods; Bartlett, E., Ed.; World Scientific: London, 1997; p. 1. 97. Komasa, J.; Rychlewski, J. Explicitly Correlated Gaussian Functions in Variational Calculations: The Ground State of Helium Dimer. Mol. Phys. 1997, 91, 909–915. 98. Komasa, J.; Cencek, W.; Rychlewski, J. Adiabatic Corrections of the Helium Dimer From Exponentially Correlated Gaussian Functions. Chem. Phys. Lett. 1999, 304, 293– 298. 99. Gdanitz, R. Accurately Solving the Electronic Schrodinger Equation of Atoms and Molecules Using Explicitly Correlated (r(12)-)MR-CI. IV. The Helium Dimer (He-2). Mol. Phys. 1999, 96, 1423–1434. 100. Gdanitz, R. Accurately Solving the Electronic Schrodinger Equation of Atoms and Molecules Using Explicitly Correlated (r(12)-)MR-CI. VI. The Helium Dimer (He-2) Revisited. Mol. Phys. 2001, 99, 923.

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101. Anderson, J. B.; Traynor, C. A.; Boghosian, B. M. An Exact Quantum Monte-Carlo Calculation of the Helium-Helium Intermolecular Potential. J. Chem. Phys. 1993, 99, 345–351. 102. Anderson, J. B. An Exact Quantum Monte Carlo Calculation of the Helium-Helium Intermolecular Potential. II. J. Chem. Phys. 2001, 115, 4546–4548. 103. Anderson, J. B. Comment on “An Exact Quantum Monte Carlo Calculation of the Helium-Helium Intermolecular Potential”. J. Chem. Phys. 2004, 120, 9886–9887. 104. Cencek, W.; Jeziorska, M.; Bukowski, R.; Jaszu nski, M.; Jeziorski, B.; Szalewicz, K. Helium Dimer Interaction Energies From Gaussian Geminal and Orbital Calculations. J. Phys. Chem. A 2004, 108, 3211–3224. 105. Hurly, J. J.; Mehl, J. B. He-4 Thermophysical Properties: New Ab Initio Calculations. J. Res. Natl. Inst. Stand. Technol. 2007, 112, 75. 106. Komasa, J. Exponentially Correlated Gaussian Functions in Variational Calculations: Energy Expectation Values in the Ground State Helium Dimer. J. Chem. Phys. 1999, 110, 7909–7916. 107. Bich, E.; Hellmann, R.; Vogel, E. Ab Initio Potential Energy Curve for the Helium Atom Pair and Thermophysical Properties of the Dilute Helium Gas. II. Thermophysical Standard Values for Low-Density Helium. Mol. Phys. 2007, 105, 3035–3049.

CHAPTER SEVEN

Nonrelativistic non-Born–Oppenheimer approach for calculating atomic and molecular spectra using all-particle explicitly correlated Gaussian functions Monika Stankea,∗ and Ludwik Adamowiczb a

Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University, Toru
n, Poland Department of Chemistry and Biochemistry, University of Arizona, Tucson, AZ, United States ∗ Corresponding author: e-mail address: [email protected] b

Contents 1. Introduction 2. Separation of the center-of-mass motion from the total nonrelativistic Hamiltonian of the system 3. Generation of the Basis set in a non-BO calculation 4. Examples of non-BO atomic and molecular calculations 5. Challenges of non-BO calculations 6. Summary and future directions Acknowledgments References

264 268 270 271 277 277 277 278

Abstract Very accurate calculations of atomic and molecular spectra require accounting for the coupling of the motions of the nuclei and electrons. To fully account for the nucleus– electron coupling, the nuclei and electrons forming the system have to be treated on an equal footing without assuming the Born–Oppenheimer approximation (non-BO). This can be done by first separating out the Hamiltonian representing the motion of the center of mass from the total nonrelativistic Hamiltonian of the system and then using in the calculation the remaining part of the Hamiltonian that represents the system’s internal state. In this work, we review some recent developments of methods by our group for non-BO calculations of atoms and molecules and the results obtained in these calculations. In particular, the review focuses on the challenges of the non-BO calculations and ways to overcome these challenges.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.007

Copyright

#

2023 Elsevier Inc. All rights reserved.

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Abbreviations BO CEGF ECG function Non-BO approach QED

Born-Oppenheimer complex explicitly correlated Gaussian function explicitly correlated Gaussian function non-Born-Oppenheimer approach quantum electrodynamics

1. Introduction Methods for calculating stationary states of atoms and molecules without assuming the Born–Oppenheimer (BO) approximation are being developed for over two decades by the Stanke group at the Nicolaus Copernicus University and by the Adamowicz group at the University of Arizona, and their collaborators.1–4 The methods allow for very accurate calculations of the spectra of small atoms including the leading relativistic and QED corrections. The non-BO calculations for S, P, D, and F states of atomic systems with 4 and 5 electrons5–10 are among the most accurate in the literature. Unlike the non-BO methods that employ Gaussian orbitals developed by others,11–13 our methods employ various types of explicitly correlated all-particles Gaussian (ECG) functions. As ECGs explicitly depend on the distance between the particles (electrons and nuclei), they very efficiently represent their coupled motion and account for the inter– particle correlation effects. These effects are indispensable in non-BO calculations, as the Coulombic interactions make particles with like charges avoid each other and particles with opposite charges to follow each other. The correlation effects are particularly strong for the nuclei because, due to their larger masses, they stay apart to a much greater extent than the electrons do. Also, the electrons, particularly the core electrons, follow the nuclei very closely, and this effect can also be also very effectively described by the ECGs. Without the BO approximation, both isolated atoms and molecules are spherically symmetric systems represented by an internal Hamiltonian, which is spatially isotropic (or atom-like). As the level of the excitation increases, more radial and angular nodes appear in the atomic or molecular non-BO wave function. The are some similarities, as well as differences of the atomic and molecular non-BO wave functions that the ECG basis functions used in the calculation should enable us to describe. In general, an ECG used in a non-BO calculation is a product of two factors, an exponential and a preexponential one. The exponent can either be single-centered, exp½
r0 Ar,

Nonrelativistic Non-Born–Oppenheimer Approach

265


 or a multiple-center, exp
ðr
sÞ0 Ak ðr
sÞ , Gaussian dependent on the distances between every pair of the N particles (including the reference particle; after separating out the center-of-mass motion, the Hamiltonian representing the internal motion of the system depends on the internal coordinates of n ¼ N
1 pseudoparticles) forming the system, r being a vector of the internal pseudoparticle coordinates, A being a symmetric 3n  3n matrix of exponential parameters, and s being the vector of the shifts of the Gaussian centers. The Gaussian exp½
r0 Ar can be alternatively represented as follows: h  i   exp
α1 r21 + α2 r22 + ⋯ + αn r2n + β12 r212 + β13 r213 + ⋯ + βðn
1Þn r2ðn
1Þn , (1)

where the first part is a product of n orbitals and the second “correlation” part shows the explicit dependency of the Gaussian on the squares of all interparticle distances, r2ij . The preexponential factor of the Gaussian may include an angular term in the form of a product of Cartesian spherical harmonics and/or powers of the lengths of the internal coordinate vectors. In a single-center or a multi-center Gaussian, Ak is an n  n (where A ¼ ðAk  I 3 Þ) symmetric matrix,  is the Kronecker product symbol, and I3 is a 3  3 identity matrix. In the Gaussians, the exponential parameters are included by the use of a quadratic form involving a vector-matrix-vector product, r0 ðAk  I 3 Þr . In non-BO calculations of atomic or molecular bound states, the Gaussians have to be square integrable which effectively imposes restrictions on the Ak matrix. The Ak matrix must be positive definite. Rather than restricting the Ak matrix elements, Ak is represented in a Cholesky factored form as: Ak ¼ L k L 0k , where Lk is a lower triangular matrix and its elements can vary from
∞ to + ∞. With this representation, Ak is automatically positive definite. It should also be mentioned that this form of Ak matrix does not limit the flexibility of the basis functions since any allowable choice of the Ak matrix can be represented by some Lk matrix. This is because any symmetric positive-definite matrix can be represented in a Cholesky factored form. As the zero-field nonrelativistic non-BO Hamiltonian commutes with the operators representing the square of the total (electron + nuclei) angular momentum and its projection on a selected axis, the Hamiltonian eigenfunctions have to transform according to irreducible representations of the SO(3) symmetry group of rotations. In particular, the ground-state non-BO molecular wave function is spherically symmetric. In highprecision calculations of isolated atomic and molecular systems, the angular symmetry, as well as permutational symmetries of identical particles, the

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symmetry properties have to be strictly enforced. If an external perturbation (e.g., interaction with an electric field) is present, the symmetry is lowered and only functions reflecting the new Hamiltonian symmetry have to be included in the basis set. The complexity of the calculation of the properly symmetrized ECG Hamiltonian and overlap matrix elements depends factorially on the numbers of identical particles. Therefore, the application of ECG methods is currently limited to systems with up to 7–8 particles. Using single-centered (i.e., s ¼ 0) ECGs with real or complex exponential parameters (i.e., the A matrix is real or we have A + iB) and with angular factors in the form of products of Cartesian spherical harmonics and with the gradient-based variational algorithms, we have performed very accurate calculations of the spectra of small atoms. The calculations have included the leading relativistic and QED corrections. In particular, the calculations of atomic systems with 4 and 5 electrons should be mentioned, as our nonBO calculations performed for S, P, D, and F states of these systems5–10 are among the most accurate in the literature. For calculating ground and excited states of atoms and molecules, one needs to accurately describe radial and angular oscillations of the non-BO wave functions. In particular, the radial oscillations due to vibrational excitations of the molecule require the use of certain types of ECGs. Treating the electrons and nuclei on an equal footing as N particles and after separating out the center of mass from the total nonrelativistic Hamiltonian of the system, the Hamiltonian representing the system’s internal motion depends on the n internal coordinates of the N
1 pseudoparticles. We denote by r  Rn the vector of the internal coordinates in the following description: 1. ECGs with preexponential multipliers in the form of powers of internuclear distances, Y m Y mij ϕðrÞ ¼ r i i r ij exp½
r0 Ar ðPECGÞ (2) i

i>j

where mi and mij are even nonnegative integers, and A is a real symmetric positive definite 3n  3n matrix.14–17 2. Complex single-center ECGs are of the form: ϕðrÞ ¼ exp½
r0 ðA + iBÞr

ðCECGÞ

(3)

CEGs have been used in Refs. 10,18–21. It has been shown that CECGs are equally if not more efficient than PECGs in describing radial oscillations of excited rovibrational states, even states located near the dissociation threshold.

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3. Real ECGs with shifted centers are of the form ϕðrÞ ¼ exp½
ðr
qÞ0 Aðr
qÞ

ðSECGÞ

(4)

and have a zero imaginary part B ¼ 0 of the width matrix and a zero momentum vector p ¼ 0. SECGs have been used in Refs. 22–26. Including real shifts in the Gaussians enables us to describe radial oscillations and the angular polarization of the wave function due to, for example, the interaction of the system with the field. These types of deformations can be also described by appropriate spherical harmonics factors, though the shifts may be a more effective way for this task. It should be noted that adding a momentum vector p is equivalent to introducing a complex shift vector q. The complex shifts allow us to describe ionization/dissociation events that may happen upon high-energy irradiation of an atom or a molecule. To describe the angular/rotational excitations, ECGs have to be multiplied by products of Cartesian spherical harmonics or by shifting the centers of the Gaussians as it is done in SECGs. Thus, the shifts quite effectively represent both the radial oscillations and the angular polarization of the wave function. The fully complex ECGs enable the accurate description of the following features of the non-BO wave function: (a) the electron–electron, electron– nucleus, and nucleus–nucleus correlation effects; (b) the angular and radial polarization and oscillation caused by the interaction with laser pulses with variable frequencies and intensities; and (c) ionization/dissociation of the system. With complex matrix Ck and complex shift vectors sk, i.e., Ck ¼ Ak + iBk and sk ¼ zk + iwk, where zk and wk are real vectors, ϕk(r) has the form: ϕk ðrÞ ¼ exp½
ðr
sk Þ0 ðAk + iBk Þðr
sk Þ,

(5)

where r is the vector of the Cartesian internal coordinates of the particles pffiffiffiffiffiffiffi forming the system, i ¼
1 and 0 denotes vector/matrix transposition. An alternative, but equivalent, form of ϕk(r) is: ϕk ðrÞ ¼ exp½
ðr
zk Þ0 ðAk + iBk Þðr
zk Þ + iw0k r:

(6)

Recent work27 has shown that such complex Gaussians possess sufficient flexibility to describe the time-evolving state of an atom or molecular system subject to strong and short laser pulses with variable frequencies and intensities. The presence of the plane-wave component in the Gaussians allows the description of the ionization/dissociation dynamics.

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2. Separation of the center-of-mass motion from the total nonrelativistic Hamiltonian of the system As the number-crunching power of computers keeps growing over the years, it becomes possible, at least for the smallest molecular systems, to reduce the number of approximations to a minimum. In the approach adopted in this work, we consider few-electron diatomic systems, LiH and LiH
, without the use of the Born–Oppenheimer approximation. This means that all particles, i.e., the nuclei and electrons forming the molecule, are treated on an equal footing. Such an approach, if it only concerns the internal bound states of the system, necessitates that the center-of-mass motion is removed from the Hamiltonian. This separation in the non-BO method we have developed is done by starting with the total nonrelativistic Hamiltonian of the molecule written in terms of laboratory Cartesian coordinates representing the kinetic and potential energies of the nuclei and the electrons. The general lab-frame nonrelativistic all-particle Hamiltonian of a system of N particles (N being a sum of the number of nuclei and the number of electrons) in atomic units is represented by operator: H lab nr ¼

N N X N X X Qi Qj P2i , + 2M |R i i
Rj | i¼1 i¼1 j6¼i

(7)

where Mi, Qi, Ri, Pi are the mass, charge, the Cartesian coordinates, and the corresponding linear momenta of the ith particle, respectively. The following nuclear masses are used in the present calculations: M(1H) ¼ 1836.15267389me, M(2H) ¼ 3670.48296785me, M(6Li) ¼ 10961.89865me, M(7Li) ¼ 12786.39228me, M(9Be) ¼ 16424.2055me, M(10B) ¼ 18247. 46879me, M(11B) ¼ 20063.73729me, M(14N) ¼ 25519.045282me, and M(15N) ¼ 27336.528712me, where mass of electron me ¼ 1. Next, Hamiltonian (7) is expressed in terms of new Cartesian coordinates that comprise the three lab-frame coordinates describing the position of the system’s center of mass in laboratory Cartesian coordinate frame and the remaining 3N
3 coordinates that are internal coordinates. In our approach, the internal coordinates ri
1 ¼ Ri
R1 are the Cartesian coordinates of vectors with the origins at a chosen reference particle located at R1 (usually the heaviest nucleus in the system, which we can call particle 1) and ending at the positions of particles 2, 3, …, N. A transformation of the total nonrelativistic Hamiltonian to the new coordinates system results

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in its rigorous separation of the lab-frame Hamiltonian into an operator that represents the kinetic energy of the center-of-mass motion and an internal Hamiltonian, as described in the next section. The center-of-mass kinetic energy Hamiltonian depends only on the center-of-mass laboratory coordinates and the internal Hamiltonian depends only on the internal coordinates.28 As mentioned, the nonrelativistic calculations in the present work are carried out using a Hamiltonian that represents the internal state of the molecule and excludes the motion of the center of mass, i.e., excludes the translational motion of the system as a whole. The internal Hamiltonian expressed in terms of the internal Cartesian coordinates, ri, i ¼ 1, …, n , where n ¼ N
1 and N is the number of particles (electrons and nuclei) forming the molecule, is as follows28: 0

1

BX C X n n n n C qi qj 1B 1 0 1 X q0 qi X 0 B C ^ ¼
r r + r r + + , H r r i jC ri ri B 2 @ i¼1 μi m0 i, j¼1 r A i¼1 ri i>j¼1 ij

(8)

i6¼j

where m0 is the mass of the reference nucleus (in the present calculations, the lithium nucleus) and q0 is its charge, qi are the charges of the other particles, μi ¼ m0 mi =ðm0 + mi Þ is the reduced mass of particle i (mi, i ¼ 1, …, n, are the particle masses), ri ¼ jrij, i ¼ 1, …, n is the distance from particle n + 1 to the reference particle, i.e., particle 1, and rij is the distance between particle j + 1 and particle i + 1. The prime symbol in (8) denotes the matrix/vector transposition. One can notice that the internal Hamiltonian represents the motion of n particles, whose charges are the original particle charges, but the masses are the reduced masses (because of that, one can use the term “pseudoparticles” to denote the particles described by the internal Hamiltonian (8)), in the central field of the charge of the reference nucleus. Thus, the internal Hamiltonian is invariant upon all rotations around the center of the internal coordinate system and one can think of it as an “atom-like” Hamiltonian. The eigenfunctions of this Hamiltonian can be classified using the same symmetries as the wave functions of atoms. These eigenfunctions and the corresponding eigenvalues (energies) represent all modes of the internal motions of the molecule including the electronic, vibrational, and rotational motions. In particular, the ground-state solution is spherically symmetric, i.e., it is invariant under rotations in 3D.

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The approach used to obtain the internal Hamiltonian (8) and to separate out the center-of-mass motion from the lab-frame Hamiltonian is a generalization of the standard textbook approach used to reduce a two-body problem to a one-body problem in quantum mechanics, e.g., in the case of an electron and proton forming the hydrogen atom.

3. Generation of the Basis set in a non-BO calculation The calculations described in this work are performed using the Rayleigh–Ritz variational scheme involving minimization of the Rayleigh quotient: c0 HðaÞc Eða; cÞ ¼ min 0 , fa,cg c SðaÞc

(9)

where H(a) is the matrix of the internal Hamiltonian H, S(a) is the overlap matrix, and c is a vector of the linear expansion coefficients of the wave function in terms of the basis functions. Notation E ða; cÞ indicates that the energy depends on the nonlinear parameters of the basis functions (the a vector) and on the c coefficients (the c vector). In the variational energy minimization, the matrix elements of the Lk matrices of the Gaussians (and the coordinates of the Gaussian shifts, sk, in the case of shifted Gaussians) that form vector a are fully optimized. Generating the basis set for the lowest-energy state of a particular spin/spatial symmetry is initiated with a small, randomly chosen set of functions and involves incremental addition of new functions and variationally optimizing them with an approach employing the analytic energy gradient. The new functions are added to the basis set one by one with Lk parameters chosen as a best guess out of several hundred candidates. The parameters of the candidate functions are generated based on the parameters of the functions already included in the set. After a new function is selected, its ik electron number index and the Lk parameters are optimized. Next the function is checked for any linear dependency with the functions already included in the basis set and, if such linear dependency appears, the function is rejected and replaced by a new function. This new function is then subject to optimization. After a certain number of new functions (usually a hundred) are added to the basis set, the whole set is reoptimized by cycling over all functions one by one and reoptimizing their Lk parameters. After the parameters of a function are reoptimized, the function is again checked for any linear dependency with all other functions in the set, and its parameters are reset

Nonrelativistic Non-Born–Oppenheimer Approach

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to their original values if the linear dependency within a certain predefined threshold occurs. The cyclic optimization of all functions is usually repeated several times. The process of basis set growing continues until satisfactory convergence is reached for each state. The optimization of a basis set for the second lowest state of the system is initiated with a basis set generated for the lowest-energy state. Usually, that basis set is significantly smaller than the largest basis set generated for this state, but still provides reasonably good representation in terms of the state’s energy and the wave function of the state. In the optimization of the basis set for the second lowest state, the initial basis is augmented by additional functions which are reoptimized along with the functions contained in the initial basis set to obtain a well-converged solution (i.e., the energy and the wave function) for the state. A similar procedure is applied to obtain the energy and the wave function for the third-lowest energy state. In this case, the optimization is initiated using a basis set generated for the second-lowest energy state. A unique feature of our approach is the use of the analytically calculated first derivatives of the energy functional determined with respect to the nonlinear Gaussian parameters in their optimizations. The derivatives that form the gradient vector are determined as:   1 ∂vechH ∂vechS 0 ∂a E ¼ 0
E ðvech½2cc0
diagcc0 Þ, (10) c Sc ∂a0 ∂a0 where vech is an operation that forms a m(m + 1)/2-dimensional vector (m is the number of the basis functions) of unique matrix elements of a symmetric matrix. The use of the energy gradient in the minimization of the energy functional considerably accelerates the convergence of the optimization process.

4. Examples of non-BO atomic and molecular calculations For atomic states with all s electrons, such as 1S states of the beryllium atom, Gaussians exp½
r0 Ar are used. For atomic states with one p electron and the remaining electrons being s electrons, such as 2P states of the lithium and boron atoms, the Gaussians are: zi exp½
r0 Ar, where zi is the z coordinate of the i electron. For atomic states with three p electrons, such as for 4S states of the nitrogen atom, the basis functions can be obtained by coupling together their

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angular momenta using the appropriate Clebsch–Gordan coefficients. Thus, a linear combination of the following nonnormalized forms of spherical harpffiffiffi pffiffiffi monic functions Y 01 ¼ zi , Y 11 ¼ 1= 2ðxi + iyi Þ, and Y
1 1 ¼ 1= 2ðxi
iyi Þ expressed in the Cartesian coordinates where the common constant factor and the 1=r lii factor were dropped (li is the angular momentum quantum number of the ith electron) is generated. Let us suppose that the orbital quantum numbers of electrons i, j, and k (k in this case is not the index for the basis function expansion but an electron index) are li, mi, lj, mj, lk, and mk. Using the bracket notation, the angular part of the basis function corresponding to particular L and M can be represented as the following linear combination: jL Mi ¼

X mi , mj , mk mi + mj + mk ¼M

ðL Mjli mi ljk mjk Þðljk mjk jlj mj lk mk Þjli mi ijlj mj ijlk mk i,

(11)









where the L Mjl i mi ljk mjk and ljk mjk jl j mj lk mk factors are the Clebsch–Gordan coefficients.29 With that, the angular part of the basis function for a state of three p electrons with L ¼ 0 and M ¼ 0 becomes:     (12) xj yi
xi yj zk + ðxi yk
xk yi Þzj + xk yj
xj yk zi , l

where again the 1=ðr lii r jj r lkk Þ factor was dropped. Therefore, the full form of the basis function ϕk(r) is obtained by multiplying together exp½
r0 Ak rand the angular component (12):      

xjk yik
xik yjk zkk + xik ykk
xkk yik zjk + xkk yjk
xjk ykk zik exp ½
r0 Ak r             x y z y y ¼ r0 vkj r0 vki r0 vzkk
r0 vxki r0 vkj r0 vzk k + r0 vxki r0 vkk r0 vkj         z y y
r0 vxkk r0 vki r0 vk j + r0 vxkk r0 vkj r0 vzki    
 x y
r0 vkj r0 vkk r0 vzk i exp
r0 Ak r : (13)

ψ k ðrÞ ¼

In the above, vector–vector products are used to replace the angular Cartesian coordinates. These replacements allow for a more generalized approach in deriving the expressions for the Hamiltonian matrix elements. The vectors y x z y y vxki , vki , vzki , vkj , vkj , vk j , vxkk , vkk , and vzk k used in (13) are sparse 3n vectors comprising of only one element of a nonzero value of “1” in the position describing the nonzero-angular-momentum particle and its angular

Nonrelativistic Non-Born–Oppenheimer Approach

273

coordinate. Note that vk vectors cannot be represented as a Kronecker product of an n-component vector vk with I3: vk 6¼ vk  I3. Before ECGs are used to expand the wave function of the considered state of the system, they have to be appropriately permutation-symmetry adapted. In the present approach, we use the spin-free formalism for this adaptation. The approach involves construction of a permutation-symmetry projector using the standard procedure based on Young operators. In the case of the doublet states of lithium, the singlet states of beryllium, the doublet states of boron, and the quartet states of the nitrogen, the corresponding Young operators are chosen as (particle one is the nucleus and particles 2, 3, … are the electrons): (1 + P23)(1
P34), (1
P24)(1
P35)(1 + P23)(1 + P45), (1
P24)(1
P26
P46)(1
P35)(1 + P23)(1 + P45), and (1+P56)(1+P78)(1
P68)(1
P57)(1
P27
P25)(1
P23
P37
P35) (1
P34
P24
P47
P45), respectively, where the Pij operator permutes the labels of the i and j electrons. More details about the implementation of the Young-operator approach in our calculations can be found in Ref. 30. It should be noted that the variational optimization of the ECG exponential parameters is only carried out for the wave functions of the leading isotope of the atom (e.g., the 7Li isotope of the lithium atom). In the calculations of the other isotopes and in the infinite-nuclear-mass calculations, the basis sets generated for the leading isotopes are used without reoptimization of their nonlinear parameters. Our experience with atomic calculations has shown that adjusting the linear expansion coefficients, c, provides a sufficiently accurate way to account for a relatively small change of the wave function caused by the change in the nuclear mass. These are the following bound states of four atomic systems considered in the calculations performed in this work: the lowest twelve 2P states of the lithium atom, the lowest ten 1S states of the beryllium atom, the lowest nine 2 P states of the boron atom, and five 1S states of the nitrogen atom. For all the considered states, except for some lowest states of the lithium atom (for these states, calculations performed with explicitly correlated Slater functions produced slightly better energies than those obtained with Gaussians) the total variational energies obtained in the calculations are the lowest ever obtained in ab initio calculations for these systems. The optimizations of the ECG basis sets for each atom are performed for the leading isotope of that atom. The generated basis are subsequently used in the calculations of other stable

274

Monika Stanke and Ludwik Adamowicz

isotopes of each atom, as well as for an atom with an infinite nuclear mass. The results for the 2P states of the 6Li, 7Li, and ∞Li isotopes of the lithium atom are shown in Table 1, the results for the 1S states of the 9Be and ∞Be isotopes of the beryllium atom are shown in Table 2, the results for the Table 1 The total nonrelativistic non-BO energies Enrel of the n 2P states of isotopes of the lithium atom 6Li, 7Li, ∞Li, where n ¼ 2, 3, …, 13. 6 7 ∞ State Basis Li Li Li

2 2P

17,000
7.409 458 110 577
7.409 557 759 019
7.410 156 532 650

2

3 P

13,100
7.336 457 285 733
7.336 556 363 709
7.337 151 708 591

2

4 P

14,000
7.311 196 254 262
7.311 295 101 661
7.311 889 060 748

5 2P

14,000
7.299 596 170 656
7.299 694 902 379
7.300 288 166 250

2

6 P

14,000
7.293 328 522 041
7.293 427 187 815
7.294 020 055 325

2

7 P

15,500
7.289 563 667 252
7.289 662 291 968
7.290 254 912 721

8 2P

16,000
7.287 127 026 011
7.287 225 623 467
7.287 818 080 395

2

16,000
7.285 460 106 102
7.287 225 623 467
7.286 151 027 247

9 P

10 P 17,000
7.284 269 825 426
7.284 368 390 101
7.284 960 650 009 2

11 2P 17,000
7.283 390 376 949
7.283 488 931 330
7.284 081 129 330 12 2P 17,000
7.282 722 235 788
7.282 820 782 270
7.283 412 932 799 13 2P 17,000
7.282 300 479 312


7.284 079 728 671

All values in hartrees.

Table 2 The total nonrelativistic non-BO energies Enrel of the n 1S states of isotopes of the beryllium atom 9Be and ∞Be, where n ¼ 2, 3, …, 11. 9 ∞ State Basis Be Be

2 1S

16,000


14.666 435 526 317


14.667 356 508 387

1

3 S

16,000


14.417 335 144 083


14.418 240 368 939

4 1S

16,000


14.369 185 514 672


14.370 087 938 507

1

5 S

16,000


14.350 610 429 797


14.351 511 737 910

1

6 S

16,000


14.341 502 934 723


14.342 403 688 992

7 1S

16,000


14.336 366 130 065


14.337 266 570 323

1

8 S

17,000


14.333 186 043 180


14.334 086 288 670

1

9 S

17,000


14.331 080 947 170


14.331 981 063 789

10 1S

17,000


14.329 614 861 820


14.330 514 888 895

17,000


14.328 550 996 542


14.329 450 958 909

1

11 S

All values in hartrees.

275

Nonrelativistic Non-Born–Oppenheimer Approach

Table 3 The total nonrelativistic non-BO energies Enrel of the of n 2P states of isotopes of the boron atom 10B, 11B, ∞B, where n ¼ 2, 3, …, 10. 10

B

∞

Basis

2 2P

16,000
24.652 502 680 636
24.652 626 315 397
24.653 868 525 181

2

3 P

16,000
24.430 973 916 386
24.431 097 899 360
24.432 343 606 383

2

4 P

16,000
24.389 170 171 127
24.389 294 047 421
24.390 538 682 291

2

5 P

16,000
24.372 547 534 026
24.372 671 358 362
24.373 915 471 635

2

6 P

16,000
24.364 219 865 377
24.364 343 660 569
24.365 587 486 912

2

7 P

16,000
24.359 451 263 777
24.359 575 039 883
24.360 818 693 430

2

8 P

16,000
24.356 464 432 488
24.356 588 195 903
24.357 831 737 404

2

16,000
24.354 455 451 626
24.354 579 209 823
24.355 822 673 751

9 P

B

11

State

B

10 P 16,000
24.353 024 913 107
24.353 148 632 914
24.354 392 079 994 2

All values in hartrees.

Table 4 The total nonrelativistic non-BO energies Enrel of the of n 2P states of isotopes of the nitrogen atom14N, 15N, ∞N, where n ¼ 2, 3, …, 6. 15 ∞ State Basis 14N N N

2 2P 128


54.548 106 795 683
54.548 244 637 939
54.550 180 140 287

2


54.070 292 960 572
54.070 430 919 917
54.072 368 063 483

2


53.899 406 696 128
53.899 541 400 451
53.901 432 831 328

2

5 P 244


53.701 687 058 152
53.706 202 762 186
53.708 150 318 836

6 2P 256


53.122 252 189 699
53.122 388 837 119
53.124 306 684 080

3 P 128 4 P 128

All values in hartrees.

P states of the 10B, 11B, and ∞B isotopes of the boron atom are shown in Table 3, and the results for the 1S states of the 14N and ∞N isotopes of the nitrogen atom are shown in Table 4. An example of non-BO molecular calculations shown in this work concerns all bound states of the HD+ ion corresponding to the zero total angular momentum. There are 23 such states and they correspond to what is conventionally called the “pure vibrational excitations.” However, due to the coupling of the motion of the nuclei and the motion of the electrons that occurs when the system is described without assuming the Born–Oppenheimer approximation, the vibrational quantum number is not, strictly speaking, a good quantum number. The challenge in calculating all 23 bound “vibrational” states of HD+ with comparable accuracy is due to the need of describing the radial oscillations of the wave functions of these states with equal accuracy. Two approaches are used for this purpose in this work. In the first one, PECGs are used, and in the second one, CECGs are used. The results obtained with these two basis sets are compared in Table 5. 2

Table 5 The total nonrelativistic non-BO energies of the pure vibrational states of HD+. ν Basis Enrel ν Basis Enrel

ν

Basis

Enrel

0

1700 *1500


0.597 897 968 59
0.597 897 968 58

1

1649 *1500


0.589 181 829 50
0.589 181 829 51

2

1700 *1600


0.580 903 700 14
0.580 903 700 12

3

1780 *1500


0.573 050 546 41
0.573 050 546 31

4

1700 *1600


0.565 611 041 73
0.565 611 041 61

5

1840 *1700


0.558 575 520 39
0.558 575 520 19

6

1740 *1700


0.551 935 947 94
0.551 935 947 84

7

1740 *1700


0.545 685 913 88
0.545 685 913 57

8

1865 *1700


0.539 820 639 73
0.539 820 638 67

9

1865 *1700


0.539 820 639 73
0.534 337 009 15

10

1900 *1700


0.534 337 010 93
0.529 233 628 50

11

1820 *1700


0.529 233 630 55
0.524 510 900 41

12

1800 *1700


0.524 510 903 52
0.520 171 133 16

13

1800 *1600


0.520 171 138 12
0.516 218 684 23

14

1800 *1600


0.516 218 695 97
0.512 660 156 98

15

1800 *1600


0.512 660 174 92
0.509 504 603 52

16

1800 *1600


0.509 504 625 62
0.506 763 816 14

17

1800 *1600


0.506 763 848 46
0.504 452 626 59

18

1800 *1600


0.504 452 661 84
0.502 589 154 43

19

1800 *1600


0.502 589 190 42
0.501 194 710 60

20

1860 *1600


0.501 194 759 49
0.500 292 362 14

21

1940 *1700


0.500 292 416 44
0.499 910 332 87

22

1980 *1700


0.499 910 347 15
0.499 865 770 26

23

2000


0.499 865 774 69

The result marked with * are taken from Ref. 31. All values in hartrees.

Nonrelativistic Non-Born–Oppenheimer Approach

277

As one can see, results obtained with the two basis sets are almost equally accurate with, perhaps, the PECG results being slightly better than the CECG result—an effect which can be attributed to the larger sizes of the PECG basis sets.

5. Challenges of non-BO calculations The computer time for an all-particle non-BO calculation scales as a product of the factorials of the numbers of identical particles. For an atom, the scaling factor is the factorial of the number of the electrons. The scaling is due to the number of terms in the permutation-symmetry operator which for an atom with n electrons is equal to n!. Thus, each H or S matrix element is a sum of n! corresponding elemental integrals. The n! time scaling is the major bottleneck of a non-BO calculation will all-particle ECGs. Another size-limiting step in the non-BO calculation is the solution of the secular equation used to determine the energy and the linear expansion coefficients of the wave function. At this point, our effort is particularly focused on creating a procedure for a very fast one-root eigen-problem solver for complex all-particle ECGs.

6. Summary and future directions In this work, the procedure for performing non-BO calculations of small atoms and molecules with all-particle explicitly correlated Gaussian functions developed in our laboratories is described. The procedure is used to calculate some Rydberg states of the lithium, beryllium, boron, and nitrogen atoms. The basis sets generated for the leading isotope of each atom are used to calculate energies and wave functions of other stable isotopes of the atom, as well as the energy of the atom with an infinite nuclear mass. The results of the calculations can be used to calculate the isotope energy shifts. Some bottlenecks of the non-BO calculations are discussed.

Acknowledgments This work has been supported by the National Science Foundation (Grant No. 1856702). L.A. acknowledges the support of the Centre for Advanced Study (CAS), the Norwegian Academy of Science and Letters, in Oslo, Norway, which funds and hosts our research project titled “Attosecond Quantum Dynamics Beyond the Born–Oppenheimer Approximation,” during the 2021–23 academic years. The authors are grateful to the University of Arizona Research Computing for providing computational resources for this work.

278

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References 1. Bubin, S.; Pavanello, M.; Tung, W. C.; Sharkey, K. L.; Adamowicz, L. BornOppenheimer and Non-Born-Oppenheimer, Atomic and Molecular Calculations with Explicitly Correlated Gaussians. Chem. Rev. 2013, 113, 36. 2. Kozlowski, P. M.; Adamowicz, L. Equivalent Quantum Approach to Nuclei and Electrons in Molecules. Chem. Rev. 1993, 93, 2007. 3. Mitroy, J.; Bubin, S.; Horiuchi, W.; Suzuki, Y.; Adamowicz, L.; Cencek, W.; Szalewicz, K.; Komasa, J.; Blume, D.; Varga, K. Theory and Application of Explicitly Correlated Gaussians. Rev. Mod. Phys. 2013, 85, 693. 4. Bubin, S.; Cafiero, M.; Adamowicz, L. Non-Born-Oppenheimer Variational Calculations of Atoms and Molecules With Explicitly Correlated Gaussian Basis Functions. Adv. Chem. Phys. 2005, 131, 377. 5. Stanke, M.; Kedziorski, A.; Adamowicz, L. Fine Structure of the Beryllium P 3 States Calculated With All-Electron Explicitly Correlated Gaussian Functions Phys. Rev. A 2022, 105, 012813. 6. Hornya´k, I.; Nasiri, S.; Bubin, S.; Adamowicz, L. S 2 Rydberg Spectrum of the Boron Atom. Phys. Rev. A 2021, 104, 032809. 7. Hornya´k, I.; Nasiri, S.; Bubin, S.; Adamowicz, L. Low-Lying S 2 States of the Singly Charged Carbon Ion. Phys. Rev. A 2020, 102, 062825. 8. Stanke, M.; Adamowicz, L. Finite-Nuclear-Mass Calculations of the Leading Relativistic Corrections for Atomic D States With All-Electron Explicitly Correlated Gaussian Functions. Phys. Rev. A 2019, 100, 042503. 9. Stanke, M.; Bralin, A.; Bubin, S.; Adamowicz, L. Relativistic Corrections for NonBorn-Oppenheimer Molecular Wave Functions Expanded in Terms of Complex Explicitly Correlated Gaussian Functions. Phys. Rev. A 2018, 97, 012513. 10. Bubin, S.; Stanke, M.; Adamowicz, L. Relativistic Corrections for Non-BornOppenheimer Molecular Wave Functions Expanded in Terms of Complex Explicitly Correlated Gaussian Functions. Phys. Rev. A 2017, 95, 062509. 11. Pavosevic, F.; Culpitt, T.; Hammes-Schiffer, S. Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects Via the Nuclear-Electronic Orbital Method. Chem. Rev. 2020, 120 (9), 4222. 12. Nakai, H.; Sodeyama, K. Many-Body Effects in Nonadiabatic Molecular Theory for Simultaneous Determination of Nuclear and Electronic Wave Functions: Ab Initio NOMO/MBPT and CC Methods. J. Chem. Phys. 2003, 118, 1119. 13. Tachikawa, M. Multi-Component Molecular Orbital Theory for Electrons and Nuclei including Many-Body Effect With Full Configuration Interaction Treatment: Isotope Effects on Hydrogen Molecules. Chem. Phys. Lett. 2002, 360, 494. 14. Kinghorn, D. B.; Adamowicz, L. Non-Born-Oppenheimer Calculations on the Lih Molecule With Explicitly Correlated Gaussian Functions. J. Chem. Phys. 2001, 114, 3393. 15. Kinghorn, D. B.; Adamowicz, L. High Accuracy Non-Born-Oppenheimer Calculations for the Isotopomers of the Hydrogen Molecule With Explicitly Correlated Gaussian Functions. J. Chem. Phys. 2000, 113, 4203. 16. Kinghorn, D. B.; Adamowicz, L. Improved Nonadiabatic Ground-State Energy Upper Bound for Dihydrogen. Phys. Rev. Lett. 1999, 83, 2541. 17. Bubin, S.; Stanke, M.; Adamowicz, L. Non-Born-Oppenheimer Calculations of the Bh Molecule. J. Chem. Phys. 2009, 131, 044128. 18. Bubin, S.; Adamowicz, L. Computer Program Atom-Mol-Nonbo for Performing Calculations of Ground and Excited States of Atoms and Molecules without Assuming the Born-Oppenheimer Approximation Using All-Particle Complex Explicitly Correlated Gaussian Functions. J. Chem. Phys. 2020, 152, 204102.

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19. Chavez, E. M.; Bubin, S.; Adamowicz, L. Implementation of Explicitly Correlated Complex Gaussian Functions in Calculations of Molecular Rovibrational J¼1 States without Born-Oppenheimer Approximation. Chem. Phys. Lett. 2019, 717, 147. 20. Bubin, S.; Formanek, M.; Adamowicz, L. Universal All-Particle Explicitly-Correlated Gaussians for Non-Born-Oppenheimer Calculations of Molecular Rotationless States. Chem. Phys. Lett. 2016, 647, 122. 21. Bednarz, E.; Bubin, S.; Adamowicz, L. Non-Born-Oppenheimer Variational Calculations of HT + Bound States with Zero Angular Momentum. J. Chem. Phys. 2005, 122, 164302. 22. Cafiero, M.; Adamowicz, L. Non-Born-Oppenheimer Calculations of the Ground State of H 3. Int. J. Quantum Chem. 2007, 107, 2679. 23. Cafiero, M.; Adamowicz, L.; Duran, M.; Luis, J. M. Nonadiabatic and BornOppenheimer Calculations of the Polarizabilites of LiH and LiD. Mol. Struct. THEOCHEM 2003, 633, 113. 24. Cafiero, M.; Adamowicz, L. Non-Born-Oppenheimer Calculations of the Polarizability of LiH in a Basis of Explicitly Correlated Gaussian Functions. J. Chem. Phys. 2002, 116, 5557. 25. Adamowicz, L.; Stanke, M.; Tellgren, E.; Helgaker, T. A Quantum-Mechanical NonBorn-Oppenheimer Model of a Molecule in a Strong Magnetic Field. Chem. Phys. Lett. 2020, 761, 138041. 26. Adamowicz, L.; Stanke, M.; Tellgren, E.; Helgaker, T. A Computational QuantumMechanical Model of a Molecular Magnetic Trap. J. Chem. Phys. 2018, 149, 244112. 27. Adamowicz, L.; Kvaal, S.; Lasser, C.; Pedersen, T. B. Laser-Induced Dynamic Alignment of the HD Molecule Without the Born-Oppenheimer Approximation. J. Chem. Phys. 2022, 157, 144302. 28. Kinghorn, D. B.; Poshusta, R. D. Nonadiabatic Variational Calculations on Dipositronium Using Explicitly Correlated Gaussian Basis Functions. Phys. Rev. A 1993, 47, 3671. 29. Zare, R. N. Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics; John Wiley and Sons: New York, 1988. 30. Bubin, S.; Adamowicz, L. Explicitly Correlated Gaussian Calculations of the 2P Rydberg Spectrum of the Lithium Atom. J. Chem. Phys. 2012, 136, 134305. 31. Bubin, S.; Adamowicz, L. Computer Program Atom-Mol-Nonbo for Performing Calculations of Ground and Excited States of Atoms and Molecules Without Assuming the Born-Oppenheimer Approximation Using All-Particle Complex Explicitly Correlated Gaussian Functions. J. Chem. Phys. 2020, 152, 204102.

CHAPTER EIGHT

Relativistic perturbative and infinite-order two-component methods for heavy elements: Radium atom Maria Barysz* Faculty of Chemistry, Nicolaus Copernicus University in Toru
n, Toru
n, Poland *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. The two-component methodology 2.1 The generalized Douglas–Kroll–Hess transformation up to arbitrary order 2.2 Exact decoupling of the Dirac Hamiltonian: The IOTC method 3. Computational details 4. Results and discussion References

281 286 286 288 291 292 293

Abstract A brief overview of the relativistic two-component methods is presented, with an emphasis on the Douglas–Kroll–Hess (DKHn) perturbation method and the exact infinite-order two-component (IOTC) method. To illustrate the performance of the methods, results of theoretical research on the ground and excited states of the radium atom are presented. Calculations were carried out using the spin-free relativistic DKH2, DKH6, DKH8 methods, and the IOTC method at high correlated level. The spin-orbit (SO) coupling is introduced via the restricted active space state interaction method with the use of the atomic mean-field SO integrals.

1. Introduction The importance of relativistic effects in chemistry and physics of heavy elements is already well recognized. Over the past three decades, originally a rather narrow area of atomic and molecular physics, has grown into separate and rapidly developing branch of quantum chemistry.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.02.001

Copyright

#

2023 Elsevier Inc. All rights reserved.

281

282

Maria Barysz

Most compounds containing heavy element atoms require a relativistic description to yield even qualitatively correct results. For highly accurate calculations of properties and energies relativistic methods also have to be applied for atoms and molecules containing light nuclei, especially if spin-orbit (SO) coupling became conclusive. The basic theory underlying all methods of relativistic quantum chemistry is essentially present in four-component Dirac’s equation.1 The Dirac Hamiltonian in its usual form is
H D ¼ cαp + βc + ðV  c ÞI ¼ 2

2

V

cσp

cσp V  2c 2

 ,

(1)

where standard symbols are used for Dirac α, β, and Pauli spin σ matrices. The Coulombic potential V is used to represent the electron–nucleus attraction. The atomic units are used throughout this chapter and c ¼ 137.0359895 a.u. denotes the velocity of light, p is the momentum operator. In order to get electronic binding energies directly comparable to the nonrelativistic theory the energy scale is shifted by c2. The methods based on the Dirac Hamiltonian (Eq. 1) are computationally very demanding and this limits their possible applications. Most of these computational limits arise from the presence of the so-called small component of the four-component Dirac’s solutions. Currently, molecules containing more than two symmetry nonredundant heavy element atoms are hardly accessible for these methods. Therefore it is not surprising that over the years a considerable effort was invested into development relativistic methods in which four-component functions are replaced by their two-component counterparts. The major step in this direction can be achieved by converting the so-called fourcomponent spinor solutions of the Dirac equation1 into what is known as the two-component spinors. Conceptually, the later resemble to a large extent the nonrelativistic solutions with spin. The transition from four-component solutions to separate twocomponent eigenfunctions of Eq. (1) means that we need a unitary transformation that makes Eq. (1) block diagonal with respect to the large- and small-component spinor spaces. Such a transformation is called the Foldy– Wouthuysen (FW) transformation2 and leads to H U ¼ U {H DU ¼




h+

0

0

h

 :

(2)

Relativistic perturbative and IOTC methods for heavy elements

283

The transformed four-component wave function is then obtained as,
 ψ+ U { Ψ ¼ U ΨD ¼ , (3) ψ with ψ  ¼ 0 for electronic states. Hence, the transformation (Eq. 2) converts the 4  4 Dirac equation into two separate 2  2 eigenvalue problems. The one which corresponds to what is called the positive (electronic) part of the spectrum is given by h+ ψ + ¼ E+ ψ + ,

(4)

where ψ + is the two-component function, and should convert into the two-component nonrelativistic Schr€ odinger equation in the limit of c !∞. It is worth noting that, the Dirac Hamiltonian HD as defined in Eq. (1) contains two even terms (V  c2)I, βc2 and one odd term αp. Even terms are block diagonal, whereas odd terms are off-diagonal. If the odd terms αp were not present in the Hamiltonian, the upper and lower components could be solved independently for the two spinors ψ + and ψ . Traditionally the general Foldy–Wouthuysen (FW) transformation2 was considered in the framework of the perturbation theory with the decoupling of positive and negative states through certain order in a suitable coupling parameters. Achieving the goal of the complete separation of the electronic and positronic spectra by the traditional FW transformation faces, however, serious problems. The two-component Hamiltonians derived as the expansion in power series of the fine-structure constant α (in atomic units c ¼ 1/α) results in two-component Hamiltonians which are essentially singular. Only the lowest α2-order approximation, known as the Pauli Hamiltonian is of certain usefulness. In this context, one should mention that in the majority of cases the methods of relativistic quantum chemistry, which are based directly on four-component spinors only the “electronic” part of the energy spectrum is explicitly considered. The so-called negative energy solutions are simply neglected. Thus, although the underlying theory is expressed in terms of four-component spinors, it can be, at least in principle, exactly transformed into a two-component representation. This feature of most of the practical four-component methods of quantum chemistry, though hardly stated explicitly, makes the background for the development of the two-component formalism. Obviously, the two-component methods which give the electronic solutions cannot account for the quantum electrodynamics (QED) effects.

284

Maria Barysz

Thus, over the past two decades, various approximate schemes have emerged allowing the separation of negative and positive Dirac energy solutions and to restrict the theoretical description to the electronic part only. These schemes may be divided into perturbative and infinite-order methods. One of the most popular perturbative two-component Hamiltonian is the Douglas–Kroll–Hess (DKH) Hamiltonian.3,4 Closely related is the Hamiltonian proposed by Barysz et al.5 In both cases, the free-particle Foldy–Wouthuysen (fpFW) transformation2 is first applied to the Dirac equation. The transformed Dirac equation undergoes further transformation that is why we call these schemes the two-step transformation methods. The subsequent coupling equation is then expanded in the external potential V or in the fine-structure constant α giving the DKH2 and Barysz– Sadlej–Snijders (BSS) Hamiltonians,5 respectively. Higher-order DKH Hamiltonians have been proposed by Nakajima and Hirao (DKH3),6 van W€ ullen,7 and Reiher and Wolf (DKHn—nth order DKH).8–11 Perturbative methods include also the above-mentioned FW transformation and the method based on analytical perturbative X-operator techniques.2 Among the other, the priority should be given to the so-called regular Hamiltonian techniques.12 The lowest, zeroth-order regular Hamiltonian approximation, which is referred to as ZORA (zeroth-order regular approach), has been originally proposed by Chang et al.12 The major advantage of this approach is that the two-component ZORA Hamiltonian is manifestly nonsingular. However, the eigenvalues are not invariant upon shifting the external potential by a constant. The method of obtaining the two-component Hamiltonian, exact to the infinite-order two-component (IOTC), was first proposed by Barysz and Sadlej.13 Shortly thereafter, Ilias˘ and Saue14 proposed a one-step IOTC method (without the initial fpFW transformation). Another version of the method for obtaining exact two-component (X2C) Hamiltonians was also proposed by Liu and Peng.15 In this context one should mention some misunderstanding concerning the two-component methods. Historically, they were developed as an approximation to the full four-component formalism and the label twocomponent theory became synonymous with approximate theory. This is true for any finite-order approach such as DKH, DKHn, ZORA, and FORA (first-order regular approach).12 However, the IOTC methods are equivalent to the four-component methods. Hence such a method “for electrons only,” is essentially as good as the Dirac theory restricted to the positive part of the eigenspectrum.

285

Relativistic perturbative and IOTC methods for heavy elements

This review is not aimed at the completeness of the presentation of different two-component methods in relativistic quantum chemistry. We shall focus our attention on the DKHn/DKH approximations and related IOTC method. In the next section we describe the general ideas of the one of the very powerful finite-order DKHn approach,8–11 but the main discussion will be based on the IOTC method by Barysz and Sadlej.13 Both methods have been implemented to MOLCAS and GAMESS-US codes.16,17 The final goal of this study is to investigate the performance of the IOTC method for case that is usually difficult to study theoretically. Radium, transition metal element, is one of such example. In the last decade, there has been renewed interest in radium element (radium atom as well its ion).18–23 Radium atom is proposed as a promising candidate for search of a permanent electric dipole moment in an atomic system. Singly ionized radium ion is proposed as a promising candidate for atomic parity violation experiments (the equation of particle physics are not invariant under mirror inversion) and as atomic clock.23 Experimental results for radium atom and its ion are available,22,24,25 and some theoretical results.19–21,23,26 The theoretical energy values of the lowest 40 states of radium were obtained quite recently by Dzuba20 in the atomic four-component method. To assess the quality of the IOTC results, calculations are also made using the DKH2, DKH6, and DKH8 methods. If both the DKH8 and IOTC methods are correct, the results should converge to the same values and this is what we see in Table 1. Table 1 The IOTC, DKH2, DKH6, and DKH8 CASSCF/CASPT2 RASSI atomic energy levels of radium neutral atom. Sym. Configuration DKH2 DKH6 DKH8 IOTC Expt.a,b,c

1. 1S0

7s2

0.000

0.000

0.000

0.000

0.000

3

2. P0

7s7p

1.629

1.639

1.639

1.639

1.622

3. 3D1

7s6d

1.659

1.671

1.671

1.671

1.701

3

4. D2

7s6d

1.720

1.732

1.731

1.731

1.735

3

5. P1

7s7p

1.707

1.717

1.717

1.715

1.736

6. 3D3

7s6d

1.825

1.836

1.836

1.836

1.824

3

7s7p

1.901

1.912

1.912

1.912

2.069

7. P2 a

National Institute of Standards and Technology (NIST).25 Ref. 24. c Ref. 22. All data are in eV. b

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Maria Barysz

2. The two-component methodology 2.1 The generalized Douglas–Kroll–Hess transformation up to arbitrary order While the unitary matrix U in Eq. (2) can be defined in an clearly defined form, what will be shown later in this review, historically the first was the strategy introduced by Douglas and Kroll,27 and followed by Hess.3,4 Hess defined the unitary transformation U as the sequence of two transformations U1U0 where the unitary transformation U0 was chosen to be the fpFW transformation.
 A αAB U0 ¼ , (5) αAB A where α is the fine-structure constant, sffiffiffiffiffiffiffiffiffiffiffiffi ep + 1 1 A¼ , B¼ σp, 2ep ep + 1

(6)

and ep ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + α2 p2 :

(7)

When applied to the free-particle Dirac Hamiltonian this transformation brings it into diagonal form:

 0 Tp 0 cσp { U0 U0 ¼ , (8) 0 2c 2  Tp cσp 2c 2 where Tp ¼ c 2 ðep  1Þ:

(9)

Thus the free-particle positive and negative energy spectra become fully separated. In other words one may say that fpFW accounts for all “kinetic” relativistic effects. Simultaneously, if the same transformation is applied to the Dirac Hamiltonian Eq. (1) the divergent operators do not appear and the transformed Hamiltonian H 1 ¼ U 0 H D U {0 ¼ E 0 + E 1 + O1

(10)

287

Relativistic perturbative and IOTC methods for heavy elements

where each term can be uniquely classified according to its diagonal (E 0, E 1—even) or off-diagonal (O1—odd) form and to its order in the external potential. The explicit form of this fpFW Hamiltonian reads E 0 ¼ βep  c 2

(11)

E 1 ¼ AðV + BV BÞA O1 ¼ βA½B, V A

(12) (13)

After this fpFW transformation the lowest-order odd term O1 is first order in the external potential, instead of zeroth-order as in the original HD. The decoupling of the Dirac Hamiltonian with the external potential V in the framework of the Douglas–Kroll transformation was achieved by an expansion of the transformed Hamiltonian in powers of the external potential V, and the off-diagonal, odd terms were removed to a certain order in V. What is referred to as the Douglas–Kroll–Hess (DKH2) method corresponds to the approximate unitary transformation U1 which makes the partly block-diagonalized Hamiltonian h+ accurate through the second order in V 2. The idea of the generalized DKHn method aims at decoupling of the Hamiltonian by a sequence of further unitary transformations following the initial fpFW step U0. The final diagonal Hamiltonian may then be written as hd ¼ …U 3 U 2 U 1 U 0 H D U {0 U {1 U {2 U {3 … ¼

∞ X

Ek:

(14)

k¼0

where each term E k comprises all contributions which are exactly of kth order in V. The Hamiltonian is thus expanded in even terms of ascending order in potential V, whereby odd terms are systematically removed step by step. The (4  4)-matrix operator E k consists of two-dimensional operators E k+ and E k . The specific form of the unitary transformations U i , i ¼ 1, 2, 3, … is not decisive. In general, a unitary transformation U can be parameterized by an odd anti-Hermitian operator W what ensures that we can clearly identify any operator sequence as being in total odd or even. Since this is the only mandatory property of W, the anti-Hermitian operator may be chosen freely in a purpose of eliminating the odd contributions in the Dirac Hamiltonian. The most general ansatz to construct a unitary transformation U ¼ f(W) as an analytical function of an anti-Hermitian operator W is a power series expansion,

288

Maria Barysz

Ui ¼ ai, 0 1 + ai, 1 Wi + ai, 2 Wi2 + ai, 3 Wi3 + ⋯ ¼ ai, 0 1 +

∞ X ai, k Wik

(15)

k¼1

which is assumed to converged within a suitable number of steps. The analytic expansion of Ui in Eq. (15) is the most general form of a parametrization of a unitary matrix with Wi being the parameter. Exploiting the antiHermiticity (W {i ¼ W i ) the power series expansion of the Hermitian conjugate transformation can be given as U {i ¼ ai,0 1  ai,1 W i + ai,2 W 2i  ai,3 W 3i + ⋯ ∞ X ¼ ai,0 1 + ð1Þk ai,k W ki

(16)

k¼1

The coefficients ai,k have to satisfy a set of constraints such that Ui is unitary, i.e., U {i U i ¼ 1 . With the requirement that different powers of W be linearly independent, the equations for the coefficients ai,k are obtained. The use of these most general parametrization of the unitary matrices Ui has led to the term generalized DKH transformation. In general the first 2n + 1 terms of transformed block-diagonal Hamiltonian depend only on the n lowest matrices Wi, i.e., they are independent of all succeeding unitary transformations. This property of the even terms originates from the main idea of DKH method to choose the latest odd operator Wi always in such a way, that the lowest of the remaining odd terms is eliminated (see the details in Refs. 8–11).

2.2 Exact decoupling of the Dirac Hamiltonian: The IOTC method The idea of the complete separation of the electronic and positronic spectra of the four-component formalism of the Dirac theory to the X2C form in the IOTC method belongs to the two-step infinite-order methods. Similarly to the DKH2 method, also in this case the unitary transformation U (Eq. 4) of the Dirac Hamiltonian with the external potential V (Eq. 1) is a sequence of two U1U0 transformations, of which the first U0 is fpFW. H 1 ¼ U {0 H D U 0 ! αA½V , BA T p + AðV + α2 BV BÞA , ¼ αA½B, V A 2α2  T p + AðV + α2 BV BÞA (17)

289

Relativistic perturbative and IOTC methods for heavy elements

General form of the second unitary transformation U1 is given by5,28:
 R  Ω Ω+ U1 ¼ , (18) R + Ω+ Ω where Ω+ ¼ ð1 + R{+ R+ Þ

1=2

, Ω ¼ ð1 + R{ R Þ

1=2

, R ¼ R{+ ,

(19)

and R+ is the root of the following operator equation28: R ¼ ½fH 1 g22 1 ½fH 1 g21 + RfH 1 g11 + RfH 1 g12 R,

(20)

which corresponds to the assumption that R+ is a “small” operator as compared with the other operators in the r.h.s. of Eq. (20). The subscripts at H1 denote the appropriate 2  2 blocks of (17). Once the appropriate solution R+ ¼ R of Eq. (20) is known, the exact two-component “electronic” Hamiltonian h+ becomes h+ ¼ Ω{+ ½fH 1 g11 + fH 1 g12 R + R{ fH 1 g21 + R{ fH 1 g22 RΩ+ :

(21)

After substituting the H1 blocks into Eq. (20) one finds that the determination of R requires that the following operator equation 2R + α2 T p R + α2 RT p ¼ ep R + Rep ¼ α3 A½B, V A + α2 ½AV A, R + α4 ½BAV AB, R + α3 R½B, AV AR, (22) is solved. Different truncated iterative solutions have been proposed,5,29 leading to explicit analytic approximations to the exact Hamiltonian h+. All of them can be classified5,29 according to what is called the leading order with respect to α2. The approximate solution which gives the Douglas– Kroll–Hess (DKH2) Hamiltonian follows immediately from Eq. (22) restricted to terms which are linear in V and of the leading order of α3. The operator equation for R is not easy to solve since it involves terms which are linear in σp. These terms cannot be handled efficiently by using the method of Hess.3,4 This problem was solved in the IOTC method13 by replacing R by another operator Y: Y ¼ p1 σpR:

(23)

290

Maria Barysz

Then, the equation for the determination of matrix elements of Y in the basis set which diagonalizes the matrix of the p2 operator reads     ep Y + Y ep ¼ α3 pAbV A  p1 AσpV σpbA + α2 p1 AσpV σpp1 AY  Y AV A   + α4 ðpAbV bApY  Y AbσpV σpbAÞ + α3 Y AbσpV σpAp1  AV Abp Y , (24)

where b¼

1 : ep + 1

(25)

The natural solution of the operator Eq. (24) would be in the momentum space due to the presence of momentum operator p which replaces the standard configuration space formulation by a Fourier transform. The success of the DKH and related approximations is mostly due to excellent idea of Bernd Hess to replace the explicit Fourier transformation by some basis set (discrete momentum representation) where momentum p2 is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy operator T ¼ p2/2m is already available. Consequently, all DKH and IOTC operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques.13 The matrix elements of the Y operator in the p2 basis set can be obtained from the iterative solution of Eq. (24) to arbitrary order of numerical accuracy. Once these solutions are known, the matrix representation of the (formally exact) h+ Hamiltonian can be determined and used as the basis for IOTC calculations. This in brief describes the basic principles of the IOTC methodology. Furthermore, both Eq. (24) and the IOTC one-electron Hamiltonian can also be used in what is called the spin-free approximation. Actually, most of molecular IOTC calculations of energies and molecular properties use the h+ Hamiltonian in its one-component form. This permits the use of most of the standard codes for Hartree–Fock and post-Hartree–Fock calculations of atomic and molecular energies. In this form the IOTC method has been implemented to the one of the most commonly used molecular codes MOLCAS30 and GAMESS.17 Within the IOTC approach in MOLCAS, the effect of SO coupling is introduced via the restricted active space state interaction (RASSI) method with the use of the atomic mean-field SO integrals (AMFI).31,32 The option of calculating lifetimes and transition probabilities is not yet available.

Relativistic perturbative and IOTC methods for heavy elements

291

3. Computational details To illustrate the accuracy of the IOTC method, the ground state and several excited states of a neutral radium atom are calculated. All calculations are carried out with the IOTC CASSCF/CASPT2 RASSI methodology. The multi-configuration complete active space self-consistent field (CASSCF) method33–35 followed by the second-order single-state multi-reference perturbation (CASPT2) scheme36,37 are used. Within the IOTC CASCF/ CASPT2 approach, the effect of SO coupling is introduced via the restricted active space state interaction (RASSI) method with the use of the atomic mean-field SO integrals (modified AMFI).31,32 The correct selection of the active space is crucial for the method and decides about the accuracy of the calculations. Ideally, this space should be as large as possible, unfortunately, in practice, it is often impossible to do. In the present research, I use the active space consists of the valence 6d, 6s, 7s, 8s, 6p, and 7p atomic orbitals of radium. In total, this gives 14 active orbitals. This part of the molecular calculations are carried out in C2 symmetry. Thus the orbital (frozen/inactive/active) subspaces are defined by the number of orbitals in irreducible representations of that group (A, B). The CASSCF wave function is formed from a complete distribution of a number of active electrons in a set of active orbitals. Inactive orbitals are always double occupied in all configurations. Frozen orbitals are not modified in the calculations. Active and inactive orbitals are optimized in the CASSCF procedure. The partition of the orbital space used in CASSCF calculations is (0,0/20,19/8,6; nel), where nel is the number of electrons in the active space and nel ¼ 10 for neutral radium atom. State-average CASSCF (SA CASSCF) calculations are performed for each state with the same symmetry and spin. The question which remains is how many roots we have to include in each of the cases. This is determined by the symmetry and the active space available. The selected active space, although not small, is not sufficient to calculate too many excited states exactly. On the other hand, in order to converge the calculated energies, it is necessary to take into account much more excited states than those of interest to us. Usually we do not know how many, which is why these calculations are so difficult. Twelve singlet states and thirty five triplet states have been necessary to obtain the converged energies of seven states in the case of the IOTC and DKHn methods. In order to obtain more excited states, the active space would have to be increased. This will be the target of our further research.

292

Maria Barysz

The CASSCF wave functions and energies need to be further improved to account for the dynamic electron correlation. This is done by using the multi-state CASPT2 method. In the CASPT2 method the corresponding calculations can be labeled as (20, 19/0, 0/8, 6;nel) partition. All calculations performed within the study for the radium atom are carried out with the Gaussian-type orbitals/contracted Gaussian-type orbitals (GTO/CGTO). We employ the atomic natural orbital relativistic core correlating (ANO-RCC) large Gaussian basis (L-ANO-RCC): Ra [28s.25p.17d.12f/12s.11p.8d.5f].38 All calculations were carried out using the MOLCAS7.316,30 packages of quantum chemistry programs.

4. Results and discussion Table 1 presents the energy levels of the first nine states of the radium atom obtained in IOTC calculations. In addition, the symmetry of atomic levels, and electron configurations are given. These results are compared with the experimental excitation energy data published by Sansonetti et al.,24 Dammalapati et al.,22 and NIST.25 All these data are taken primarily from Moore 1958 compilation.39 and other sources where improved experimental values were available.22 The only theoretical spectrum of the radium atom available so far was calculated by an atomic four-component CP + SD + CI method that was the combination of configuration interaction and many-body perturbation theory. The agreement between the theory and experiment was very good.19–21 However, the comparison of the results obtained with the atomic four-component method with the basis set dependent molecular IOTC method does not seem appropriate. In the IOTC calculations, the ground state of radium atom is the 1S0 level and the next two excited states are the 3P0 and 3D1 levels which agrees very well with the experimental analysis. The one significant difference between the IOTC and experimental data concerns the third and fourth excited states. In the IOTC method, the third excited state (No. 5 in Table 1) is the 3P1 level and the fourth excited state (No. 4 in Table 1) is the 3D2 level. The identical order of these levels was obtained by Dzuba et al.20 The experimental results indicate the reverse order of these levels (see Table 1). However, it can be noticed that the calculated and experimental excitation energies of these two levels are very close to each other. The energy difference between these two states is 0.016 eV for the IOTC method. The corresponding

Relativistic perturbative and IOTC methods for heavy elements

293

experimental difference is 0.001 eV which is definitely smaller than indicated by the IOTC method. These results may suggest that the measurements and analysis should be repeated. To assess the quality of the IOTC results, also the DKH2, DKH6, and DKH8 CASSCF/CASPT2 RASSI calculations have been performed. It can be seen that the DKH6 and DKH8 results converge to the IOTC data. This further confirms the correctness of the results of our calculations and demonstrates the excellent performance of both IOTC and DKH8 methods.

References 1. Moss, R. J. Advanced Molecular Quantum Mechanics; Chapman and Hall: London, 1973. 2. Foldy, L. L.; Wouthuysen, S. A. On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit. Phys. Rev. 1950, 78, 29. 3. Hess, B. A. Applicability of the No-Pair Equation With Free-Particle Projection Operators to Atomic and Molecular Structure Calculations. Phys. Rev. A 1985, 32, 756. 4. Hess, B. A. Relativistic Electronic Structure Calculations Employing a TwoComponent No-Pair Formalism with External-Field Projection Operators. Phys. Rev. A 1986, 33, 3742. 5. Barysz, M.; Sadlej, A. J.; Snijders, J. G. Nonsingular Two/One Component Relativistic Hamiltonians Accurate Through Arbitrary High Order in α2. Int. J. Quantum Chem. 1997, 65, 225. 6. Nakajima, T.; Hirao, K. The Higher-Order Douglas-Kroll Transformation. J. Chem. Phys. 2000, 113, 7786. 7. van W€ ullen, C. Relation Between Different Variants of the Generalized Douglas-Kroll Transformation Through Sixth Order. J. Chem. Phys. 2004, 120, 7307. 8. Reiher, M.; Wolf, A. Exact Decoupling of the Dirac Hamiltonian. I. General Theory. J. Chem. Phys. 2004, 121, 2307. 9. Reiher, M.; Wolf, A. Exact Decoupling of the Dirac Hamiltonian. II. The Generalized Douglas-Kroll-Hess Transformation up to Arbitrary Order. J. Chem. Phys. 2004, 121, 10945. 10. Reiher, M.; Wolf, A. Relativistic Quantum Chemistry. The Fundamental Theory of Molecular Science; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, 2009. 11. Reiher, M. Sequential Decoupling of Negative Energy States in Douglas-Kroll-Hess Theory. In Handbook of Relativistic Quantum Chemistry; Liu, W., Ed.; Vol. 1; Springer Berlin: Heidelberg, 2015. 12. Dyall, K. G.; Faegri, K., Jr. Introduction to Relativistic Quantum Chemistry; Oxford University Press: New York, 2007. 13. Barysz, M.; Sadlej, A. J. Infinite-Order Two-Component Theory for Relativistic Quantum Chemistry. J. Chem. Phys. 2002, 116, 2696. 14. Ilias˘, M.; Saue, T. An Infinite-Order Two-Component Relativistic Hamiltonian by a Simple One-Step Transformation. J. Chem. Phys. 2007, 126, 064102. 15. Liu, W.; Peng, D. Exact Two-Component Hamiltonians Revisited. J. Chem. Phys. 2009, 131, 031104. 16. The IOTC method was implemented in the version of the Molcas 7.3 system of programs by Slovakia Group of Quantum Chemistry (Comenius University, Bratislava, Slovakia 2010), The corresponding patches for Molcas 7.3 release can be obtained directly from M. Barysz (e-mail: [email protected]) or Vladimir Kell€ o (e-mail: [email protected]).

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17. GAMESS System of Quantum Chemistry Programs 2020, from Iowa State University, USA. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K.A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J.A. General Atomic and Molecular Electronic Structure System. J. Comput. Chem., 1993, 14, 1347–1363. 18. Guest, J. R.; Scielzo, N. D.; Ahmad, I.; Bailey, K.; Greene, J. P.; Holt, R. J.; Lu, Z.-T.; O’Connor, T. P.; Potterveld, D. H. Laser Trapping of 225Ra and 226Ra with Repumping by Room-Temperature Blackbody Radiation. Phys. Rev. Lett. 2007, 98. 003901-1–003901-4. 19. Dzuba, V. A.; Ginges, J. S. M. Calculations of Energy Levels and Lifetimes of Low-Lying States of Barium and Radium. Phys. Rev. A 2006, 73. 032503-1–032503-8. 20. Dzuba, V. A.; Flambaum, V. V. Calculation of Energy Levels and Transition Amplitudes for Barium and Radium. Journal of Physics B: At. Mol. Opt. Phys. 2007, 40, 227–236. 21. Ginges, J. S. M.; Dzuba, V. A. Spectra of Barium, Radium, and Element 120: Application of the Combined Correlation-Potential, Single-Doubles, and Configuration-interaction Ab Initio Methods. Phys. Rev. A 2015, 91. 042505-1–042505-9. 22. Dammalalati, U.; Jungmann, K.; Willmann, L. Compilation of Spectroscopic Data of Radium (Ra I and Ra II). J. Phys. Chem. Ref. Data 2016, 45. 013101-1–013101-14. 23. Fleig, T.; DeMille, D. Theoretical Aspects of Radium-Containing Molecules Amenable to Assembly from Laser-Cooled Atoms for New Physics Searches. New J. Phys. 2021, 23, 113039. 24. Sansonetti, J. E.; Martin, W. C. Handbook of Basic Atomic Spectroscopic Data. J. Phys. Chem. Ref. Data 2005, 34, 1559–2259. 25. Kramida, A.; Ralchenko, Y.; Reader, J.; Team, N. I. S. T. A. S. D. NIST Atomic Spectra Database (ver. 5.9); National Institute of Standards and Technology: Gaithersburg, MD, 2021. https://doi.org/10.18434/T4W30F. Available: https://physics.nist.gov/asd (19 September 2022). 26. Bieron, J.; Fischer, C. F.; Fritzsche, S.; Pachucki, K. Lifetime and Hyperfine Structure of the 3D2 State of Radium. J. Phys. B: At. Mol. Opt. Phys. 2004, 37, L305–L311. 27. Douglas, M.; Kroll, N. M. Quantum Electrodynamical Corrections to the Fine Structure of Helium. Ann. Phys. 1974, 82, 89–155. 28. Heully, J. J.-L.; Lindgren, I.; Lindroth, E.; Lundqvist, S.; Møortensson-Pendril, A. J.-M. Diagonalisation of the Dirac Hamiltonian as a Basis for Relativistic Many-Body Procedure. J. Phys. B: Atom. Mol. Phys. 1986, 19, 2799. 29. Barysz, M. Systematic Treatment of Relativistic Effects Accurate Through Arbitrarily High Order in α2. J. Chem. Phys. 2001, 114, 9315–9324. 30. Aquilante, L.; de Vico, N. MOLCAS: A Program Package for Computational Chemistry. Comput. Mater. Sci. 2003, 28, 222. 31. Schimmelpfennig, B.; Maron, L.; Wahlgren, U.; Fagerli, H.; Gropen, O. On the Combination of ECP-based CI Calculations With All-Electron Spin-Orbit MeanFeld Integrals. Chem. Phys. Lett. 1998, 286, 267–271. 32. Malmqvist, P. A.; Roos, B. O.; Schimmelpfennig, B. The Restricted Active Space (RAS) State Interaction Approach With Spin-Orbit Coupling. Chem. Phys. Lett. 2002, 357, 230–240. 33. Siegbahn, P.; Heiberg, A.; Roos, B. O.; Levy, B. A Comparison of the Super-CI and the Newton-Raphson Scheme in the Complete Active Space SCF Method. Phys. Scr. 1980, 21, 323–327. 34. Roos, B. O.; Taylor, P. R.; Sigbahn, P. E. M. The Complete Active Space SCF Method in a Fock-Matrix-Based Super-CI Formulation. Chem. Phys. 1980, 48, 157–173. 35. Roos, B. O. The Complete Active Space SCF Method in a Fock-Matrix-Based SuperCI Formulation. Int. J. Quantum Chem. Symp. 1980, 14, 175–183.

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36. Andersson, K.; Malmqvist, P.-A.; Roos, B. O.; Sadlej, A. J.; Woli
nski, K. Second Order Perturbation Theory With a CASSCF Reference Function. J. Phys. Chem. 1990, 94, 5483–5488. 37. Andersson, K.; Malmqvist, P.-A˚.; Roos, B. O. Second-Order Perturbation Theory With a Complete Active Space Self-Consistent Field Reference Function. J. Chem. Phys. 1992, 96, 1218–1226. 38. Roos, B. O.; Lindh, R.; Malmqvist, P.; Veryazov, V.; P.-O., W. New Relativistic ANO Basis Sets for Transition Metal Atoms. J. Phys. Chem. 2005, 109, 6575–6579. 39. Moore, C. E. Atomic Energy Levels. In NBC Circular, Vol. 467; U.S. Government Printing Office: Washington, DC, 1958; pp. 231–232.

CHAPTER NINE

Physically meaningful solutions of optimized effective potential equations in a finite basis set within KS-DFT framework Aditi Singh, Vignesh Balaji Kumar, Ireneusz Grabowski, and Szymon Śmiga* Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Toru
n, Toru
n, Poland *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Theory 3. Computational details 4. Results 5. Conclusions Acknowledgments Author contributions Data availability References

298 300 304 304 311 312 312 312 312

Abstract The numerical evidence is provided showing that the physical solutions of the optimized effective potential equations (OEP) in a finite basis set, in the context of the Kohn–Sham (KS) Density Functional Theory, can only be obtained by employing the proper regularization procedure in the OEP method and a judicious choice of basis sets used in the KS OEP calculations. The regularization relies on the truncated singular value decomposition procedure to obtain the pseudoinverse of the density–density response matrix. We are showing that this is a critical aspect in determining the stable and numerically accurate solutions of the KS OEP equations for the exchange-only and correlated cases.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.003

Copyright

#

2023 Elsevier Inc. All rights reserved.

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1. Introduction There has been a strenuous interest in using the orbital-dependent exchange-correlation (XC) functionals in the context of the Kohn–Sham (KS) density functional theory (KS-DFT),1–19 since they have proven to provide a systematic way to overcome the limitations of conventional density-dependent approximations20 such as the presence of self-interaction error, qualitatively incorrect exchange, and correlation KS potentials,21,22 description of dispersion interactions,23,24 and the KS energy gaps.25,26 In addition to providing one of the most effective tools to develop accurate XC KS-DFT potentials and functionals, they are also emerging as an appealing tool for computational chemistry studies.27–30 Nevertheless, the use of explicit orbital-dependent functionals within the KS scheme is significantly hindered by the fact that the KS orbitals cannot be expressed directly as the function of the electron density. For this reason, in the true self-consistent KS scheme, the optimized effective potential (OEP) method must be employed,31 to determine the associated local XC potential, which is formally defined as the functional derivative of the XC functional Exc[ρ] with respect to the electronic density ρ(r). For the exchange-only case, the OEP method has a long history, starting with the work of Sharp and Horton32 to determine a multiplicative potential variationally minimizing the Hartree– Fock (HF) energy expression followed by its practical realization proposed by Talman and Shadwick.33 This method, denoted as OEPx (or EXX), was immediately identified as the exact exchange-only KS method. There exist many realizations of OEPx with different efficiency, accuracy, or numerical stability34–48 applied for both molecular29 and solid-state systems.28,30 The OEPx method possesses many important features, e.g., it removes the oneelectron self-interaction error, the exchange potential posseses a correct
1/r asymptotic behavior,33 is exact in the homogeneous-electron-gas limit, exhibits an integer discontinuity49–51 upon addition of an infinitesimal fraction of an electron to the highest occupied orbital, and obeys the exchange virial theorem,52,53 improving at the same time many other KS quantities.3,4,7,29,36,37 For the XC case and fully self-consistent correlated KS OEP (KS-OEP) calculations, usually, the OEPx method is supplied with correlation energy expressions coming from the perturbation theory which in most cases is restricted to second-order correlation energy expression. In such a case, the most straightforward choice is the utilization of energy expression from the second-order G€ orling–Levy perturbation theory (GL2)54 which has the

Physically meaningful solutions of optimized effective potential equations

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same form as a functional defined from the many-body perturbation theory (MBPT).6,55 This choice of XC energy expression constitutes the OEPGL2 method.6,55 Alternatively, the random-phase-approximation correlation can be used.14,17,48,56,57 The most successful forms of the KS-OEP correlation functionals have emerged from the ab initio DFT approach of Bartlett et al.6,8,58 in which the OEP-correlated energy expressions are based on many-body coupled-cluster and perturbation theories. They are applied to the KS system so that systematic improvement can be achieved, in analogy with wave function theory (WFT)-based methods. Among the secondorder ab initio DFT approaches, the OEP2-sc8 method has proven to be one of the most stable, providing very good results in many investigated cases.8,22,59–64 It gives reasonable correlation and total energies, correlation KS-OEP potentials, and relaxed electron densities, correctly describes selfinteraction, has the correct long-range behavior, and provides good ionization potentials and excitation energies.59 Most of the quantities obtained from OEP2-sc calculations are often better than the corresponding ab initio second-order Moller–Plesset (MP2)65 and even coupled-cluster singles and doubles (CCSD)66 and certainly much better than the ones obtained using the existing standard density-dependent DFT functionals22. On the contrary, the OEP-GL2 approach leads to a large overestimation of correlation effects,6,8,67–69 like correlation energy, correlation potentials, or even correlated density. It undoubtedly causes a problem with convergence in many cases.8,22,60,61,63,70–72 It makes us understand that the KS-OEP method depends on the efficient approach for solving OEP integral equations. The numerical gridbased concept can be used, but with limitations to the atomic and molecular systems with high symmetry. In practical realization, to ensure that KS-OEP calculations can be made in the standard Gaussian basis sets for quantum chemical applications, the linear combination of atomic orbitals (LCAO) OEP method is used.36,37 The finite basis set implementation of the OEP approach incorporates a projection method37,73 for solving the required integral equation, and by construction, all XC potentials could be expanded in terms of auxiliary Gaussian functions. Some computational difficulties have been faced in the application of the LCAO OEP calculations, both to the exchange-only energy functional31,41,42,45,74–80 and with correlation included60,72,81 which is a usual manifestation of the well-known instability associated with numerical solutions of Fredholm integral equations of the first kind.82 To obtain numerical solutions of reasonable accuracy for this class of equations (KS-OEP), cautious selection of Gaussian basis sets is required, and what is even more important is the utilization of the proper

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Aditi Singh et al.

regularization techniques in obtaining the inverse of the density–density response matrix in the OEP procedure. Previous experience has shown that using the same basis set to represent both the orbitals and potentials is desirable and that the uncontraction of the basis set often minimizes numerical difficulties.8,31,60,61,73 These issues have been discussed extensively in the literature and several schemes have been proposed for managing this problem.41,42,45,7274–8083 In addition, for correlated OEP calculations, such as those considered in this work, we must also ensure that the basis sets used are suitable for describing the correlation effects akin to those employed in standard WFT-based calculations. Mixing the problems mentioned above of solving the OEP equations, with the convergence issues in the case of OEP-GL2 for some atomic and molecular systems,67,69,84,85 together with the wrong choice of Gaussian basis sets and the underestimation of the use of the regularization procedure,72 introduces some confusion in the context of the application of KS-OEP methods. Therefore, it still affects the lack of confidence in the implementation of KS-OEP methods, as well as in the quality of the results obtained using these methods. In this study, we provide numerical evidence that the physically meaningful solutions of the KS-OEP equations in a finite basis can only be obtained by employing the proper regularization procedure in the solution of the OEP equations.

2. Theory In the KS-OEP method, the KS XC potential vxc,σ (r) ¼ δExc/δρσ (r) corresponds to a given orbital-dependent energy expression Exc ¼ Exc[{ϕpσ }, {εpσ }], where {ϕpσ } and {εpσ } are sets of KS orbitals and eigenvalues, respectively, obtained through the self-consistent solution of the KS equations h i 1
r2 + vs,σ ðrÞ½ ρ ϕpσ ðrÞ ¼ εpσ ϕpσ ðrÞ, (1) 2 with the local effective KS potential vs,σ ðrÞ½ ρ ¼ vext ðrÞ + vJ ðrÞ + vxc,σ ðrÞ,

(2)

where vext is the external (nuclear) potential, vJ is the classical Hartree potential, and X X ρðrÞ ¼ ρσ ðrÞ ¼ jϕiσ ðrÞj2 (3) σ i, σ

Physically meaningful solutions of optimized effective potential equations

301

being the electron density. All through this section, we use the convention to label with σ, τ the spin degrees of freedom and with i, j the occupied KS orbitals, with a, b the unoccupied ones, with p, q, r, s the general (occupied or unoccupied) ones. It is useful and a common practice in orbital-dependent approaches to divide the XC energy functional as Exc ¼ Ex + Ec, separating the exchange and the correlation contributions. The exchange energy functional has the form of the usual HF exchange energy E x ½fϕqτ g ¼


1 XX ði j j j i Þ, 2 σ ij σ σ σ σ

(4)

with (pσ qσ jrσ sσ ) being two-electron integrals in the Mulliken notation computed from KS orbitals. For the correlation energy part, we limit ourselves to the ab initio DFT functional and expression obtained from the second-order, namely OEPGL2 method E ð2Þ c ¼

jðiσ aσ j jτ bτ Þj2 1 XX 1 X X ðiσ aσ j jτ bτ Þðiσ bσ j jτ aτ Þ
2 στ ijab εiσ + εjσ
εaτ
εbτ 2 στ ijab εiσ + εjσ
εaτ
εbτ +

X X j f σ j2 ia ε
εaσ iσ σ ia (5)

with f pqσ ¼ εpσ δpq
h pσ jK^ + vxc jqσ i being the Fock matrix elements defined in terms of the KS spin orbitals. We also present results for the OEP2-sc method, in which the KS orbitals additionally undergo a semicanonical transformation that makes the second-order energy expression invariant with respect to orbital rotations (mixing of occupied or virtual orbitals among themselves).8,60 For a given orbital- and eigenvalue-dependent XC energy functional (Exc), the OEP equation for the XC KS potential can be written as1,3,4,6,32–34 Z

0 0 σ X σ ðr, r0 ÞvOEP xc,σ ðr Þdr ¼ Λxc ðrÞ,

(6)

which is an integral equation (Fredholm of the first kind) with the inhomogeneity given by

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Λσxc ðrÞ

Aditi Singh et al.

¼

( X Z p

 X ϕqσ ðrÞϕqσ ðr0 Þ ∂Exc ∂E xc 2 0 jϕ ðrÞj dr + ϕpσ ðrÞ εpσ
εqσ ∂ϕpσ ðr0 Þ ∂εpσ pσ q6¼p (7)

where X σ ðr, r0 Þ ¼

X ϕ ðrÞϕ ðrÞϕ ðr0 Þϕ ðr0 Þ iσ aσ iσ aσ + c:c:,
ε ε iσ aσ ia

(8)

is the static KS linear density–density response function. The explicit form of the OEP XC potential corresponding to the functionals of Eqs. (4) and (5) can be found elsewhere.6,8,61,81,86 In practice, to solve the OEP equations (6) and calculate the OEP exchange and correlation potentials, one employs the finite basis set LCAO procedure, of Refs. 31,36,37,73, which directly transforms the OEP equation into an algebraic problem. This is done by expanding the exchange and correlation potential and the KS linear response function (8) on an auxiliary, orthonormal, M-dimensional basis set fgp ðrÞgM as p¼1 vOEP xc,σ ðrÞ

¼

M X

c σp gp ðrÞ,

(9)

p¼1

and matrix representation of the response function reads X X σ ðr; r’Þ ¼ ðXÞσpq g*p ðrÞgq ðr’Þ,

(10)

pq

where ðXÞσpq

Z ¼

g*p ðrÞX σ ðr, r0 Þgq ðr0 Þdr0 dr

 Xðiσ aσ jpÞðiσ aσ jqÞ* ¼ + c:c: , (11) εiσ
εaσ ia

and Z ðr σ sσ jqÞ ¼

drϕsσ ðrÞϕ*rσ ðrÞgq ðrÞ,

is the overlap integral of the KS orbitals ϕsσ (r), ϕrσ (r) and the auxiliary function gq(r). The expansion coefficients (c σp ) are obtained from the solution of OEP equation in the form ðXÞσqp c σp ¼ Yσq ,

(12)

Physically meaningful solutions of optimized effective potential equations

303

and Yσq is defined by choice of the form of exchange and correlation functionals, e.g., EXX, OEP-GL2, OEP2-sc. For the explicit expressions of Yσq , see Refs. 6,8,60. To ensure the correct (
1/r) asymptotic behavior of the whole XC potential, the Slater87 or Fermi–Amaldi (FA)88 potential is incorporated into Eq. (9) σ vOEP xc,σ ðrÞ ¼ v Slater=FA ðrÞ +

M X

c σp gp ðrÞ,

(13)

p¼1

where the expansion coefficients c σp are determined from the solution of the OEP equation (see Eq. 12). Moreover, because Eq. (12) determines the OEP potential only up to an additive constant, the so-called HOMO condition at the exchange and correlation level89 must be imposed in KS-OEP calculations to constrain the XC potential to vanish asymptotically. We remark that since the density–density response matrix is in general singular,31,73 to solve Eq. (12) in finite basis set, one needs to employ a truncated singular value decomposition (SVD) in the OEP procedure to calculate the pseudoinverse of the density–density response matrix σ ðX
1 Þqp .37,70,73 The singularities occur because the auxiliary basis set gq(r) used in Eq. (9) can represent the functions in the null space of the matrix representation of Eq. (8).31,73 In consequence, Eq. (6) cannot be uniquely solved by generating a set of XC potentials that differ from each other by null-space functions of Xσ (r,r’).31 The SVD procedure is usually performed in the following two-steps. In the first one, the real symmetric matrix X is diagonalized through an orthogonal transformation U, i.e., X ¼ UΩU
1 where Ω is a diagonal matrix that contains the ωp eigenvalues and U contains the eigenvectors. In the second step, the pseudoinverse is calculated by X
1 ¼ UΩ
1U
1 where each element of the diagonal Ω
1 is the reciprocal (1/ωp) of the corresponding eigenvalue when the eigenvalue is greater than a specified threshold—the SVD cutoff ¼ 10
SVDC and is set to zero if the eigenvalue is smaller than or equal to the threshold. This replacement with zero is similar to excluding the eigenfunctions with zero eigenvalues, i.e., constant functions, from the representation given by Eq. (9). This regularization is the critical and essential step in determining the stable solutions of the OEP equations in the finite basis set LCAO representation.8,59,70,86 Usually the SVD cutoff ¼ 10
6

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Aditi Singh et al.

is chosen for LCAO OEP calculations, in order to remove all singularities from ðXÞσqp matrix. This parameter allows for achieving a compromise between good convergence and numerical stability.29,60,86,90

3. Computational details All OEP calculations have been performed using a locally modified ACES II package91 where the finite basis set LCAO code of OEP36,37,73 is implemented. As mentioned above, in most cases, we use the same basis set to represent both the orbitals and the XC OEP potential, which has been shown to give reasonable accuracy of results.92 The only exception from that is made for two cases in Table I of the current work (rows denoted as cc-pVTZ/P1 and cc-pVTZ/P2) where we have performed the calculations using the identical computational setup as in Ref. 72 with auxiliary basis set taken from Table I of the aforementioned work. All details about the basis set used in the present work are given in the text. As noted before, the numerical instability in the solution of OEP equations31,42,60,75,7678–8183 was minimized by employing a truncated SVD method to construct the pseudoinverse of the linear response function in the OEP procedure. This regularization is an essential step in determining stable solutions of the OEP equation, which, in combination with the proper choice of basis set, ensures stable and physically sound solutions, avoiding in particular variational collapse observed in Ref. 72. Finally, we underline that in all LCAO OEP calculations, we have employed FA potential as a seed potential in Eq. (13) (N is a number of electrons). Z ρðr’Þ 1 FA vxc ðrÞ ¼
dr’ (14) N jr
r’j Moreover, the tight convergence criteria were enforced in the KS-OEP procedure, i.e., at the convergence point, the gradient of the total electronic energy with respect to variations of the cσp coefficients in Eq. (13) was less than 10
9. Finally, the cutoff for truncated SVD was set to 10
6, and results were carefully checked to ensure convergence with respect to this parameter.

4. Results To start our discussion, first, in Table 1, we report the results obtained for OEPx, OEP-GL2, and OEP2-sc for He atom and several basis sets listed

Physically meaningful solutions of optimized effective potential equations

305

Table 1 The OEP results for the He atom as obtained using various basis sets and methods. SVD5OFFb SVD5ONa FA

Basis set

NONE

FA

NONE

OEPx cc-pVTZ

c

cc-pVTZ/P1d cc-pVTZ/P2

e

aug-cc-pVQZ UGBSg 20S10P2D

h

f

2.8612

0

0

diverge

2.8612

0

0

diverge

2.8612

0

0

diverge

2.8615

0

0

diverge

2.8617

0

0

diverge

2.8617

0

0

diverge

OEP-GL2 cc-pVTZ

2.9052

10


5
5

cc-pVTZ/P1

2.9052

10

cc-pVTZ/P2

2.9052

10
6

aug-cc-pVQZ

2.9085

10


8
9

UGBS

2.8801

10

20S10P2D

2.9078

10
9

diverge

diverge

diverge

diverge

diverge

diverge

diverge

diverge

diverge

diverge

diverge

diverge

OEP2-sc cc-pVTZ

2.8949

0

diverge

diverge

cc-pVTZ/P1

2.8949

0

diverge

diverge

cc-pVTZ/P2

2.8949

0

diverge

diverge

aug-cc-pVQZ

2.8976

0

diverge

diverge

UGBS

2.8752

0

diverge

diverge

20S10P2D

2.8970

0

diverge

diverge

The SVD cutoff was set to 10
6. The SVD procedure was omitted. c Uncontracted basis set from Ref. 93. d Uncontracted basis set from Ref. 93 was used for the orbitals and basis set from Table I of Ref. 72 (Potential 1) for representing the XC potential. e Uncontracted basis set from Ref. 93 was used for the orbitals and basis set from Table I of Ref. 72 (Potential 2) for the XC potential. f Uncontracted basis set from Ref. 94. g Universal Gaussian basis set from Ref. 95. h Even-tempered basis set from Ref. 6. The second column contains the total energies (in Ha, sign reversed) obtained within the given method with FA guiding potential and SVD regularization turned on. The third and following columns contain the total energy differences calculated concerning data in the second column. a

b

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Aditi Singh et al.

in the first column of Table 1. The second column contains the total energies (in Hartree, sign reversed) obtained within the given method with FA guiding potential (see Eq. 13) and SVD regularization turned on. One can note that for these settings, all methods converged without problems, regardless of the utilized basis set. Additionally, we observe that at the OEPx level, we can converge to numerical results (2.8617 Ha) with the proper choice of basis set (UGBS95 and 20S10P2D6). This is not a surprising result, because the FA guiding potential provides the exact solution of the OEPx equation for this system. Thus, the remaining part of Eq. (13) is exactly zero. A similar situation can also be observed in the case of OEP-GL2 and OEP2-sc methods. In the former method, the results for the 20S10P2D and uncontracted aug-cc-pVQZ basis sets (2.9078 and 2.9085 Ha) are in line with the numerical data (2.9099 Ha) from Ref. 67, showing the numerical stability of the LCAO OEP procedure. In turn, the OEP2-sc method for the 20S10P2D basis set gives almost the same total energy as in the case of the MP2 method, i.e., 2.8970 Ha. A similar situation was observed for the uncontracted aug-cc-pVQZ basis set. This behavior is not surprising for the He atom and was already observed in some studies.71,86,89 One important fact to note is that the stable solution was also obtained for the basis set reported in Table I of Ref. 72, which has been specially designed to recover neardegeneracy of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) in the case of OEP-GL2 method leading to a variational collapse of this method. This is the first important finding of the present study, showing that with proper regularization, i.e., in our case, the truncated SVD, one can obtain physically meaningful OEP solutions. The third column of Table 1 reports the total energy differences calculated concerning data in the second column for the case where the guiding FA potential is not included in Eq. (13). Therefore, in this situation, the OEP procedure must fully catch the XC potential for a given orbitaldependent functional. One can note that for OEPx and OEP2-sc methods, the total energy difference error is exactly zero. Unfortunately, this is not the case for the OEP-GL2 method, where the error is between 10
5 Ha for the smallest basis set used and 10
9 Ha for the largest variant. This is traced back to the overestimation of the correlation effects in the OEP-GL2 method, which is even more pronounced with the lack of FA guiding potential. Nonetheless, these errors are sufficiently small at the convergence threshold for the largest basis set. The fourth and fifth columns of Table 1 provide the same data for the case where SVD regularization of the density–density response matrix

Physically meaningful solutions of optimized effective potential equations

307

was switched off. In this situation, the convergence is only reached for the OEPx method with FA guiding potential. As was noted before, this is not surprising because the FA potential is the exact OEP solution for twoelectron systems in the singlet state.96 The neglection of the FA term leads to divergence of the OEPx method regardless of the basis set used. Again, it confirms the necessity of using regularization in solving the LCAO OEP KS equations. For the OEP-GL2 method with and without FA guiding potential, the results are diverging, being in line with the one reported in Ref. 72. One important fact to note here is that the variational collapse is observed for all basis sets we employed in our calculations. In the case of OEP2-sc, we observe similar behavior. In both cases, the calculations diverge. Although the construction of OEP2-sc is based on different choices of H0 in secondorder perturbation theory,8 it leads to the same numerical problems as the one encountered for the OEP-GL2 method when regularization of (Xσ )pq is omitted. This confirms that the divergence of correlated OEP equations is not related to the second-order correlation energy expression itself but mostly to the procedure of solving the OEP equations. To conclude, we first note that to avoid the variational collapse reported in the literature72 of all tested OEP methods, one needs to employ the SVD regularization of the density–density response matrix. Second, to obtain OEP results that are close to their numerical counterparts, one needs to utilize a sufficiently large basis set. To support the second conclusion, we report in Table 2 the total energies obtained with numerical67,97 and LCAO OEP implementation36,37 for several closed- and open-shell atoms at OEPx and OEP-GL2 level of theory. In the case of LCAO OEP code for the He atom, we have employed the augcc-pV6Z98 basis set to solve OEP equations, whereas for Li and Be, we have used aug-cc-pCVQZ99–101 basis set. For all other systems, we utilized the aug-cc-CV5Z93,102 basis set. In all cases, the basis sets were fully uncontracted to provide the best possible representation of KS orbitals and XC potential within the LCAO OEP procedure. At the exchange-only level of theory (OEPx method), the agreement between both approaches is almost perfect. The mean absolute error (MAE) and mean absolute relative error (MARE) of the LCAO OEPx method calculated with respect to the numerical counterpart give MAE ¼ 0.1 mHa and MARE ¼ 0.002%, respectively. Moreover, the mean error (ME) shows that we always approach the numerical results from above, possibly meaning that at the complete basis set limit (CBS), we can approach the perfect correspondence of results. In the case of the OEP-GL2 method,

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Aditi Singh et al.

Table 2 Total ground-state energies (in Ha, sign reversed) obtained for a few atoms with the OEP-GL2 and OEPx methods. OEP-GL2 OEPx Atom

Numerical

LCAO

Numerical

LCAO

He

2.9099

2.9097

2.8617

2.8617

Li

7.4818

7.4805

7.4325

7.4325

Be

14.6965a

14.6948a

14.5724

14.5724

N

54.6216

54.6206

54.4034

54.4032

Ne

129.0265

129.0223

128.5454

128.5452

Na

162.3195

162.3081

161.8566

161.8564

Mg

200.1286

200.1190

199.6116

199.6115

P

341.3382

341.3300

340.7150

340.7150

Ar

527.6608

527.6515

526.8122

526.8121

ME[mHa]


5.2153

ME[mHa]


0.1000

MAE[mHa]

5.2153

MAE[mHa]

0.1000

MARE[%]

0.0063

MARE[%]

0.0002

a

From perturbative calculation based on the OEPx density. The bottom lines report the mean error (ME), mean absolute error (MAE), and mean absolute relative error (MARE) calculated with respect to numerical OEP-GL2 and OEPx data. The numerical OEPGL2 results have been taken from Ref. 67, whereas numerical OEPx data have been generated using the exact code.97

the situation is slightly worse. Here, the MAE and MARE (computed w.r.t. OEP-GL2 numerical data) yield 5.1253 mHa and 0.0063%, respectively, meaning that the impact of the basis set used in the correlated OEP calculation is more pronounced. Thus, the CBS limit is harder to achieve (also confirmed by the value of ME). Finally, we note that for all investigated atoms, the LCAO OEP-GL2 method converged without any significant problem. One exception is only for Be atom,8,22,67,69 which due to its nature (quasidegeneracy of (1s)2(2s)2 and (1s)2(2p)2 ground-state configurations103), does not converge, oscillating around some local minima. This is a well-known fact reported and discussed many times in the previous works.8,22,60,61,67,69,71 To support our findings for the total energies, we also analyzed the quality of the ionization potentials obtained from our LCAO OEP KS calculations. In Table 3, we report the ionization potentials (in Ha) obtained from HOMO orbital energies of the neutral atoms from numerical and

309

Physically meaningful solutions of optimized effective potential equations

Table 3 Ionization potentials (in Ha) obtained from the HOMO orbital energies of the neutral atoms from OEP-GL2 and OEPx methods. OEP-GL2 OEPx Atom

Numerical

He

0.893

0.892

0.918

0.918

Li

0.198

0.198

0.196

0.196

Be

0.367a

0.359a

0.309

0.309

N

0.499

0.499

0.571

0.571

Ne

0.656

0.655

0.851

0.851

Na

0.191

0.191

0.182

0.182

Mg

0.305

0.301

0.253

0.253

P

0.385

0.384

0.392

0.392

Ar

0.557

0.556

0.591

0.591

ME[mHa]

LCAO


1.7006

Numerical

ME[mHa]

LCAO


0.0493

MAE[mHa]

1.7323

MAE[mHa]

0.0599

MARE[%]

0.0063

MARE[%]

0.0109

a

From perturbative calculation based on the OEPx density. The ME, MAE, and MARE are calculated with respect to numerical OEP-GL2 and OEPx data. The numerical OEP-GL2 results have been taken from Ref. 67, whereas numerical OEPx data have been generated using the exact code.97

LCAO implementations of OEPx and OEP-GL2 methods. The findings are very similar to the ones reported for total energies. Both OEP-GL2 and OEPx methods provide very close results to their numerical counterparts, yielding MAE of about 0.0599 Ha (MARE ¼ 0.0109%) and 1.7323 Ha (MARE ¼ 0.0063%), respectively. These results indirectly also confirm the quality of XC potentials obtained within the LCAO OEP procedure since the HOMO energy is very sensitive to their quality.63,89,104,105 We have to stress here, that in the case of the OEP-GL2 method, like in the OEPx method,31,37,106 the correlated HOMO condition89 must be incorporated to correct HOMO energies obtained within the LCAO OEP procedure. As the last point of our analysis, we analyze the convergence of the total energies of all investigated atoms with respect to the SVDC parameter (the eigenvalues of density–density response matrix (see Eq. 11) ωp < 10
SVDC are set to zero). In Figs. 1 and 2, we report two representative examples for

310

Aditi Singh et al.

'Etot[Ha]

1e-2

OEPx

1e-3 1e-4 1e-5

'Etot[Ha]

1e-1

OEP-GL2

1e-2 aug-cc-pV6Z cc-pVTZ 20S10P2D

1e-3

1e-4 4

8

12

SVDC

16

20

Fig. 1 Total energy difference between numerical and LCAO OEP(x and -GL2, respectively) data for He atom as a function of SVD cutoff parameter (ωp < 10
SVDC are set to zero in density–density response matrix (see Eq. 11)). All basis sets are fully uncontracted.

OEPx

'Etot[Ha]

1e-2

1e-3

1e-4

OEP-GL2

'Etot[Ha]

1e0

aug-cc-pCV5Z cc-pVTZ ROOS-ATZP

1e-1

1e-2 4

8

12

SVDC

16

20

Fig. 2 Total energy difference between numerical and LCAO OEP(x and -GL2, respectively) data for Mg atom as a function of SVD cutoff parameter (ωp < 10
SVDC are set to zero in density–density response matrix (see Eq. 11)). All basis sets are fully uncontracted.

Physically meaningful solutions of optimized effective potential equations

311

He and Mg atoms obtained in various uncontracted basis sets and with FA seed guiding potential used in the solution of OEP equations. In the case of the He atom and OEPx method, the convergence is reached regardless of the SVD cutoff parameter since, as was noted, FA potentials can be considered as exact exchange potential for two-electron systems. One can also note that for sufficiently large basis sets (i.e., 20S10P2D and aug-cc-pV6Z), we can reach almost numerical accuracy (error less than 10
4 Ha). In the case of the OEP-GL2 method, in turn, for SVDC > 16, the calculation diverges for uncontracted cc-pVTZ and 20S10P2D basis sets. Again, this is the manifestation of the necessity of incorporating the SVD regularization in all LCAO OEP KS calculations. For the Mg atom at OEPx and OEP-GL2 level of theory, the calculations diverge for SVDC > 5 or 6 depending on the basis set used. This indicates that for a relatively small basis, the SVD regularization is essential to obtain a stable solution within the given basis set. Because the value of total energy does not depend on the value of the SVDC parameter, in most cases, it is sufficient to remove from the density–density response matrix all eigenvalues below 10
6. This value of the SVD cutoff parameter was already recommended in several prior studies.29,31,73 A different situation is encountered for the He atom with the uncontracted aug-cc-pV6Z basis set, where stable convergence is reached up to SVDC ¼ 20. A similar situation occurs for the Mg atom with an uncontracted aug-cc-pCV5Z basis set. This might indicate that when the basis set is sufficiently large (close to the CBS limit), one can obtain a stable solution of the OEP method without employing any regularization procedure in the LCAO OEP calculations. This is a very interesting finding confirming that at CBS limit Eq. (8) is essentially well spanned, and hence, the density– density response has only one singularity related to the additive constant uniquely determining the XC potential.31 To confirm this observation, we have performed the OEP calculations in the same basis set with fully switched-off regularization for the OEP-GL2 equations obtaining the same results as previously.

5. Conclusions By performing several finite basis set LCAO OEP calculations for a set of close- and open-shell atoms, we have shown that the physical solutions of the OEP equations in the context of KS-DFT at the exchange-only and XC level can be only obtained by employing the truncated SVD regularization

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of Eq. (11) together with a proper choice of basis sets. Moreover, we have shown that by using this regularization, we can avoid the variational collapse reported in the literature72 of all tested OEP methods. Therefore, regularization is the critical aspect in determining the stable and accurate solutions of the KS-OEP equations in LCAO implementation of OEP. Additionally, we have also shown that in order to obtain the KS-OEP results that are close to their numerical counterparts, the utilization of a sufficiently large basis set is necessary. Finally, our results prove that KS-OEP LCAO calculations can be safely used in test and routine applications.

Acknowledgments S.S´. thanks the Polish National Science Center for the partial financial support under Grant No. 2020/37/B/ST4/02713. I.G. and V.B.K. thank the Polish National Science Center for the partial financial support under Grant No. 2020/39/O/ST4/00005.

Author contributions A.S. and V.B.K. contributed equally to this work.

Data availability The data that support the findings of this study are available upon request.

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CHAPTER TEN

Methane activation and transformation to ethylene on Mo-(oxy)carbide as a key step of CH4 to aromatics Dorota Rutkowska-Zbik, Renata Tokarz-Sobieraj, and Małgorzata Witko* Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Krakow, Poland *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Computational methods 3. Results and discussion 3.1 Mechanistic studies of methane coupling to ethylene 3.2 Influence of catalyst particle size and its composition on methane activation 4. Conclusions Acknowledgments References

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Abstract The process of methane coupling to ethylene occurring on the small molybdenumcontaining cluster mimicking the (oxy)carbide phase introduced to the ZSM-5 zeolite was investigated. The focus was to investigate which properties of the active centers of the reaction, their size and chemical composition, determine the high activity of the catalyst in the studied process. Considering that the CdH bond breaking in methane is the rate limiting step of the studied process, the cluster composed of two or three molybdenum octahedra was found to be the optimum size of the active phase. Further, the transformation of the molybdenum oxide into molybdenum (oxy)carbide was beneficial for the reaction.

1. Introduction Quantum chemistry and theoretical methods of solid-state physics have long been used to study the mechanisms of chemical reactions at surfaces as well as electronic structure of materials. At present, they are Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.008

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2023 Elsevier Inc. All rights reserved.

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complementary and parallel techniques for a whole range of sophisticated experimental methods. One of the chemical problems for which quantum chemistry has been applied is the phenomenon of catalysis. Research in this direction was started in 1973 by the theoretical group in Krakow directed by Prof. Alojzy Gołębiewski in cooperation with Prof. Jerzy Haber. Prof. Gołębiewski was a specialist in the field of quantum chemistry methods. Together with other Polish theoreticians (Włodzimierz Kołos and Wiesław Woz´nicki) he contributed to the emergence of quantum chemistry as an independent field of theoretical chemistry. He was a specialist not only in ab initio methods, such as Self-Consistent Field Linear Combination of Gaussian Orbitals Molecular Orbitals (SCF LCGO MO), Configuration interaction (CI), and Multiconfiguration Self-Consistent Field (MC SCF), but also was a pioneer in the application of quantum chemistry methods (both semiempirical and ab initio) to solve, among others, problems of catalysis in cooperation with Prof. Haber, an outstanding authority on catalysis. In theory group in Krakow, at first, the semiempirical method of molecular orbitals Self-Consistent in Charge and Configuration (SCCC MO) was developed, as a generalization of the extended H€ uckel method for systems containing transition elements [1]. The work published in the Journal of Molecular Catalysis in 1975 [1] was probably the first in the world presenting the application of semiempirical quantum chemistry methods to the problems of catalysis where in the calculations the explicit model catalyst with a metal center was considered. Then, the original Scaled Intermediate Neglect of Differential Overlap (SINDO) method [2,3] was elaborated which explicitly treated electron repulsion at the level of the Intermediate Neglect of Differential Overlap (INDO) approximation. This method was widely used in model catalytic systems. The direction of using quantum chemical methods to the catalytic problems has been continued and developed further by Małgorzata Witko, who concentrated on catalytic reactions taking place at the surfaces of transition metal oxides (see, for example, Refs. [4,5]). Among them vanadium and molybdenum oxides, both pure or mixed with other metals/oxides or promoted by various metals, were of particular interest because they are active and selective in a wide spectrum of chemical processes belonging to different types of reactions. Resolving mechanism of catalytic reaction that takes place at the metal oxide surface requires a detailed knowledge about physical and chemical properties of the surface, characterization of the intermediate species composed of reacting molecule and active surface species, and a description of

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the reaction elementary steps of the reaction between adsorbates. Electronic structure of the surface accounts for the interaction and binding with an adsorbate(s), their bond changes, reaction between adsorbed reactants and the desorption process. Knowledge of electronic characterization of the surface allows to define surface active species responsible for a desired reagent transformation. In addition, it enables to trace influence of geometrical and chemical arrangements on active species properties. It is also of a prior importance to study the role of steric factor (i.e., mutual orientation of the reacting molecule and the surface) in order to get information about influence on the bond modification in reactant, what directly determines the type of product. Article below shows application of theoretical methods to the exemplary catalytic problem, which is the process of methane coupling. An increased production of raw natural gas, composed mostly of methane, stimulates intensive research on new ways for its transformation to higher value chemicals, which can replace or compliment reforming process. As such, selective oxidation of methane to methanol, oxidative coupling of methane, or direct aromatization of methane are under scrutiny [6–9]. The main difficulty in methane transformation lies in the high energy needed to split its CdH bond (105 kcal/mol), which requires very active catalytic sites and/or high temperature. Methane dehydroaromatization: 6CH4 ! C6 H6 + 9H2 was reported for a number of systems, e.g., Ga/HZSM-5 [10], Pt-Bi/ ZSM-5 [11], Fe/HZSM-5, Mo-Fe/HZSM-5 [12], but by far the most studied example is a bifunctional catalysts consisting of ZSM-5 zeolite impregnated with molybdenum oxide/carbide [12–17]. The detailed description of the process is not straightforward, as the study is hindered by the harsh experimental conditions, mostly high temperature (650–700 °C). It is generally assumed that the methane coupling to ethylene proceeds on the molybdenum-containing site, whereas ethylene cyclization toward benzene runs on the acid sites of the zeolite support. During the catalytic process, methane reduces the initial molybdenum(VI) oxide to yield MoO3x, further molybdenum (oxy)carbide, and finally molybdenum carbide [18,19]. Consequently, the active site changes considerably during reaction time. The process of the active phase formation and transformations was modeled by both cluster and periodic Density Functional Theory (DFT)

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calculations [20]. It is known that the molybdenum phase is highly dispersed [18,19,21,22]—the active phase clusters are small enough to fit in the pores of the ZSM-5 matrix. In the current paper we aim to study how the CdH bond breaking in methane depends on the size and composition of the active phase (whether it occurs on molybdenum oxide, molybdenum oxycarbide, or molybdenum carbide cluster). Accordingly, methane activation process leading to its coupling to yield ethylene is studied and a set of reactions is proposed. The calculations are done assuming the probable structure of the catalyst being a small cluster composed of molybdenum(VI) oxide fragment in agreement with experimental indications [18,21,23]. Next, an analysis on how the initial step of methane to aromatics conversion, which is the CdH bond activation, depends on the size and composition of the catalyst applied is presented. In literature, one finds scattered studies on the methane dehydroaromatization reaction mechanism employing different models of the active site composed of molybdenum oxide or molybdenum carbide of different sizes. Authors propose various compositions and geometries of the active phase. Jie Gao and coworkers considered Mo4C2 cluster anchored in different sites of ZSM-5 as active site models for their DFT calculations (revised Perdew–Burke–Ernzerhof (RPBE) functional, the double numerical plus polarization function (DNP) basis set) [24,25]. Guanna Li et al. studied the methane to ethylene reaction on [MoO2]2+, [Mo2O5]2+ and [Mo2C2]2+ clusters by DFT using Perdew–Burke–Ernzerhof (PBE) functional and plane waves [26]. The process was further analyzed by ab initio thermodynamic approach. Authors considered two mechanisms: one involving direct CdC bond coupling to ethylene, and the radical hydrocarbon-pool mechanism involving the presence of aromatic hydrocarbon. The reaction barriers over Mo-oxo sites were much higher (>84 kcal/mol) than when molybdenum carbide models were considered (ca. 48 kcal/mol for center comprised of single Mo site and below 60 kcal/mol for center comprised of two molybdenum centers). The rate limiting step of the process was the dehydrogenation of the C2H5 fragment formed on the bimolybdenum center. The reaction was favored when proceeded on the reduced center. Suvra Khan and coworkers investigated methane coupling to ethylene over Mo4C2 and Mo4C6 clusters within DFT method using Perdew–Wang (PW91) functional and the double-numeric quality basis set with polarization functions (DNP) [27]. The authors found that two steps limit the reaction kinetics: CdH bond breaking and CH3dCH3 coupling. The decisive factor in

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the studied reaction was the reduction state of the active site modeled there by the net charge accumulated on the MoxCy clusters—the energy barriers were lowered when the investigated clusters were reduced. Feng Yin et al. also considered Mo4C2 as the active site for the reaction [28]. Their periodic DFT studies assumed the CdH bond breaking and creation of surfacebound CH3 groups, which can recombine to form *C2H6 (where * denotes the surface active site). This species can react further to yield longer hydrocarbons or undergo dehydrogenation and desorb as ethylene. The methane CdH bond breaking was found to be the step requiring overcoming the highest energy barrier (42 kcal/mol). Other energy-consuming steps were two methyl species coupling (28 kcal/mol) and the cyclization step of the C6H8 chain (27.8 kcal/mol). Recently, Zhang et al. also considered methane coupling on molybdenum-terminated MoC surfaces with DFT using PBE functional and plane-wave basis functions with VASP [29]. They reported that CH4 was preferentially activated on bare Mo atoms, where it sequentially dehydrogenated to *CH moieties. Two such species can recombine to *C2H2 being the precursors for longer chain hydrocarbons. Further, the same authors considered coupling of CH4 on Mo-terminated Mo2C (001) and (100) surfaces [30] to check how the Mo/C ratio and the number of adjacent carbon atoms to the methane adsorption site influences the catalytic performance of the molybdenum carbide. They demonstrated that the carbon species resulting from the consecutive dehydrogenation of CHx (x ¼ 0, 3) species diffuse into the interstitial sites and alter the activity and selectivity of the catalyst in the studied process. When the carburization is low, the catalyst exhibits high activity in CdH bond breaking leading to the complete dissociation of the CHx ⁎ species and insertion of the resulting *C atoms to the interstitial fcc hollow sites of the subsurface layer of the catalyst (carburization). Such a modified system proved to be very active in the CdC coupling step resulting in the formation of C2H2* species bound to the surface. Our results follow these considerations.

2. Computational methods Quantum chemical method based on Density Functional Theory (DFT) with nonlocal Perdew–Burke–Ernzerhof functional [31–34] was applied. This functional is often used to study molybdenum-based systems [26,35]. The calculation consisted of geometry optimizations of the studied structures with further confirmation by vibrational analysis. The Resolution-of-Identity (RI)

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algorithm was considered in order to accelerate computation [36,37]. Allelectron Gaussian type orbitals of def-TZVP quality were used to define atomic orbitals [38]. The present results were obtained with Turbomole v. 6.3 [39].

3. Results and discussion At first, the theoretical modeling of activation of methane and its further transformations to yield C2H4 on MoO3/ZSM-5 catalyst was undertaken. The theoretical model of catalyst for the possible reactions was cut from the lattice of MoO3 [40] and was based on our previous studies [20]. The investigated structure is depicted in Fig. 1. It consists of two molybdenum–oxygen octahedra in which the terminal oxygen atoms are saturated with hydrogen atoms. As can be seen, there exist three structurally different oxygen atoms: singly (O1), doubly (O2), and triply (O3) coordinated ones. The singly coordinated oxygen atoms form molybdenyl groups ˚ pointing upward the cluster. The doubly coordinated (Mo]O) of 1.70 A oxygens join two adjacent molybdenum atoms forming short ModO ˚ ) and long (2.22 A ˚ ) bonds in alteration. The triply coordinated atoms (1.77 A ˚ form two short (1.95 A) and one long (2.37 A˚) bonds with molybdenum atoms.

3.1 Mechanistic studies of methane coupling to ethylene The first considered step of the reaction is the physisorption of methane over the catalyst. The physisorption state lies 1 kcal/mol lower in energy than the substrates. The geometry parameters describing this step are shown in Fig. 2. O

O

O

O Mo

Mo O

O

O

O

O

O

O

Fig. 1 Model of the active site used in the current studies, following the data presented in Ref. [41]. Color code: blue—molybdenum, white—oxygen, gray—hydrogen.

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Fig. 2 Methane activation at the MoO3 cluster. Lengths of the selected bonds are given in Å.

Fig. 3 Methane activation at a reduced MoO3 cluster (with O1 vacancy). Lengths of the selected bonds are given in Å.

Next, the CdH bond in methane is activated. In the transition structure ˚ (in comparison to 1.10 A ˚ in the isolated CH4) its length is equal to 1.19 A and the distance between one of hydrogen atoms of methane and oxygen ˚ . The energy needed to split atom of the active phase is equal to 1.43 A the CdH bond of methane in the studied system is equal to 63 kcal/mol. This value is taken as a reference value for further investigations—see Section 3.2. The product of methane dissociation, i.e., the system, in which CH3 species interacts with the OH group on the molybdenum atom, lies 56.9 kcal/mol above the energy of the substrates. Taking into account highly reducing environment of the reaction, a situation when methane is activated over the reduced system is then considered. At first, CH4 activation over MoO3 with one vacant site in position of the singly coordinated oxygen (O1) is investigated—see Fig. 3. Similarly to the reaction catalyzed by the undefected molybdenum oxide, the first step consists of methane physisorption over bare molybdenum ion. The structure is stabilized by 1 kcal/mol. Next, CdH bond is activated and, as a result, both H and CH3 fragments bind to molybdenum atom. The overall formation of this system is thermodynamically privileged—it is characterized by total energy lower by 15 kcal/mol than the sum of total energies of substrates. Our studies indicate that the activation of CdH in methane should be easier in the reduced systems: methane dissociation product lies higher in energy for pure MoO3 as compared to

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partially reduced molybdenum oxide. In the created intermediate state, the ˚ with the molybdenum atom. resulting CH3 group forms a bond of 2.21 A ˚ The similar ModCH3 bond length (2.16 A) was found also in Ref. [22], ˚ ), depending on the while slightly longer bond length range (2.33–2.41 A binding site and the level of the catalyst carburization, were recently reported by Zhang et al. [30], although in the latter study they showed that the carbon atom of the methyl species interacts simultaneously with three molybdenum ions. Our conclusions agree with data presented in Ref. [26], which indicates the reduced form of the catalyst to have higher activity in methane activation. They are also in line of the experimental observation, that the reaction accelerates once the catalyst undergoes reduction in the stream of reactants [13,14]. On the other hand, the methyl radical species formed on the oxidized form of the catalyst can enter the radical hydrocarbon pool whose existence is also evoked as an important pathway for higher hydrocarbon formation during methane coupling process [42]. Next, the second hydrogen atom from CH3 group is abstracted and the resulting H2 molecule desorbs from the catalyst leaving CH2 species double bound to Mo ion—see Fig. 4. As a result, the molybdenum–carbon bond is ˚. strengthened, and its lengths decreases to 1.92 A Ethylene might be then formed because of a recombination of two CH2 groups present on the adjacent molybdenum centers—see Fig. 5.

Fig. 4 Hydrogen elimination step. Lengths of the selected bonds are given in Å.

Fig. 5 Ethylene formation step. Lengths of the selected bonds are given in Å.

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Fig. 6 Ethylene desorption step. Lengths of the selected bonds are given in Å.

The energy barrier of such a process is 24 kcal/mol and the reaction involves ˚ , slightly shorter the formation of the new carbon–carbon bond of 1.53 A ˚ than the typical single CdC bond (1.54 A), with a simultaneous elongation of both molybdenum–carbon bonds to 2.19 and 2.13 A˚. Ethylene desorption needs 44 kcal/mol, but no barrier associated with the ethylene desorption step is found (Fig. 6). This finding should be view in light of the calorimetric results reported by Kazansky et al. [43] who demonstrated that the cyclic hydrocarbon intermediate bound to the surface Mo atoms are not stable. Although their conclusion was drawn for propene metathesis reaction mechanism, one can see analogy to the formation of ethylene as discussed in the current paper.

3.2 Influence of catalyst particle size and its composition on methane activation To study the influence of the size and chemical composition of the active phase of the catalyst, different models of molybdenum oxide and molybdenum (oxy)carbides, which might mimic ZSM-5 crystal structure, were constructed. In the first instance, the effect of the size of the MoO3 cluster and the location of the methane activation event was examined. Linear clusters of varying size (containing from 1 to 6 molybdenum–oxygen octahedra denoted Model x where x indicates the number of Mo atoms in the structure) were constructed—see Fig. 7. The energy of CdH bond activation in methane was calculated for all clusters. Next, the computed values were referred to the CdH bond activation energy found for the reference MoO3 cluster consisting of two molybdenum–oxygen octahedra as described in Section 3.1. The differences between the CdH activation energies (ΔE ) on the studied catalyst clusters and the reference value of 63 kcal/mol when the activation occurs at the edge of the cluster and in its center are depicted in Figs. 8 and 9, respectively. In general, when methane activation takes place at molybdenum oxide, the CdH activation is preferred at the edge of the cluster built from multiple

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Fig. 7 Various models of molybdenum-containing active sites considered in the current paper. Color code: blue—molybdenum, white—oxygen, gray—hydrogen.

Fig. 8 Dependence of the CdH bond activation energy (in kcal/mol) in CH4 process occurring at the edge of the cluster.

MoOx octahedra. When the process occurs in the center of the cluster, the energy needed to split CdH bond increases by ca. 20 kcal/mol. Clusters consisting of two or three molybdenum octahedra are of optimum size for the process—see Fig. 8. It should be noted that the CH4 activation energy is lowered by 1.6 kcal/mol when occurring on the Model 3 cluster with respect to the reference value. The accuracy of the preformed calculations being approximately 2.5 kcal/mol does not allow for definite conclusion whether Model 2 or Model 3 would be the most appropriate for the reaction. Further increase of the size of the catalyst cluster increases the CdH activation barrier.

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Fig. 9 Dependence of the CdH bond activation energy (in kcal/mol) in CH4 process occurring at the center of the cluster.

Fig. 10 Dependence of the CdH bond activation energy (in kcal/mol) in CH4 on the basic cluster composition.

Next, the effect of the composition of the reference cluster, containing two molybdenum atoms, was studied. To do so, particular oxygen atoms of the model catalyst were replaced by CH2, CH3, or carbon atoms depending on their structural position. The process of the transformation of the active phase of the catalyst from pure molybdenum oxide to molybdenum (oxy) carbide is modeled in the same way, as in our previous studies—see Ref. [20]. Fig. 10 presents the dependence of CdH bond activation energy on the content and the position of the introduced carbon atoms. It is shown that while the content of carbon, especially substituting doubly coordinated oxygen atoms (O(2)), is increased, the CdH activation

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barrier is lowered. When the outer oxygen atoms are replaced by the CH3 groups methane activation becomes more difficult. The obtained results clearly show that the presence of the (oxy)carbide phase has a beneficial effect on the methane activation process. Similar conclusions were drawn by Zhang et al. [30] who pointed out that the carburization of the (100) surface of β-Mo2C, i.e., the introduction of carbon atoms into the subsurface layer, lowered the catalyst ability to activate methane. The effect was attributed to the decrease of the d-band center of the system. As a result, the carburized (100) β-Mo2C exhibited a very similar reactivity to (001) α-Mo2C, (001) α-MoC, and (111) δ-MoC, all of them having either different Mo/C ratio and/or different filling of subsurface interstitial sites. While discussing the dependence of the methane activation energy on the composition of the catalyst one should bear in mind that the Mo/C/O ratio is not constant during the methane coupling process. The catalyst composition can be changed as reaction proceeds affecting catalyst reactivity toward methane by modifying CdH bond activation energy. Such a dynamic nature of the catalyst under operating conditions was also underlined in Ref. [30].

4. Conclusions Our studies shed light on the optimum size and composition of the molybdenum (oxy)carbide phase for methane coupling to ethylene. It was found that the CdH bond breaking in methane is a rate limiting step of the studied process. The coupling of two methyl groups requires lower energy input than the methane CdH breaking, and thus does not hinder the kinetics of the process. Methane activation is favored when the active phase is composed of two or three molybdenum octahedra, because the CdH bond activation energy is higher for smaller and bigger MoOx clusters. The composition of the active phase is found crucial too. Once carbon atoms replace singly and doubly coordinated oxygen atoms in the active phase, the methane activation requires lower energy. Altogether, our results highlight the importance of carefully designing the size and composition of the active phase particles, when developing molybdenum-based catalyst for methane aromatization.

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Acknowledgments This work was in part supported by the project EC 7th FP 2009-229183 NEXT-GTL “Innovative catalytic technologies & materials for next gas to liquid processes” and in part by the statutory funds of the Jerzy Haber Institute of Catalysis and Surface Chemistry Polish Academy of Sciences.

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CHAPTER ELEVEN

Molecular systems in spatial confinement: Variation of linear and nonlinear electrical response of molecules in the bond dissociation processes Wojciech Bartkowiak*, Paweł Lipkowski, and Marta Chołuj Department of Physical and Quantum Chemistry, Wrocław University of Science and Technology, Wrocław, Poland ⁎ Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. The spatial confinement models and methodology of quantum chemical calculations 3. Results and discussion 4. Concluding remarks Acknowledgment References

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Abstract The (hyper)polarizabilities (α, β and γ) of lithium hydride (LiH) as a function of internuclear distance in the free space and under the spatial confinement were investigated through quantum chemical calculations. The spatial confinement effects were simulated using a two-dimensional harmonic oscillator potential, expected to capture the exchange repulsion within the confining environments of cylindrical symmetry. The most important observation is connected with the fact that the (hyper)polarizabilities of LiH during the dissociation process (at large internuclear distances) are remarkable enhanced. Moreover, we found that when the bond length of the LiH molecule embedded in an external repulsive potential is strongly stretched (in the intermediate dissociation region), the nonlinear optical response strength may increase by several orders of magnitude. Additionally, the obtained results for LiH were confronted with the predictions, without spatial confinement, for the H2 molecule received by other authors.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.009

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1. Introduction Traditionally (in the textbooks and original scientific papers), the dipole moment (μ) and energy (E) of molecular system in presence of an external field (F) is expressed as power series in the following way1–3: μα ðFÞ ¼ μα + ααβ Fβ + ð1=2Þ βαβγ Fβ Fγ + ð1=6Þγ αβγδ Fβ Fγ Fδ + …,

(1)

EðFÞ ¼ E0  μα Fα  ð1=2Þ ααβ Fα Fβ  ð1=6Þβαβγ Fα Fβ Fγ ð1=24Þγ αβγδ Fα Fβ Fγ Fδ + …:

(2)

Through these expressions, the components of linear polarizability as well as first- and second-order hyperpolarizability tensors are defined: ααβ, βαβγ and γ αβγδ. The comparison of the Eqs. (1) and (2) with the Taylor expansion shows that the static electric dipole moment is the first derivative of the energy with respect to the electric field, the dipole polarizability is given by the second derivative of the energy and the first derivative of the dipole moment with respect to the electric field, and so on. Of course, in general the external field (F) may be non-uniform, there would be additional terms from higher multipoles in the energy expansion. The molecular hyperpolarizabilities play central role in the field of nonlinear optics and constitute the measure of a response of chemical systems to the external electric fields generated by lasers. From a theoretical point of view, knowledge of these quantities allows quantitative interpretation many experimental measurements. Even though that the correct calculations of hyperpolarizabilities still remain a challenge for quantum chemical methods, during the past decades, because of development of theoretical formalisms, software and hardware substantial progress has been made in this field. We refer the interested reader to available recent reviews and books.1–5 The investigations of nonlinear optical (NLO) properties of molecular systems including environmental effects have been conducted in our research group from many years. The first paper devoted to these issues was published in 1996.6 From that time our results of theoretical and experimental studies have been presented in about 100 articles released in different journals devoted to physical chemistry, chemical physics as well as quantum chemistry. Recently, the dominant direction of our quantum chemical investigation is connected with the influence of the spatial confinement effects on the NLO response of molecular systems (atoms, anions, molecules and molecular complexes). Recently, the theoretical results received on this

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field were summarized in chapter of the book.7 In this work, the directions of changes of the molecular dipole moments and (hyper)polarizabilities as a function of the strength of spatial confinement for extended collection of the molecular systems where presented. In particular, it was reported that the values of effective linear polarizability (α) as well as second hyperpolarizability (γ) always decrease together with the increasing strength of orbital compression. In contrast, the behavior of dipole moment (μ) and first hyperpolarizability (β) differs depending on the topology of the confining environment and the system under consideration. It should be noticed that the beginnings of our investigations in this field were inspired by papers of Sadlej et al.8 These authors have studied the atom of beryllium confined by helium clusters of different symmetry within supermolecular approach and showed that the spatial confinement leads to the orbital compression effect which causes α and γ to diminish. Moreover, based on the similar models of the confinement it was possible to explain the color of the tetramethylammonium auride crystals containing the auride ion.9 This simple model (helium atoms as chemical environment) was also applied by some of us for description of the confinement effects on the NLO properties of the series of small molecules (see, e.g., Refs. 10–12). It is well established based on theoretical and experimental considerations that the structural changes of different type of molecules, without interference into chemical composition of molecules, lead to significant modification of their NLO properties. In this context, the important examples are the bond-length alternation (BLA), internal rotation between structural fragments of the molecular systems as well as stretched and compressing chemical bonds.13–21 These changes at the molecular level (and consequently in macroscopic materials) can be induced by applied the external electric fields, spatial confinement (external pressure) as well as effects of chemical environment (solvent effects).14,15,21–26 The tuning of the NLO properties of molecular systems based on above observations and paradigms still seems to be a very promising feature for possible applications in the field of nonlinear optics. The present contribution provides a theoretical description of the (hyper)polarizabilities of the LiH molecule embedded in a two-dimensional harmonic oscillator potential, expected to capture the exchange repulsion within the confining environments of cylindrical symmetry. The most important aspect of this work was to explore the bond-length dependence of the investigated molecular quantities, for both free and spatially restricted of LiH. In many contexts the present work is inspired our previous investigations related to behavior of the two-photon absorption (TPA) response

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(resonance phenomenon) of LiH during the dissociation process.21 The bond-length dependence of the TPA strength including spatial confinement was analyzed for the first time in the scientific literature.21 The obtained results, among others, provide evidence that at large internuclear distances the TPA response of LiH may be significantly enhanced and this effect is much more pronounced upon embedding of the LiH molecule in an external confining potential. More precisely, when the bond length of the LiH molecule embedded in an external potential is strongly stretched, the TPA strength could increase by several orders of magnitude. This finding was both interesting and unexpected.

2. The spatial confinement models and methodology of quantum chemical calculations The box models (with hard and soft boundaries) of confinement have been widely used in the past and in present, particularly to survey the effect of spatial limitation on electronic structure of molecular systems and to mimic high pressure conditions. This approach has became popular in the field of theoretical physics and chemistry since the classic paper by Michels et al.27 in 1937 in which the confinement of one-electron atom in spherical box was considered in order to describe local pressure effect on electronic structure and polarizability of this object. However, the pioneering work in this field was published by Wigner in 1934.28 Following these ideas, there have been many ways of describing the molecular systems in the spatial confining environments (in the cage potential). As pointed out by Karwowski, basically every external potential, having different spatial symmetry, may be considered as confining potential.29 From a formal point of view, the effect of spatial confinement is usually modeled by including analytical external potential in the clamped nuclei Hamiltonian of b 0 ): an isolated n-electron systems (H   b¼H b 0 + Vb c ! H r ,

(3) ! is the external confining potential, being the sum of where Vb c r one-electron operators: ! Xn   bc ! V Vb c r ¼ (4) ri , i¼1

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! ! ! !  ! where r ¼ r 1 , r 2 , …, r n means the position of all electrons and r i ¼ ðxi , yi , zi Þ denotes the position of i-th electron. This repulsive potential acts directly only on electrons. Hence, the effect of confinement on nuclei is accounted for indirectly by its effect on the electrons. The following analytical functionals, among others, are usually applied: impenetrable or penetrable spherical and spheroidal boxes, the confining harmonic oscillator, Morse potential, the linear combination of shifted Gaussian functions of spherical symmetry and the effective potentials connected with the plasma environment.7,26,30–35 It should be emphasized that such repulsive potentials accounts mainly for the effects arising from the Pauli exclusion principle (valence repulsion) (see, e.g., Refs. 12, 30, 35–39 and references therein). Therefore, this way of the spatial confinement represented by the external potentials may correspond to the case in which the studied molecular system interacts with chemically and electronically inert (non-polarizable) environment. It should be noticed that in real situations, the picture of intermolecular interactions in extreme conditions is more complicated, but this approximation is useful for description essentials aspect of the spatial confinement effects. It was shown, based on theoretical considerations, that the spatial restriction can significantly modify various chemical and physical properties of molecular systems. For example, the shape (curvature) of potential energy curves (in the ground state and in the excited states) of the molecules are noticeably change in the presence of spatial confinement.34,40–44 The most important result is the shift toward smaller values of the equilibrium distance (in the ground state): this means that the molecule contracts due to the increase in the density of the electron cloud. These facts are reflected in the change of the dissociation limits, binding energies and spectroscopic parameters.34,40–46 Moreover, the spatial confinement leads to increase of the total energy (because of the kinetic energy of electrons rapidly increases upon spatial confinement) and the separation of HOMO and LUMO orbitals. These findings have important meaning for chemical reactivity of confined systems (also under high pressure).33,37 The changes of the structure and energetics of the hydrogen bonds in molecular complexes are also observed in such extreme conditions.45,47–49 As was mentioned in the introduction, the spatial confinement, in the form of analytical potential, causes also significant changes the linear and nonlinear optical response of atoms and molecules.7 In this place, it should be noticed that there are alternative approaches to the description of the properties of molecular matter under extreme conditions (directly related to spatial confinement effects) on the molecular structures. Most of them, namely the

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extreme pressure polarizable continuum model,26,50 the generalized force-modified potential energy surface,51 and the hydrostatic compression force field,52 approach has been recently reviewed by Stauch.53 In this work, similarly as in the case of some our earlier studies, the two-dimensional harmonic oscillator potential in the form: !  1 ! 2 1   Vb c r i ¼ ω2 r i ¼ ω2 x2i + y2i 2 2

(5)

was applied to mimic the effect of orbital compression. In Eq. (5) the ω parameter, which is the frequency of oscillator with the mass equal to me, describes the strength of spatial confinement. According with the aims of this work, we investigate spatial confinement effects on the (hyper)polarizabilities of the LiH molecule in its ground state (X1Σ+) and in the situation, when the internuclear separation is changed. The curves of these quantities for LiH where generated by repeating the calculations at a number of internuclear separation distances. The confinement effect was modeled by the external cylindrical harmonic oscillator potential (Eq. 5). In all computations, the principal axis of the harmonic oscillator potential overlaps with the molecular axis of LiH, assumed to be the z-axis. As mentioned before, the ω parameter controls the strength of spatial confinement, which is obtained by changing the curvature of the harmonic potential. Because of its symmetry, the cylindrical harmonic oscillator potential can be used as simplified representation of the nanotube-like confining cages. The bond-length dependence of the molecular (hyper)polarizabilities (in particular, the hyperpolarizabilities in spatial restriction) was analyzed, to the best of our knowledge, for the first time in the literature, using the highly accurate multiconfiguration self-consistent field (MCSCF) method and response function formalism,4 as implemented in the Dalton package.54 The computations were performed in C2v symmetry, using the ANO-L basis set.55 All electrons were correlated and all orbitals included in the active space during the MCSCF calculations.

3. Results and discussion At the beginning, to clarify the discussion presented in this work, it would be advisable to show what we know about the linear and NLO response of molecular systems (without spatial confinement) in the context of internuclear distance. Our discussion will be restricted to ground state of diatomic molecules (with stable covalent bond).

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A systematic investigations of the behavior of multipole moments and static (hyper)polarizabilities of the H2 molecule as a function of the internuclear distance (ranging from 0.567 to 10.0 a.u.) were conducted by Miliordos and Hunt20 (see also references therein). The calculations were performed at the CISD/d-aug-cc-pV6Z level of theory for all of the independent components of the property tensors. It should be noticed that the behavior of α and γ is not monotonic as a function of the H-H distance. The increasing of the H-H separation (from equilibrium distance re ¼ 1.400 a.u.) leads to increase values of α and γ. The maximum values are reached at the H-H distances away from equilibrium (αzz ¼ 18.2882 a.u. for r ¼ 3.400 a.u and γ zzzz ¼ 15,739 a.u. for r ¼ 4.250 a.u., where αzz and γ zzzz are longitudinal components of α and γ tensors, respectively). It means that the maximum values are larger by an order of magnitude in the case of polarizability and two orders in magnitude in the case of second hyperpolarizability in comparison with the equilibrium distance. In contrast, on shorter the H-H separations (r < re) the values of α and γ are significantly reduced. A similar observations, for example, comes from theoretical investigations of Kołos nad Wolniewicz (for the polarizability of H2),56 Bishop and Pipin57 (for second hyperpolarizability for H2 and D2), Lo and Klobukowski (for the polarizability of LiH),43 Maroulis (for the (hyper)polarizability of F2),58 Dykstra (for the (hyper)polarizability of HF),59 Nakano et al.60,61 (for the (hyper)polarizabilities of H2). The publications58,59 consider the problem not far from equilibrium distance. These cited investigations, at least from theoretical point of view, show that it is possible to influence on the electrical response of molecules by manipulation of the internuclear distance. Our knowledge about behavior (in particular, in presence of the spatial restriction) of the liner and NLO properties (molecular (hyper)polarizabilities) of the diatomic polar molecules as a function of the internuclear separation is very restricted. Except of our investigations of the TPA cross section of LiH,21 the systematic studies, in this context, of the linear electrical response of the same molecule were conducted by Lo and Klobukowski.43 It should be noticed that these authors applied the same type of the model potential (Eq. 5) for simulations of the spatial confinement effect. The calculations in this context for the free LiH molecule were also made by Nakano et al.60 I should be noticed that the linear polarizabilities and second hyperpolarizabilities of atoms and molecules (forming the strong covalent bond) in the ground states at the equilibrium distance are always reduced in the presence of the spatial confinement.7 Of course, this is also true for the LiH molecule. As was mentioned above, the most extended theoretical

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investigations (and also interpretation) of the linear electrical response of the LiH molecule subjected to the cylindrical harmonic confining potential were presented in the work of Lo and Klobukowski.43 What is important from our point of view, in this work the behavior of the longitudinal component (αzz) of the polarizability tensor (the largest components of α tensor due to symmetry of the investigated molecule) as a function of the internuclear distance was analyzed in the case of the free and spatially confined of LiH. Hence, the results of calculations presented here, for αzz ore not strictly original. However, the numerical results obtained in the cited paper may be treated as the reference for the computational methodology applied in our investigations. Dependence of the αzz of LiH on the internuclear distance is shown in Fig. 1. The obtained picture is in good agreement with the results published in Ref. 43 In short, the behavior of αzz as a function of the Li-H distance (r) is not monotonic and the maximum value (457.12 a.u.) of this quantity is obtained at r  7.0 a.u. in the case of LiH in the free space. The value of αzz at re ¼ 3.015 a.u. (the experimental value) is equal to 25.66 a.u. The influence of the confining potential on αzz is very significant. In opposite to the equilibrium distance, αzz is strongly enhancement by the presence of the spatial confinement at larger internuclear separations. In the case of the strongest ω=0.0

ω=0.1

ω=0.2

10

12

1000 900 800 700

αzz [a.u.]

600 500 400 300 200 100 0

0

2

4

6

8

14

16

18

20

r [a.u.]

Fig. 1 Polarizability of the free and spatially confined LiH molecule as a function of the internuclear distance (r).

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confinement applied here (ω ¼ 0.2 a.u.), the maximum value of αzz (975.20 a.u.) is reached at r ¼ 8.5 a.u. In general, the increasing of the strength of spatial confinement leads to the shift of the maximum for αzz for larger Li-H distances. Lo and Klobukowski indicated that the αzz is enhanced by the polarization of the 2σ molecular orbital of the LiH molecule. It is interesting that behavior of the longitudinal components (the γ zzzz component) of γ tensor as a function of the internuclear separation is significantly different in the case of the isolated H2 and LiH. For H2 one maximum (far from equilibrium distance, see discussion above) of γ is observed for the isolated molecules.20,57,59,60 The situation for LiH is completely different. It indicates that NLO response of simple polar and nonpolar molecules on the internuclear distance far from the equilibrium value may be substantially different. Inspection of Fig. 2 shows that the γ values increase significantly and reaches the maximum value. Then for the even bigger Li-H separation the decreasing values of γ are observed and this quantity reached minimum. Noticed changes of the values of γ are in orders of magnitude. One of the most important finding is the fact that in the intermediate dissociation region the change in the sign of γ is observed. Results of our calculations are in the line in the predictions presented by Nakano et al.59 In this study, the variations in the longitudinal ω=0.0

ω=0.1

ω=0.2

8.0E+07 6.0E+07 4.0E+07 2.0E+07

γzzzz [a.u.]

0.0E+00

0

2

4

6

8

10

12

14

16

18

20

-2.0E+07 -4.0E+07 -6.0E+07 -8.0E+07 -1.0E+08 -1.2E+08

r [a.u.]

Fig. 2 Second hyperpolarizability of the free and spatially confined LiH molecule as a function of the internuclear distance (r).

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component of γ in the bond-dissociation processes of the LiH molecule in the free space were investigated using high corelated methods (UCCSD and RCCSD, for example). In particularly, the authors observed the remarkable electron correlation dependency of the variation in γ in the intermediate and large bond-distance region, corresponding to intermediate and strong correlation regime, respectively. Additionally, in the case of the H2 molecule it has been suggested that the polarization in the ionic structure contributes primarily to the enhancement of (hyper)polarizability in the intermediate dissociation region, while the polarization in the diradical structure causes the decrease of the (hyper)polarizability at a large internuclear distance.60 These studies strong support in the interpretation of our predictions. In the comparison with α, the influence of confining potential on γ is more drastic (see Fig. 2). Especially, it is seen for the strongest spatial confinement (ω ¼ 0.2 a.u.). In this case, the maximum value of γ (at r  7.0 a.u.) is four order of magnitude larger in comparison with their value at the equilibrium distance (γ for re is equal to 5401 a.u). Even more drastic changes for the negative values of γ are seen. The shift of the extreme points of the curves (presented in Fig. 2) toward larger values of the Li-H separation with the increasing of the confinement strength is also observed. For the sake of clarity, we present the results of calculation for the longitudinal component (the βzzz component) of the first hyperpolarizability tensor (β) at the end. It is mainly connected with the fact that β is not equal to zero in the case of the noncentrosymmetric molecular systems—the LiH molecule meets this requirement. Hence, the discussion about β has not general nature as in the case of α and γ. In the context of our investigations, the first hyperpolarizability of LiH is rather specific case for obvious reasons. First, in the limit of dissociation, in opposite to α and γ, β goes to zero: the separated spherical atoms (Li and H) are the results of this process. Moreover, the first bonded excited state of the LiH exhibits the charge-transfer (CT) character. One of the most important paradigm in the molecular nonlinear optics states that the presence of the CT excited state determines, to a great degree, the NLO response of molecular systems described, in terms of quantity, by β. This is reflected in the two-state model commonly used, since the publication of Oudar and Chemla,62 for analyzing of a sources of the NLO activity of investigated molecules.24,25 In this model:

β

  μ2ge μe  μg E2ge

,

(6)

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where μg and μe are the dipole moments of ground and excited state (CT), μge is the transition dipole moment between two states and Ege means the excitation energy of the excited (CT) state. It should be noticed that meaning of this model is due to its simplicity and the quantities in Eq. (6) are easily extracted from the spectroscopic measurements or quantum chemical calculations. The similar two-state model was the key for the explanation of behavior of the TPA cross section of LiH as a function of the Li-H separation.21 Hence, the see through the prism of Eq. (6) on the results obtained here is justified. In Fig. 3, the variation of β of LiH with the internuclear separation is presented. A closed inspection of the presented plots allows one to conclude that the course of β curves is very similar to γ, as far as its quality is concerned. Here, we can also observe the maxima (extrema), which are shifted to the larger internuclear separations in the presence of the cylindrical harmonic confinement as well as inversion of the β sign is seen. However, as opposed to γ the only absolute values of β are of significance in the nonlinear optics. The strong influence of the spatial confinement on the values of β is also observed. What is more, this quantity exhibits the similar direction of changes as a function of the strength of spatial confinement. The quality interpretation of the presented results is quite simple on the basis of the two-level approximation ω=0.0

ω=0.1

ω=0.2

150000

100000

βzzz [a.u.]

50000

0

-50000

-100000

-150000

-200000

0

2

4

6

8

10

12

14

16

18

20

r [a.u.]

Fig. 3 First hyperpolarizability of the free and spatially confined LiH molecule as a function of the internuclear distance (r).

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(Eq. 6). In fact, this interpretation was made in our earlier study for the TPA cross section.21 In this work, the dependence of all spectroscopic parameters contributing to Eq. (6) on the bond length of LiH was shown (see Table 1 and Figure 2 in Ref. 21). It would not be appropriate to repeated the discussion, which was conducted in cited paper. However, it is visibly that the changes of the β values (and change of the sign) are mostly governed by variation of (μe  μg). The remaining quantities (μge and Ege) have less impact on the nature of the β changes in the function of internuclear separation. This statement applies to both free and spatially limited LiH. The similar simple interpretation for γ is very difficult. The few-level models based on perturbation theory for this quantity are more complicated than in the case of the first hyperpolarizability. However, it is rather clear that the CT excited state plays also a crucial role in the description of behavior of γ. This is important difference between LiH and the homonuclear molecules like H2. Hence, in this work we present numerical results as preliminary study (the pilot calculations) of the problem of behavior of γ as a function of the Li-H distance for the free molecule and especially for the molecule in the spatial confinement. We hope that in the future, it would be possible more deeper and many-side explanation in this picture of these behavior at the quantitative and qualitative level. Moreover, it would be possible to obtain more general conclusions, when extended collections of molecules will be taken into consideration.

4. Concluding remarks In this work, the consequences of the spatial confinement on the molecular (hyper)polarizabilities in the bond dissociation process were investigated based on quantum chemical calculations. The lithium hydride (LiH) molecule was examined as an important example of the model polar molecule. Additionally, the results obtained by other authors were also discussed (for the H2 and LiH molecule). The large and nonmonotonic changes of the (hyper)polarizabilities were observed due to the variation in the internuclear separation. As it was found that at distances larger than the equilibrium bond length (in the intermediate dissociation region) a substantial enhancement of α, β and γ might occur. In particular, it is visible when the bond length of the LiH molecule embedded in the external potential is strongly stretched. In this case, the linear and nonlinear optical response of the investigated molecule could increase by several orders of magnitude.

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Our results confirm predictions of the earlier works as well as shining a new light on the structure–property relationship in the context of molecular nonlinear optics. However, it should be remembered that that presented conclusions are based on numerical estimation and rather simple model of the confinement. Investigations in this direction should be in-depth in the future for more extended collections of polar and nonpolar molecules.

Acknowledgment We thank Dr. Robert Zale
sny for helping us in quantum chemical computations performed using the molecular electronic structure program (Dalton).

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nska-Wa¸z, D.; Diercksen, G. H. F.; Klobukowski, M. Quantum Chemistry of Confined Systems: Structure and Vibronic Spectra of a Confined Hydrogen Molecule. Chem. Phys. Lett. 2001, 349, 215–219. 41. Sarsa, A.; Alcaraz-Pelegrina, J. M.; Le Sech, C.; Cruz, S. A. Quantum Confinement of the Covalent Bond Beyond the Born-Oppenheimer Approximation. J. Phys. Chem. B 2013, 117, 7270–7276. 42. Labet, V.; Hoffmann, R.; Ashcroft, W. A Fresh Look at Dense Hydrogen Under Pressure. II. Chemical and Physical Models Aiding our Understanding of Evolving H-H Separations. J. Chem. Phys. 2012, 136, 074502. 43. Lo, J. M. H.; Klobukowski, M. Computational Studies of One-electron Properties of lithium Hydride in Confinement. Chem. Phys. 2006, 328, 132–138. 44. LeSar, R.; Herschbach, D. R. Electronic and Vibrational Properties of Molecules at High Pressures. Hydrogen Molecule in a Rigid Spherical Box. J. Phys. Chem. 1981, 85, 2798–2804. 45. Chołuj, M.; Luis, J. M.; Bartkowiak, W.; Zale
sny, R. Infrared Spectra of Hydrogen-Bonded Molecular Complexes under Spatial Confinement. Front. Chem. 2002, 9, 801426. 46. Zale
sny, R.; Go´ra, R. W.; Luis, J. M.; Bartkowiak, W. On the Particular Importance of Vibrational Contributions to the Static Electrical Properties of Model Linear Molecules under Spatial Confinement. Phys.Chem.Chem.Phys. 2015, 17, 21782–21786. 47. Lipkowski, P.; Kozłowska, J.; Roztoczy
nska, A.; Bartkowiak, W. Hydrogen-Bonded Complexes Upon Spatial Confinement: Structural and Energetic Aspects. Phys. Chem. Chem. Phys. 2014, 16, 1430–1440. 48. Roztoczy
nska, A.; Kozłowska, J.; Lipkowski, P.; Bartkowiak, W. Does the Spatial Confinement Influence the Electric Properties and Cooperative Effects of the Hydrogen Bonded Systems? HCN Chains as a Case Study. Chem. Phys. Lett. 2014, 608, 264–268. 49. Zale
sny, R.; Chołuj, M.; Kozłowska, J.; Bartkowiak, W.; Luis, J. M. Vibrational Nonlinear Optical Properties of Spatially Confined Weakly Bound Complexes. Phys. Chem. Chem. Phys. 2017, 19, 24276–24283. 50. Cammi, R. Quantum Chemistry at the High Pressures: The eXtreme Pressure Polarizable Continuum Model (XP-PCM). In Frontiers of Quantum Chemistry; Wo´jcik, M. J., Nakatsuji, H., Kirtman, B., Ozaki, Y., Eds.; Springer: Singapore, 2018. 





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51. Subramanian, G.; Mathew, N.; Leiding, J. A Generalized Force-Modified Potential Energy Surface for Mechanochemical Simulations. J. Chem. Phys. 2015, 143, 134109. 52. Stauch, T.; Chakraborty, R.; Head-Gordon, M. Quantum Chemical Modeling of Pressure-Induced Spin Crossover in Octahedral Metal-Ligand Complexes. Chem. Phys. Chem. 2019, 20, 2742–2747. 53. Stauch, T. Quantum Chemical Modeling of Molecules Under Pressure. Int. J. Quantum Chem. 2021, 121, e26208. 54. Dalton, a molecular electronic structure program, Release v2015.0 (2015), see http:// daltonprogram.org. 2015. 55. Widmark, P. O.; Malmqvist, P.A˚.; Roos, B. O. Density Matrix Averaged Atomic Natural Orbital (ANO) Basis Sets for Correlated Molecular Wave Functions. Theor. Chim. Acta 1990, 77, 291–306. 56. Kołos, W.; Wolniewicz, L. Polarizability of the Hydrogen Molecule. J. Chem. Phys. 1967, 46, 1426–1432. 57. Bishop, D. M.; Pipin, J. Ab Initio Study of Third-Order Nonlinear Optical Properties of the H2 and D2 Molecules. Phys. Rev. A Gen. Phys. 1987, 36, 2171–2181. 58. Maroulis, G. On the Bond-Length Dependence of the Static Electric Polarizability and Hyperpolarizability of F2. Chem. Phys. Lett. 2007, 442, 265–269. 59. Dykstra, C. E. Electrical Polarization in Diatomic Molecules. J. Chem. Educ. 1988, 65, 198–200. 60. Nakano, M.; Yamada, S.; Yamaguchi, K. Third-Order Nonlinear Optical Responses of Molecules in the Intermediate and Strong Correlation Regime: Variation of Second Hyperpolarizability in the Bond Dissociation. J. Comput. Methods Sci. Eng. 2004, 4, 677–701. 61. Nakano, M.; Nagao, H.; Yamaguchi, K. Many-electron Hyperpolarizability Density Analysis: Application to Dissociation Process of One-Dimensional H2S. Phys. Rev. A 1997, 55, 1503–1513. 62. Oudar, J. L.; Chemla, D. S. Hyperpolarizabilities of the Nitroanilines and Their Relations to the Excited State Dipole Moment. J. Chem. Phys. 1977, 66, 2664–2668.

CHAPTER TWELVE

Interparticle correlations and chemical bonding from physical side: Covalency vs atomicity and ionicity Ewa Brocławika, Maciej Fidrysiakb, Maciej Hendzelb, and Józef Spałekb,* a

Polish Academy of Arts and Sciences, Krako´w, Poland Institute of Theoretical Physics, Jagiellonian University, Krako´w, Poland *Corresponding author: e-mail address: [email protected] b

Contents 1. Motivation 2. Method: First and second quantization combined 3. True covalency, ionicity, atomicity: H2 molecule 3.1 Two-particle wave function and its basic properties—Analytic solution 3.2 Toward complementary characterization of the chemical bond: The case of H2 molecule 3.3 Atomicity as the onset of localization and consistent characterization of the chemical bond 4. Many-body covalency in related systems 4.1 LiH and HeH+ 4.2 Essential extension: The hydrogen bond—An outline 5. Outlook Acknowledgments References Further reading

352 354 359 359 361 363 366 366 369 371 372 372 373

Abstract In this chapter we reexamine the concept of covalency and ionicity on example of the simplest molecules. First, starting from the exact expression for the two-particle wave function in the case of H2 molecule within the Heitler–London model, we demonstrate an unphysical behavior of the covalency at large interatomic distance which, within standard definition, reaches the maximal value in the limit of separated atoms. Second, we correct this deficiency by introducing the concept of atomicity, with the help of which, we define the true (intrinsic) covalency, as well as retain the precise concept of ionicity. We connect the introduced atomicity to the onset of Mott–Hubbard

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.02.002

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localization, adopted here from the well-established notion in the condensed matter. The evolution from the molecular to atomic states develops rapidly with interatomic distance beyond the localization threshold. This brief overview is intended as pedagogical in nature, nonetheless is analyzed quantitatively for the case of H2 molecule. At the end, we outline a similar-type of model for the related case of the hydrogen bond on example of adenine–thymine pair. Methodologically, the approach is based on combining the first (wave mechanics) and second quantization into a single scheme of formal description.

1. Motivation The quantum mechanical concept of the chemical bond was introduced in the quantitative way on example of H2 by Heitler and London (HL)1 (see also related papers2–4). This theory was formulated only a year after the wave mechanics had been rigorously established by Schr€ odinger for the discrete states of single hydrogen atom and subsequently extended by Dirac5 by introducing the indistinguishability principle to the manyparticle wave function. The HL approach was based on what is now known as the Hartree–Fock two-particle wave function (strictly speaking, solely on its orbital part). This wave function is at present taken of the following spinsinglet form 1 ψ 12 ðr,rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½φ1 ðr1 Þφ2 ðr2 Þ + φ1 ðr2 Þφ2 ðr1 Þ½χ " ð1Þχ # ð2Þ  χ # ð1Þχ " ð2Þ: 2ð1 + S2 Þ (1)

The wave function originally contained hydrogen atomic wave functions φ1(r) ¼ φ(r  R1) and φ2(r) ¼ φ(r  R2) centered at H2 nuclei positions R1 and R2, respectively. They represent the hydrogen 1s orbital wave functions, whereas χ σ (i) are the corresponding spin functions with spin σ ¼ ", # and i ≡ Ri. This form (1) is a result of an ad hoc assumption and expresses the principle of antisymmetry with respect to transposition of the quantum numbers (i, σ); S ¼ hφ1jφ2i is the overlap of the functions. In general, the evolution of the wave function form in time is schematically summarized in Fig. 1 (see Fig. 1 caption for detailed description). The ansatz taken to define the form (1) was later extended by selecting more involved wave functions and has resulted in the full configuration interaction (FCI) formulation,6 involving also the most important virtually excited states. Nowadays, it comprises a whole discipline of advanced computational chemistry, including also the density functional theory

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Fig. 1 The evolution of the form of the two-particle wave function starting from original (Hartree–Fock I), through Heitler–London (originally without the spin part), followed by spin-coupled valence bond from Coulson and Fisher, as well as the configuration interaction form (the last represents linear combination of Slater determinants for possible occupancies). The bottom line composes our result obtained on the basis of our EDABI method.

(DFT) calculations, etc.7 It is this stage at which the unique calculations of Kołos and Wolniewicz resulted in practically exact results for the H2 molecule ground state.8,9 Such a procedure also requires, as a must, replacement of the original atomic wave functions {φi} in (1) by appropriate molecular orbitals {Φi}.4 Our approach begins from a different starting point.10–12 Namely, we combine the wave (1st quantization) and particle (2nd quantization) languages of electron states description, composing a single chemical bond (i.e., of two-electron states) and show how the ionicity arises in a natural manner and, what is crucial, the importance of the introduced concept of atomicity in the bonding state, so that the true (or intrinsic) covalency can be

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quantitatively defined. In this manner, an intrinsic inconsistency in the standard definition of covalency is removed. Furthermore, the full Hamiltonian with all two-particle interactions can be written in a compact form, as well as an accurate analytic expression for the two-particle wave function is explicitly provided. This formulation represents the exact solution of the HL model and constitutes a crucial step forward in a systematic analysis of the bonding properties of more complex molecular systems. In particular, the limiting situation of separate atoms is recovered correctly, i.e., when the state of indistinguishable (bound) electrons transform into that of their distinguishable (atomic) correspondents. Some of the possible extensions of our approach, discussed here in detail for H2 molecule, are briefly characterized at the end.

2. Method: First and second quantization combined The principal features of our original exact diagonalization ab initio (EDABI) method and interpretation of physical results are carried out here on example of H2 molecule. Thus the starting point are the electronic states of the two electrons that originate from parent hydrogen atoms, each placed in 1s atomic Slater state. We neglect a possibility of virtual electronic transitions to 2s and higher excited states and would like to study an adiabatic evolution of those atomic states into the two-particle molecular state (forming the bond) with the decreasing interatomic distance R. In the particle (second quantization) language, the situation can be described by starting from the field operator language in which the particles are represented by the field operator of the form ψ^σ ðrÞ ¼ w 1 ðrÞχ σ ð1Þ^a1σ + w 2 ðrÞχ σ ð2Þ^a2σ

(2)

in which the molecular (Wannier) orthogonalized and normalized wave functions {wiσ (r) ≡ wσ (r  Ri)}i¼1,2 are defined as w iσ ðrÞ ¼ β½φiσ ðrÞ  γφjσ ðrÞ ¼ β½ϕi ðrÞχ σ ðiÞ  γϕj ðrÞχ σ ðjÞ,

(3)

where β and γ are mixing coefficients for the state centered on site i with that centered on the neighboring site j. We should stress that the selection of the hybridized (Wannier) basis is a natural choice here as only then the anticommutators between the creation and annihilation operators have a universal form (see below).

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For the starting atomic states we select Slater-type orbitals ϕiσ ¼ pffiffiffiffiffiffiffiffiffiffi α3 =π exp½αjr  Ri jχ σ, in which α1 is an adjustable in correlated state size of the orbital (see below). The explicit form of the mixing coefficient is   1 1 1 + pffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ pffiffiffi 2 1  S2 1  S2

(4)

where γ 2 ≡ (1  β2)/β. One methodological remark is in place here. Namely, the form (2) of the field operator is approximate, as the sum over states {wi(r)} should, in principle cover all higher excited states, i.e., {ϕi(r)} should represent a complete set of hydrogen atomic states or the corresponding full basis of molecular states. For the sake of argument clarity the basis is limited to (2). However, to optimize the truncated energy of the interacting (correlated) two-particle state we minimize the ground state energy EG additionally with respect to α. In this way, the orbital size is readjusted in the resultant interacting (correlated) state. Note that incompleteness of the basis {wi(r)} in defining the field operator (2) means that we reduce here the problem to the HL type of model, albeit we amend it with the singleparticle basis optimization (through adjustment of α in the resultant state) and reanalyze the exact solution of the model in the second quantization representation (i.e., going systematically beyond the original Hartree– Fock-type solution). For the selected starting single-particle orthogonalized basis {wσ (r)} we construct the system Hamiltonian in the second-quantized representation which is ^¼ H

  ħ d 3 rψ^{σ ðrÞ  r2 + V ðrÞ ψ^σ ðrÞ 2m σ Z X 1 e2 + d3 rd3 r0 ψ^{σ ðrÞψ^{σ 0 ðr0 Þ ψ^ 0 ðrÞψ^σ ðr0 Þ 2 σσ 0 jr  r0 j σ XZ

(5)

where the first term represents the single-particle part containing both kinetic energy of electron and the original atomic potential V(r) coming from both protons. The second term expresses two-particle Coulomb repulsion. Upon inserting expression (2) for ψ^σ ðrÞ and the corresponding one for ψ^{σ ðrÞ≡ ½ψ^σ ðrÞ{ we obtain the full Hamiltonian in terms of creation   ^a{iσ and annihilation ð^aiσ Þ operators, that takes the following form

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X

X0

X

1 X0 K n^ n^ 0 2 ijσσ 0 ij iσ jσ iσ ijσ i     1 X0 H ^ ^ 1 1 X0 0 { {  J ij Si  Sj  n^i n^j + J ij ^ai" ^ai# a^j# ^aj" + H:c: 2 ij 4 2 ij   1 X0 + Vij ð^ niσ + n^jσ Þ ^a{iσ^ajσ + H:c: + Hionion , 2 ij Ei n^iσ +

tij ^a{iσ ^ajσ +

U i n^i" n^i# +

(6)

In this expression the first term of electrons,  expresses the atomic energy  { { { ^i ≡ ^ai" ^ai# , ^ai# ^ai" , 1 ð^ ^i# Þ are the particlewhereas n^iσ ≡ ^aiσ ^aiσ and S 2 ni"  n number- and spin-operators, respectively. The second term represents the hopping energy of electrons between the sites (with i 6¼ j); it expresses the energy contribution coming from resonant hops of individual carriers between the sites (prime means i ¼ 6 j). The third term represents the intraatomic part of the Coulomb repulsion between the electrons on the same atom with opposite spins, whereas the fourth contains the corresponding part of the intersite Coulomb (Hubbard) interaction. The next three terms represent direct (Heisenberg) exchange interaction, pair-particle hopping between the sites and the so-called correlated hopping, respectively. The last three terms are usually of lesser importance in correlated systems. The microscopic parameters are integrals containing the Slater single-particle Wannier functions in a standard manner (cf. Appendix in Ref. 13) Finally, Hionion expresses the classical Coulomb interaction between the nuclei (protons in the case of H2 molecule). Note also that the microscopic parameters Ea, tij, U, Kij, 0 JH ij , J ij , and Vij can be calculated analytically for 1s orbitals, as shown by Slater14 (see also Ref. 10); they will be evaluated explicitly here in the interacting (correlated) state (for i.e., readjusted orbital size). The next step in process of solving the Hamiltonian is the diagonalization of (6) in the Fock space for given of microscopic parameters. For that purpose, we select a set of trial orthogonal and normalized states for two electrons which in this case of H2 molecule are 8 { { > > > j1i ¼ ^a1" ^a2" j0i, > > { { > > > j2i ¼ ^a1# ^a2# j0i, > >   > > 1 { { { { > > p ffiffi ffi j3i ¼ ^ a + ^ a ^ a ^ a > 1# 2" j0i, > > 2 1" 2# <   1 { { { { (7) p ffiffi ffi ^ a  ^ a ^ a ^ a j4i ¼ > 1" 2# 1# 2" j0i, > 2 > > >   > 1 > { { { { > p ffiffi ffi > ^ a + ^ a ^ a ^ a j5i ¼ > 1" 1# 2# 2" j0i, > 2 > > >   > 1 > { { { { > : j6i ¼ pffiffiffi ^a1" ^a1#  ^a2# ^a2" j0i: 2

357

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 Note that for N sites with Ne electrons, we have

2N

 two-particle states,

Ne which is equal to six for N ¼ 2 sites and Ne ¼ 2 electrons. The diagonalized state thus contains three spin-triplet states (with the total-spin SZ component equal to SZ ¼ 1, 1, and 0, respectively) and three spin-singlet states. The basis is complete for this model with two 1s-type states (HL model) and for Hamiltonian (6) can be brought to the 6  6 matrix form, which splits into three 1  1 irreducible parts representing three separate spin-triplet states with eigenenergy λ1 ¼ λ2 ¼ λ3 ¼ E1 + E2 + K  J H ,

(8)

as well as 3  3 submatrix of mixed spin-singlet states, which is of the form 1 0 E + K + JH 2ðt + V Þ 0 C B 1 2E + J + U ðU 1  U 2 Þ C ^ ¼B H C, (9) B 2ðt + V Þ 2 A @ 1 0 2E + U  J H ðU 1  U 2 Þ 2 with Ea ≡ (E1 + E2)/2 and U ≡ (U1 + U2)/2. In the case of two identical atoms (H2 molecule) this matrix can be diagonalized analytically (since U1 ¼ U2 ¼ U), with the eigenvalues λ1 ¼ λ2 ¼ λ3 ¼ 2Ea + K  J for the states j1i ¼ jλ1i, j2i ¼ jλ2i, and j3i ¼ jλ3i and λ6 ¼ 2Ea + U  J for the state j6i ¼ jλ6i. However, the states j4i and j5i are intermixed and in the case E1 ¼ E2, U1 ¼ U2 (the case of H2 molecule) the eigenvalues can be written in an analytic form, namely15 λ4,5 ≡ λ ¼ 2Ea +

1 1 1 ðU + KÞ  ½ðU  KÞ2 + 16ðt + V Þ2 2 : 2 2

(10)

The corresponding two eigenstates, in turn, have the following explicit form jλ4,5 i ≡ jλ i ¼ ½2DðD  U  KÞ2 f4ðt + V Þj4i  ðD  U  KÞj5ig, 1

(11) 1

with D ≡ ½ðD  KÞ2 + 16ðt + V Þ2 2. The last two states are the most interesting here, because jλi is the ground state; all the remaining states are excited states. We see that the intersite spin-singlet states jλi have an admixture of symmetric ionic state j5i. This is understood that the bonding λ state for H2 has an admixture of the ionic states H+H and HH+. This fact will be interpreted further later. To complete the approach we sketch the whole procedure from numerical point of view. First, we take the

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Fig. 2 Flowchart of the EDABI method. The method is initialized by selection of a trial single-particle basis of wave functions (3), and subsequent diagonalization of the many-particle Hamiltonian (6). Optimization of the single-particle-state size leads to an explicit determination of the trial-wave function parameters, microscopic interaction and hopping parameters, as well as ground state energy, and explicit form of the many-particle wave functions (II), all in the correlated interacting state for a given interatomic distance R.

eigenvalue λ as our starting point and minimize this energy with respect to the Slater-orbital inverse size α, contained in the assumed functions {wi(r)}. This means that we iterate the procedure depicted in Fig. 2 until it converges for optimal α ¼ α0 and for a given interatomic distance. The optimal value of α ¼ α0(R) is subsequently the same for all the eigenstates λ1, λ2, λ3, λ4, λ5, and λ6. The ground state and the next five excited electron energies (for fixed R ¼ jRj) are exhibited in Fig. 3. Those results are for H2 molecule; the other examples, LiH and HeH+, are discussed briefly later. They are of limited absolute accuracy, but these results are only a starting point to a deeper analysis discussed next.

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359

Fig. 3 The lowest energy levels composed of three singlet and three triplet states, with the marked Mott regime and associated with it strong-correlation limit (shaded area). The scale U  K represents the effective repulsive Coulomb interaction between electrons, i.e., the HOMO–LUMO splitting. The atomic character of the states shows up gradually becoming atomic with the increasing interatomic distance R.

3. True covalency, ionicity, atomicity: H2 molecule 3.1 Two-particle wave function and its basic properties—Analytic solution Further analysis of the bonding requires a calculation of the two-particle wave function in the position (Schr€ odinger) representation. For that purpose, we quote a general theorem about the N-particle wave function representation in the Fock space (see, e.g., Ref. 16) Z 1 jψ N i ¼ pffiffiffiffiffiffi d3 r1 d3 rN ψ N ðr1 , …, rN Þψ^{1 ðr1 Þ…ψ^{M ðrN Þj0i, (12) N! where ψ N ðr1 , …, rN Þ is the desired wave function for state N in the position representation and j0i represents the vacuum state in the Fock space. We can extract the wave function by reversing the above relation. This leads to the following general explicit expression 1 ψ α ðr1 , …, rN Þ ¼ pffiffiffiffiffiffi h0jψ^1 ðr1 Þ…ψ^N ðrN Þjλα i, N!

(13)

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where jλi is the eigenstate, for which the wave function ψ σ is explicitly determined by replacing jψ Ni with the corresponding Fock-space eigenstate jλαi. For the spin-conserving interaction here, σ ≡ ðσ 1 , …, σ N Þ is fixed N-spin configuration. In what follows we discuss the situation for N ¼ 2 and σ ≡ (σ 1, σ 2) spin-singlet ground state configuration for the H2 molecule. Namely, the wave function is obtained by utilizing the expression (2) for the field operators and the anticommutation relations of the creation and annihilation operators, namely ( ^aiσ ^a{jσ 0 + ^ajσ 0 ^a{iσ ¼ δij δσσ 0 , (14) ^aiσ ^ajσ 0 + ^ajσ 0 ^aiσ ¼ 0: Additionally, we make use of the vacuum-state property ^aiσ j0i ≡ 0 for every state jii (cf. Eq. 7). In effect, the two-particle wave function representing a single bond in the ground state has the following explicit form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðt + V Þ 1 DU +K Ψ0 ðr1 , r2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ψcov ðr1 ,r2 Þ  Ψion ðr1 , r2 Þ, 2 2D 2DðD  U + KÞ (15) where the covalent (Ψcov) and ionic (Ψion) components are Ψcov ðr1 , r2 Þ ¼ ½w1 ðr1 Þw2 ðr2 Þ + w1 ðr2 Þw2 ðr1 Þ½χ " ðr1 Þχ # ðr2 Þ  χ # ðr1 Þχ " ðr2 Þ, (16) Ψion ðr1 ,r2 Þ ¼ ½w1 ðr1 Þw1 ðr2 Þ + w2 ðr1 Þw2 ðr2 Þ½χ " ðr1 Þχ # ðr2 Þ  χ # ðr1 Þχ " ðr2 Þ: (17) From Eqs. (16) and (17) it is evident that the part (16) represents the covalent (cov) part, whereas (17) represents the combination of the two ionic (ion) configurations of the electrons. Also, note that those wave functions have an involved form; spatially symmetric and spin antisymmetric, as it should be for the spin-singlet state. Parenthetically, these are not single Slater determinant states. Also, their factorization into space and spin parts reflects the fundamental properly that they represent separate orbital and internal (spin) symmetries (cf. Ref. 17). The coefficients before the term contain all interactions present in the two-particle single-orbital quantum state. The wave function (15) represents the exact wave function for the HL model, with additional readjustment of orbital size in the resultant (correlated) eigenstate.

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In order to interpret the wave function (15) in terms of the original Slater (atomic) wave functions ψ i(r) ≡ ψ(r Ri), we make use of transformation (3) and obtain



ψ 0 ðr1 , r2 Þ ¼ Cβ2 1 + γ 2  2γIβ2 ϕatcov ðr1 , r2 Þ



+ Iβ2 1  γ 2  2γCβ2 ϕation ðr1 , r2 Þ (18) at at ~ cov + I~ϕion , ≡ Cϕ where the two-particle functions ϕatcov ðr1 , r2 Þ and ϕation ðr1 , r2 Þ have the same formal expressions as (16) and (17), respectively, but with molecular functions {wi(r)} being replaced by atomic orbitals {φi(r)}. This last expression coincides with the spin-valence bond wave function, except that the postulated coefficients A and (1-A) before the component functions (cf. Fig. 1) are here calculated microscopically. Additionally, as said above, the Slater-orbital size is optimized in the resultant (interacting) ground state. Therefore, one can say that the presented solution exemplifies a direct and exact solution of the original formulation of HL model with subsequent optimization of the initial atomic states. More importantly, the intrinsic covalent bonding (see below) will be defined as a contribution calculated beyond that obtained from HL approach. Achieving these goals is the principal aim of the article so far. Note that the solution in its fully analytic form, but requires the utilization of both first and second quantization (wave and particle) aspects of the problem. In the following subsection, we interpret in detail the obtained results and revise the covalency concept.

3.2 Toward complementary characterization of the chemical bond: The case of H2 molecule The wave function (18) reduces to the HL form after taking I~ ≡ 0 and neglecting the spin part, as well as disregarding the readjustment the orbital size in the correlated state. But, most importantly, neglecting to mix the Slater orbitals, i.e., putting in Eq. (3) β ≡ 1, and γ ≡ 0. Such an assumption leads to the two-particle wave function (18) in terms of atomic Slater orbitals and without bare covalency (single-particle mixing).18 This leaves us with the ~ in (18) describes the real principal question to what extent the factor C at covalency, as the ϕcov ðr1 , r2 Þin the original HL approach describes the whole covalency. The fact that it is not the whole story, even when γ 6¼ 0, can be demonstrated explicitly by taking limR!∞ γ cov ¼ limγ!0 γ cov ¼ 1. This is a clearly unphysical result. The reason why it is so is coded in choosing the

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Fig. 4 Two-particle covalency vs corresponding ionicity for H2 molecule, calculated within EDABI method and compared with the results of Ref. 19. Shaded regime marks a gradual evolution toward atomicity, as determined from the Mott–Hubbard criterion (see in the main text). The vertical dotted line marks equilibrium interatomic distance, whereas the horizontal dotted lines illustrate the dominant character of the covalency in that state (with the ratio r ¼ 1.43  2 : 1).

symmetric form of ψ atcov ðr1 , r2 Þ, which amounts to selecting that function in the form for indistinguishable particles, whereas in the limit R ! ∞ the electrons are located on separated atoms and hence are clearly distinguishable in the quantum mechanical sense. The unphysical behavior of the covalency factor γ cov ≡ γ cov(R) is exhibited explicitly in Fig. 4. This behavior is quite astonishing in view of the fact that the same form of postulated (ad hoc) wave function (15) provides a good semiquantitative estimate of the binding energy and the bond length. In retrospect, the definition of covalency in the HL fashion may be expected as insufficient, since it contains as factors the pure atomic functions, which in turn, overestimate the atomicity mixed up with the covalency, as elaborated next. In order to remove this inconsistency, appearing practically in every approach of selecting the two-particle form based on the indistinguishability of the particles, irrespective of the interatomic distance R, we have the following bold proposal. Namely, to define intrinsic or true covalency we extract from the covalency γ cov in (18) (or, equivalently in (15)) the part γ cov taken in the limit γ ¼ 0. This is carried out for each R. This step does not

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mean that we mix up indistinguishable and distinguishable states, as it would be principally incorrect. It merely helps to define the intrinsic covalency (and degree of atomicity) in any nominally covalent system. To summarize, we formally define the true covalency, atomicity, and ionicity for each R as ~ 2  jAj ~2 jCj , ~ 2 + jI~j2 jCj jI~j2 ionicity: γ ion ≡ , ~ 2 + jI~j2 jCj

covalency: γ cov ≡

~2 jAj atomicity: γ at ≡ : ~ 2 + jI~j2 jCj

(19) (20) (21)

Note that the sum of contributions (two-particle probabilities) is equal to unity. The atomicity is incorporated on the classical level (through the probability). Next, we incorporate quantitatively the introduced characteristics into the interpretation scheme, but first relate the atomicity to the Mott–Hubbard localization onset, the latter transferred here from the condensed-matter states and adopted to the molecular (finite-size) systems.

3.3 Atomicity as the onset of localization and consistent characterization of the chemical bond In general, the atomicity here is purposefully associated with the Mott– Hubbard localization, as it provides a physical rationale behind our concept. The localization on parent atoms of electrons in delocalized (band or other) states takes place under a gradually increasing repulsive interaction among the involved particles. In other words, it represents atomization of particles from collective states in lattice systems experiencing an increasing magnitude of repulsive interaction between them. This can be achieved either by increasing the interatomic distance and/or decreasing particle density in the system. In our case here the atomization can be gradual with the increasing interatomic distance. To elucidate how this happens we relate it to the well known Mott20 and Hubbard21 criteria for onsets of localized (atomic) behavior of electrons. This is because, we would like to interpret the evolution of molecular H2 (electron-paired) state into individual singly occupied (atomic) as a gradual process of the Mott–Hubbard localization. Namely, we define the Hubbard and Mott onset criteria adapted to the present situation as

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2jt + V j 1 1 ¼ 1, or n1=d ≡ ’ 0:5, c α U K α0 RMott

(22)

respectively, where α1 0 is the readjusted orbital size at R ¼ RMott (note, the criteria define, explicitly RMott). The first of them means that, for R ¼ RMott, the kinetic (hopping) energy, arising from (6) is equal to the interaction energy. That is, for R < RMott the ratio is greater than unity, whereas for R > RMott it is smaller than unity and approaches relatively rapidly zero with R ! ∞ (beyond RMott). Physically, the kinetic energy predominates in the former case; this hopping of electrons, in practice, gets gradually frozen out on atoms as R increases beyond RMott. On the other hand, the Mott criterion expresses the localization onset when the diameter of the renormalized orbital (2α1 0 ) crosses the value of interatomic distance; this happens at R ’ RMott. Quasiclassicaly, the latter criterion means that the collective electron behavior is established when their orbitals start overlapping strongly. To visualize this physical and formal reasoning we have plotted in Fig. 5 the left parts of (22) vs R, as well as have marked explicitly their values at

Fig. 5 Hubbard (orange) and Mott (blue) characteristics of atomicity vs interatomic distance R (dashed horizontal lines). The dots mark the points corresponding to Hubbard and Mott criteria. The vertical dotted line marks the onset of Mottness at RMott. The inset: R dependence of the orbital size of the renormalized atomic wave functions composing the molecular (Wannier) single-particle states. The dotted line marks the equilibrium distance Rbond.

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R ¼ RMott. The blue shaded area may be called the Mott regime. Additionally, in the inset we display the R dependence of the readjusted orbital size α1 0 ; it ˚ upon entering the Mott approaches rapidly the atomic value a0 ¼ 0.53 A regime. This feature illustrates the rate at which the atomic states are being established with the increasing R beyond RMott. The connection of the above criteria with the true covalency (γcov), atomicity (γat), and ionicity (γion) is shown explicitly in Fig. 6, where the R dependence of those quantities is drawn. Quite remarkably, at R ¼ RMott the true covalency and atomicity acquire the same value, what illustrates decidedly that the point RMott expresses a crossover point from covalency to atomicity-dominated regime. In the complementary regime R ! 0, the true covalency and ionicity (atom double occupancy) gradually coalesce. Both figures provide a combined picture in accord with our physical intuition. Simply, the criteria depicted in Figs. 5 and 6 are mutually consistent and complementary to each other. As we concluded in Ref. 11: This agreement leads to the conclusion that the introduced entities (19)–(21) and (22) are not only relevant in condensed-matter (extended) systems, but also appear as a crucial incipient feature in molecular systems. Such characterization is possible only with introducing

Fig. 6 Intrinsic properties of the chemical bond: atomicity (green), true covalency (blue), and ionicity (red), all as a function of interatomic distance R. They represent the relative weights in the total two-particle wave function. In the R ! 0 limit the atomicity practically disappears and is the only contribution in the separate-atom limit R !∞. The solid circle defines the onset of localization effects (Mottness) due to interelectronic correlations. If the atomicity is disregarded, the covalency exhibits a drastic nonphysical behavior with increasing R > Rbond. The figure illustrates a systematic evolution of molecular states into separate atoms and vice versa, formation of molecular states out of separate atoms. The Slater states have a renormalized size α1 αB.

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microscopically derived two-particle wave function (15) (or 18) as the proper characteristic of a single bond, which, after all, is composed of electron pair. We can illustrate further the onset of partial electron localization by calculating directly the density of electrons in the ground state from the formula taken from second quantization scheme, i.e. nσ ðrÞ ¼ hλ5 jψ^{σ ðrÞψ^σ ðrÞjλ5 i:

(23)

Note that nσ (r) represents the electron density with spin σ; it is related to the total particle density by n(r) ¼ 2nσ (r) in the spin-singlet state. This density represents complementary quantity to the probability density jψ 0(r1, r2)j2 in the sense that the former represents the physical-particle-occupancy density profile of nσ (r) that has been displayed in the panel in Fig. 7 for char1 acteristic interatomic distances R ¼ Rbond ¼ 1:43a1 0 , R ¼ RMott ¼ 2:3a0 , and R ¼ 4a1 0 . The parts centered at the nuclei are practically disjoint for R ≳ 4a0. For comparison, the results from the valence bond (VB) approach have been also displayed. In both approaches the behavior of the density profiles with the increasing distance is qualitatively similar. The reason is that in both cases the atomicity part has not been excluded. From this point of view, a direct definition (21) of atomicity is the only explicit characteristic.

4. Many-body covalency in related systems 4.1 LiH and HeH+ One can apply the same method of approach (EDABI) to other simple molecular systems. In Fig. 8A and B we have plotted the ground state energy vs R for HeH+ and LiH and compare is with the results obtained from other methods. As one can see, the results from different methods converge relatively quickly, even before reaching the Mottness (shaded) asymptotic regime. This trend is also observed in the R dependence of the size of orbitals in those two systems, as depicted in Figs. 9A and B. The 1s orbital size approaches very rapidly the asymptotic (atomic) limiting value in HeH+, whereas the second, 2s electron, is responsible for the main part of the bonding with the increasing R beyond Rbond. In LiH the situation is different as with the increasing R the atomicity is enhanced and is signaling a robust formation of an almost purely ionic state. Note that the extraction of the atomicity requires a further analysis in this case, as we have to introduce a separate atomicity factor for 1s and 2s original states. A detailed analysis, as well as the results depicted in Fig. 10 are of numerical character only,

1 1 Fig. 7 Electron density nσ (r) for different interatomic distances: R ¼ 1:43a1 0 , R ¼ 2:3a0 , and R ¼ 4a0 . The parts centered at nuclei are practically disjoint for R≳4a0, illustrating the robustness of atomic behavior in that situation. This density contains also the double-occupancy (ionicity) contribution which is becoming rapidly negligible with the increasing R beyond RMott. The evolution toward the atomic states beyond RMott is more rapid in the EDABI approach, even so the Heitler–London wave function overestimates the atomicity, as it is composed of bare atomic wave functions.

Fig. 8 The HeH+ and LiH binding energies vs relative interatomic distance, obtained using EDABI method and compared with restricted Hartree–Fock (RHF) and full configuration interaction (full CI) approach taking the minimal basis set for both calculations. a0 ¼ 0.53 Å is the Bohr radius.

Fig. 9 Renormalized 1s and 2s orbitals size α1 (in Bohr units a0) vs relative interatomic distance for the H2 molecule. Note that after crossing the Mott–Hubbard point R ¼ RMott, α1 approaches rapidly its atomic-limit value α1 ¼ a0.

Fig. 10 Illustrative picture of the molecular orbitals (isosurface probability density cut ¼ 0.02) on the right; relative covalency and ionicity contributions, without the atomicity extracted, on the left.

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as in this case with two different orbitals an analytic discussion is not possible [e.g., we should use the full matrix form of (9)]. To summarize, the related molecular systems such as those discussed briefly above, require a further analysis to extract the atomicity in heterogeneous systems, independently of the degree of covalency in the physical ground state. This statement is illustrated in Fig. 10 with the starting degree of covalency/ionicity for those systems considered. Additionally, the values of calculated microscopic parameters of Hamiltonian (6), as well as of other microscopic characteristics, are provided in our original papers10,11 (see section Supporting Information in Ref. 11).

4.2 Essential extension: The hydrogen bond—An outline Recently, application of a similar model based on the second quantization representation to description of the hydrogen bond has been formulated.22–26 In that model the resonating fermion is proton and is taking between two more complex molecules. In particular, the question of defining covalency in that case has been raised.27–29 Simply speaking, the proton hops between two (in)equivalent positions, in an analogical situation to that for electrons between two protons in H2 molecule. In the simplest picture of such a hydrogen bond one has the situation with quantum tunneling of a proton regarded now as a quantum particle between two positions with energy E1 and E2, created by two larger donor/acceptor groups X1 and X2 as shown in Fig. 11. If we ignore the explicit electron charge-density shift in X1 and X2, associated with the proton hopping, then the proton between the two positions labeled with E1 and E2 can be modeled by Hamiltonian   ^ ¼ E1 ^a{ ^a1 + E2 ^a{ ^a2 + t ^a{ ^a2 + ^a{ ^a1 H (24) 1 2 1 2

Fig. 11 Schematic representation of proton resonant hopping of magnitude t, as the simplest model of hydrogen bond. X1 and X2 symbolize donor and acceptor molecules. Different reference orbital energies E1 and E2 refer to two possible proton positions.

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where t is the hopping and ^a{i (^ai ) is spinless proton creation (annihilation) operator at site Ei. The energy shift jE1  E2j 6¼ 0 appears as the proton hops between the two sites. This simple two-site model leads to bonding and antibonding states of the form x1,2

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ðE1 + E2 Þ  ðE1  E2 Þ2 + 4t2 : 2

(25)

The eigenenergies are degenerate with respect to proton spin direction. Those energies are depicted as a function of ΔE ≡ E1  E2 in Fig. 12 for different t values. Such a simple model provides the bonding-antibonding. The corresponding eigenfunctions can be obtained easily will not be reproduced here. The more realistic situation is drawn in Fig. 13 for the adenine–thymine exemplary case. In this situation there are two hydrogen bonds as shown, as well as two proton transfers. The resultant schematic model is drawn at the bottom, where now both the protons spins and transverse (vertical) hopping processes  t0 have been marked. This bottom part leads to the model   8 with two fermions (protons) in 4-site system. It leads to ¼ 24 2 spin-singlet and -triplet states. The Hamiltonian is of the form similar to expression (6), which can be analyzed formally rigorously. We omit here a detailed examination of the resultant states. This example is to show only that

Fig. 12 Bonding and antibonding states for resonating fermion (proton) between two sites as a function of the atomic energy difference. Three values of the hopping t have been selected.

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Fig. 13 Top row: Two resonating proton configuration before (left) and after correlated pair hopping. Adenine (A) and thymine (T) pair has been selected. Middle row: Schematic representation of the processes depicted in the top row. Bottom row: 4-site model with two protons with their resonating between the sites. Dashed vertical lines represent virtual hopping between the rungs. An inclusion of the electron density shift may be included by vertical hops of the electrons toward the hopping proton.

the model constructed for simple molecules can be readily constructed and adopted also to more involved situations. Namely, the hopping of the proton to the particular site can be associated with a motion of charge compensating electron toward the incoming proton to the closest site, i.e., the vertical electron hopping toward the proton (hopping of two electrons in total, in opposite direction, to that of each of the proton). Such a simple model  4 4 requires determination of ¼ 1296 -fermion states. In essence, the 2 double hydrogen bond in that case is composed of both the intersite proton resonance of a covalent type and associated with it electrostatic attraction of electron to the hopping proton in its final state. We should see a progress along this lines in near future.

5. Outlook In this chapter we have analyzed in detail the nature of covalency and have elaborated on it by defining an intrinsic (true) covalency, in addition to

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ionicity, as well as have defined the degree of atomicity in nominally covalent state. With introduction of the atomicity we have defined a degree of atomic character in the resultant bound state. Our analysis starts with the exact solution of the HL model of bonding in H2 molecule as well as extends it. The analyzed solution and bonding properties are carried out explicitly and this possible by starting from the analytic solution of the model. This solution, in the closed form, is in turn, possible with the help of our original EDABI approach. The method detailed for the H2-molecule complementary characterization can also be applied to other simple molecular systems such as LiH and HeH+, each discussed briefly. Additionally, a similar modeling can also be carried out for the analysis of hydrogen bond, as briefly outlined at the end. We should be able to see an important progress in understanding these systems along these lines in the near future.

Acknowledgments This work was supported by Grants OPUS No. UMO-2018/29/B/ST3/02646 and No. UMO-2021/41/B/ST3/04070 from Narodowe Centrum Nauki. We thank our colleagues from the Chemistry Department of the Jagiellonian University for numerous discussions and general remarks during the course of this project.

References 1. Heitler, W.; London, F. Wechselwirkung Neutraler Atome und Hom€ oopolare Bindung nach der Quantenmechanik. Z. Phys. 1927, 44, 455–472. 2. Condon, E. U. Coupling of Electronic and Nuclear Motions in Diatomic Molecules. Proc. Natl. Acad. Sci. 1927, 13, 962–965. 3. Pauling, L. The Shared-Electron Chemical Bond. Proc. Natl. Acad. Sci. 1928, 14, 359–362. 4. Coulson, E. U.; Fischer, I. Notes on the Molecular Orbital Treatment of the Hydrogen Molecule. Phil. Mag. 1949, 40, 386–393. 5. Dirac, P. A. M. On the Theory of Quantum Mechanics. Proc. Roy. Soc. A 1926, 112, 661–677. 6. Sherill, C. D.; Schaeffer, H. F. The Configuration Interaction Method: Advances in Highly Correlated Approaches. Adv. Quant. Chem. 1999, 34, 143–269. 7. Becke, A. D. Perspective: Fifty Years of Density-Functional Theory in Chemical Physics. J. Chem. Phys. 2014, 140, 18A301. 8. Kołos, W.; Wolniewicz, L. Improved Theoretical Ground-State Energy of the Hydrogen Molecule. J. Chem. Phys. 1968, 49, 404–410. 9. Kołos, W.; Wolniewicz, L. Vibrational and Rotational Energies for the B1 Σ+u , C1Πu, and a3 Σ+g States of the Hydrogen Molecule. J. Chem. Phys. 1968, 48, 3672–3680. 10. Hendzel, M.; Fidrysiak, M.; Spałek, J. Toward Complementary Characterization of the Chemical Bond. J. Phys. Chem. Lett. 2022, 13, 10261–10266. 11. Hendzel, M.; Fidrysiak, M.; Spałek, J. Many-Particle Covalency, Ionicity, and Atomicity Revisited for a Few Simple Example Molecules. J. Phys. B: At. Mol. Opt. Phys. 2022, 55, 185101–185113.

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12. Hendzel, M.; Spałek, J. Degree of Atomicity in the Chemical Bonding: Why to Return to the H2 Molecule? arXiv:2210.06524. 13. Spałek, J. S.; Podsiadły, R.; Wo´jcik, W.; Rycerz, A. Optimization of Single-Particle Basis for Exactly Soluble Models of Correlated Electrons. Phys. Rev. B 2000, 61, 15676–15687. 14. Slater, J. C. Quantum Theory of Molecules and Solids, Vol. I; McGraw-Hill: New York, 1963; p. 50. 15. Spałek, J.; Oles, A. M.; Chao, K. A. Magnetic Phases of Strongly Correlated Electrons in a Nearly Halg-Filled Narrow Band. Phys. Stat. Solidi B 1981, 108, 329. 16. Robertson, B. Introduction to Field Operators in Quantum Mechanics. Am. J. Phys. 1973, 43, 678–690. 17. Dirac, P. The Principles of Quantum Mechanics; Oxford Science Publications, 1982. 18. Pauling, L. The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry; Cornell University Press, 1960. 19. Penda`s, M. A.; Francisco, E. Decoding Real Space Bonding Descriptors in Valence Bond Language. Phys. Chem. Chem. Phys. 2018, 20, 12368–12372. 20. Mott, N. F. Metal-Insulator Transitions, 2nd ed.; Taylor & Francis: London, 1991. 21. Hubbard, J. Electron Correlations in Narrow Energy Bands III. An Improved Solution. Proc. Roy. Soc. 1964, 281, 401–419. 22. Pusuluk, O.; Farrow, T.; Deliduman, C.; Vedral, V. Emergence of Correlated Proton Tunnelling in Water Ice. Proc. Roy. Soc. A 2019, 475, 20180867. 23. Pusuluk, O.; Farrow, T.; Deliduman, C.; Burnett, K.; Vedral, V. Proton Tunneling in Hydrogen Bonds and Its Implications in an Induced-Fit Model of Enzyme Catalysis. Proc. Roy. Soc. A 2018, 474, 20180037. 24. Brovarets, O.; Hovorun, D. M. Proton Tunneling in the A-T Watson-Crick DNA Base Pair: Myth or Reality? J. Biomol. Struct. Dyn. 2015, 33, 2716–2720. 25. Witkowski, A. The Schr€ odinger Group in Molecular Quantum Mechanics: Beyond the Born-Oppenheimer Approximation. Mol. Phys. 2011, 109, 1423–2432. 26. Grabowski, S. J. What is the Covalency of Hydrogen Bonding? Chem. Rev. 2011, 111, 2597–2625. 27. Dereka, B.; Yu, Q.; Lewis, N. H. C.; Carpenter, W. B.; Bowman, J. M.; Tokmakoff, A. Crossover from Hydrogen to Chemical Bonding. Science 2021, 371, 6525. 28. van der Lubbe, S. C. C.; Guerra, C. F. The Nature of Hydrogen Bonds: A Delineation of the Role of Different Energy Components on Hydrogen Bond Strengths and Lengths. Chem. Asian. J. 2019, 14, 2760–2769. 29. McKenzie, R. H.; Bekker, C.; Athokpam, B.; Ramesh, S. G. Effect of Quantum Nuclear Motion on Hydrogen Bonding. J. Chem. Phys. 2014, 140, 174508.

Further reading 30. Spałek, J. Mott Physics in Correlated Nanosystems: Localization-Delocalization Transition by the Exact Diagonalization Ab Initio Method. In Topology, Entanglement, and Strong Correlations; Pavarini, E., Koch, E., Eds.; Vol. 10; Forschungszentrum J€ ulich GmbH, Institute for Advanced Simulation, 2020; pp. 7.1–7.38.

CHAPTER THIRTEEN

ETS-NOCV and molecular electrostatic potential-based picture of chemical bonding Olga Żurowska, Mariusz P. Mitoraj, and Artur Michalak* Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Krakow, Poland *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Theory 3. Computational details and models 4. Results and discussion 5. Concluding remarks Acknowledgments References

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Abstract The main goal of the present work was to compare the Extended Transition State— Natural Orbitals for Chemical Valence (ETS-NOCV) description of chemical bond with the picture based on deformation of Molecular Electrostatic Potential (MEP), within a similar differential (molecule vs promolecule) perspective. The set of analyzed systems include: the bond in N2 molecule, the CH3-R bond, in a series of systems with different R (R ¼ CH3, CH2dCH3, CH(CH3)2, C(CH3)3, CH2dC6H6, CH2I, CH2Br, CH2Cl, CHCl2, CCl3, CH2dOH, COOH, CH2NH2), and examples of hydrogen bonds in water dimer, and the adenine-thymine complex. A special attention was paid to the donation-/backbonding between the considered fragments, which is one of typical features of the ETS-NOCV analysis. The results show that with use of the same promolecule, the main characteristics of the chemical bonding disclosed by the ETS-NOCV analysis are well reflected in deformation of MEP. However, some details can be more clearly visible in one of the two approaches (i.e., in the results of deformation density/NOCV-approach, or in the results of MEP-deformation analysis). In particular, the deformation of MEP seems to more clearly indicate the shift in electron density (charge transfer) between the fragments. What follows, the analysis of the MEP-deformation, used in combination with the ETS-NOCV approach can give a more complete picture in a simple way.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.005

Copyright

#

2023 Elsevier Inc. All rights reserved.

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1. Introduction Detailed description of chemical bonding in various molecular systems is crucial for understanding and predicting structures, as well as reactivity.1–4 The groundbreaking concept of Lewis electron-pair bond5 was followed by the pioneering works6–10 that built the quantum-mechanical foundations of chemical-bond theory. During decades of development of modern quantum chemistry, a number of concepts, quantities, and methods suitable for in depth bonding description have been introduced and utilized. As examples one can provide alternative localized-orbital/bond-orbital-based description,11–16 valence bond (VB) approaches,17 Domain Averaged Fermi Holes (DAFH),18 Localizedorbital Locator (LOL),19 Conceptual Density Functional Theory approaches,20–22 Reaction Fragility Spectra-based approach by Ordon and Komorowski,23,24 FALDI descriptors by de Lange, Cukrowski et al.,25 or quasi-atomic orbitals by Ruedenberg.26,27 Further, a group of real-space approaches based on electron density includes Bader’s Quantum Theory of Atoms in Molecules (QTAIM),28 Electron Localization Function (ELF),29,30 reduced density gradient (NCI),31 Interacting Quantum Atoms (IQA) method,32 and its IQA-FAMSEC extension,33 or Density Overlap Regions Indicator (DORI).34 The information-theory based description of chemical bonding has been developed by Nalewajski within the Information Theory of Molecular Systems.35,36 Various energydecomposition schemes (EDA) proposed in literature provide physical insight into the nature of a chemical bond by extraction interaction-energy components (i.e., electrostatic, polarization, etc.), e.g., Symmetry Adapted Perturbation Theory (SAPT) by Jeziorski, Moszynski and Szalewicz,37 Kitaura and Morokuma EDA scheme,38 and the related Ziegler and Rauk Extended-Transition-State (ETS) method,39,40 BSSE corrected EDA by Sokalski et al.,41 Head-Gordon’s EDA based on Absolutely Localized Molecular Orbitals (ALMO),42 populational space approach by de Silva and Korchowiec,43 or Local Energy Decomposition based on CCSD(T) by Neese and coworkers.44 Finally, there exist quite a few alternative ways to characterize bond-orders/bond-multiplicity indices45–59 It should be pointed out that the practical use of so many alternative models, methods and quantities focused on chemical bonding is desirable since it permits to understand a chemical bond from different perspectives, although this may sometimes lead to some controversies.

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In connection to the Nalewajski and Mrozek theory of chemical valance and bond multiplicities,54–59 we have defined two sets of orbitals useful in interpretation of chemical bonding: Natural Orbitals for Chemical Valence (NOCV) and Localized Orbitals from Bond-Order Operator (LOBO).60,61 Both, LOBO and NOCV are localized in the bond region. However, while LOBO exhibit properties typical to other localized orbitals, the interpretation of NOCV is quite different.60 Namely, when concerning the A-B bond between two molecular fragments A, and B, NOCV in a natural way, without any external constraints, allow for separation of donation A ! B, and back-donation B ! A, thus addressing in a natural way the Dewar-Chatt-Duncanson model of chemical bonding,62,63 well established in inorganic chemistry. The donation/back-bonding picture of bond formation is an intrinsic property of NOCV, not limited to transition-metal complexes. Furthermore, it has been shown, that NOCV decompose overall deformation density (Δρ ¼ ρAB  ρA  ρB) into diagonal contributions, which reveal formation of separated, chemically meaningful bonding components, e.g., σ, π or δ, etc.60,61 At a next stage we have merged NOCV approach with the Ziegler-Rauk, ETS energy decomposition scheme,39,40 resulting in the combined charge-and-energy decomposition scheme ETS-NOCV (also called in literature EDA-NOCV).64 This approach was recently extended to discuss specific bonding contributions in terms of changes in their kinetic and potential energies.65 Examples of the applications of NOCV/ETS-NOCV were reviewed in several articles.66–69 Initially, NOCV have been applied to describe metal–ligand bonding in transition metal complexes61,66–79 due to aforementioned connection to the Dewar-Chatt-Duncanson model. It has been further proven, that NOCV based methodology is also suitable for covalent bond description in a series of organic species.64,71 It was also shown, that NOCV resolution allows to extract qualitatively and quantitatively intra-molecular agostic CH-metal bonding in transition metal complexes, as well as hydrogen bonds in adenine-thymine base pair.64 Utility of ETS-NOCV was later demonstrated for other weak, intra-, and inter-molecular interactions, as well as for a description of changes in chemical bonding during chemical reactions.65–69,72–80 Molecular Electrostatic Potential (MEP)81–84 is another quantity that has been widely used in the context of description of charge distribution in molecular systems, and in particular, chemical reactivity. MEP was demonstrated as well to be a very useful tool for interpretation of chemical bonding, especially in the case of non-covalent interactions; in this context, it should

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be emphasized that MEP-based explanation of halogen bonding has led to the formulation of the σ-hole concept,85 opening new areas of description of various non-covalent interactions.85–93 The topology of MEP has been also shown to provide important insight, in the context of chemical bonding.94–96 ETS-NOCV scheme rely on differential approach which attempts to understand formation of a final molecule/molecular system from a promolecule (which can be either built of isolated atoms or larger molecular fragments). The main goal of this paper is to compare for a few examples a picture emerging from ETS-NOCV method with changes in MEP due to bonding between the fragments, depicted within a similar differential (molecule vs promolecule) perspective.

2. Theory Deformation density, Δρ(r) ¼ ρAB(r)  ρA(r)  ρB(r) , corresponds to the difference in the electron density of the molecular system AB (ρAB(r)), and of a promolecule built of isolated fragments A, B (placed in the same positions as in AB), ρ0(r) ¼ ρA(r) + ρB(r). In the basis set {φi; i ¼ 1,n}, deformation density can be expressed as: Δρðr Þ ¼

n X n X α

ΔP αβ φα ðr Þφβ ðr Þ

β

(1)

where ΔP 5 P 2 P0 stands for to the corresponding difference in the charge and bond-order matrix in the molecule (P) and promolecule (P0), representing the ΔP^ operator in this basis set; for simplicity we assumed here that the basis set includes the functions of real variables. Natural Orbitals for Chemical Valence (NOCV) are defined60,61 as eigen^ vectors of ΔP: ^ i ¼ νi ψ i ΔPψ

(2)

obtained from a diagonalization of ΔP, and thus leading to the diagonal representation of Δρ(r) in the basis of NOCV: Δρðr Þ ¼

n X

νi ψ 2i ðr Þ

(3)

i

It should be mentioned that the original name of NOCV refers to relation of ΔP^ to the total valence operator from the Nalewajski and Mrozek theory of chemical valence and bond-order indices.53–59

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Further, since the eigenvectors of ΔP^ can be coupled in pairs of complementary orbitals characterized by eigenvalues with the same absolute value and the opposite sign,60,61,97,98 ^ +k ¼ νk ψ +k ; ΔPψ

^ k ¼ νk ψ k , ΔPψ

k ¼ 1, …, n=2

(4)

the deformation density can be finally expressed as sum of NOCV-pair contributions Δρk(r): n=2 n=2 X  2  X 2 νk ψ k ðr Þ  ψ k ðr Þ ¼ Δρk ðr Þ Δρðr Þ ¼ k

(5)

k

It is important that only few NOCV-pairs exhibit non-negligible contribution to Δρ(r), as for vast majority of NOCV pairs, the corresponding eigenvalues are close to zero. Within those pairs corresponding to large νk, ψ k, and ψ k exhibit usually antibonding, and bonding character, respectively. Electron populations of the two orbitals are fractional in both, promolecule and molecule, and the eigenvalue measures the respective changes in the population (i.e., charge flow from bonding and antibonding NOCV, accompanying the bond formation process).60,61 Furthermore, the main NOCV-pair contributions, Δρk(r) represent easily interpreted, meaningful components of chemical bonds (e.g., donation/back-bonding, σ, π or δ, etc.)60,61,64 In the Ziegler-Rauk ETS bond-energy decomposition method,39,40 the interaction (bonding) energy of a bond A-B between the two fragments A, B is decomposed into the electrostatic (ΔEelstat), Pauli repulsion (ΔEPauli) and orbital interaction (ΔEorb) components: ΔEint ¼ ΔE elstat + ΔE Pauli + ΔE orb + ΔE disp

(6)

The first term, ΔEelstat, corresponds to the electrostatic interaction between the frozen charge distributions of the two fragments in the geometry of the whole molecular system considered, ΔEPauli is the repulsive interaction between occupied orbitals of the two fragments. It is common to combine ΔEelstat and ΔEPauli into the steric interaction energy as: ΔEsteric ¼ ΔEelstat + ΔEPauli. The orbital interaction term is the stabilizing component due to the bond formation, i.e., charge flows from the occupied orbitals of one fragment to virtual orbitals of the other subsystem and vice versa; this term includes also the intra-fragment polarizations (mixing of occupied and virtual orbitals of the same fragment). It must be emphasized that in the original Ziegler-Rauk ETS analysis, only the above mentioned terms (first three in

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Eq. 6) were considered. In the dispersion corrected DFT calculations, typical nowadays, an additional dispersion term (ΔEdisp) is used in addition. It should be also added that Eq. (6) defines interaction energy between molecular fragments in their distorted geometries (as in AB). Thus, when addressing bond energy defined as negative energy of dissociation into the fragments, an additional term must be considered corresponding to the energy change due to geometry preparation/distortion (from optimum geometry of the isolated fragments in to their molecular geometry). In the combined ETS-NOCV (EDA-NOCV) method,64 the orbitalinteraction term of ETS (depending on Δρ) is further decomposed into the Pn=2 contributions from NOCV-pairs (defined in Eq. 5), ΔE orb ¼ k ΔE orb,k . Thus, the ETS-NOCV approach provides compact description of chemical bond in terms both, deformation density NOCV contributions, and their energy measures.64 Molecular electrostatic potential (MEP) represents the interaction energy of the molecular system with the unitary point charge located at position r, and thus contains contributions from nuclei, and electrons: Z X ZA ρðr 0 Þ 3 0 V ðr Þ ¼ (7)  d r |RA  r| |r 0  r| A where ZA is the charge of nucleus A located at RA. In the present account we will discuss deformation in MEP of the molecular system AB due to bonding between the fragments, A and, B (i.e., in molecule vs promolecule): ΔV ðr Þ ¼ V AB ðr Þ  V A ðr Þ  V B ðr Þ

(8)

It should be mentioned that all the atomic position are the same in the molecular system, and the corresponding promolecule, and thus, the deformation in MEP includes only the electronic part: Z Δρðr 0 Þ 3 0 ΔV ðr Þ ¼  d r (9) |r 0  r|

3. Computational details and models All the results of DFT calculations were obtained with the Amsterdam Density Functional (ADF) package (version 2017.103).99–101 The BeckePerdew (BP86) exchange-correlation functional102,103 was applied with

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the D3 dispersion corrections with the Becke-Johnson damping.104 Scalar relativistic corrections were included within the ZORA approximation.105–109 The Slater-type TZP basis sets110 were used. In the present account we will analyze bonding in a few typical organic molecules, as well as in examples of hydrogen-bonded complexes, using the ETS-NOCV analysis, and deformation in MEP (Eq. 8), using the same promolecule. We start our discussion with the bond in N2 molecule, in order to demonstrate the main features, and the typical interpretation of the ETS-NOCV results. The terminal CdC bond will be analyzed for a series of molecules CH3-R, including a few hydrocarbons (R ¼ CH3, CH2dCH3, CH(CH3)2, C(CH3)3, CH2dC6H6), and molecules containing various types of substituents: halogens (R ¼ CH2I, CH2Br, CH2Cl, CHCl2, CCl3), alcohol OH group (R ¼ CH2dOH), aldehyde group (R ¼ CHO), carboxylic group (R ¼ COOH), and amine group (R ¼ CH2NH2). Finally, examples of hydrogen bonds will be analyzed for water dimer, and the adenine-thymine complex. It should be emphasized that in the description of the NdN and CdC bonds, the promolecule built of two open-shell fragments will be considered (two N atoms, or CH3, and R radicals, respectively). For these systems the spin-resolved ETS-NOCV analysis was performed, in which the spin-α and spin-β NOCV {ψ σi ; i ¼ 1,n; σ ¼ α, β} are obtained from separate diagonalizations of ΔPα and ΔPα matrices, and thus, deformation-density decomposition of Eq. (5) includes summation over two spins: Δρðr Þ ¼

n=2 X X σ¼α, β

k

υσk

n=2 n=2 n X X X 2  2 o ψ σk ðr Þ  ψ σk ðr Þ Δρσk ðr Þ ¼ Δρk ðr Þ ¼ k σ¼α, β k (10)

and the orbital-interaction energy can be decomposed into decomposed as: ΔE orb ¼

n=2 X X σ¼α, β

ΔE σorb,k ¼

k

n=2 X

ΔEorb,k

(11)

k

4. Results and discussion In the beginning, we will present a typical interpretation of NOCV for a simple case of N2 molecule; here a promolecule built of two open shell nitrogen atoms was used. Fig. 1 presents the NOCV orbitals, grouped in pairs (see Eq. 4), and the corresponding contributions to the deformation density, Δρ, as defined in Eq. (10). In spin resolution, there are six pairs

Fig. 1 ETS-NOCV description of the bond in N2 molecule. In row (A) the dominating (spin-resolved) pairs of NOCV are presented, row (B) collects the corresponding NOCV-pair contributions to deformation density, in row (C) the respective σ- and π-contributions (summed up over spins) are shown, and finally, part (D) presents the total deformation density. The isocontour values are j0.01j a.u for orbitals, and j0.005j a.u. for Δρ and its contributions.

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of NOCV, corresponding to non-negligible eigenvalues v, they are presented in row A of Fig. 1. The first two pairs (jvj ¼ 0.59 a.u) exhibit σ-symmetry, the remaining two NOCV pairs (jvj ¼ 0.57) are the π-type orbitals. Within each pair, the orbital corresponding to positive eigenvalue exhibits antibonding character, while that corresponding to positive v is a bonding orbital. Further, due to symmetry of the molecule, the spin-α orbitals are mirror images of the corresponding spin-β NOCV. As an obvious consequence, the corresponding Δρ contributions (shown in part B of Fig. 1) for two-spins are mirror images as well. Further, it is important to notice, that the two corresponding Δρ contributions for the two spins show accumulation of electron density in the bond region, but as well, the charge transfer from one fragment to the other and vice versa (N1 ! N2, and N1 N2); this demonstrates the fact that donation/back-bonding picture of the Dewar-Chutt-Duncanson model, is a typical feature of NOCV analysis, not limited to the transition metal complexes. The Δρ components summed up for two spins are presented in row C of Fig. 1. Here, the first contribution shows a typical σ-bond formation picture, with charge depletion in the atomic regions, and charge accumulation in the bond-region and in the lone-pair regions. The two π-symmetry contributions show accumulation of electron density characteristic to the π-bond formation, with charge-depletion in the atomic p-orbital region. Finally the sum of the discussed σ- and π-NOCV contributions practically recovers the total Δρ picture shown in Fig. 1D. A comparison of the deformation density, Δρ, and the deformation of MEP, ΔV, for N2 is presented in Fig. 2, using isocontour- (Fig. 1A–C), and color-coded-representation (Fig. 1D and E) of the two quantities. For ΔV we present two isocontours (jVj ¼ 0.01 a.u., and jVj ¼ 0.001 a.u.) focused on “high,” and “low” jΔVj values, as it is known that often not all the features can be seen at one isocontour plot, and in particular, for ΔV the experience-based, “optimum” value has not been worked out yet. In the case of the “high” ΔV (Fig. 2C), only the negative values of ΔV in the bond area are visible; this reflects accumulation of electrons seen in Δρ. In the case of the “low” ΔV contour, in addition to the negative values in the bond region are accompanied by negative values (seen as much “smaller” contours) at the extension of the bond (lone-pair regions), reflecting the positive Δρ in this area. Thus, a picture of bonding, emerging from Δρ and ΔV is qualitatively similar. This is even more clearly seen in color-coded representation of Δρ, and ΔV (Fig. 2D and E, respectively): the two pictures for the two quantities are “negatives” of each other. In Fig. 3 the results of the NOCV analysis are presented for ethane molecule (with promolecule built of two methyl radicals), together with

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Fig. 2 A comparison of the deformation density, Δρ, and the deformation of MEP, ΔV, for N2 molecule. Parts (A–C) shows isoncontour representation; the contour values are 0.005 a.u for Δρ (part A), while for ΔV they are j0.01j, and j0.011j a.u (part B and C, respectively). Parts (D) and (E) show Δρ, and ΔV respectively, color-coded on the electron density isocontour (ρ ¼ 0.01 a.u).

Fig. 3 ETS-NOCV description of the σ-symmetry components the CH3dCH3 bond in ethane, and comparison of the corresponding deformation density, and deformation in MEP. In row (A) the dominating (spin-resolved) pairs of NOCV are presented, row (B) collects the corresponding NOCV-pair contributions to deformation density, part (C) presents the total deformation density (contour value: j0.005j a.u.) and parts (D) and (E)—deformation in MEP (contour values: j0.1j a.u., and j0.011j a.u., respectively).

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a comparison of the deformation density, Δρ, with the deformation of MEP, ΔV. The interpretation of NOCV for ethane was already discussed in details in the initial ETS-NOCV paper.64 Despite that, we decided to present in detail part of the ETS-NOCV analysis for ethane, focusing at the σ-bond component, since it is a basis for understanding the results for other CH3-R systems discussed in the present account. In Fig. 3A the two pairs of spin alpha, and spin-beta NOCV with σ-symmetry are presented. The corresponding Δρ components, shown in Fig. 2B, show the shifts in electron density from one methyl radical to the other, and vice versa; the sum of this two components represent the largest part of Δρ, shown in Fig. 3C. Concerning the orbital-interaction energy, the contribution from the σ-bond component (sum for two spins, ΔEorb, 1 ¼ 172.0 kcal/mol) represents ca. 93% of the total ΔEorb ¼ 184.8 kcal/mol (see Table 1). The major part of the difference between the total ΔEorb, and ΔEorb, 1 (12.8 kcal/mol) corresponds to the two π-type contributions (not shown here), describing the electron-density shifts between the methyl hydrogen atoms and the π-bonding orbitals (methyl hyperconjugation).64 A comparison of the deformation density, Δρ, and the deformation of MEP, ΔV, for ethane is presented in Fig. 3C–E. Again, for ΔV we present two isocontours (jVj ¼ 0.01 a.u., and jVj ¼ 0.001 a.u.). In the case of the “high” ΔV (Fig. 3D), similarly to ΔV for N2 (cf. Fig. 2B) only the negative values of ΔV in the bond area are visible; reflecting accumulation of electrons in this area, also observed in Δρ. In the case of the “low” ΔV contour (Fig. 3E), the positive ΔV contours appear, at the extensions of the CdC bond. This reflects a relatively large decrease in electron density in the region of the carbon atoms of two CH3 groups. There is one feature of Δρ is not represented in the ΔV plot; namely, the increase in the electron density at the extensions of the CdC bond (i.e., between the methyl hydrogen atoms) is not reflected by the negative ΔV, as a result of the long-range character of MEP (i.e., concerning ΔV values at the extension of the CdC bonds, the positive ΔV-part originating from relatively large, negative values of Δρ in the carbon-atomic regions, dominate over the negative MEP originating from relatively small, positive values of Δρ observed at the CdC bond extensions). Let us now discuss the systems containing CH3-R, with various R, all containing carbon atom directly connected to the terminal methyl group. The ETS-NOCV energy components are summarized in Table 1, and the pictures presenting a comparison of the deformation density, Δρ, and the deformation of MEP, ΔV, are collected in Fig. 4.

Table 1 The ETS-NOCV energy components (see Eqs. 6 and 11) for the CH3-R bond in a series of CH3-R molecules. Molecule Total interaction energy components [kcal/mol] ΔEPauli

ΔEsteric

ΔEorb

(CH3 ! R) ΔEβorb,

R

ΔEint

Ethane

CH3

116.0 1.4

131.0 201.2

70.2

184.8

86.0

86.0

Propane

CH2dCH3

112.1 2.2

142.5 223.0

80.5

190.4

89.4

86.9

Isobutane

CH(CH3)2

110.6 3.0

151.9 236.8

85.0

192.6

89.6

88.2

Neopentane

C(CH3)3

108.0 3.7

156.3 244.1

87.8

192.1

89.3

87.7

Ethylobenzene

CH2dC6H5 115.7 2.9

141.7 218.4

76.7

189.5

88.4

86.5

Ethanol

CH2dOH

111.7 1.8

156.4 252.3

95.9

205.8 100.1

89.8

Acetaldehyde

CHO

97.1 1.6

169.9 295.9

126.0

221.5 107.4

87.5

Acetic acid

COOH

107.5 1.8

159.9 272.0

112.2

217.8 107.7

81.3

Ethyloamine

CH2dNH2

112.2 2.0

154.5 248.1

93.6

203.8

99.0

88.2

Iodoethane

CH2dI

114.3 2.7

159.9 264.7

104.8

216.4 111.9

82.7

Bromoethane

CH2dBr

114.0 2.4

153.7 255.9

102.2

213.8 109.8

83.5

Chloroethane

CH2dCl

113.0 2.2

152.1 250.4

98.3

209.1 106.7

83.5

1,1-Dichloroethane

CH(Cl)2

108.8 3.0

162.2 278.1

115.9

221.7 119.7

79.5

104.6 3.8

164.2 289.2

125.0

225.9 126.7

74.8

1,1,1-Trichloroethane C(Cl)3

ΔEdisp ΔEelstat

ΔEαorb,

CH3-R

1

1

(CH3

R)

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Fig. 4 A comparison of deformation density, Δρ, and the deformation of MEP, ΔV, contours for the CH3-R bond in propane (A), isobutane (B), neopentane (C), ethylobenzene (D), ethanol (E), acetaldehyde (F), acetic acid (G), ethyloamine (H), iodoethane (I), bromoethane (J), chloroethane (K), and 1,1,1-trichloroethane (L) .The contour values are j0.005j a.u., and j0.011j a.u., for Δρ, and ΔV, respectively.

The first group are hydrocarbons: propane, isobutane, neopentane, and ethylobenzene (R ¼ CH3, CH2dCH3, CH(CH3)2, C(CH3)3, CH2dC6H6, respectively). Compared to ethane, the orbital interaction energy for all these systems is more stabilizing (more negative values), by ca. 5–8 kcal/mol. However, the total interaction energy indicates strongest interaction for ethane; for the other hydrocarbons it is weaker by ca. 1–8 kcal/mol. This is a result of increasing Pauli repulsion, and as follows—the steric repulsion, due to the presence of the larger substituents. In the last two columns of Table 1, the two dominating, spin-resolved contributions to Δρ are presented, describing the σ-symmetry component of the bond; the first spin-component describes CH3 ! R shift in electron density, while the second the electron transfer in the other direction CH3 R (see Fig. 2B for the corresponding Δρ contributions in ethane). The results shown in Table 1 indicate that in all the analyzed hydrocarbons, compared to ethane, both ΔEαorb, 1(CH3 ! R), and ΔEβorb, 1(CH3 R) are slightly more negative (by. ca. 0.9–3.5 kcal/mol). As a results of relatively minor variation in the NOCV characteristic for these systems, the Δρ contour plots in Fig. 4A–D, look qualitatively similar to ethane (Fig. 3C). In the case of ΔV, the large negative ΔV contour is present in the bond region, due to the dominating accumulation of electron density, corresponding to

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the bond formation. However, the positive ΔV contour at the extension of the CH3-R bond on the side of R, is visibly larger, with a contribution from carbon atoms of all the methyl groups in propane, isobutane, and neopentane (and ipso phenyl-carbon atom in ethylobenzene). The second group of the CH3-R systems include ethanol, acetaldehyde, acetic acid, and ethyloamine (R ¼ CH2dOH, CHO, COOH, and CH2dNH2, respectively), i.e., the systems including electronwithdrawing, polar groups. For these systems, the total orbital interaction is substantially more negative (by between ca. 19.0 and 36.7 kcal/mol) then for ethane, with the strongest orbital interaction for acetaldehyde, ΔEorb ¼ 221.5 kcal/mol. It should be noticed, however, that also in the case of these systems, due to increase in the Pauli repulsion and the steric repulsion, the total interaction energies, ΔEint, are less negative than in ethane. The increase in the strength of the orbital interaction energy compared to ethane, can be explained by visibly higher transfer of electron density form CH3 to R. Values of ΔEαorb, 1(CH3 ! R) vary between 99.0 kcal/mol for ethyloamine, and 107.7 kcal mol for acetic acid (compared with 86.7 kcal/mol for ethane). At the same time, the changes in ΔEorb, β R) are much smaller (between 1.5 kcal/mol, and +4.7 1 (CH3 kcal/mol, relative to the value for ethane). The deformation density plots in Fig. 4E–G, in addition to the electron density accumulation in the bond region, reveal as well rearrangement of electron density in the vicinity of the functional groups, i.e., decrease in the vicinity of the N/O atom and increase in the region of corresponding lone pair(s); the effect is strongest for carbonyl oxygen in acetaldehyde and acetic acid. This effect is nicely reflected by large, negative ΔV contour, extending from the CdC bond region to the functional group region. This, together with the relatively large, positive ΔV contours at the side of CH3 group confirms the substantial shift in electron density CH3 ! R (manifested by a relatively large ΔEorb, α 1 (CH3 ! R) values). This is even more clearly seen in the color-coded representation of ΔV, presented for ethanol, and acetic acid examples in Fig. 5A and B. In the group of considered halogen-substituted ethane (R ¼ CH2I, CH2Br, CH2Cl, CHCl2, CCl3) the overall picture is qualitatively similar. The orbital interaction energy is substantially more negative than for ethane, varying between 209.1 and 225.9 kcal/mol (thus, the difference with respect to the ethane value varies between 24.2 and 41.2 kcal/mol). Among the studied systems, the strongest orbital interaction is observed for trichloroethane. As in the previous examples, the Pauli repulsion

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Fig. 5 Examples of the deformation in MEP, ΔV, plots, color-coded on the electron density isosurface for ethanol (A), acetic acid (B), chloroethane (C), 1,1,1-trichloroethane (D), water dimer (E), adenine-thymine complex (F). The density isocontour value is 0.01 a.u. for (A–D), and 0.05 a.u. for (E, F).

and the steric energy counterbalance the orbital interaction energy, resulting in the total interaction energy being slightly less negative, compared to ethane. The CH3 ! R donation is much larger then CH3 R back-donation, which is reflected by the corresponding ΔEαorb(CH3 ! R), and ΔEβorb(CH3 R) contribution. It is worth noticing, that the increase in the number of halogen atoms causes relatively large effect in changes of the ΔEαorb(CH3 ! R): for chloroethane, dichloroethane, and trichloroethane, the corresponding values are: 106.7, 119.7, and 126.7 kcal/mol, respectively. At the same time, the opposite electron transfer is decreasing; for chloroethane, dichloroethane, and trichloroethane, the ΔEβorb(CH3 R) values are: 83.5, 79.5, 74.8 kcal/mol. The effect of increasing (compared to ethylene) CH3 ! R donation is more clearly visible in the ΔV contour then in the deformation density contours, see Fig. 4I–L (certainly, for smaller isocontour values the effect might be more pronounced). In particular, in Fig. 4L for trichloroethane, a substantial increase is observed (compared to the other systems) in the size of the positive ΔV contour on the side of the CH3 group, and the negative ΔV contour spreading over the CCl3 group; this reflects the most negative value of ΔEαorb(CH3 ! R) among the studied systems (largest “charge transfer”). Also, the color-coded representation of ΔV clearly reflects the large CH3 ! R shift in the electron density, see Fig. 5C and D for chloroethane, and trichloroethane, respectively.

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Fig. 6 A comparison of deformation density, Δρ, and the deformation of MEP, ΔV, contours for the hydrogen bonds in water dimer (A), and the adenine-thymine complex (B). The contour values are j0.0003j a.u., and j0.015j a.u., for Δρ, and ΔV, respectively.

Finally, we would like to present a comparison of the deformation density and deformation in MEP contours for two examples of hydrogenbonded systems: water dimer and adenine-thymine complex. We will not discuss all the ETS-NOCV energy contributions, as a detailed ETSNOCV analysis was presented in Ref. 64 A comparison of Δρ, and ΔV for these systems is presented in Fig. 6. In the case of water dimer (Fig. 6A), Δρ indicates accumulation of electron density in the hydrogen bond region, at the expense of the lone-pair of the hydrogen-acceptor water molecule. Polarization within both water molecules is visible as well. In the corresponding ΔV plot the negative contour in the hydrogen bond region reflects accumulation of electron density. However, the shift in the electron density from the hydrogen-acceptor water molecule to the hydrogendonor water molecule is clearly reflected by the positive ΔV in the region of the former, and the negative ΔV in the region of the latter water molecule. The shift in electron density between the two water molecules is even more strongly emphasized in the color-coded plot of ΔV presented in Fig. 5E. In the case, of the adenine-thymine complex (Fig. 6B), the Δρ plot again indicates accumulation of electron density in the hydrogen bond region, and the decrease in electron density in the areas corresponding to the lone-pairs of hydrogen-acceptor N/O atoms, as well as in the areas

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of the hydrogen atoms forming the hydrogen bonds. In the corresponding ΔV plots, the contours of the negative ΔV are pronounced in the hydrogenbond region, reflecting density accumulation. Emphasizing other details would require analyzing contour-plots for other isocontour values. However, the color-coded representation of ΔV presented in Fig. 5F, emphasize polarization within the adenine and thymine fragments. This example clearly shows that the analysis of different contours and/or different representations of ΔV may be needed to obtain more complete picture, especially in the case of weak interactions.

5. Concluding remarks The main goal of the present work was to compare the ETS-NOCV description of chemical bond with the picture based on deformation of MEP, within a similar differential (molecule vs promolecule) perspective. To demonstrate the main features, and the typical interpretation of the ETS-NOCV results, we discussed in detail the bond in N2 molecule. Further, we presented the results of ETS-NOCV approach for a series of CH3-R molecules with different R, as well as for example of hydrogenbonded systems. In the case of the N2, and CH3-R systems, the promolecule was built of the open shell fragments (CH3, and R radicals), and a special attention was paid to the intrinsic property of the NOCV description, which is donation, and back-bonding, CH3 ! R, and CH3 R. The results show that with use of the same promolecule, the main features of the chemical bonding disclosed by the analysis of the deformation density (and its ETS-NOCV-decomposition) are well reflected in deformation of MEP. Concerning the dominating changes, Δρ-, and ΔV-based pictures are “negatives” with respect to each other. However, some details can be more clearly seen in the results of Δρ-/NOCV-approach, or in the results of ΔV analysis. In particular, the deformation of MEP seems to more clearly indicate the shift in electron density (charge transfer) between the fragments. What follows, the analysis of the MEP-deformation, used in combination with the ETS-NOCV approach can give a more complete picture in a simple way. Further, for the sake of more detailed analysis, it may be useful in future to decompose ΔV into the NOCV-pair contribution.

Acknowledgments We thank the PL-Grid Infrastructure and the Academic Computational Centre Cyfronet of the University of Science and Technology in Krakow for providing computational resources.

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CHAPTER FOURTEEN

From bulk to surface—Transferability of water atomic charges Anna Stachowicz-Ku
snierz and Jacek Korchowiec⁎ Faculty of Chemistry, Jagiellonian University, Krako´w, Poland ⁎ Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Computational details 3. Results and discussion 4. Summary Acknowledgments References

398 400 403 410 411 411

Abstract The purpose of many molecular dynamics simulations is to describe molecular systems at the interface, for example, water/air interface. The force fields applied often encounter problems with correct reproduction of the surface tension of water, which affects the surface pressure isotherms of monolayer films. In this work, it was checked to what extent the charge distribution of a water molecule is modified during its movement from the bulk phase to the interface area, and whether such a change may affect description of monolayer properties. To describe the transfer process, the method of self-consistent polarization for subsystems was used. The studies utilized a hundred of hemispherical clusters with different radii cut from molecular dynamics trajectories. It was demonstrated that, both on the surface and in bulk, charge distribution is significantly modified due to the presence of explicit water molecules (modeled with isolated-molecule charges) as compared to vacuum. The polarization of environment is also important, though the changes in charge distribution were smaller in magnitude then those involved by direct environment effect. Polarization upon transfer of the molecule from the surface to bulk was c.a. 5% relative to the surface charge value, which corresponds to c.a. 0.04 e
. The modified, interface-derived, charge distribution in water molecule improved its surface tension and had qualitative influence on the results of MD simulations of a DPPC monolayer spread on pure water. Employment of bulk-derived charges did not have such an improving effect on the simulations.

Advances in Quantum Chemistry, Volume 87 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2023.01.006

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2023 Elsevier Inc. All rights reserved.

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1. Introduction Description of dynamical behavior of large molecular systems requires the use of simplified theoretical models. For systems composed of hundreds of thousands of atoms, classical molecular dynamics (MD) and the Monte Carlo (MC) simulations with non-polarizable force fields1–3 seem to be the only choice. On one hand, preserving the atomic resolution is desirable, since it gives atomic-level information about processes appearing in molecular systems, and thus enriches the experiment with details inaccessible from direct measurements. On the other hand, more sophisticated, i.e., polarizable4–7 and reactive8,9 force fields, are troublesome due to computational cost, which is prohibitive for large systems and/or long simulation times. The same reasons exclude more accurate quantum-chemical calculations. Even fragmentation techniques10–14 designed to determine the electronic structure of large systems are beyond the realm of dynamical applications. Despite these problems, quantum-chemical calculations can always be used in conjunction with classical methods. Hybrid calculations of the QM/MM type15,16 can supplement our knowledge with respect to those effects that escape the classical description. The potential functions (force fields) used in standard molecular dynamics simulations include bonding and non-bonding contributions.17 The former are associated with changes in the length of chemical bonds, angles between bonds, and dihedral angles. If necessary, they are supplemented with improper angles in order to enforce appropriate configuration of selected groups, consistent with quantum-chemical results or experimental structural data. The latter contributions include van der Waals and electrostatic interactions. These contributions are responsible for mutual orientation of molecules, prevent atoms from overlapping and influence the conformation of interacting molecules/functional groups. Numerical parameters of such force fields (equilibrium structural parameters, atomic charges, van der Waals parameters) are nowadays derived mostly by fitting to quantum-chemical data. The properties of the condensed and surface phases depend on the non-bonding contributions. Here, the long-range electrostatic contribution is of particular importance. This contribution is based on partial charges assigned to atoms in molecules. The total charge of a molecule is a physical observable but its decomposition is not unique. There is a variety of charge decomposition schemes proposed in the literature. They can be divided into three main categories: (i) partitioning the electron density in functional

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space,18–20 (ii) partitioning the electron density in physical space,21,22 (iii) fitting charges to molecular electrostatic potential.23–26 The latter may differ in grid construction and constraints imposed on charges. The most popular force fields used in MD simulations are based on different assumptions in respect of decomposing charges to atoms. For example, AMBER charges are derived by reproducing the molecular electrostatic potential.1 CHARMM charges27 should reproduce quantum-mechanical interaction energies between solute and a solvent (water) molecule. Yet another strategy is applied in OPLS force-field,28,29 namely, charges are fitted to reproduce experimentally measured condensed phase properties. Recently, we have proposed a simple scheme30 that combined the above mentioned approaches in a methodically and numerically consistent way. The model takes into account the environment effects - the solvent molecules are explicitly taken into account. The charges for a water molecule were derived by fitting to molecular electrostatic potential. Interaction energies and experimental radial pair distribution function were also correctly reproduced. This approach is conceptually similar to the electrostatic embedding approach used in QM/MM15,16 and elongation techniques,31,32 namely electronic structure of the central molecule is determined in the external electrostatic field produced by point atomic charges of the environment. The difference lies in the fact that here the external charge distribution, instead of being fixed, is determined iteratively on the QM level. Water models are crucial in proper description of many molecular systems.33–36 In this respect, description of phenomena occurring at the air/ water interface is a good example. Modeling surface pressure-area isotherms of phospholipid monolayers faces the problem of surface tension.34,36–38 In fact, none of the most popular 3-point water models reproduces the surface tension of pure water. In addition, formation of unphysical pores in the monolayer systems were observed in laterally relaxed states. The 4-point models behave better, but the somehow artificial location of the additional charge on the bisector of the HOH angle does not improve the spatial distribution of charges. Therefore, in principle, 3-point models with modified description of the non-bonding interactions, should be able to lead to similar predictions. In this paper we applied the self-consistent polarization (SCP) method30 to trace the changes accompanying transfer of a water molecule from bulk phase to the interface. The SCP approach can be regarded as simplification of fragment molecular orbital method of Kitaura and Fedorov39,40 or an extension of self-consistent charge and configuration method for subsystems.41,42 The proposed computational scheme is a combination of

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preliminary MD simulations with quantum-chemical calculations performed on clusters cut from MD trajectories. In this study the clusters contained a central molecule picked from the water/vacuum phase boundary and its’ hemispherical solvation envelope. Like in MD/MC simulations, each water molecule in the clusters maintained its’ identity, i.e., charge transfer between them was not allowed.

2. Computational details The molecular systems studied in this work were built from MD simulations. A cubic box of c.a. 4000 TIP3P water molecules (box length of 50 A˚) was constructed using the vmd program.43 The system was equilibrated for 50 ns in canonical NVT ensemble at 298 K, with z dimension of the periodic box extended (200 nm) to accommodate for the interface. Next, 50 ns of production run was recorded. Hemisphere-shaped clusters for quantum-chemical calculations were cut from the interface trajectory every 50 ps. A reference (central) water molecule was picked from the central region of the phase boundary. All water molecules within the assumed radius (Rcutoff) from the centre of the hemisphere were included in the clus˚ were taken into account. In our previous study30 ter. Rcutoff ¼ 5, 6, 7, and 8 A we have shown that such Rcutoff values, including 2–3 solvation spheres, enable saturation of the polarization response in bulk water. Therefore, it can be expected that such hemispherical clusters should adequately represent water/air interface. Examples of clusters with Rcutoff ¼ 5 and 8 A˚ are shown in Fig. 1. The average number of molecules in the clusters varied form 16 for ˚ to 51 for Rcutoff ¼ 8 A ˚ . Analogous systems but without the interRcutoff ¼ 5 A face (same number of water molecules, initial box length in all three dimensions equal to 50 A˚), was built to model the bulk phase. NpT simulations at 298 K and 1 atm were performed. Next clusters for SCP calculations were chosen in the same fashion as for the interface system. The central molecule was now chosen from the middle of the box. Here, the average number ˚ to 98 for Rcutoff ¼ 8 A ˚ . Similar of molecules varied form 28 for Rcutoff ¼ 5 A 30 calculations were done in our previous study. Yet the results there, were obtained with differed computational procedure (system size, simulation time and conditions, averaging protocol, etc.). Those calculations are therefore repeated, in order to enable direct comparison between the systems. QM calculations were performed using Gaussian 09 suite of programs.44 Three electronic structure approaches were taken into account:

From bulk to surface—Transferability of water atomic charges

a) Rcutoff = 5 Å

401

b) Rcutoff = 8 Å

Fig. 1 Side view (top row) and top view (bottom row) visualization of hemispherical water clusters used in the self-consistent polarization method for the interfacial water. The central water molecule is highlighted by overlapping van der Waals radius spheres. The molecules selected for each cluster are shown in green. The remaining water molecules in the simulation box are shown in gray. Blue lines mark the simulation box edges. Panels a and b, correspond to Rcutoff equal to 5, and 8 Å, respectively.

DFT/B3LYP, DFT/M062X,45 and MP2. The same methods were also used in our previous work.30 cc-pVQZ46 basis set was applied. The charges were calculated from fitting to electrostatic potential. Merz-Kolman (MK)23 approach was used. It was shown previously that the choice of fitting procedure is of little importance in SCP calculations. A more important was the choice of the level of theory and basis-set. The cc-pVQZ basis was shown to give the best performance when deriving charges within the SCP scheme, i.e., guaranteed the proper convergence behavior. All calculations were performed within polarization approximation using SCP scheme. The scheme is an extension of the self-consistent charge and configuration method for subsystems (SCCCMS)41,42 to many interacting molecules or simplification of FMO of Kitaura and Fedorov.39,40 The SCP scheme may be summarized by a cycle of single-point energy calculations performed on each of the N molecules composing a molecular system M ¼ (1j2j…jij…j N):

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snierz and Jacek Korchowiec


 8 M M > E ð q , q , …, q Þ≡E > 2 3 N 1 1 q6¼1 > > > > > ⋮ < M ð q , … , q , q EM 1 i
1 i+1 , …, qN Þ≡E i ðq6¼i Þ i > > ⋮ > > >
 > > : E M ðq , q , …, q Þ≡E M q N

2

3

N
1

N

(1)

6¼N

The notation EM i (q6¼i) means that the energy of i-th molecule of M was computed in the presence of charges distributed on all remaining N
1 molecules: q1, …, qi
1, qi+1…, qN. The zero vector, q6¼i ¼ 0, corresponds to an isolated molecule in vacuum. The solid lines in M ¼ (1j2j…jij…j N) denote that charge-transfer between subsystems is not allowed (polarization approximation). The SCP cycle is carried out up to self-consistency in the total electron density, energy, and charge distribution. The charge distribution computed after the first iteration (charges derived in the field of isolated molecules) denotes polarized molecule in a non-polarized environment. The geometry of all water molecules was fixed (TIP3P structure). After determining SCP charges, surface tension was calculated from NVT simulations (100 ns, 298 K) of the interfacial system with various water models. The models tested include: TIP3P model47 with its’ original atomic charges, 6 models with TIP3P geometry and atomic charges from the SCP scheme corresponding to DFT/B3LYP, DFT/M06-2X, and MP2 levels of theory, in the interface and bulk systems. Two models from the OPC family were also includes: 3-point OPC348 model, and a 4-point OPC model.49 The latter system was chosen as a reference, since OPC model has be shown to correctly reproduce numerous structural and thermodynamic characteristics of water, in particular surface tension. Surface tension was calculated from variations between the lateral and normal pressure tensor components. Additionally, a 100 ns long NpT simulation (1 atm., 298 K) of the bulk system (without the interface) was run to calculate bulk properties. Final set of simulations included symmetric models of a DPPC monolayers (area per lipid equal to 100 A˚2) on pure water subphase at 298 K. The systems contained 100 lipids per monolayer and 30,000 water molecules. The same water models as in the γ calculations were employed and all-atom CHARMM force field was applied for lipids.27,50 All MD calculations were carried out using NAMD package.51

From bulk to surface—Transferability of water atomic charges

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3. Results and discussion The results of SCP calculations for the interface system are summarized in Fig. 2, which shows the charge distribution in the central water molecule as a function of simulation time. Straight horizontal lines in blue and red colors correspond to the average values. The shaded areas mark mean absolute deviations. As might be expected, the fluctuations of the oxygen charge are larger than those of the hydrogen atoms. The charges on the two hydrogen atoms average to the same value. Such result is obvious since none of hydrogen is distinguished and they are related by symmetry. The curves obtained for different methods are qualitatively similar. They differ in the average values (see Table 1). Charge fluctuations were similar in the bulk system. As expected, the averaged values were lower than in the interface system (Table 1). Table 1 gathers average values of the oxygen charge (qO) obtained for ˚ , as well as some literature data for comparcutoff radius (Rcutoff) equal to 8 A ison. The dependence of qO values on Rcutoff is shown in Fig. 3. Three plots obtained for various levels of theory are drawn. Two lines are depicted for each approach. The dashed lines correspond the central molecule in the field of non-polarized environment while the solid lines show the same for polarized environment. Rcutoff value equal to zero corresponds to vacuum. As expected the environment has a strong influence on the charge distribution. Results presented in Table 1 and Fig. 3 show that, in all environments the highest polarization is obtained with MP2 method, and the lowest with DFT/B3LYP method. The biggest effect is obtained in the first “step” of increasing polarization degree, i.e., when an isolated molecule is placed in the field of non-polarized environment of the smallest radius. The magnitude of oxygen charge further increases with Rcutoff, but relative differences between consecutive points is diminishing. This indicates that the curves are approaching asymptotic values. In the interface system (Fig. 3A), placing a water molecule in a non-polarized environment modifies qO by c.a. 20.0% for all three levels of theory. Environment polarization is a more subtle effect. The oxygen charge is further modified by 6–7%. In the bulk phase (Fig. 3B) polarization due to the presence of non-polarized environment is higher: the magnitude of qO is increased by c.a. 25%. The influence of environment polarization increases slightly to 8–9%. The total

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a) −0.5

H1

H2

Haver

0.6

−0.6

0.5

−0.7

0.4

qH

qO

B3LYP/ cc-pvqz Rcutoff = 5 Å

O

−0.8

0.3

−0.9

0.2

−1.0

0.1

−1.1

0.0 0

10

20

t [ns]

30

40

50

b) −0.5

H1

H2

Haver

0.6

−0.6

0.5

−0.7

0.4

qH

qO

M06-2X/ cc-pvqz Rcutoff = 7 Å

O

−0.8

0.3

−0.9

0.2

−1.0

0.1

−1.1

0.0 0

10

20

t [ns]

30

40

50

c) −0.5

H1

H2

Haver

0.6

−0.6

0.5

−0.7

0.4

qH

qO

MP2/ cc-pvqz Rcutoff = 8 Å

O

−0.8

0.3

−0.9

0.2

−1.0

0.1

−1.1

0.0 0

10

20

t [ns]

30

40

50

Fig. 2 Variations in ESP/MK charge distribution in the central water molecule during a 50 ns of MD simulation, for selected combinations of Rcutoff and SCP scheme. Panels a, b and c correspond to B3LYP, M062X and MP2 levels of theory (cc-pVQZ basis) and Rcutoff equal to 5, 7 and 8 Å, respectively. Color code: oxygen atom (left axes)—blue, hydrogen atoms (right axes)—red, yellow, and orange. Shaded areas represent mean deviation.

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From bulk to surface—Transferability of water atomic charges

Table 1 MK charges on the water oxygen atom calculated by SCP scheme. The results for the isolated molecules, interface system and bulk systems are given. Computational scheme B3LYP M062X MP2


0.70  0.00


0.71  0.00


0.73  0.00

Non-polarized environment


0.84  0.03


0.85  0.03


0.88  0.03

Polarized environment


0.89  0.04


0.90  0.04


0.92  0.04

Non-polarized environment


0.87  0.03


0.88  0.03


0.90  0.03

Polarized environment


0.93  0.04


0.94  0.04


0.96  0.04

TIP3


0.83

OPC3


0.89

OPC (MW site)


1.36

Vacuum Interface:

Bulk:

The charges are averaged over clusters cut from 50 ns long trajectory (100 frames, Rcutoff ¼ 8 A˚). Charges from TIP3P, OPC3, and 4-point OPC water models are also included for comparison.

polarization (relative to vacuum) reaches 28% in the interface system, and 34% in the bulk system. The polarization accompanying transfer of a water molecules from surface to bulk is therefore equal to c.a. 5% relative to the surface charge value, which corresponds to c.a. 0.04 e. Both bulk and interfacial charges obtained here are clearly different from the TIP3P model charges. As a consequence, when they are used in an MD simulation the description of intermolecular interactions will change. For this reason, it seems interesting to see how these differences translate onto surface tension (γ) of water. To check this, 100 ns long NVT simulations were performed for a water box at the water/vacuum interface. SCP charges from B3LYP, M06-2, and MP2 were tested. Plots of the surface tension along the simulations are presented in Fig. 4. Results for TIP3P charges, 3-point OPC3 and 4-pioint OPC water models are also included for comparison. Numerical values of the averaged surface tensions are gathered in Table 2. The experimental value of γ for water at 298 K equals to 72 mN/m.52 The value from TIP3P (52 mN/m) is largely underestimated, that of the 3-point OPC3 models is larger (64 mN/m), while that of the 4-point OPC model (72 mN/m) agrees with the experiment, as previously reported.33,35 When SCP charges determined for the interface system are employed in the simulation, the value of γ is corrected. The average values

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a) −0.65 −0.70 −0.75

qO

−0.80 −0.85 Interface B3LYP M06-2X MP2

−0.90 −0.95

dashed lines/open symbols: nonpolarized environment solid lies/symbols: polarized environment

−1.00 0

1

2

3

4

5

6

7

8

Rcutoff [Å]

b) −0.65 −0.70 −0.75

qO

−0.80 −0.85 Bulk: −0.90

B3LYP M06-2X MP2

−0.95 −1.00

dashed lines/open symbols: nonpolarized environment solid lies/symbols: polarized environment 0

1

2

3

4

5

6

7

8

Rcutoff [Å] Fig. 3 Dependence of central oxygen charge (qO) on cutoff radius, qO ¼ qO(Rcutoff), for different computational schemes: B3LYP/cc-pVQZ (green), M062X /cc-PVQZ (red), and MP2/cc-pVQZ (blue) levels of theory. Solid and dashed lines correspond to polarized and nonpolarized environment, respectively. Panel (A)—interface system, panel (B)—bulk system.

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From bulk to surface—Transferability of water atomic charges

a)

80 75

J [mN m-1]

70 65 60 55 50 SCP/B3LYP SCP/M06-2X SCP/MP2

45 40 0

10

20

30

TIP3 OPC OPC3

40

experimental

50 60 t [ns]

70

80

90 100

b) 90 85

J [mN m-1]

80 75 70 65 60

SCP/B3LYP SCP/M06-2X SCP/MP2

55

TIP3 OPC OPC3

experimental

50 0

10

20

30

40

50 60 t [ns]

70

80

90 100

Fig. 4 Variations of surface tension (γ) along 100 ns simulations of interfacial water with various charge schemes: SCP charges from B3LYP (green), M06–2X (red), and MP2 (blue) levels of theory, TIP3P charges (yellow), 3-point OPC3 (violet), and 4-point OPC water model (gray). Averages over 1 ns (light colored lines), 5 ns (points), and the last 20 ns of the simulations (horizontal lines) are shown. Experimental value is also shown as a dashed black line. Simulations shown in panel (A) were done with SCP charges determined for the interface. Panel (B) corresponds to SCP charges calculated for the bulk.

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Table 2 Surface tension obtained from MD simulations with varying water charges. Values for pure water surface and a DPPC monolayer at APL ¼ 100 Å2, T ¼ 298 K are presented. γ of pure water [mN/m] π of DPPC Interface-derived Bulk-derived γ of DPPC monolayer charges monolayer [mN/m] [mN/m] Water model charges

SCP/B3LYP 66.5  0.9

75.9  1.6

69.7  1.5


3.2

SCP/M062X

69.0  1.1

79.8  2.0

72.5  1.6


3.5

SCP/MP2

73.0  1.3

84.2  2.4

76.7  1.4


3.6

TIP3P

53.5  0.7

56.8  1.1


3.3

OPC3

63.8  1.2

66.8  1.9


3.2

OPC

71.7  1.2

74.3  1.8


2.6

Experiment

71.8

1 1

Surface pressure was calculated as π ¼ γ H2O
γ. Experimental values for γ H2O and π are from references,52,53 respectively.

obtained with SCP charges vary from 66 to 73 mN/m. The increase of γ follows the trend of increasing atomic charges. While the results from SCP/B3LYP or SCP/M06 schemes are slightly underestimated, the value obtained for SCP/MP2 procedure agrees with the experiment within the dynamical fluctuations (standard deviation of 1-ns block averages equals to 1 nN/m in all cases). When SCP charges determined from the bulk were inserted in MD simulation, the surface tension becomes overestimated. This is in line with the observed trend that γ increases with increased polarization of the water molecule. Also note, that employing charges derived from the bulk system increases fluctuations of γ and leads to larger computational error. Underestimated surface tension of water is also responsible for unphysical perforation of phospholipid monolayers in laterally relaxed states.36,38 In all-atom simulations with TIP3P water, premature appearance of pores in the monolayers (transition from liquid expanded to gaseous phase) was observed. In DPPC monolayers at 298 K stable pores were observed at areas ˚ 2. Changing the water model to one that per lipid (APL) as low as 63 A correctly reproduces the surface tension of water, e.g., OPC, leads to a ˚ 2, which is in agreement continuous monolayer for APL values up to 110 A 54 with experimental findings. Simulations of a DPPC monolayer

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From bulk to surface—Transferability of water atomic charges

˚ 2, pure water subphase, 298 K) were performed to examine (APL ¼ 100 A the behavior of the model with SCP charges derived for the interface. The APL value of 100 A˚2 was chosen because it corresponds to the transitory region of the π—A isotherm near the lift-up area. Behavior of the system at this APL should therefore be a sensitive test of possible imbalance between the water and lipid force fields. The results are depicted in Fig. 5. It can be seen that increasing the magnitude of atomic charges results in pore closure. The pore is present when TIP3P, OPC, and SCP/B3LYP models are used. It is closed in systems with OPC and SCP/MP2 models. The system with SCP/M06 charges exhibits an intermediate behavior—the pore is partially closed, but the monolayer is not completely continuous. Surface tension and surface pressure (π ¼ γ H2O
γ) values obtained from the simulations of DPPC system are listed in Table 2. It can be seen that all of the water models produce unphysical, negative values of π, i.e., surface tension in the DPPC system is higher than the surface tension of pure water. Such uniform trend suggests that this problem may be related to the parameterization of the lipids and possible imbalance between the lipid and water force fields. Increasing the value of γ H2O increases the energy cost of exposing water surface to vacuum, which results in pore closure. On the other (a) Initial structure

(b) TIP3P

(c) OPC3

(d) OPC

(e) SCP/B3LYP

(f) SCP/M06-2X

(g) SCP/MP2

Fig. 5 Initial (panel A) and final (panels B–G) structures from simulations of DPPC monolayers on pure water subphase at 298 K. The simulations were performed with the TIP3P water model with its’ original charges (B), 3-point OPC3 model (C), 4-point OPC water model (D), and 3 SCP models with charges from B3LYP (E), M06–2X (F), and MP2 (G) level of theory, (F). Color code: water—silver, lipids (no hydrogens)—green, N atoms from lipid headgroups—blue balls.

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hand, in order to close the pore, lipids attain highly extended conformations. Because the APL is large, some of the hydrophobic lipid chains come to direct contact with water. This kind of interaction is somehow overlooked in parameterization of lipid force fields, which is more concerned with the water-headgroups and chain-chain interactions. Another explanation for the negative π values may be the finite size effects. The pores observed in the simulations are in the scale of nm. The lipids lining the pore edges curl their zwitterionic headgroups towards the vacuum. Such system can produce large γ values since there is little bare water surface left without the headgroups. On the other hand, reproduction of experimental π in MD simulations is rarely regarded as quantitatively precise. In light of the error estimates included in Table 2, both experimental and calculated values can be described as “close to zero”. Also note, that the chosen APL corresponds to the region close to phase transition between liquid expanded (LE) and gaseous (G) phases. While it is useful as a test case, it is often of less importance in real isotherm calculations. The later are oriented on reproducing the whole isotherm, especially transition between LE and liquid condensed (LC) phases, where the surface tension is much lower due to lipid packing, and tail-water contacts are scarce.

4. Summary The aim of this work was to demonstrate the usefulness of selfconsistent polarization (SCP) approach in determining atomic charges for MD simulations. SCP was used to study behavior of water molecules on the air/water interface. It was shown that this method captures the changes in the molecule’s charge distribution upon transfer from the surface to bulk. The charge-corrected water model improved the value of the surface tension of water obtained from MD simulation. Furthermore, it was shown, that such adjustment in γ H2O removes some of the non-physical behavior of a DPPC monolayer at the air/water interface. Namely, it precludes premature pore formation in a laterally relaxed state of the monolayer. Notably, this improvement was gained without the need to introduce virtual sites in the water model. SCP therefore allows to keep the 30% lower computational cost of 3-point models. This is important in monolayer studies which require including large number of water molecules in the system in order to properly model the interface. Here, a traditional 3-point water model was safely used in simulations, since it accurately described the effect of the environment and its polarization.

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The computational protocol adopted in this study for charge determination involved preliminary MD simulations with fixed geometry and a standard general-purpose force field (TIP3P) and subsequent SCP calculations for clusters extracted from MD trajectories. The charges were then inserted into existing force fields without any further changes in the remaining parameters. As demonstrated, the model works surprisingly well for the water molecule used in this study, both in pure water and mixed lipid/water systems. This approach presents a number of advantages, which could be important in deriving charge distribution in novel molecules. Firstly, the polarization approximation adopted in the scheme maintains the identity of each molecule, which is a natural treatment for classical MD and MC simulations. Secondly, by computing atomic charges in the field of the environment molecules, the scheme takes many-body effects into account. This feature is especially desirable when studying systems with inhomogeneities, such as phase boundaries or membranous systems. As illustrated in the present study, in such systems underestimation of polarization effects, may lead to qualitative errors in the simulations, such as changing the phase of a monolayer. SCP approach is conceptually similar to the current CHARMM force field parameterization strategy, i.e., the charges are determined on the ground of modeling intermolecular interactions with the solvent. However, in SCP scheme numerous solvent molecules are included, instead of only one water molecule, as it is done in CHARMM. As shown here, environmental effect lead to 20–30% increase of oxygen charge in water. Therefore, charges obtained from interactions with a single molecule may be underestimated and thus falsify the simulation picture. SCP approach has the additional advantage that, by altering the composition of the clusters used for charge derivation, it can easily be used with solvents other than water and/or various physical environments (e.g., surface/bulk).

Acknowledgments The calculations were carried out at ACK CYFRONET with the support of the project “HPC Infrastructure for Grand Challenges of Science and Engineering”.

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Index Note: Page numbers followed by “f ” indicate figures and “t” indicate tables.

A Active-space coupled-cluster approaches, 157–158 Alkali metal dimers, 87 AMBER charges, 399 Amsterdam density functional (ADF) package, 380–381 Annihilation hole lines, 76 Annihilation particle lines, 76 ANO-L basis set, 340 Anti-Hermitian operator, 287–288 Atomic mean-field SO integrals (AMFI), 290 Atomic orbital (AO), 192

B Basis set superposition error (BSSE), 240–242 Becke-Perdew (BP86) exchange-correlation functional, 380–381 Bethe logarithm, 21 Bethe–Salpeter equations, 53 Bloch equation, 90–93 Bond-dissociation processes, 343–344 Bond orders (BOs), 193 Born–Handy method, 10–11 Born–Oppenheimer (BO) approximations, 4, 232 Born–Oppenheimer energy curves, 5–10 Born–Oppenheimer interaction energy, 240–241t Breit-Pauli Hamiltonian, 18–20 Brillouin–Wigner scheme, 93 Bulk helium, 233

C Center-of-mass motion, 268–270 Charge-transfer, 128–133, 344–345 CHARMM charges, 399 Chemical bond, H2 molecule, 361–363 Chemical-bond theory, 376

Chemical potential descriptors, 133–136 Classical Fisher (gradient) information, 121 Clebsch–Gordan coefficients, 271–273 Cluster operator, 92–93 Complete active space self-consistent field (CASSCF), 62, 198, 214, 291–292 Complete basis set (CBS), 40–41, 59, 195, 234, 237, 239–240, 243 Conditional probabilities, 126–128 Configuration interaction (CI) method, 232 Convection velocity, 118 Correlation energy, 75–76 Coulombic potential, 282 Coulomb interaction, 355–356 Coupled-cluster (CC) theory, 74, 103–104, 142–148, 160–161 Coupled Kohn–Sham (CKS), 42–43 Covalency and iconicity atomicity, 359–366

D Davidson procedure, 83 Deformation density, 378 Density functional theory (DFT), 38, 119, 194–195, 352–353, 380–381 Dewar-Chatt-Duncanson model, 381–383 Dimensionality reduction techniques, 142 Dirac Hamiltonian, 282–283, 286–290 Dirac’s equation, 282, 284 Direct nonadiabatic (DNA) approach, 4 Discrete momentum representation, 290 Double electron-attachment (DEA), 82, 88–89, 102 Double ionization potentials (DIP), 82, 88–89, 102 Double unitary coupled-cluster Ansatz (DUCC), 143–144, 150–152 Douglas–Kroll–Hess (DKH) transformation, 286–288 Downfolding techniques, 142–143, 157–158, 160–161 415

416

Index

E

F

Effective core potentials (ECPs), 98–100 Effective Hamiltonian operator, 90–91 Electron-attachment (EA), 82, 86–87 Electronegativity equalization (EE) principle, 129 Electronically excited (EE) states, 82 Electronic binding energies, 282 Electronic convection, 117–120, 128–136 Electronic ECG wave functions, 234–235 Electronic excited states, 84–85 Electron–nucleus attraction, 282 Energy decomposition schemes (EDA), 376 Entropic descriptors, chemical bonds in communication theory, 126–128 Equation-of-motion coupled cluster (EOMCC), 82–83, 158 DIP-EOM-CC and DEA-EOM-CC approaches, 88–90, 88t EE-EOM-CC, 84–85 IP-EOM-CC and EA-EOM-CC approaches, 86–88, 86t Equivalence theorem, 148, 157–158 Exact diagonalization ab initio (EDABI) method, 354 Exchange-correlation (XC) potentials, 42–43 Excitation energies, 84, 85t Excitation orders (EOs), 198–205 Expectation-value CC theory (XCC), 49–52 Expectation value expression, 101–102 Explicitly correlated Gaussian (ECG) function, 7, 232, 234, 264 Explicitly correlated Gaussians (ECGs) wave functions, 7–8 Extended random phase approximation (ERPA), 61–62 Extended Transition State—Natural Orbitals for Chemical Valence (ETS-NOCV), 377–378, 391 charge and bond-order matrix, 378 deformation density, 378, 385, 387f Ziegler-Rauk ETS bond-energy decomposition method, 379 External-ensemble analysis, 134

Factorizable CCD (FCCD), 243 Factorization procedure, 78–80 Fermi–Amaldi (FA) potential, 303 Fermi vacuum, 94–96 Feshbach resonance phenomenon, 209–210, 223–224 First-order regular approach (FORA), 284 Fock matrix elements, 301 Fock-space multireference coupled-cluster approach, 94–97 Foldy–Wouthuysen (FW) transformation, 282–283 Four-body Schrodinger equation, 5 Four-component wave function, 283 Four-electron antisymmetrizer, 234–235 Four-electron ECG functions, 236–237 Four-electron singlet spin function, 234–235 4-point OPC model, 402, 405–408 Free-particle Dirac Hamiltonian, 286–287 Free-particle Foldy–Wouthuysen (fpFW) transformation, 284, 286–287 Full configuration interaction (FCI), 40–41, 81, 84–85, 85t, 195, 232–233, 242–243, 352–353

G Gaussian functions, 236–237 Gaussian-type geminal (GTG), 233–234, 251–253 Gaussian-type orbitals/contracted Gaussiantype orbitals (GTO/CGTO), 292 Generalized Douglas–Kroll–Hess transformation, 286–288 Generalized exciton (GE), noninteger particle and hole charges first-order reduced density matrix (1RDM), 192–193 Kohn–Sham (KS) molecular orbitals (MOs), 192–195, 197 Global density-dependent (GDD) scheme, 53 G€ orling–Levy perturbation theory (GL2), 298–299 Green-function Monte Carlo (GFMC), 251 Green’s function applications, 155–157 Ground-state wave function, 74–75, 82–83

417

Index

H Hamiltonian matrix, 14–15, 172–174 Hartree–Fock (HF) equations, 74–75, 232 Hartree–Fock two-particle wave function, 352–353 He–He interaction energy, 243 Heitler and London (HL) approach, 352–353 Helium atom, 233–236 Helium dimer, 233–236, 238t, 255 Hellmann–Feynman theorem, 49–52 Hermitian CC downfolding techniques, 146–148 Hermitian CC flows, 149–152 Hermitian conjugate transformation, 287–288 Highest occupied molecular orbital (HOMO), 304–306 Hilbert-space multireference coupled cluster method, 90–94 H2Spectre program, 21–22 Hydrogen molecule, 4 Hylleraas functional, 59–60 Hylleraas–Slater bases, 232

I Infinite-order two-component (IOTC) method, 284–285, 285t, 288–293 Information theory (IT), 116–117 Interaction energies, 240–242, 244–246t, 248t, 249–250f, 252f Intermediate Hamiltonian (IH), 97–101 Intermediate neglect of differential overlap (INDO) approximation., 320 Internuclear separation, 240–241t Interparticle correlation effects, 264 Ionization potentials (IP), 82, 86–87

J Jacobi coordinate system, 214, 217–218 James–Coolidge (JC) wave function, 2, 5–7, 10

K Kato’s cusps, 2, 19–20 Kohn–Sham (KS) density functional theory (KS-DFT), 298

Kohn-Sham (KS)-Optimized effective potential equations (KS-OEP) LCAO OEP procedure, 304–309, 311 OEP-GL2 method, 307 Kołos and Wolniewicz calculations, 13, 18–19 Kołos–Wolniewicz wave function, 5–7, 10 Koopmans’ theorem, 173 Kronecker symbol, 80

L Lagrangian approach, 49–52 LAPACK numerical library, 173–174 Legendre polynomials, 215, 217–218 Lewis electron-pair bond, 376 Linear combination of atomic orbitals (LCAO), 299–300, 304–306, 311–312 Linear operator, 82–83 Linear parameters, 234–235 Linear response theory (LRT), 82 Linked cluster theorem, 142 Li–O2 system, cold collisions, 210–211 Feshbach resonance phenomenon, 209–210 high-spin interactions, 215–218 lithium superoxide, 212–213 low-spin potential, 219 magnetic Zeeman decelerator, 210–211 NaLi triplet molecules, 211 nonmagnetic molecules, 211–212 paramagnetic molecules, 211–212 potential anisotropy, 211–212 s-wave scattering, 209–210 sympathetic cooling, 211–212 thermalization, collisional cooling, 209–210 ultracold collision calculations, 219–224 velocity-controlled experiments, 209–210 Localized Orbitals from Bond-Order Operator (LOBO), 377 Long-range-corrected (LRC) density functional, 42–43 Lowest unoccupied molecular orbital (LUMO), 304–306

418

M Many-body perturbation theory (MBPT), 77–78, 159, 298–299 Many-electron wave functions, 232 Mean scattering length, 219–221 Methane dehydroaromatization, 321–323 Modulus-dynamics, 124 Molecular dynamics (MD), 398 Molecular electrostatic potential (MEP), 377–378, 380 Molecular hyperpolarizabilities, 336 Molecular orbitals (MOs), 172, 192 Molecular probability distribution, 125 Møller–Plesset partition, 243 MOLPRO program package, 198 Monomer-contraction (MC), 234–236 Monte Carlo (MC) simulations, 398 Mott–Hubbard localization, 363–364 Mrozek theory, 377 Mulliken’s interpolation, 135 Multiconfiguration self-consistent field (MCSCF) method, 340 Multireference coupled-cluster (MRCC) approach, 94–97

N Nalewajski theory, 377 Natural occupation numbers (NONs), 193–195 Natural Orbitals for Chemical Valence (NOCV), 377–385 Negative energy solutions, 283 Nicolaus Copernicus University, 104 Noisy intermediate-scale quantum devices (NISQ), 142, 159 Nonadiabatic perturbation theory (NAPT), 4 Nonapproximated spinorbital expression, 56–57 Non-Hermitian CC downfolding techniques, 144–146 Non-Hermitian CC flows, 148–149 Non-Hermitian matrices, 83 Nonlinear optical (NLO) properties, 336–337 Nonlinear optimization, 234–235 Nonlinear parameter optimization, 236–237

Index

Nonlinear variational parameters, 234–235 Nonparallel error (NPE), 158 Nonadiabatic basis explicitly correlated Gaussian (naECG) wave functions, 13–14 Nonadiabatic James–Coolidge wave functions, 14–18 Nonadiabatic perturbation theory (NAPT), 11–13 Nonrelativistic Hamiltonian, 268–270 Nonrelativistic non-Born–Oppenheimer (BO) calculation, 264–277 Nonrelativistic quantum electrodynamics (NRQED), 3 Nonstandard realizations, of coupled-cluster theory, 101–103

O One-electron reduced density matrix (1-RDM), 44–46 OPLS force-field, 399 Optimal fit parameters, 237 Optimal orbital-based strategy, 232–233 Optimized diagonal Ansatz (ODA), 60–61 Orbital calculations, 242–251, 244–246t, 248t, 249–250f Orbital communication theory (OCT), 116–117

P Pair couple-cluster doubles (pCCD), 103 Particle annihilation operators, 76 Perdew–Burke–Ernzerhof functional (PBE), 198 Perdew–Wang functional (PW91), 322–323 Phase-equalization problem, 124–125 Phase flow, 119 Phase supplements of classical entropic descriptors, 120–126 Physisorption of methane, 324 Polarization approximation, 39–40 Post-Hartree–Fock calculations, 290 Potential energy curves (PEC), 87, 89–90, 98–100

Q Quantum Chemistry Laboratory, 103–104 Quantum computing, 159–160, 160t

419

Index

Quantum electrodynamic corrections, 21–22 Quantum electrodynamics (QED), 3, 283 Quantum flows (QF), 144, 148–152, 157–158 Quantum Monte Carlo energies, 253 Quasilinear form, 78–80 Quasilinearization, 80–81

R Radium atom, 285 Raman spectroscopy, 213 Rayleigh–Ritz variational principle, 234–235 Real-time time-dependent configuration interaction singles and doubles (RT-TDCISD), 176–177, 180 bremsstrahlung-like radiation, 180–181 configuration state functions (CSFs), 170–171, 173 Crank–Nicolson propagator, 171–172 Dunning orbitals, 180 fixed-nuclei approximation, 175–176 Gaussian basis sets, 180–181 heuristic finite lifetime model, 177 high-harmonic generation (HHG), 168–169, 174–175, 180, 184f, 185 Kohn–Sham orbitals, 168–169 real-time propagation methods, 169 restricted Hartree–Fock (RHF) equations, 170–171 Slater determinants, 170–171 time-dependent Schr€ odinger equation, 168, 170–171 virtual-orbital truncation scheme, 181–182 Reference function, 143–144 Relativistic corrections, hydrogen molecule, 18–20 Relativistic quantum chemistry, 282–283, 285 Resolution-of-identity (RI) algorithm, 323–324 Response function formalism, 340 Restricted active space state interaction (RASSI) method, 290–291

S Scaled intermediate neglect of differential overlap (SINDO) method, 320 Schr€ odinger equation, 4, 11–13, 74–76, 82–83, 90, 283 Second-quantized formalism, 74–75 Second unitary transformation, 289 Self-consistent charge and configurationmethod for subsystems (SCCCMS), 400–402 Self-consistent polarization (SCP) method, 399–400, 410–411 Self-consistent subalgebra flow coupledcluster method (SCSAF-CC), 157–158 Semiempirical method of molecular orbitals self-consistent in charge and configuration (SCCC MO), 320 Shannon entropy, 128 Single-exchange approximation, 55–58 Single reference coupled cluster (SRCC) approach, 77–81 Single reference (SR) formulation, 74 Singular value decomposition (SVD), 303–304, 309–311 Slater determinant, 74–76 Spatial confinement box models, 338 Spectroscopic measurements, hydrogen molecule, 3 Spin-free approximation, 290 Spin-free formalism, 273 Spin-orbit (SO) coupling, 282 State-specific (SS) approaches, 93 Static correlation effects, 161 Stationary equilibrium in quantum mechanics, 123 String-based FCI algorithm, 81 Subsystem embedding subalgebras coupled-cluster (SES-CC) theorem, 143–144, 146 Symmetrized Rayleigh–Schrodinger (SRS) method, 39–40 Symmetry adaptation/symmetry forcing, 39–40 Symmetry-adapted perturbation theory (SAPT) energy decomposition analysis (EDA) methods, 38

420 Symmetry-adapted perturbation theory (SAPT) (Continued ) explicitly correlated SAPT, 59–61 interaction energy decomposition, 38 SAPT(CC), 49–52 SAPT(DFT), improvements to, 53–54 spin-flip SAPT, multiplet splittings, 63–66

T Tailored coupled-cluster (TCC), 101 Taylor expansion, 336 3-point OPC model, 402, 405–408 Time-dependent CC extensions, 153–154 TIP3P model, 402, 405–408 Transition density matrix (TDM), 44–46 Transition exciton (TE), 192 Trotter formula, 150–152 Two-dimensional harmonic oscillator potential, 340 Two-electron ECG function, 234–235 Two-electron reduced density matrices (2-RDMs), 46–47 Two-photon absorption (TPA) response, 337–338

Index

U Unitary coupled-cluster (UCC) approach, 102 Unitary matrix, 286–288 Unitary transformation, 287–289 Unrestricted HF (UHF), 77

V Valence bond (VB) approach, 366 Valence-universal (VU) method, 90–91 Variational optimization, 273 Variational quantum eigensolvers (VQE) approach, 142 Vertical excitation energies, 99t

W Wave function expansion, 2 Wave function theory (WFT)-based methods, 298–299 Wave operator, 94–95

Z Zeroth-order regular approach (ZORA), 284, 380–381 Zhao–Morrison–Parr scheme, 53