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Peter R. Surjan Second Quantized Approach to Quantum Chemistry An Elementary Introduction With 11 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Peter R. Surj8n Chinoin Pharmaceutical and Chemical Works Ltd. Research Centre P.O. Box 110 H-1325 Budapest ISBN-13:978-3-642-74757-1 e-ISBN-13:978-3-642-74755-7 DOl: 10.1007/978-3-642-74755-7 Library of Congress Cataloging-in-Publication Data Surjan, Peter R., 1955- Second quantized approach to quantum chemistry: an elementary introduction with 11 figures / Peter R. Surjan. p. cm. Includes bibliographical references. 1. Quantum chemistry. 1. Title. QD462.S871989 541.2'8-dc20 89-37150 ISBN-13 :978-3-642-74 757-I(U .S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustration, recitation, broad casting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1st edition 1989 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Asco Trade, Hong Kong 215113020-543210 - Printed on acid-free paper To my wife Zsuzsa and to my parents Foreword The aim of this book is to give a simple, short, and elementary introduction to the second quantized formalism as applied to a many-electron system. It is intended for those, mainly chemists, who are familiar with traditional quantum chemistry but have not yet become acquainted with second quantization. The treatment is, in part, based on a series of seminars held by the author on the subject. It has been realized that many quantum chemists either interested in theory or in applications, being educated as chemi~ts and not as physicists, have never devoted themselves to taking a course on the second quantized approach. Most available textbooks on this topic are not very easy to follow for those who are not trained in theory, or they are not detailed enough to offer a comprehensive treatment. At the same time there are several papers in quantum chemical literature which take advantage of using second quantization, and it would be worthwhile if those papers were accessible for a wider reading public. For this reason, it is intended in this survey to review the basic formalism of second quantization, and to treat some selected chapters of quantum chemistry in this language. Most derivations will be carried out in a detailed manner, so the reader need not accept gaps to understand the result. Due to space limitations, however, the whole of quantum ~hemistry is not translated into second quantized language; the treat ment is kept as short as possible so as not to deter the reader by a voluminous book, while the material presented here is fully sufficient to completely under stand the approach. This allows everyone to continue hislher studies alone or by more advanced textbooks, some of which will be referred to at the appropriate places. Again to save space, some other related methods such as Green's function formalism or the diagrammatic perturbation theory, which are usually treated with second quantization on an equal footing, are not presented here. Merely the second quantized approach (particle number representation) will be elaborately discussed. However, a short review of some recent developments partly connected to the author's own work is included to illustrate the value and actuality of second quantization. A limited number of problems (excercises) is also included to help the reader to gain more experience in the formalism; the solutions are given at the end of the book. The text is detailed enough to make the inclusion of a larger number of problems unnecessary. Only the most important basic references are given, with the exception of the last few sections which contain more references since they discuss recent develop ments. VITI Foreword The author is convinced that second quantization is the simplest approach to the many-electron problem, and it should be available not only for physicists but for every chemist engaged in quantum chemistry. If this does not come about from the present treatment, it is the fault of the presentation and not of the second quantized approach. Budapest, April 1989 Peter R. Surjan Acknowledgements The author is very indebted to Prof. R. A. Poirier for his hospitality at the Memorial University of Newfoundland as well as to Prof. M. Kertesz for his hospitality at Georgetown University, Washington, D.C.; the majority of the contents of the present treatment is based on a series of seminars held at these universities. It is a pleasant task to thank all thos~ who agreed to review parts of this book, their suggestions contributed significantly to improving the presentation. Especially the kind efforts of J. Angyan, Gy. Ferenczy, I. Lukovits, I. Mayer, R. A. Poirier, M. Urban and C. Valdemoro are gratefully acknowledged. Last not least, the author wishes to express his gratitude to all senior and younger colleagues who have continuously helped in his pursuit towards the present publication. Writing the last chapters definitely would not have been possible without the extensive co-operation with Profs. Gabor Naray-Szab6, Istvan Mayer and others. Some of the research project reviewed in Sects. 10, 13 and 16 were partly supported by the Hungarian Academy of Sciences, under grants No. 64/1987 and 145/1985. Contents List of Symbols . XIII 1 Introduction . 1 1.1 Importance of Second Quantization 1 1.2 The One-Electron Model . . . . . 2 2 Concept of Creation and Annihilation Operators 6 2.1 The Vacuum State ...... . 6 2.2 Creating Electrons . . . . . . . 6 2.3 Particle Number Representation. 9 2.4 Annihilating Electrons ..... 10 2.5 Commutator Relation between Creation and Annihilation Operators ...................... . 12 2.6 The Adjoint Relation - Role of Orthogonality of One-Particle States ........................... . 14 2.7 Summary of the Properties of Creation/Annihilation Operators 16 3 Particle Number Operators . . . . . . . . . . . . . . . . . . . . 18 4 Second Quantized Representation of Quantum Mechanical Operators 22 4.1 General ....... . 22 4.2 One-Electron Operators .................. . 23 4.3 Two-Electron Operators .................. . 26 4.4 Second Quantized Form of the Born-Oppenheimer Hamiltonian 29 4.5 Hermiticity of Second Quantized Operators . . . . . . . . . . 30 5 Evaluation of Matrix Elements . . 33 5.1 Basic Matrix Elements .... 33 5.2 Concept of the Fermi Vacuum. 35 6 Advantages of Second Quantization - mustrative Examples 40 6.1 General ......... . 40 6.2 Overlap of two Determinants 41 6.3 Htickel Energy Expression . 42 6.4 Interaction of Two Electrons 44 Contents XI 7 Density Matrices . . . . . . . . 46 7.1 First-Order Density Matrix . 46 7.2 Second-Order Density Matrix 49 7.3 Hartree-Fock Energy Expression 51 8 Connection to "Bra and Ket" Formalism 53 9 Using Spatial Orbitals 59 10 Some Model Hamiltonians in Second Quantized Form 66 10.11t-Electron Hamiltonians 66 10.1.1 Huckel Level ........ . 66 10.1.2 Hubbard Model ....... . 70 10.1.3 Pariser-Parr-Pople (PPP) Model 72 10.2 Particle-Hole Symmetry ..... . 73 10.3 All-Valence Electron Hamiltonians 80 10.4 The Hartree-Fock Hamiltonian 83 11 The Brillouin Theorem ..... 87 12 Many-Body Perturbation Theory 93 13 Second Quantization for Nonorthogonal Orbitals . 103 13.1 Anticommutation Rules .......... . 103 ·13.2 The Hamiltonian in Nonorthogonal Representations . 108 13.3 Extended Huckel Theory . . . . . . . . . . . . . 111 14 Second Quantization and HeUmann-Feynman Theorem 114 14.1 General .................. . 114 14.2 Variation of Energy-Orthogonal Basis Set . . 116 14.3 Variation of Energy-Nonorthogonal Basis Set 117 14.4 Special Case: The SCF Gradient Formula 119 15 Intermolecular Interactions ........ . 121 15.1 The Operator for Interaction ..... . 121 15.2 Symmetry-Adapted Perturbation Theory 131 16 Quasiparticle Transformations . . 137 16.1 One-Particle Transformations 137 16.2 Two-Particle Transformations 141 16.3 A Theory of the Local Chemical Bond 145 xn Contents 17 Miscellaneous Topics Related to Second Quantization 153 17.1 Spin Operators and Spin Hamiltonians 153 17.2 Unitary Group Approach ........... . 159 18 Problem Solutions . 163 19 References 175 Index .... 181