Table Of ContentPeter 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.)
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© 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
