Simple Theorems, Proofs, and Derivations in Quantum Chemistry

Transcription

1 Simple Theorems, Proofs, and Derivations in Quantum Chemistry

2 MATHEMATICAL AND COMPUTATIONAL CHEMISTRY Series Editor: PAUL G. MEZEY University of Saskatchewan Saskatoon, Saskatchewan FUNDAMENTALS OF MOLECULAR SIMILARITY Edited by Ramon Carb6-Dorea, Xavier Girones, and Paul G. Mezey MANY-ELECTRON DENSITIES AND REDUCED DENSITY MATRICES Edited by Jerzy Cioslowski SIMPLE THEOREMS, PROOFS, AND DERIVATIONS IN QUANTUM CHEMISTRY Istvan Mayer A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further infonnation please contact the publisher.

3 Simple Theorems, Proofs, and Derivations in Quantum Chemistry Istvan Mayer Chemical Research Center Hungariall Academy of Sciences Budapest, HUllgary Springer Science+Business Media, LLC

4 Library of Congress Cataloging-in-Publication Data Mayer, Istvan, Simple theorems, proofs, and derivations in quantum chemistryllstvan Mayer. p. cm. - (Mathematical and computational chemistry) ISBN ISBN (ebook) DOI / l. Quantum chemistry. I. Title. II. Series. QD462.M '8-dc ISBN Springer Science+Business Media New York Originally published by Kluwer Academic/Plenum Publishers, New York in 2003 Softcover reprint of the hardcover 1st edition A c.i.p. record for this book is available from the Library of Congress All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

5 To my wife Marti; and to the memory of my mother and aunts, who saved me and raised me in difficult times.

6 Preface Since 1983 I have been delivering lectures at Budapest University that are mainly attended by chemistry students who have already studied quantum chemistry in the amount required by the (undergraduate) chemistry curriculum of the University, and wish to acquire deeper insight in the field, possibly in preparation of a master's or Ph.D. thesis in theoretical chemistry. In such a situation, I have the freedom to discuss, in detail, a limited number of topics which I feel are important for one reason or another. The exact coverage may vary from year to year, but I usually concentrate on the general principles and theorems and other basic theoretical results which I foresee will retain their importance despite the rapid development of quantum chemistry. I commonly organize my lectures by treating the subject from the beginning, without referring explicitly to any actual previous knowledge in quantum chemistry-only some familiarity with its goals, approaches and, to a lesser extent, techniques is supposed. I concentrate on the formulae and their derivation, assuming the audience essentially understands the reasons for deriving these results. This book is basically derived from the material of my lectures. The special feature, distinguishing it from most other textbooks, is that all results are explicitly proved or derived, and the derivations are presented completely, step by step. True understanding of a theoretical result can be achieved only if one has gone through its derivation. Accordingly, this book may be considered a textbook "from the beginning but not for beginners." This means that to fully understand the book, only a familiarity with the simplest notions of quantum mechanics are required. These notions are the concepts of wave functions, operators and their matrix elements, the time-dependent and time-independent Schrodinger equation, as well as some basic knowledge of calculus, linear and matrix algebra. Any material which I suppose that may be absent from a chemist's curriculum is included either in the text or in an appendix. Therefore, anybody to whom the Vll

7 Vlll THEOREMS, PROOFS, AND DERIVATIONS IN QUANTUM CHEMISTRY book appears to be of interest based upon its title should have the necessary knowledge to read it. The book, however, is not intended as a first introduction to quantum chemistry (except, possibly, for a professional physicist); much emphasis is put on the fonnulae and their derivation and little on their practical utilization. (For these practical aspects I can recommend studying Veszpremi and Feher's Quantum Chemistry, Kluwer Academic/plenum Publishers 1999.) Thus, despite the limited amount of knowledge assumed, the book does not represent very easy reading: it requires the ability to follow the (sometimesnot-so-short) derivations and proofs. I think the title of the book reflects these aspects-relative simplicity and the emphasis put on fonnalism. The book contains many results supposed to be familiar to professionals working in the field, but not included in introductory textbooks. At the same time, it may be rather difficult to find sources containing detailed derivations or proofs of these "commonplaces," especially such derivations which do not use a more refined fonnalism, e.g., second quantization. No second quantization, no density matrices, no diagrams and practically no group theory are used in this book, so it is also suited well to the needs of those who have their background in chemistry, rather than in theoretical physics. Whereas the book is mainly intended for future professionals (students and young specialists) in theoretical chemistry and related fields (spectroscopy, material research, molecular physics etc.) I hope my fellow quantum chemists and molecular physicists will also find the book a useful source of simple proofs and derivations when they prepare their lectures. The material in the book is presented as simply as possible. The treatment remains at the so-called "physical level of rigor". I did not follow the conventional presentation if I could develop a better one. In particular, I treat all the variational problems based on the same equivalent refonnulation of the variation principle Eq.(2.13), which leads, for instance, to a very simple derivation of the Hartree-Fock equations via the Brillouin theorem. (However, in some cases the alternative derivations by using the Lagrangian multiplier technique are also shown for completeness.) The book is self-contained and for the most part avoids questions of priority. Therefore it does not contain extensive literature surveys, only short bibliographical notes quoting some basic references and giving credit to the sources where I have learned some non-standard material. Last, but not least, I should express my gratitude to my colleague Dr. Andrea Hamza who has read the entire manuscript and made substantial contributions to its improvement, I also thank Professor Peter R. Surjan, Professor Dalia Satkovskiene, Dr. Imre Papai and Dr. Gabor Schubert for their useful comments on some parts of the manuscript. The preparation of the manuscript has been done, for the most part, by Zsuzsa Kertesz. I am indebted to her for her meticulous work in placing the bulky fonnulae into LaTeX.

8 CONTENTS 1. THE BORN-OPPENHEIMER HAMILTONIAN 1. Separating the center of mass motion in quantum mechanics 1.1. Reducing the two-body problem to two one-body ones 1.2. The center of mass in quantum mechanics 1.3. Free atoms and atomcules The Born-Oppenheimer approximation Introductory remarks The Born-Oppenheimer separation 2.3. Why the Born-Oppenheimer separation is not exact 204. Approximate decoupling A note on the Born-Oppenheimer separation Notes 2. GENERAL THEOREMS AND PRINCIPLES The variation principle The Ray leigh quotient The variation principle for the ground state The variation principle as an equivalent of the Schrodinger equation: A useful formulation of the variation principle Eckart's inequality Excited states The Hellmann-Feynman theorem The differential Hellmann-Feynman theorem The integral Hellmann-Feynman theorem The virial theorem in quantum mechanics Notes Time dependence of a physical quantity. The virial theorem Scaling-a connection with the variation principle The virial theorem in the Born-Oppenheimer approximation.... The virial theorem and chemical bonding 1 I IX

9 x THEOREMS. PROOFS. AND DERIVATIONS IN QUANTUM CHEMISTRY 3. THE LINEAR VARIATIONAL METHOD AND LOWDIN'S ORTHOGONALIZATION SCHEMES 45 l. The linear variational method (Ritz method) Lowdin's symmetric orthogonalization Matrix S-1/ The S-1/2 transformation The Lowdin basis The stationary property of Lowdin 's symmetric orthogonalization scheme Lowdin orthogonalization: A two-dimensional example Linear independence of the basis and Lowdin's canonic orthogonalization l. Eigenvalues of the overlap matrix: A measure for the linear independence of the basis Lowdin's canonic orthogonalization Notes 4. PERTURBATIONAL METHODS Nondegenerate Rayleigh-Schrodinger perturbation theory The problem "Algebraic" expansion The use of the reduced resolvent in the Rayleigh-Schrodinger perturbation theory Wigner's 2n + 1 theorem The accuracy of the energy expectation value calculated by the nth order perturbational wave function Computing the exact energy corrections up to order 2n + 1 by using the first n wave function corrections Variational-perturbational method: The Hylleraas functional Degenerate Rayleigh-Schrodinger perturbation theory Brillouin-Wigner perturbation theory The size-consistency problem Size consistency of the Rayleigh-Schrodiger perturbation theory Formal considerations based on the properties of power series Size consistency of the perturbational expansions 6. Lowdin's partitioning method.. Notes DETERMINANT WAVE FUNCTIONS 1. Spin orbitals Many-electron spin states 3. Slater determinants

10 CONTENTS xi Two-electron examples The antisymmetrizing operator The projection character of the antisymmetrizing operator Commutation properties of the antisymmetrizing operator Antisymmetrization between distant subsystems Invariance of the determinant wave function with respect to "mixing" the occupied orbitals Matrix elements between determinant wave functions: Lowdin's general formulae for non orthogonal orbitals 6.1. Overlap General case Factorization Special cases for orthonormalized orbitals (Slater rules) 6.2. One-electron operators General case Special cases for orthonormalized orbitals (Slater rules) 6.3. Two-electron operators General case Special cases for orthonormalized orbitals (Slater rules) Lowdin's pairing theorem and its generalization 7.1. Lowdin's pairing theorem Karadakov's extended pairing theorem.... Theorem about the existence of orbitals of special structure 8.1. Existence theorem A posteriori determination Notes ,J THE HARTREE-FOCK METHOD The variation principle for single determinant wave functions: The Brillouin theorem The Brillouin theorem for the determinant with the absolute minimum of the energy The Brillouin theorem for a determinant with a stationary energy Algorithm of calculations based on the Brillouin theorem The Hartree-Fock equations Unrestricted Hartree-Fock equations Alternate derivation by using Lagrangian multipliers Alternate derivation by using specified variations Koopmans theorem The orbital energies and the total energy 184

11 Xli THEOREMS, PROOFS, AND DERIVATIONS IN QUANTUM CHEMISTRY The RHF method The RHF and UHF schemes Symmetry and the RHF method 4.3. The "dissociation catastrophe"; different Hartree-Fock methods Singlet and triplet excitations... Finite basis Hartree-Fock theory Hartree-Fock -Roothaan equations 5.2. The P-matrix Example of using projectors in the LCAO framework: The UHF equations for successive optimization of the orbitals The Fock matrix and the energy.. Restricted open-shell Hartree-Fock method 6.1. The ROHF equations The coupling coefficients for some open-shell systems The gradient of the energy Notes POPULATION ANALYSIS, BOND ORDERS, AND VALENCES 227 l. Mulliken's population analysis Electron density Population analysis Bond orders and valences 2.1. Bond order index Valence indices Exchange density and bond order 2.4. Bond orders in three-center bonds Notes 8. THE ELECTRON CORRELATION 1. The CI expansion The correlation energy and Nesbet's theorem 3. Methods of electron correlation calculations 3.1. The CI method MC-SCF methods The MBPT or M011er-Plesset method 3.4. The Coupled Cluster Approach Notes 9. MISCELLANEOUS 1. The liz expansion 2. The cusp condition 2.1. The nuclear cusp 2.2. The electronic cusp

12 CONTENTS Xlll 3. The asymptotical behavior of the wave function at very large distances Basis functions: The Gaussian product theorem The integral transformation problem 290 Notes Appendices 293 I. Separating the motion ofthe center of mass in classical mechanics 295 II. Reducing the two-body problem to two one-body ones in classical mechanics III. Analogy between differentials and variations 303 IV. Euler's theorem for homogenous functions 305 V. The virial theorem in classical mechanics VI. The electronic Schrodinger equation in atomic units 309 VII. The "bra-kef' formalism Dirac's "bra" and "kef' vectors Analogy with the matrix formalism The use of an overlapping basis Example of using the bra-ket formalism: The hypervirial theorem Projection operators Resolution of identity Spectral resolution of Hermitian operators The case of non-hermitian operators-biorthogonal sets of functions The trace of the projector VIII. Collection of formulas for Ray leigh-schrodinger perturbation theory (nondegenerate case) 323 IX. Direct products of matrices X. Permutations XI. An orthogonalization algorithm 331 Index 333