Brillouin-Wigner Methods for Many-Body Systems

BRILLOUIN–WIGNER METHODS FOR MANY-BODY SYSTEMS Progress in Theoretical Chemistry and Physics VOLUME 21 Honorary Edito...

Author: Ivan Hubac | Stephen Wilson

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BRILLOUIN–WIGNER METHODS FOR MANY-BODY SYSTEMS

Progress in Theoretical Chemistry and Physics VOLUME 21 Honorary Editors: W.N. Lipscomb (Harvard University, Cambridge, MA, U.S.A.) Yves Chauvin (Institut Français du Pétrole, Tours, France ) Editors-in-Chief: J. Maruani ( formerly Laboratoire de Chimie Physique, Paris, France) S. Wilson ( formerly Rutherford Appleton Laboratory, Oxfordshire, U.K.) Editorial Board: V. Aquilanti (Università di Perugia, Italy) E. Brändas (University of Uppsala, Sweden) L. Cederbaum (Physikalisch-Chemisches Institut, Heidelberg, Germany) G. Delgado-Barrio (Instituto de Matemáticas y Física Fundamental, Madrid, Spain) E.K.U. Gross (Freie Universität, Berlin, Germany) K. Hirao (University of Tokyo, Japan) R. Lefebvre (Université Pierre-et-Marie-Curie, Paris, France) R. Levine (Hebrew University of Jerusalem, Israel) K. Lindenberg (University of California at San Diego, CA, U.S.A.) M. Mateev (Bulgarian Academy of Sciences and University of Sofia, Bulgaria) R. McWeeny (Università di Pisa, Italy) M.A.C. Nascimento (Instituto de Química, Rio de Janeiro, Brazil) P. Piecuch (Michigan State University, East Lansing, MI, U.S.A.) S.D. Schwartz (Yeshiva University, Bronx, NY, U.S.A.) A. Wang (University of British Columbia, Vancouver, BC, Canada) R.G. Woolley (Nottingham Trent University, U.K.)

Former Editors and Editorial Board Members: ˆ I. Prigogine (†) I. Hubac (*) J. Rychlewski (†) M.P. Levy (*) Y.G. Smeyers (†) G.L. Malli (*) R. Daudel (†) P.G. Mezey (*) N. Rahman (*) H. Ågren ( *) S. Suhai (*) D. Avnir (*) O. Tapia (*) J. Cioslowski (*) P.R. Taylor (*) W.F. van Gunsteren (*) † : deceased; * : end of term For other titles published in this series go to www.springer.com/series/6464

Brillouin–Wigner Methods for Many-Body Systems

IVAN HUBACˇ Comenius University Bratislava Slovakia and Silesian University Opava Czech Republic

STEPHEN WILSON University of Oxford UK and Comenius University Bratislava Slovakia

123

Ivan Hubacˇ Comenius University Faculty of Mathematics, Physics and Informatics Mlynska Dolina F 1 842 48 Bratislava Solvakia Institute of Physics Silesian University 74601 Opava Czech Republic [email protected]

Stephen Wilson Theoretical Chemistry Group Physical & Theoretical Chemistry Laboratory University of Oxford South Parks Road Oxford OX1 3QZ United Kingdom [email protected]

ISBN 978-90-481-3372-7 e-ISBN 978-90-481-3373-4 DOI 10.1007/978-90-481-3373-4 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009940111 c Springer Science+Business Media B.V. 2010 No part of this work 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. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

PROGRESS IN THEORETICAL CHEMISTRY AND PHYSICS

A series reporting advances in theoretical molecular and material sciences, including theoretical, mathematical and computational chemistry, physical chemistry and chemical physics Aim and Scope Science progresses by a symbiotic interaction between theory and experiment: theory is used to interpret experimental results and may suggest new experiments; experiment helps to test theoretical predictions and may lead to improved theories. Theoretical Chemistry (including Physical Chemistry and Chemical Physics) provides the conceptual and technical background and apparatus for the rationalisation of phenomena in the chemical sciences. It is, therefore, a wide ranging subject, reflecting the diversity of molecular and related species and processes arising in chemical systems. The book series Progress in Theoretical Chemistry and Physics aims to report advances in methods and applications in this extended domain. It will comprise monographs as well as collections of papers on particular themes, which may arise from proceedings of symposia or invited papers on specific topics as well as initiatives from authors or translations. The basic theories of physics – classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics – support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the spectroscopic models employed in the determination of structural information from spectral patterns. Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry: it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions); molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, v

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Progress in Theoretical Chemistry and Physics

clusters and crystals; surface, interface, solvent and solid-state effects; excited-state dynamics, reactive collisions, and chemical reactions. Recent decades have seen the emergence of a novel approach to scientific research, based on the exploitation of fast electronic digital computers. Computation provides a method of investigation which transcends the traditional division between theory and experiment. Computer-assisted simulation and design may afford a solution to complex problems which would otherwise be intractable to theoretical analysis, and may also provide a viable alternative to difficult or costly laboratory experiments. Though stemming from Theoretical Chemistry, Computational Chemistry is a field of research in its own right, which can help to test theoretical predictions and may also suggest improved theories. The field of theoretical molecular sciences ranges from fundamental physical questions relevant to the molecular concept, through the statics and dynamics of isolated molecules, aggregates and materials, molecular properties and interactions, and the role of molecules in the biological sciences. Therefore, it involves the physical basis for geometric and electronic structure, states of aggregation, physical and chemical transformation, thermodynamic and kinetic properties, as well as unusual properties such as extreme flexibility or strong relativistic or quantum-field effects, extreme conditions such as intense radiation fields or interaction with the continuum, and the specificity of biochemical reactions. Theoretical chemistry has an applied branch – a part of molecular engineering, which involves the investigation of structure–property relationships aiming at the design, synthesis and application of molecules and materials endowed with specific functions, now in demand in such areas as molecular electronics, drug design or genetic engineering. Relevant properties include conductivity (normal, semi- and supra-), magnetism (ferro- or ferri-), optoelectronic effects (involving nonlinear response), photochromism and photoreactivity, radiation and thermal resistance, molecular recognition and information processing, and biological and pharmaceutical activities; as well as properties favouring self-assembling mechanisms, and combination properties needed in multifunctional systems. Progress in Theoretical Chemistry and Physics is made at different rates in these various research fields. The aim of this book series is to provide timely and in-depth coverage of selected topics and broad-ranging yet detailed analysis of contemporary theories and their applications. The series will be of primary interest to those whose research is directly concerned with the development and application of theoretical approaches in the chemical sciences. It will provide up-to-date reports on theoretical methods for the chemist, thermodynamician or spectroscopist, the atomic, molecular or cluster physicist, and the biochemist or molecular biologist who wishes to employ techniques developed in theoretical, mathematical or computational chemistry in their research programmes. It is also intended to provide the graduate student with a readily accessible documentation on various branches of theoretical chemistry, physical chemistry and chemical physics.

This book is dedicated to the late J. M´asˇik and to our children, Michelle, Jonathan & James, & grandchildren, Nathael & Kierann.

PREFACE

The purpose of this book is to provide a detailed description of Brillouin–Wigner methods and their application to the many-body problem in atomic and molecular physics and quantum chemistry. Recently there has been a renewal of interest in Brillouin–Wigner methods. This interest is fuelled by the need to develop robust, yet efficient multireference theoretical approaches to the electron correlation problem in molecules together with associated algorithms. Such theories are an essential ingredient of the quantum mechanical description of most dissociative processes in molecules, of excited states, and of ionization and electron attachment processes. This volume contains a concise, systematic and self-contained account of the Brillouin–Wigner methods. Chapter 1 is introductory giving the historical background to the Brillouin–Wigner methods and their application to atomic and molecular structure. Chapter 2 uses the partitioning technique to develop both single reference and multireference Brillouin–Wigner methods in a systematic fashion. The corresponding Rayleigh–Schr¨odinger expansions are also considered since it has been known for many years that they form the basis of a valid many-body theory (in the post-Brueckner sense). The many-body problem in atoms and molecules is discussed in Chapter 3. The properties of a valid many-body theory are elaborated and common approaches to the electron correlation problem are considered in some detail. The linked diagram theorem of many-body perturbation theory is described and the perturbative approach to the correlation problem considered alongside the configuration interaction and cluster expansion ansatz. In Chapter 4, the application of Brillouin–Wigner methods to many-body systems is described in some detail. This chapter deals with the central purpose of this monograph – the development of many-body Brillouin–Wigner methods in particular for applications to the problem of describing molecular electronic structure using ab initio methods – methods which start from first principles and can be systematically refined. (Semi-empirical methods and density functional theory, which in practice involves parametrization, will not be considered.) The application of Brillouin– Wigner theory to the configuration interaction and cluster expansion techniques is described as well as perturbation theory based methods. The use of Brillouin–Wigner ix

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Preface

methods in obtaining many-body corrections to theoretical approaches which are not valid many-body theories, such as limited configuration interaction, is addressed. Finally, Chapter 5 contains a summary and considers the prospects for future progress in the application of Brillouin–Wigner methods to the correlation problem in atoms and molecules and elsewhere. A number of colleagues have participated in the development of many-body Brillouin–Wigner methods over recent years. Without their dedicated efforts this work would not have been possible. We single out the late J. M´asˇik who contributed substantially to the recent development of Brillouin–Wigner methods in electronic structure theory before his untimely death at the age of only 33. We would also ˇ arsky, P. Mach, P. Neogr´ady, like to mention (in alphabetical order): P. Babinec, P. C´ P. Papp, J. Pittner, M. Pol´asˇek, H.M. Quiney and J. Urban. We are most grateful for their enthusiasm and their friendship. We are grateful to our wives for their encouragement and support during the writing of this book, without which this book would not have been completed. We thank Mr Radovan Javorˇc´ık, Charg´e dAffaires at the Embassy of the Slovak Republic, for facilitating many of our meetings in London. We thank Mrs Kathryn Wilson who undertook the labour of reading the whole book and corrected many careless slips and helped us with points of style. Nevertheless any errors in the present volume are ours alone and we would be grateful to any reader who takes the trouble to write to us on such matters. We have established a webpage at quantumsystems.googlepages.com/brillouin–wigner where we can both collect errors in the present volume and details of further developments in Brillouin–Wigner many-body theory. We are most grateful to the Royal Society for their permission to reproduce material from J.E. Lennard-Jones, Proceedings of the Royal Society of London A 129, 598, 1930, and to Mrs. Eileen Hamilton Wigner for her permission to reproduce material from E.P. Wigner, Math. u. Naturwiss. Anzeig. d. Ungar. Akad. Wiss LIII, 475, 1935. IH acknowledges support from the VEGA Grant agency, Slovakia, under project number 1/3040/06 and support by the Grant Agency of the Czech Republic under project number MSM 4781305903. SW is a senior academic visitor in the Physical and Theoretical Chemistry Laboratory, University of Oxford, and is grateful to the hospitality extended to him there. Bratislava and Oxford, July, 2009

Ivan Hubaˇc Stephen Wilson

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 PREAMBLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 HISTORICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Lennard-Jones’ 1930 paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Brillouin’s 1932 paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Wigner’s 1935 paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Studies in perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 PERTURBATION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Rayleigh–Schr¨odinger perturbation theory . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Comparison of Brillouin–Wigner and Rayleigh–Schr¨odinger perturbation theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.1 Advantages of the Brillouin–Wigner theory . . . . . . . . . . . . . . 1.3.3.2 Disadvantages of the Brillouin–Wigner theory . . . . . . . . . . . 1.4 THE MANY-BODY PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Linear scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The re-emergence of Brillouin–Wigner methods . . . . . . . . . . . . . . . . . 1.4.3 An overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 18 22 25 25 26 32 33

2 Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 THE PARTITIONING TECHNIQUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The partitioning technique for a single-reference function . . . . . . . . . 2.1.1.1 Model function and projection operators . . . . . . . . . . . . . . . .

37 37 38 38

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1 1 5 5 7 8 10 12 12 14

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2.1.1.2 An effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.3 The wave operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.4 The reaction operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The partitioning technique for a multi-reference function . . . . . . . . . 2.1.2.1 Model functions and projection operators . . . . . . . . . . . . . . . . 2.1.2.2 An effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.3 The wave operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.4 The reaction operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 PERTURBATION EXPANSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Single-reference function perturbation expansions . . . . . . . . . . . . . . . 2.2.1.1 Expansions of the inverse operator . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 Rayleigh–Schr¨odinger perturbation theory . . . . . . . . . . . . . . . 2.2.1.3 Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . . . 2.2.1.4 Generalized Brillouin–Wigner perturbation theory . . . . . . . . 2.2.1.5 Derivation of Rayleigh–Schr¨odinger perturbation theory from the Brillouin–Wigner perturbation expansion . . . . . . . . 2.2.2 Multi-reference function perturbation expansions . . . . . . . . . . . . . . . . 2.2.2.1 Multi-reference Rayleigh–Schr¨odinger perturbation theory . 2.2.2.2 Multi-reference Brillouin–Wigner perturbation theory . . . . . 2.2.2.3 An n-state system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The many-body problem in atoms and molecules . . . . . . . . . . . . . . . . . . . 3.1 LINEAR SCALING IN MANY-BODY SYSTEMS . . . . . . . . . . . . . . . . . . . 3.1.1 The exact electronic Schr¨odinger equation . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Independent particle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Many-body theories of electron correlation . . . . . . . . . . . . . . . . . . . . . 3.2 MANY-BODY PERTURBATION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Second-quantization formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Second-quantization and the many-body problem . . . . . . . . . 3.2.1.2 Creation and annihilation operators and the occupation number representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.3 Normal products, contractions and Wick’s theorem . . . . . . . 3.2.1.4 Particle–hole formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.5 The Hamiltonian operator in normal form . . . . . . . . . . . . . . . 3.2.2 Many-body Rayleigh–Schr¨odinger perturbation theory . . . . . . . . . . . 3.2.2.1 The many-body problem and quasiparticles . . . . . . . . . . . . . . 3.2.2.2 The algebraic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.3 Many-body Rayleigh–Schr¨odinger perturbation theory . . . . 3.2.2.4 Second-order contribution to the correlation energy . . . . . . . 3.2.2.5 Third-order contributions to the correlation energy . . . . . . . . 3.2.2.6 Fourth-order contributions to the correlation energy . . . . . . . 3.2.3 Many-body perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 43 44 44 45 47 48 48 48 48 50 51 53 56 58 58 64 66 68 69 69 70 73 76 77 78 78 81 87 90 94 94 94 96 98 101 108 109 110

Contents

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3.3 MANY-BODY THEORIES FOR ATOMS AND MOLECULES . . . . . . . . 3.3.1 The full configuration interaction method and limited configuration interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Cluster expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

4 Brillouin–Wigner methods for many-body systems . . . . . . . . . . . . . . . . . 4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 BRILLOUIN–WIGNER COUPLED CLUSTER THEORY . . . . . . . . . . . . . 4.2.1 Single-reference Brillouin–Wigner coupled cluster theory . . . . . . . . . 4.2.2 Multi-reference Brillouin–Wigner coupled cluster theory . . . . . . . . . 4.2.2.1 Multi-root formulation of multi-reference Brillouin–Wigner coupled cluster theory . . . . . . . . . . . . . . . . 4.2.2.2 Multi-root multi-reference Brillouin–Wigner coupled cluster: Hilbert space approach . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.3 Basic approximations employed in the multi-reference Brillouin–Wigner coupled cluster method . . . . . . . . . . . . . . . 4.2.3 Single-root formulation of the multi-reference Brillouin–Wigner coupled-cluster theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3.1 Single-root formulation of multi-reference Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . . . 4.2.3.2 Single-root multi-reference Brillouin–Wigner coupled cluster theory: Hilbert space approach . . . . . . . . . . . . . . . . . . 4.2.3.3 Single-root multi-reference Brillouin–Wigner coupled cluster single- and double-excitations approximation . . . . . . 4.2.3.4 Linear scaling corrections in Brillouin–Wigner coupled cluster theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 BRILLOUIN–WIGNER CONFIGURATION INTERACTION THEORY 4.3.1 Single-reference Brillouin–Wigner configuration interaction theory . 4.3.1.1 Brillouin–Wigner perturbation theory and limited configuration interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1.2 Rayleigh–Schr¨odinger perturbation theory and limited configuration interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1.3 ‘Many-body’ corrections for Brillouin–Wigner limited configuration interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Multi-reference Brillouin–Wigner configuration interaction theory . 4.3.2.1 Multi-reference Brillouin–Wigner perturbation theory for limited configuration interaction . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.2 A p-state system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.3 Multi-reference configuration interaction in Brillouin–Wigner form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.4 A posteriori Brillouin–Wigner correction to limited multi-reference configuration interaction . . . . . . . . . . . . . . . .

133 133 137 137 143

114 119 129

145 148 152 155 156 158 159 162 164 167 167 169 170 171 171 174 175 176

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4.4 BRILLOUIN–WIGNER PERTURBATION THEORY . . . . . . . . . . . . . . . . . 4.4.1 Single-reference Brillouin–Wigner perturbation theory . . . . . . . . . . . 4.4.2 Multi-reference Brillouin–Wigner perturbation theory . . . . . . . . . . . . 4.4.2.1 Multi-reference second-order Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 A posteriori correction to multi-reference Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 179 183 184

5 Summary and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 A L¨owdin’s studies in perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 201 B PROOF of the time-independent Wick’s theorem . . . . . . . . . . . . . . . . . . 211 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 C Diagrammatic conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 D Rayleigh–Schr¨odinger perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

NOMENCLATURE

We present a summary of the nomenclature employed in this volume. H H0 H1 Ψ Ψ0 Ψα E E0 Eα Φ Φ0 Φα Φm ΨαP E E0 Eα λ (p) E0 (p) Eα Ω Ω0 Ωα V

hamiltonian operator zero order hamiltonian operator perturbation operator exact wave function exact ground state wave function exact wave function for the state α exact energy exact ground state energy exact energy for state α zero order wave function zero order ground state wave function zero order wave function for state α model space model function energy of the reference system ground state energy of the reference system energy of the state α of the reference system perturbation parameter pth order energy coefficient for the ground state pth order energy coefficient for the state α wave operator wave operator for the ground state wave operator for the state α reaction operator

xv

xvi V0 Vα χp (p) χ0 (p) χα G R R0 B B0 Bα Bg P Q P Q Heffective Xi+ Xi δpq [A, B] [A, B]+ n[...] Yi+ Yi N [...] T T1 T2 T3 T4 Tp

Nomenclature reaction operator for the ground state reaction operator for the state α basis function pth order wave function for the ground state pth order wave function for the state α general resolvent resolvent for Rayleigh–Schr¨odinger theory resolvent for Rayleigh–Schr¨odinger theory resolvent for Brillouin–Wigner theory resolvent for Brillouin–Wigner theory resolvent for Brillouin–Wigner theory resolvent for generalized Brillouin–Wigner theory reference or model space orthogonal or complementary space projector onto the reference space P projector onto the orthogonal or complementary space Q effective hamiltonian operator creation operator in the particle formalism annihilation operator in the particle formalism Kronecker delta commutator AB − BA anticommutator AB + BA normal product in the particle formalism creation operator in the particle–hole formalism annihilation operator in the particle–hole formalism normal product in the particle–hole formalism cluster operator single excitation cluster operator single excitation cluster operator triple excitation cluster operator quadruple excitation cluster operator p-fold excitation cluster operator

ABBREVIATIONS

We present here a definition of the abbreviations used in this volume. CI BW RS CC PT MR FCI CISD MR - CI BWPT RSPT MBPT MPPT MR - BWPT MR - RSPT MR - MBPT MR - MPPT CCSD CCSD ( T )

configuration interaction Brillouin–Wigner Rayleigh–Schr¨odinger coupled cluster perturbation theory multi-reference full configuration interaction single-reference configuration interaction with single and double replacements multi-reference configuration interaction Brillouin–Wigner perturbation theory Rayleigh–Schr¨odinger perturbation theory many-body perturbation theory Møller–Plesset perturbation theory multi-reference Brillouin–Wigner perturbation theory multi-reference Rayleigh–Schr¨odinger perturbation theory multi-reference many-body perturbation theory multi-reference Møller–Plesset perturbation theory single-reference coupled cluster with single and double replacements single-reference coupled cluster with single and double replacements plus a perturbative estimate of the triple replacement component

xvii

xviii CCSDT MR - CCSD MR - CCSD ( T )

MR - CCSDT BWCC BWCCSD BWCCSD ( T )

MR - BWCC MR - BWCCSD MR - BWCCSD ( T )

CPA CPMET

Abbreviations single-reference coupled cluster with single, double and triple replacements multi-reference coupled cluster with single and double replacements multi-reference coupled cluster with single and double replacements plus a perturbative estimate of the triple replacement component multi-reference coupled cluster with single, double and triple replacements Brillouin–Wigner coupled cluster Brillouin–Wigner single-reference coupled cluster with single and double replacements Brillouin–Wigner single-reference coupled cluster with single and double replacements plus a perturbative estimate of the triple replacement component multi-reference Brillouin–Wigner coupled cluster multi-reference Brillouin–Wigner coupled cluster with single and double replacements multi-reference Brillouin–Wigner coupled cluster with single and double replacements plus a perturbative estimate of the triple replacement component coupled-pair approximation coupled-pair many-electron theory

ATOMIC UNITS

Atomic units are employed throughout this volume. In Table 1, we give the experimentally determined values of the basic atomic units.

Table 1. Basic atomic units.

Quantity

Symbol

Value

Planck’s constant = h/2π 1.05457267 × 10−34 Elementary charge e 1.60217733 × 10−19 9.1093897 × 10−31 Electron mass me

S.I. Units Js C kg

Relative uncertainty (ppm) 0.60 0.30 0.59

The atomic unit of length is the first Bohr radius a0 =

2 me e 2

The unit of energy is the Hartree EH =

e2 a0

In Table 2, we give numerical values of these atomic units together with various other derived atomic units.

xix

xx

Atomic Units Table 2. Some derived atomic units

Quantity Bohr radius, a0 Energy, EH Time Electric dipole moment Electric quadrupole moment Electric octopole moment Electric ﬁeld Electric ﬁeld gradient Polarizability (dipole) Hyperpolarizability Magnetic moment Magnetizability Magnetic vector potential Force constant (harmonic) Force constant (cubic) Force constant (quartic) Probability density

Value 5.2917726 × 10−11 4.3598 × 10−18 2.4189 × 10−17 8.4784 × 10−30 4.4866 × 10−40 2.3742 × 10−50 5.1423 × 1011 9.7174 × 1021 1.6488 × 1041 3.2063 × 10−53 1.8548 × 10−23 7.8910 × 10−29 1.2439 × 10−5 1.5569 × 103 2.9421 × 1013 5.5598 × 1023 6.7483 × 1030

S.I. Units m J s Cm Cm2 Cm3 Vm−1 Vm−2 C2 m2 J−1 C3 m3 J−2 JT−1 JT−2 mT Jm−2 Jm−3 Jm−4 m−3

1 INTRODUCTION

Abstract

An overview of the many-body problem in atomic and molecular physics and in quantum chemistry is given. Some historical background on the Brillouin–Wigner methodology is presented together with basic derivations of the Rayleigh–Schr¨odinger and the Brillouin– Wigner perturbation theories. The elementary formulations of the two theories are compared. The recent resurgence of interest in Brillouin–Wigner methodology, particularly in studies of the multi-reference correlation problem, is explained.

1.1. PREAMBLE With its roots in astronomy and classical physics, perturbation theory is the most general and systematic technique for the approximation of the solutions of quantum mechanical eigenvalue problems. The Schr¨odinger eigenvalue equation describes the motion of the electrons and the nuclei which are the fundamental ingredients of atoms and molecules. The solution of the appropriate Schr¨odinger equation is the key to understanding atomic and molecular structure and properties, as well as the interactions between atomic and molecular systems. The exact solution of the Schr¨odinger equation is associated with considerable mathematical and computational difficulties. Very few of the problems which occur in quantum mechanics can be solved exactly, often because they involve many particles. The development of systematic methods of approximation is, therefore, central to the theoretical models employed in the quantum many-body problem in studies of molecular physics and quantum chemistry. Perturbation theory relates the actual problem in hand to some simplified system for which solutions are known – the difference being treated as a ‘perturbation’. In quantum mechanics, this perturbation gives rise to an expansion in terms of a small parameter: an idea which has been employed in classical mechanics and, in particular, in solving problems in celestial mechanics. For example, in the case of a manyelectron atom or molecule, the simplified system (or reference model) might be that in which all electron–electron interactions are completely neglected – each electron moves in the field of the nucleus or nuclei – and the perturbation is taken to be the interactions between the electrons. An improved approximation scheme would take into account the averaged interactions between the electrons in the reference model – each electron moves in some mean field, treating only the instantaneous interactions 1

2

I. Hubaˇc and S. Wilson

between the electrons by means of a perturbation expansion. These instantaneous interactions are usually referred to as ‘electron correlation effects’. The component of the total energy associated with electron correlation effects is but a small fraction of the energy resulting from electron–electron interactions, if a suitably chosen mean field is employed in defining the independent particle model used as a reference. The accurate description of many-electron correlation effects remains one of the central problems of molecular physics and quantum chemistry [1–4]. Although independent particle models, such as the widely used Hartree–Fock theory, can account for all but a fraction of the total energy of a molecular system, this fraction is often as large or indeed larger than the energy changes which accompany chemical processes: binding energies, interaction energies, activation energies, barriers to rotation, and the like. The determination of the effects of electron correlation is therefore of crucial importance Molecular quantum mechanics was born in the late 1920s, but for many years the use of perturbation theory in the study of molecules was rather limited. Perturbation theory was used to describe the interactions between molecules at long range, but for the description of the structure and properties of isolated molecular systems, methods based on the variation theorem found favour. The reason for this preference is that many of the series encountered in atomic and molecular quantum mechanical problems were found to be poorly convergent or even divergent. Although as early as 1930, Dirac [5] had recognized that Even when the [perturbation] series does not converge, the first approximation obtained by means of it is usually fairly accurate. In early applications of non-relativistic quantum mechanics to atomic and molecular systems, variational techniques were preferred for the majority of practical applications. The non-relativistic Hamiltonian operator is a semi-bounded operator and the variation principle ensures that the energy expectation value corresponding to any approximate wave function lies above the corresponding exact energy. Approximate solutions of the Schr¨odinger equation could be constructed which depend on certain arbitrary parameters, which can be refined by invoking the variation principle. However, the theoretical description of heavy (and superheavy) atoms (and molecules containing them) demands a fully relativistic formulation. The development of quantum field theory in the late 1940s and the early 1950s saw perturbation theory as the only technique for obtaining information about the eigenvalues of the pertinent Hamiltonians for relativistic problems. The prospects for the approximate solution of the quantum mechanical equations governing the properties of atoms and molecules changed radically with the advent of the electronic digital computing machine around the middle of the twentieth century. The systematic implementation of approximation techniques could be automated. However, it was soon recognized that these first computers were too slow and too short on memory to seriously consider applications to molecules containing more than a few light atoms. Furthermore, theoretical progress was required both to

Brillouin–Wigner Methods for Many-Body Systems

3

cast existing theories in a form amenable to automatic computation and to tackle the formidable electron correlation problem. In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree–Fock equations – the ubiquitous independent particle model – in their matrix form. The Hartree– Fock equations describe the motion of each electron in the mean field of all the electrons in the system. Hall and Roothaan invoked the algebraic approximation in which, by expanding molecular orbitals in a finite analytic basis set, the integro-differential Hartree–Fock equations become a set of algebraic equations for the expansion coefficients which are well-suited to computer implementation. McWeeny [8] and, independently, Boys [9] introduced the Gaussian basis function, which held the key to practical applications to arbitrary polyatomic molecules because of the ease and accuracy with which the associated molecular integrals could be evaluated. The (cartesian) Gaussian basis functions have the form (1.1)

χi = χi (ζi , i , mi , ni ) = xi y mi z ni exp ζi r2

r = (x, y, z) where x, y and z are the cartesian coordinates. ζi is a screening constant. The integers i , mi and ni determine the nodal structure of the Gaussian basis functions. The single particle state functions, φμ , of the independent particle model are approximated as φμ = (1.2) χi cμ,i i

where the coefficients are determined by iterative solution of the matrix Hartree–Fock equations – the self-consistent field method. However, it has to be noted [10] that the use of Gaussian basis functions was not initially greeted with enthusiasm and initially they were not used extensively – even by their proponents. The second half of the twentieth century saw a sustained attack on the correlation problem in atoms and molecules. By the mid-1950s the basic structure of correlated wavefunctions was understood, thanks mainly to developments in solid state physics and nuclear physics. The linked diagram theorem of the many-body perturbation theory and the connected cluster structure of the exact wavefunctions were firmly established. Goldstone [11] exploited diagrammatic techniques developed in quantum field theory [12–15], to complete Brueckner’s work [16] on the scaling of energies and other expectation values with the number of electrons in the system. Hugenholtz [17] provided an alternative approach. The exponential ansatz for the wave operator suggested by Hubbard [18] in 1957 was first exploited in nuclear physics by Coester [19] and by Coester and K¨ummel [20]. In applications to the atomic and molecular electronic structure problem – the problem of describing the motion of the electrons in the field of clamped nuclei – there was, as Paldus [21] describes, an initial hope that the configuration interaction approach limited to doubly excited configurations, originating from a single-reference state, [would] provide a satisfactory description of correlation effects.

4

I. Hubaˇc and S. Wilson

However, this approach, which exploits the variation theorem to determine the correlated wavefunction for the non-relativistic problem, was soon thwarted by the slow convergence of the configuration interaction expansion. The 1960s saw the applications of the many-body perturbation theory developed during the 1950s by Brueckner [16], Goldstone [11] and others, to the atomic structure problem by Kelly [22–31]. These applications used the numerical solutions to the Hartree–Fock equations which are available for atoms, because of the special coordinate system. Kelly also reported applications to some simple hydrides in which the hydrogen atom nucleus is treated as an additional perturbation. At about the same ˇ ızˇ ek [32] developed the formalism of the coupled cluster approach for use in time, C´ the context of molecular electronic structure theory. By the early 1970s, both the coupled cluster theory and the many-body perturbation theory had been implemented in the algebraic approximation and applications to ˇ ızˇ ek and Shavitt [33] arbitrary molecular systems became a reality. In 1972, Paldus, C´ initiated applications of ab initio coupled cluster theory using finite basis set expansions, whilst in the following year Kaldor [34] first invoked the algebraic approximation in an application of the many-body perturbation theory to the hydrogen molecule ground state. In 1976, Wilson and Silver [35] compared finite order many-body perturbation theory with limited configuration interaction calculations when both methods are formulated in the algebraic approximation. Progress was particularly rapid during the late 1970s and early 1980s with the introduction of a new generation of ‘high performance’ computing machines which enabled the realization of practical schemes of calculation which in turn gave new levels of understanding of the nature of the electron correlation problem in atoms and, more particularly, in molecules. It became widely recognized that a successful description of correlation effects in molecules must have two key ingredients: 1. It must be based either directly or indirectly on the linked diagram theorem of many-body perturbation theory, so as to ensure that the calculated energies and other expectation values scale linearly with particle number. 2. It must be based on a careful and systematic realization of the algebraic approximation (i.e. the use of finite basis set expansions), since this can often be the dominant source of error in calculations which aim to achieve high precision. The past 20 years have witnessed a relentless increase in the power of computing machines. It has been observed1 that the processing power of computers seems to double every eighteen months. As J.M. Roberts [36] points out No other technology has ever improved so rapidly for so long. This ever-increasing computing power has led to both higher accuracy in molecular electronic structure calculations, often because larger basis sets can be utilized, and has opened up the possibility of applications to larger molecules and molecular systems. 1

By Mr. Gordon Moore of Intel. This has been dubbed “Moore’s Law”.

Brillouin–Wigner Methods for Many-Body Systems

5

Today, there remain a number of problems in molecular electronic structure theory. The most outstanding of these is undoubtedly the development of a robust theoretical apparatus for the accurate description of dissociative processes which usually demand the use of multi-reference functions. This requirement has recently kindled a renewal of interest in the Brillouin–Wigner perturbation theory and its application to such problems. This volume describes the application of Brillouin–Wigner methods to many-body systems and, in particular, to molecular systems requiring a multi-reference formalism. This introductory chapter is organized as follows: In Section 1.2, we give some historical background to the Brillouin–Wigner perturbation theory and its application in the study of problems in molecular quantum mechanics. The Brillouin–Wigner and the more familiar Rayleigh–Schr¨odinger perturbation theories are presented in elementary form in Section 1.3 and the advantages and disadvantages of the Brillouin– Wigner expansion enumerated. In Section 1.4, we consider the re-emergence of Brillouin–Wigner methods in recent years particularly in handling problems requiring the use of a multi-reference formalism. Section 1.4 concludes with an outline of the remaining chapters of this volume. 1.2. HISTORICAL BACKGROUND The theory which is today called “Brillouin–Wigner perturbation theory” was introduced in three seminal papers published in the 1930s. The first of these, which appears to be not so well known as the other two, was published in 1930 by J.E. Lennard-Jones [37]. (Indeed, some authors [38, 39] refer to the method as “Lennard-Jones–Brillouin–Wigner perturbation theory”.) Subsequently, L. Brillouin published his famous paper in 1932 [40] whilst E.P. Wigner’s paper appeared some 2 years later [41]. In this section, we provide a brief synopsis of each of these important papers. During the period 1963–1971, P.-O. L¨owdin published2 a series of papers [42–53] with the general title “Studies in Perturbation Theory”, which afforded deep insight into perturbation theory expansions, the relation between different expansions and their application to quantum mechanical problems. We conclude this section with a brief overview of L¨owdin’s work on perturbation theory. 1.2.1. Lennard-Jones’ 1930 paper The seminal paper on what is nowadays called “Brillouin–Wigner perturbation theory” by Lennard-Jones3 was published in the Proceedings of the Royal Society of London in 1931. It was communicated by R.H. Fowler and received on 1st September, 1930. Here we reproduce Lennard-Jones’ introduction: 2 3

The last paper in the series was coauthored by O. Goscinski. Sir John Edward Lennard-Jones (1894–1954).

6

I. Hubaˇc and S. Wilson

Perturbation problems in quantum mechanics J.E. Lennard-Jones Department of Theoretical Physics, The University, Bristol (Communicated by R.H. Fowler, F.R.S. – Received September 1, 1930) Introduction One of the great achievements of the Schr¨odinger wave-mechanics is the elegance of its perturbation theory, which has brought many problems, formerly considered intractable, within the range of highly-developed mathematical techniques. It is not necessary at this stage to review the numerous applications which have been made of this perturbation theory or to dwell upon its many advantages. The important advance towards an understanding of chemical forces which it has made possible is in itself a considerable achievement. There are, however, certain disadvantages in the perturbation theory in its present form, which limit the extent of its applications to complex problems of atomic and molecular structure. If the interaction of atoms, for instance, is to be calculated, as is most desirable, improved methods will have to be found. One such improvement is considered in this paper. In its present form, it is easy to calculate the first approximation of the energy of a system, subject to small perturbations, but difficult to proceed further. This is a considerable disadvantage in those problems where the first approximation vanishes as in calculating the Stark effect or the van der Waals attraction of two atoms at large distances apart. Moreover, the theory expresses the perturbed eigenfunction in terms of all the unperturbed eigenfunctions of the system and these are not always known. This paper shows how, by a slight modification of the usual method, these difficulties may be overcome and the energy and eigenfunction of a perturbed system calculated to higher approximations with comparative ease. As an illustration the method is applied to calculate the van der Waals fields of two hydrogen atoms at large distances, and this is done with more ease and directness than the usual form of the theory permits. It is possible, too, to extend the theory to calculate the van der Waals fields of more complicated atoms, as it is hoped to show in a later paper. The results of the Schr¨odinger perturbation theory are also obtained by another method, which has the advantage of exhibiting exactly what is neglected in the usual successive approximations. This alternative method is not limited to small perturbations; in fact, it is shown that certain perturbation problems can be solved, however strong the perturbation. One such problem is that of rotating polar molecules under the influence of an external electric field, which is a necessary step in the theory of gaseous dielectrics. A solution to this problem is given and the range of validity of the usual dielectric theory is thus determined.

(Taken from Proceedings of the Royal Society of London A 129, 598, 1930)

Brillouin–Wigner Methods for Many-Body Systems

7

We should emphasize the point made by Lennard-Jones that the alternative method is not limited to small perturbations; in fact, it is shown that certain perturbation problems can be solved, however strong the perturbation. As we have noted already, some authors, such as Dalgarno [38] and Wilson [39], term what is nowadays called “Brillouin–Wigner perturbation theory” “Lennard-Jones– Brillouin–Wigner perturbation theory” in recognition of the seminal contribution of Sir John Lennard-Jones. 1.2.2. Brillouin’s 1932 paper Brillouin’s4 seminal 1932 paper on perturbation theory was entitled “Les Probl`emes de Perturbation et les Champs self-consistents”. It was published in 1932 in Le Journal de Physique et Le radium. It would appear that Brillouin was not aware of the earlier work by Lennard-Jones, since the earlier work is not cited. Here we reproduce the abstract of his paper, which is in French and which was submitted for publication on 22nd July, 1932.

Les probl` emes de perturbations et les champs self-consistents L. Brillouin Manuscript rec¸u le 22 juillet 1932 Sommaire Le probl`eme des perturbations, en mecanique ondulatoire, peut eˆ tre traite rigoureusement et aboutit a` une equation seculaire, e´ crite sous forme de determinant; on retrouve facilement les formules de Schr¨odinger pour un probl`eme non d´egen´er´e ou d´egen´er´e; mais cette equation permet aussi d’´etudier le cas d’une grosse perturbation agissant sur un syst`eme qui poss`ede des niveaux d’´energie tr`es voisins. La m´ethode des champs self-consistents est expos´ee, sous une forme tr`es directe qui donne non seulement la premi`ere approximation, mais aussi toute la matrice de perturbation qui subsiste apr`es cette premi`ere approximation. Deux types de champs self-consistents sont compar´es, celui de Hartree et celui de Fock. Le second donne une matrice de perturbation plus faible, mais sans pouvoir annuler les termes complet`ement. Les formulas e´ tablies dans cet article seront ult´erieurement utilis´ees pour l’etude des electron libres dans les m´etaux.

(Taken from Le Journal de Physique et le Radium S´eries VII, Tome III, 373, 1932)

4

L´eon Brillouin (1877–1972).

8

I. Hubaˇc and S. Wilson

Brillouin points out that the new perturbation series converges much more rapidly than the power series of Schr¨odinger. 1.2.3. Wigner’s 1935 paper Wigner’s5 contribution to Brillouin–Wigner perturbation theory appeared some 2 years after Brillouin’s paper in 1935. It was published in Mathematischer und Naturwissenschaftlicher Anzeiger der Ungarischen Akademie der Wissenschaften. Wigner cites the earlier work of Brillouin but not that of Lennard-Jones. The first part of Wigner’s paper is in Hungarian:

A Rayleigh–Schr¨ odinger-f´ ele perturb´ aci´ o-elm´ elet egy m´ odos´ıt´ as´ ar´ ol Wigner Jen˝ o-t˝ ol Ismeretes, hogy m´ıg a Rayleigh–Schr¨odinger-f´ele perturb´aci´os elm´elet els˝o k¨ozel´ıt´ese mindig t´ul magas eredm´enyt ad a legals´o energian´ıv´ora, semmi hasonl´o nem a´ ll´ıthat´o a m´asodik e´ s magasabb k¨ozel´ıt´esekre. A jelen dolgozat c´elja egy olyan k¨ozelit˝o elj´ar´as kidolgoz´asa, mely a legals´o energianiv´ot mindig fel¨ulr˝ol k¨ozel´ıti meg, e´ pp´ugy, mint a vari´aci´os m´odszer, melynek mindenik k¨ozel´ıt´ese a k´erd´eses H + V operatornak egy k¨oz´ep´ert´eke eqy normaliz´alt hull´amf¨uggv´enyre. Ha H + V k¨oz´ep´ert´ek´et a (4a) alatt adott f¨uggv´enyre kisz´am´ıtjuk (φk H-nak Ek hoz tartoz´o saj´atf¨uggv´enye), a feladat nem m´as, mint az ´ıgy kapott (4b) kifejez´esnek minimumm´a val´o t´etele. Ha a sz´aml´al´oban lev˝o kett˝os o¨ sszeget elhagyjuk, az ´ıgy egyszer˝us´ıtett kifejez´es az a-knak (5a) alatt adott e´ rt´eke mellett lesz legkisebb. Ebben (2) a kifejez´esben F1 az eg´esz minimaliz´aland´o kifejez´es e´ rt´eke az a-k ezen e´ rt´eke mellett e´ s meghat´aroz´as´ara a (6) implicit egyenlet szolg´al. Ezen implicit egyenlet megold´asa teh´at egy - a Rayleigh–Sch¨odinger egyenlet harmadik k¨ozel´ıt´es´enek megfelel˝o, - de mindig t´ul magas e´ rt´eket ad az energi´ara. A magasabb k¨ozel´ıt´eseket m´ar most u´ gy nyerj˝uk, hogy az eml´ıtett kett˝os o˝ sszegben, mely az a-knak biline´aris, kifejez´ese, az egyik faktort az el˝oz˝o k¨ozel´ıt´esnek megfelel˝o kifejez´essel helyettes´ıtj¨uk, mig a m´asik faktort valtozatlanul ismeretlennek tekinj¨uk. Az ´ıgy nyert kifejez´eseket a (7a) alatt adott a-k teszik legkisebb´e. A (7a)-ben (3) szerepl˝o F1 megint az eg´esz minimaliz´aland´o kifejez´es e´ rt´eke ezen a-rendszer mellett (meghat´aroz´as´ara a (7) egyenlet szolg´al) e´ s ´ıgy ism´et egy fels˝o hat´ar´at k´epezi a legkisebb energia´ert´eknek. Az e´ ppen le´ırt k¨ozelit˝o kifejez´esek a Rayleigh–Schr¨odinger-f´ele elm´elet megfelel˝o kifejez´esein´el valamivel egyszer˝uebbek e´ s ´ıgy k¨onnyen a´ ltal´anos´ıthat´ok n = ∞-re, mely esetben a saj´at´ert´ekegyenletnek egy form´alis megold´as´at adj´ak egy v´egtelen sor alakj´aban. Ezen v´egtelen sort el˝osz¨or L. Brillouin adta meg, a n´elk¨ul azonban, hogy a r´eszlet¨osszegek viszony´at a val´odi megold´ashoz megvizsg´alta volna. Bizonyos esetekben k¨onnyen ki lehet mutatni, hogy Brillouin v´egtelen sorai a val´odi megold´ashoz knoverg´alnak, e´ s vil´agos, hogy a jelen elj´ar´as sok esetben konverg´al, amikor az eredeti nem haszn´alhat´o. V´eg¨ul a nyert k¨ozelit˝o kifejez´es egy Mathieu egyenletre alkalmaztatik p´eldak´eppen. 5

Eugene P. Wigner (1902–1995), Nobel Prize in Physics, 1963.

Brillouin–Wigner Methods for Many-Body Systems

9

Wigner’s paper then continues in English:

On a modification of the Rayleigh–Schr¨ odinger perturbation theory Eugene Wigner From the meeting of the IIIrd class of the Hungarian Academy of Sciences on the 12th November 1934 1. The Rayleigh–Schr¨odinger perturbation theory6 gives an explicit power series in λ for the characteristic values Fn and the characteristic functions φn of a Hermitean operator H + λV (1)

(H + λV )φn = Fn φn

if the corresponding quantities En and ψn for the unperturbed operator H are known (1a)

Hψn = En ψn .

If the so-called matrix elements of V are denoted, as usual, by (2)

∗ Vnm = (ψn , V ψm ) = Vnm

the first terms in the aforementioned series read (3a)

Fn(2) = En + λVnn + λ2

(3b)

φ(1) n

= φn + λ

k

k

|Vnk |2 En − Ek

Vkn ψ. En − Ek

Generally only these first terms of the series are used in actual calculations, the higher order terms become increasingly complicated. We shall fix our attention on the lowest energy value, F1 . While it is evident that (1) the first approximation for this F1 = E1 + λV11 is always greater than the real amount – since it is the expectation value of a normalized wave function ψ1 ; nothing like this holds for the second and higher approximations. It even happens quite often that the last series in (3a) diverges in cases when the lowest energy value is finite itself. In these cases, of course, Rayleigh–Schr¨odinger perturbation theory is inapplicable to the problem. The aim of the present paper is to give an approximation formula for F1 which always yields values that are too high, and which can be proved to converge at least in certain simple cases. Such an expression is naturally provided by the variational method which had been used frequently indeed in cases for which the general shape of the characteristic functions could be obtained by physical considerations. 6

J.W.S. Rayleigh, The Theory of Sound, London and New York, 1894, vol. 1, p. 113; E. Schr¨ odinger, Collected Papers on Wave Mechanics, London and Glasgow, 1928, p. 64.

10

I. Hubaˇc and S. Wilson The final result, the ∞-approximation, will appear in the form of an infinite series. This infinite series was first found by L. Brillouin7 who obtained it by an intuitive consideration of the usual scheme. He has already pointed out in his important paper that his series converges much more rapidly than the power series of Schr¨odinger. He has not investigated, however, the successive approximations and their relations to the actual problem.

(Taken from Math. u. Naturwiss. Anzeig. d. Ungar. Akad. Wiss LIII, 475, 1935)

Wigner repeats Brillouin’s observation that the new series converges much more rapidly than the power series of Schr¨odinger. The papers by Lennard-Jones, by Brillouin and by Wigner represent the genesis of what is now termed “Brillouin–Wigner perturbation theory”. 1.2.4. Studies in perturbation theory During the 1960s, a series of papers with the general title Studies in perturbation theory were published by P.-O. L¨owdin8 [42–53]. The first paper in the series appeared in the Journal of Molecular Spectroscopy in 1963 [42]. A final and 14th paper in the series was co-authored by O. Goscinski and appeared in 1971 in the International Journal of Quantum Chemistry [53]. This series of papers used the partitioning technique to develop perturbation series in a very general way. The series provided a deep insight into the use of perturbation methods in the study of quantum eigenproblems. We shall consider the partitioning technique in detail in Chapter 2. Here we list the titles of the 14 papers with the general title “Studies in perturbation theory” authored by L¨owdin and, in the case of the fourteenth, Goscinski. The abstracts of these papers are reproduced in Appendix A. For further details, the reader should consult the original publications. 1. Journal of Molecular Spectroscopy 10, 12 (1963) Studies in Perturbation Theory. I. An Elementary Iteration-Variation Procedure for Solving the Schr¨odinger Equation by Partitioning technique 2. Journal of Molecular Spectroscopy 13, 326 (1964) Studies in Perturbation Theory. II. Generalization of the Brillouin–Wigner Formalism 7

L. Brillouin, Le Journal de Physique et le Radium S´eries VII, Tome III, 373, 1932. 8 Per-Olov L¨ owdin (1916–2000).

Brillouin–Wigner Methods for Many-Body Systems

11

3. Journal of Molecular Spectroscopy 13, 326 (1964) Studies in Perturbation Theory. III. Solution of the Schr¨odinger equation under a variation of a parameter 4. Journal of Mathematical Physics 3, 969 (1962) Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism 5. Journal of Mathematical Physics 3, 1171 (1962) Studies in Perturbation Theory. V. Some Aspects on the Exact Self-Consistent Field Theory 6. Journal of Molecular Spectroscopy 14, 112 (1964) Studies in Perturbation Theory. VI. Contraction of secular equations 7. Journal of Molecular Spectroscopy 14, 119 (1964) Studies in Perturbation Theory. VII. Localized perturbation 8. Journal of Molecular Spectroscopy 14, 131 (1964) Studies in Perturbation Theory. VIII. Separation of the Dirac equation and study of the spin-orbit coupling and Fermi contact terms 9. Journal of Mathematical Physics 6, 1341 (1965) Studies in Perturbation Theory. IX. Connection Between Various Approaches in the Recent Development – Evaluation of Upper Bounds to Energy Eigenvalues in Schr¨odinger’s Perturbation Theory 10. The Physical Review 139, A357 (1965) Studies in Perturbation Theory. X. Lower Bounds to Energy Eigenvalues in Perturbation-Theory Ground State 11. The Journal of Chemical Physics 43, S 175 (1965) Studies in Perturbation Theory. XI. Lower bounds to energy eigenvalues, ground state, and excited state 12. in Perturbation theory and its applications in quantum mechanics, ed. C.H. Wilcox, p.255, Wiley, New York (1966) The Calculation of Upper and Lower Bounds of Energy Eigenvalues in Perturbation Theory by means of Partitioning Technique 13. International Journal of Quantum Chemistry 2, 867 (1968) Studies in Perturbation Theory. XIII. Treatment of Constants of the Motion in Resolvent Method, Partitioning Technique, and Pertubation Theory 14. International Journal of Quantum Chemistry 5, 685 (1971) Studies in Perturbation Theory. XIV. Treatment of Constants of the Motion, Degeneracies and Symmetry Properties by Means of Multidimensional Partitioning

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I. Hubaˇc and S. Wilson

1.3. PERTURBATION THEORY Brillouin–Wigner perturbation theory has a number of advantages over the more widely used Rayleigh–Schr¨odinger theory. It also has significant disadvantages which have resulted in its neglect as the basis for many-body theories over the past 50 years. Before describing these advantages and disadvantages, let us recall the basic structure of the Brillouin–Wigner and Rayleigh–Schr¨odinger perturbation theories. In this section, we shall first provide an ‘elementary’ derivation of the Brillouin– Wigner perturbation theory and then present a comparable introduction to the Rayleigh–Schr¨odinger perturbation theory. The section concludes with a comparison of the advantages and disadvantages of the two theories. 1.3.1. Brillouin–Wigner perturbation theory We seek to develop approximate solutions of the time-independent Schr¨odinger eigenvalue equation (1.3)

ˆ μ = Eμ Ψμ , μ = 0, 1, 2, . . . HΨ

ˆ is the Hamiltonian operator, Eμ is the energy eigenvalue, and the wave where H function, Ψμ is the eigenfunction for the μth state. For present purposes we shall not concern ourselves with the precise form of the Hamiltonian. A perturbation theory is developed by starting from some reference or model defined by the equation (1.4)

ˆ 0 Φk = Ek Φk , H

k = 0, 1, 2, . . .

for which the solutions are known. We write the total Hamiltonian operator in eq. (1.3) as the sum of the zero-order Hamiltonian operator from eq. (1.4) and a perturbation operator (1.5)

ˆ =H ˆ 0 + λH ˆ1 H

where the parameter λ is used to interpolate between the unperturbed problem (λ = 0) and the perturbed problem (λ = 1). Perturbation theories are developed by making expansions for the exact eigenvalue, Eμ , and the corresponding eigenfunction, Ψμ . Equation (1.5) allows the Schr¨odinger equation (1.3) to be written in the form ˆ 0 Ψμ = λH ˆ 1 Ψμ (1.6) Eμ − H which can then be rewritten in matrix form (1.7)

ˆ 1 |Ψμ (Eμ − Em ) Φm |Ψμ = λ Φm | H

(1.8)

Φm |Ψμ =

or ˆ 1 |Ψμ λ Φm | H (Eμ − Em )

In eqs. (1.7) and (1.8), the state |Φm is the unperturbed eigenket of H0 with eigenenergy Em . We find it convenient to put

Brillouin–Wigner Methods for Many-Body Systems (1.9)

13

Φμ |Ψμ = 1

in order to simplify the following formulae. Thus, |Ψμ is not normalized to unity. Equation (1.9) is termed the ‘intermediate normalization condition’. Setting m = μ in eq. (1.7) and using eq. (1.9), the exact energy eigenvalue is given by ˆ 1 |Ψμ (1.10) Eμ = Eμ + λ Φμ | H The exact eigenstate for the μth state is constructed using the completeness of the unperturbed basis, |Φm , so that (1.11) |Ψμ = |Φm Φm |Ψμ m

or (1.12)

|Ψμ = |Φμ Φμ |Ψμ +

|Φm Φm |Ψμ

m(=μ)

where we note that the summation excludes the case m = μ. Using eq. (1.8) for Φm |Ψμ and eq. (1.9), we can write eq. (1.12) as 1 ˆ 1 |Ψμ Φm | H (1.13) |Ψμ = |Φμ + λ |Φm (Eμ − Em ) m(=μ)

We have included the parameter λ in eq. (1.13) which is set equal to unity in order to recover the perturbed problem. Equation (1.13) is the basic formula of the Brillouin– Wigner perturbation theory for a single-reference function. From eq. (1.13) we can develop a series expansion for the exact eigenfunction |Ψμ , in powers of λ with coefficients depending on the perturbed energy Eμ , rather than the energy of the model Em , as would be the case in the more familiar Rayleigh– Schr¨odinger perturbation series. Iterating this basic formula, we find 1 ˆ 1 |Φμ |Ψμ = |Φμ + λ Φm | H |Φm (Eμ − Em ) m(=μ)

+ λ2

|Φj

j(=μ),m(=μ)

(1.14)

1 1 ˆ 1 |Φm Φj | H (Eμ − Ej ) (Eμ − Em )

ˆ 1 |Φμ + · · · × Φm | H

The above procedure resembles the Lippmann–Schwinger method [106] for constructing the incoming state in scattering theory. By introducing the Brillouin–Wigner type propagator (resolvent) 1 Φm | , (1.15) B = B (μ) = |Φm (Eμ − Em ) m(=μ)

eq. (1.14) for the exact wave function for the μth state can be written in the more compact form: ˆ 1 + λ2 BH ˆ 1 BH ˆ 1 + · · · |Φμ (1.16) |Ψμ = 1 + λBH

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I. Hubaˇc and S. Wilson

Substituting the expansion (1.14) for the exact wave function into eq. (1.10) for the exact energy eigenvalue, the perturbed energy values are given by ˆ 1 |Φμ Eμ = Eμ + λ Φμ | H ˆ 1 |Φm Φμ | H + λ2 m(=μ)

+ λ3

ˆ 1 |Φj Φμ | H

j(=μ),m(=μ)

(1.17)

1 ˆ 1 |Φμ Φm | H (Eμ − Em ) 1 1 ˆ 1 |Φm Φj | H (Eμ − Ej ) (Eμ − Em )

ˆ 1 |Φμ + · · · × Φm | H

or, in terms of the Brillouin–Wigner-type resolvent defined in eq. (1.15),

(1.18)

ˆ 1 |Φμ + λ2 Φμ | H ˆ 1 BH ˆ 1 |Φμ Eμ = Eμ + λ Φμ | H ˆ 1 BH ˆ 1 BH ˆ 1 |Φμ + · · · + λ3 Φμ | H

It should be noted that the expansion (1.18) is not a simple power series expansion in λ, since the Eμ which appears in the denominators of B also depends on λ. However, if the terms 1/ (Eμ − Em ) are expanded in powers of λ, then the usual Rayleigh– Schr¨odinger series for Eμ is recovered which is, as is well known, a simple power series in λ. In Brillouin–Wigner perturbation theory, we obtain the following expansion for the ground state energy of the perturbed systems (1.19)

[1]

[2]

[3]

E0 = E0 + E0 λ + E0 λ2 + E0 λ3 + · · ·

where E0 is the ground state energy of the reference or model system and the energy [1] [2] [3] coefficients, E0 , E0 , E0 , . . . , are given by p−1 [p] ˆ1 ˆ 1 BH (1.20) E0 = Φ0 | H |Φ0 and B = B (0) is the Brillouin–Wigner resolvent for the ground state. We use square brackets to distinguish the Brillouin–Wigner energy coefficients from the Rayleigh– Schr¨odinger coefficients for which round brackets are used. We reiterate our observation that the expansion (1.19) is not a simple power series in λ, since each of the (p) energy coefficients, E0 [p > 1] depends on the exact energy E. 1.3.2. Rayleigh–Schr¨odinger perturbation theory In Rayleigh–Schr¨odinger perturbation theory, the expansion for the exact ground state energy of the perturbed system can also be written in the form (1.21)

(1)

(2)

(3)

E = E0 + E0 λ + E0 λ2 + E0 λ3 + · · ·

where λ is again the perturbation parameter. An expansion in powers of λ is also made for the exact ground state wave function Ψ0 , i.e.

Brillouin–Wigner Methods for Many-Body Systems (1.22)

(1)

(2)

15

(3)

Ψ0 = Φ0 + χ0 λ + χ0 λ2 + χ0 λ3 + · · ·

(p)

(p)

where χ0 is the pth order wave function. The energy coefficients, E0 , are given by p d E (p) (1.23) E0 = dλp λ=0 (p)

whilst the pth order wave function, χ0 , is given by p d Ψ0 (p) (1.24) χ0 = . dλp λ=0 The Rayleigh–Schr¨odinger perturbation theory is developed by substituting the power series for the wave function (1.22) and that for the energy (1.21) into the Schr¨odinger equation (1.3) in the form ˆ 1 Ψ0 = E0 Ψ0 ˆ 0 + λH (1.25) H and then equating powers of λ. Explicitly, this equation then takes the form ˆ 1 Φ0 + χ(1) λ + χ(2) λ2 + χ(3) λ3 + · · · = ˆ 0 + λH H 0 0 0 (1) (2) 2 (3) 3 E0 + E0 λ + E0 λ + E0 λ + · · · (1) (2) (3) Φ0 + χ0 λ + χ0 λ2 + χ0 λ3 + · · · . (1.26) Equating powers of λ leads to the basic equations of Rayleigh–Schr¨odinger perturbation theory. The zero-order equation is simply the eigenvalue problem for the reference or model system with respect to which the perturbation expansion is developed: (1.27)

ˆ 0 Φ0 = E0 Φ0 H

The first-order equation takes the form (1.28)

ˆ 0 χ(1) + H ˆ 1 Φ0 = E0 χ(1) + E (1) Φ0 H 0 0 0

the second-order equation is (1.29)

ˆ 0 χ(2) + H ˆ 1 χ(1) = E0 χ(2) + E (1) χ(1) + E (2) Φ0 H 0 0 0 0 0 0

and so on. In general, the pth order Rayleigh–Schr¨odinger perturbation equation takes the form q=p−1 (q) (p−q) (p) ˆ 1 χ(p−1) = ˆ 0 χ(p) + H E0 χ0 + E0 Φ0 . (1.30) H 0 0 q=0

Without loss of generality, we can require that the pth order perturbed wave func(p) tions χ0 be orthogonal to the reference function, Φ0 : (1.31)

(p)

Φ0 |χ0 = 0, ∀p.

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I. Hubaˇc and S. Wilson

Multiplying the zero-order equation (1.27) from the left by Φ0 and integrating gives ˆ (1.32) E0 = Φ0 H 0 Φ0 which defines the zero-order energy. Similarly, multiplying the first-order equation (1.28) from the left by Φ0 and integrating gives the first-order energy ˆ (1) (1.33) E0 = Φ0 H 1 Φ0 . Thus, the first-order energy is given by the matrix element of the perturbation operator for the reference wave function. An expression for the second-order energy can be obtained by multiplying the second-order equation (1.29) by Φ0 and integrating giving ˆ (1) (2) (1.34) E0 = Φ0 H 1 χ0 . The second-order energy therefore depends on the first-order wave function. An expression for the first-order wave function may be obtained by rearranging the first-order perturbation equation (1.28) as ˆ 0 χ(1) = H ˆ 1 − E (1) Φ0 E0 − H (1.35) 0 0 so that (1.36)

(1) ˆ 1 − E (1) Φ0 ˆ 0 −1 H χ0 = E0 − H 0

(1.37)

(1) ˆ 1 − E (1) Φ0 χ0 = R H 0

or

where (1.38)

ˆ 0 −1 R = E0 − H

is the Rayleigh–Schr¨odinger resolvent. Substituting eq. (1.37) for the first-order wave function into expression (1.34), gives the following expression for the second-order energy

(2) ˆ 1 RH ˆ 1 Φ0 . (1.39) E0 = Φ0 H ˆ 1 RH ˆ1 Thus the second-order energy is given by the matrix element of the operator H for the reference wave function. Multiplying the third-order equation by Φ0 and integrating gives

(3) ˆ 1 χ(2) (1.40) E0 = Φ0 H 0 and, in general, (1.41)

(p)

E0

ˆ 1 χ(p−1) . = Φ0 H 0 (2)

An expression for the second-order wave function χ0 , can be obtained from the second-order perturbation equation (1.29) and then substituted in expression (1.40).

Brillouin–Wigner Methods for Many-Body Systems

17

(p−1)

can be obtained from the (p−1)th In general, the (p−1)th order wave function χ0 order perturbation equation and then substituted in eq. (1.41). The energy coefficients in the Rayleigh–Schr¨odinger perturbation expansion have a more complicated structure than those in the Brillouin–Wigner expansion and the first few orders take the form (a) Zero-order energy coefficient ˆ 0 |Φ0 E0 = Φ0 | H

(1.42)

(b) First-order energy coefficient (1)

E0

(1.43)

ˆ 1 |Φ0 = Φ0 | H

(c) Second-order energy coefficient (2)

E0

(1.44)

ˆ 1 RH ˆ 1 |Φ0 = Φ0 | H

(d) Third-order energy coefficient (3)

E0

(1.45)

(1)

ˆ 1 RH ˆ 1 RH ˆ 1 |Φ0 − E Φ0 | H ˆ 1 (R)2 H ˆ 1 |Φ0 = Φ0 | H 0

(e) Fourth-order energy coefficient (4)

E0

ˆ 1 RH ˆ 1 RH ˆ 1 RH ˆ 1 RH ˆ 1 |Φ0 − E (1) Φ0 | H ˆ 1 (R)2 H ˆ 1 |Φ0 = Φ0 | H 0 (1)

(1)

ˆ 1 |Φ0 + (E )2 Φ0 | H ˆ 1 |Φ0 ˆ 1 RH ˆ 1 (R)2 H ˆ 1 (R)3 H − E0 Φ0 | H 0 (1.46)

(2) ˆ 1 (R)2 H ˆ 1 |Φ0 . − E0 Φ0 | H

In these expressions, R is the Rayleigh–Schr¨odinger resolvent for the ground state, which can be written in sum-over-states form as 1 Φr | . (1.47) R = R (0) = |Φr (E0 − Er ) r(=0)

This should be compared with the Brillouin–Wigner resolvent (1.15) which for the ground state takes the form 1 Φr | (1.48) B (0) = |Φr (E0 − Er ) r(=0)

and which depends on the exact ground state energy E0 . 1.3.3. Comparison of Brillouin–Wigner and Rayleigh–Schr¨odinger perturbation theories In the preceding section, we have given elementary presentations of both the Brillouin–Wigner perturbation theory and the Rayleigh–Schr¨odinger perturbation

18

I. Hubaˇc and S. Wilson

theory. We are now in a position to make an elementary comparison of the two methods. We approach this comparison by first presenting the advantages of the Brillouin– Wigner perturbation theory with respect to the Rayleigh–Schr¨odinger perturbation theory, and then enumerating the disadvantages. 1.3.3.1. Advantages of the Brillouin–Wigner theory We consider the advantages of the Brillouin–Wigner perturbation theory on the basis of the elementary derivations given in Sections 1.3.1 and 1.3.2. Brillouin–Wigner perturbation theory has five main advantages over the Rayleigh– Schr¨odinger perturbation theory. These are: (i) Rayleigh–Schr¨odinger theory can be regarded as an approximation to Brillouin– Wigner theory. This perspective on Brillouin–Wigner perturbation theory was mentioned by Lennard-Jones in his seminal 1930 paper [37]. It was also described in the review by Dalgarno [38] published in 1961. [2] Consider, for example, the second-order energy, E0 , in the Brillouin–Wigner perturbation expansion for the ground state energy. From the third term on the right-hand side of eq. (1.17), we see that this may be written in sum-over-states form as (1.49)

[2]

E0 =

Φ0 | H ˆ 1 |Φr Φr | H ˆ 1 |Φ0 r=0

E0 − Er

.

If we make the approximation (1.50)

E0 ∼ E0 ,

then we are led immediately to the Rayleigh–Schr¨odinger second-order energy component (1.51)

(2)

E0

=

Φ0 | H ˆ 1 |Φr Φr | H ˆ 1 |Φ0 r=0

E0 − Er

which is identical to eq. (1.44) upon substituting (1.47) for the resolvent. If we write (1.52)

E0 = E0 + ΔE0

(1.53)

ΔE0 = E0 − E0

or

then the denominator in (1.49) becomes

(1.54)

1 1 = E0 − Er E0 − Er + ΔE0 1 1 1 = + (−ΔE0 ) E0 − Er E0 − Er E0 − Er + ΔE0

Brillouin–Wigner Methods for Many-Body Systems

19

and, if ΔE0 is small, 1 1 (1.55) ∼ E0 − Er E0 − Er and the second term on the right hand side of (1.54) is negligible.

(ii) Brillouin–Wigner theory is formally much simpler than the Rayleigh– Schr¨odinger theory. The relative simplicity of Brillouin–Wigner perturbation theory was noted by Wigner in his original publication [41]. The simplicity of Brillouin–Wigner perturbation theory was emphasized by Dalgarno in his review [38]. This simplicity is evident from a comparison of eqs. (1.15), (1.16) and (1.18) defining Brillouin–Wigner perturbation theory with eqs. (1.27)–(1.30), (1.42)–(1.46) and (1.47) for the Rayleigh–Schr¨odinger theory. In fact, eqs. (1.15), (1.16) and (1.18) provide a complete definition of Brillouin–Wigner perturbation theory through all orders. The Brillouin–Wigner perturbation series is a simple geometric series, whereas in every order beyond second-order in the energy, the Rayleigh–Schr¨odinger perturbation theory gives rise to a very much more complicated structure. In every order of the Rayleigh–Schr¨odinger perturbation theory, there is a principal term of the form p−1 (p) ˆ1 ˆ 1 RH (1.56) E0 (principal) = Φ0 | H |Φ0 , p = 1, 2, . . . which is analogous to the Brillouin–Wigner term defined in (1.20) p−1 [p] ˆ1 ˆ 1 BH (1.57) E0 = Φ0 | H |Φ0 together with other terms in third and higher orders which are often called ‘renormalization’ terms for reasons that will be elaborated below. There is only one ‘renormalization’ term in third-order, four in fourth-order, thirteen in fifth order, rising to, for example, 4,861 in tenth order of perturbation [54].

(iii) Convergence of the Brillouin–Wigner perturbation theory is often more rapid than that of the Rayleigh–Schr¨odinger theory for a given problem. This advantage of the Brillouin–Wigner perturbation theory was recognized in the original papers by Lennard-Jones [37] and of Brillouin [40]. It is also described in the review by Dalgarno [38]. For example, it is well known that energy eigenvalue for a two-state problem is given exactly by second-order Brillouin–Wigner perturbation theory. However, the Rayleigh–Schr¨odinger perturbation expansion, in general, must be taken to infinite order to solve this simple problem. Specifically, taking a zero-order matrix

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I. Hubaˇc and S. Wilson

(1.58)

H0 =

and a perturbation (1.59)

00 , 0α

0 β H1 = , β0

and then solving the secular problem −ε β =0 (1.60) β α − ε or (1.61)

ε2 − αε − β 2 = 0,

gives an exact solution 1 α ± α2 + 4β 2 . (1.62) ε = 2 The Brillouin–Wigner perturbation expansion through second-order for this problem is β2 ε−α which is seen to be equivalent to eq. (1.61) and thus the exact solution, eq. (1.62). For the two-state problem defined above, second-order Brillouin–Wigner perturbation theory provides an exact result. The exact energy is supported by second-order Brillouin–Wigner perturbation theory no matter what the specific values of α and β are assumed. The Rayleigh–Schr¨odinger perturbation series for this problem can be obtained by writing eq. (1.62) in the form (1.63)

ε=

√ α 4β 2 1± 1+x , x= 2 2 α and using the identity √ 1 1 1 5 4 7 5 21 6 x + x − x ... (1.65) 1 + x = 1 + x − x2 + x3 − 2 8 16 128 256 1026 to obtain an expansion in powers of β. The Rayleigh–Schr¨odinger expansion therefore has the form (1.64)

ε=

β2 β4 β6 + 3 − 2 5 + ··· α α α and summation to all orders is required to obtain the exact energy eigenvalue. (1.66)

ε=−

Brillouin–Wigner Methods for Many-Body Systems

21

(iv) Brillouin–Wigner perturbation theory may converge for problems for which the Rayleigh–Schr¨odinger theory does not. Wigner [41] noted this advantage of the Brillouin–Wigner perturbation theory in his 1935 paper. Dalgarno [38] also identified this advantage of Brillouin–Wigner perturbation theory in his 1961 review. An example of this advantageous feature of Brillouin–Wigner perturbation theory is again provided by the two-state problem defined by eqs. (1.58) and (1.59). The Brillouin–Wigner perturbation expansion leads to an exact solution by secondorder, irrespective of the particular values of the parameters defining the zero-order problem and the perturbation, i.e. α and β. For this problem, the Brillouin–Wigner perturbation expansion has an infinite radius of convergence. Let us explicitly introduce the perturbation parameter, λ, and determine the radius of convergence of the Rayleigh–Schr¨odinger perturbation expansion. We write the Hamiltonian matrix for the perturbed problem as 00 0 β (1.67) H0 + λH1 = +λ 0α β0 which has exact solutions 1 α ± (α2 + 4β 2 λ2 ) . (1.68) ε = 2 The Rayleigh–Schr¨odinger perturbation expansion now takes the form β6 β2 2 β4 4 λ + 3 λ − 2 5 λ6 + . . . . α α α The radius of convergence may be obtained directly by setting the discriminant in eq. (1.68) to zero, i.e. α2 +4β 2 λ2 = 0. We are thus led to the result that the Rayleigh– Schr¨odinger perturbation expansion for the two-state model converges for values of λ, satisfying 1 α (1.70) |λ| > . 2 β (1.69)

ε=−

For values of λ which do not satisfy the condition (1.70), the expansion (1.69) will diverge. It is important to note that, by forming certain approximants, it is usually possible to obtain valuable estimates of the energy eigenvalue. For example, by employing Pad´e approximants in cases where the Rayleigh–Schr¨odinger perturbation theory diverges, useful energy values can be obtained, because such approximants are usually able to handle a wider class of functions than the power series assumed in perturbation theory.

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I. Hubaˇc and S. Wilson

(v) Brillouin–Wigner perturbation theory is often more convenient to use for degenerate problems. This is one of the properties of the method which was emphasized by Wigner [41] in his original paper. Indeed the Brillouin–Wigner perturbation theory can be formally applied to degenerate problems without modification. For example, setting α = 0 in the two-state problem defined by eqs. (1.58) and (1.59) above, so as to give rise to a degenerate problem, leads immediately from expressions (1.60) and (1.61) to the exact solution ε2 = β 2 . The Brillouin–Wigner second-order energy (1.64) becomes ε = β 2 /ε which clearly agrees with the exact solution. On the other hand, the non-degenerate Rayleigh–Schr¨odinger perturbation theory breaks down, because the radius of convergence, condition (1.70), is now zero. 1.3.3.2. Disadvantages of the Brillouin–Wigner theory In spite of the five advantages listed in the preceding section, the Brillouin–Wigner perturbation theory also has some disadvantages. Specifically, Brillouin–Wigner perturbation theory has two disadvantages in comparison with the Rayleigh–Schr¨odinger perturbation theory. These disadvantages have prevented the widespread use of Brillouin–Wigner methods over the past 50 years or so. We will consider first the least problematic of the disadvantages of the Brillouin– Wigner perturbation theory and then the major negative feature of the theory in the modelling of many-body systems. (i) Brillouin–Wigner perturbation theory is iterative, since the exact energy is contained in the denominators arising in the expressions for the energy components. Consider, for example, the Brillouin–Wigner perturbation expansion for the ground state energy truncated at second-order which takes the form Φ0 | H1 |Φr Φr | H1 |Φ0 (1.71) E(2) = Φ0 | H0 |Φ0 + Φ0 | H1 |Φ0 + . E(2) − Er r=0 (We omit the subscript 0 to simplify the notation here, since we are concerned exclusively with the ground state.) Equation (1.71) is solved iteratively. The right-handside of this equation may be written f (2) (E(2) ) and eq. (1.71) itself can then be put in the form (1.72)

(2)

(2)

Ei+1 = f (2) (Ei ). (2)

Taking an initial value E0 successive, application of the recursion (1.72) defines (2) a sequence of approximations Ei , i = 0, 1, 2, . . . , which (if they are convergent) converge to the second-order Brillouin–Wigner energy through second order, which we denote E(2) in the present discussion. In general, of course, E(2) is not necessarily equal to the exact energy, E, but an approximation to it. Through the third-order of the Brillouin–Wigner perturbation theory, we have for the ground state energy:

Brillouin–Wigner Methods for Many-Body Systems E(3) = Φ0 | H0 |Φ0 + Φ0 | H1 |Φ0 + (1.73)

+

23

Φ0 | H1 |Φr Φr | H1 |Φ0 E(3) − Er r=0

Φ0 | H1 |Φr Φr | H1 |Φs Φs | H1 |Φ0 . (E(3) − Er ) (E(3) − Es ) r=0 s=0

The right-hand side of this equation may be written as f (3) (E) and eq. (1.73) itself can then be put in the form (1.74)

(3)

(3)

Ei+1 = f (3) (Ei ). (3)

As for the second-order energy, we take an initial value E0 and generate, by succes(3) sive application of (1.74), a sequence of approximations Ei , i = 0, 1, 2, . . . , which (if they are convergent) converge to the Brillouin–Wigner energy through third-order, which we denote E(3) . Again, in general, E(3) is not necessarily equal to the exact energy, E. When we substitute the converged value of E(3) into (1.73), the third term on the right-hand side is not necessarily equal to the second-order Brillouin–Wigner energy obtained by iterative solution of (1.71). In contrast to the Brillouin–Wigner approach, the Rayleigh–Schr¨odinger perturbation theory is manifestly non-iterative. The denominators in Rayleigh–Schr¨odinger perturbation theory depend on the energies obtained by solution of the reference model equations. Furthermore, extending, say, a second-order Rayleigh–Schr¨odinger perturbation theory to third-order or higher, does not change the second-order energy component.

(ii) Brillouin–Wigner perturbation theory is not explicitly a many-body theory, in that the energy expressions in each order do not scale linearly with particle number. For example, the application of second-order Brillouin–Wigner perturbation theory to an array of n well separated, i.e. non-interacting, ground state helium atoms, does not yield a total energy which is n times that of a single helium atom. This is a direct consequence of the presence of the exact energy in the denominator in the second-order energy expression. This property of the Brillouin–Wigner expansion which was first pointed out by Brueckner [16] in the mid-1950s, is the main reason for the paucity of applications of the method to the atomic and molecular electronic structure problem until recent years. In their treatise on the many-body problem in quantum mechanics, March, Young and Sampanthar [55] write that the Brillouin–Wigner form of the [many-body] theory is completely inappropriate.

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I. Hubaˇc and S. Wilson

They explicitly show that . . . the first term ∝ N and that all succeeding terms [are] individually negligible compared to the first term, whatever the strength of the perturbation. However, it would be wrong to conclude that the above argument proves that the Brillouin–Wigner series is convergent and that only the first term need be considered. Among other reasons, we know that [the first-order term] does not include dynamical correlations between particles and these must be important (on physical grounds) for a strongly interacting system. The situation therefore must be that many (N ) of the small terms beyond the first are of roughly comparable size, and add up to change the energy/particle by a finite amount. Thus it will be completely misleading to apply many-body perturbation theory in the Brillouin–Wigner form, short of considering an infinite number of terms in the limit of large N . In contrast to the Brillouin–Wigner perturbation theory, it is well known that Rayleigh–Schr¨odinger perturbation theory in its “many-body” form does afford a theoretical basis for the description of many-body systems. Again, March, Young and Sampanthar [55] write: the Rayleigh–Schr¨odinger perturbation theory . . . if carefully and systematically used, can yield an energy/particle proportional to N as required, in spite of the appearance of spurious terms proportional to N 2 , etc., in any given order. In fact, Brueckner showed that the non-physical terms cancel up to fourth-order, and the generalization to all orders was effected by Goldstone. They continue: To third-order (and to any higher order) one can in fact verify that the terms of the Rayleigh–Schr¨odinger perturbation expansion either vanish or increase as N in the limit N , the non-physical terms proportional to N 2 , N 3 , etc., all neatly cancelling each other. and then conclude: . . . the Rayleigh–Schr¨odinger series is more useful than the Brillouin–Wigner form in that it yields an expansion leading to the energy per particle as independent of the size of the system for large N . It is for these reasons, coupled with the efficiency of the corresponding computer algorithms in low-order, that Rayleigh–Schr¨odinger perturbation theory, particularly the Møller Plesset form [63], has been regarded as the method of choice in describing electron correlation effects in atoms and molecules for more than 30 years [40, 5863].

Brillouin–Wigner Methods for Many-Body Systems

25

1.4. THE MANY-BODY PROBLEM 1.4.1. Linear scaling The essential property of any true many-body theory of electronic structure is a linear scaling of the energy components, E, with the number of electrons, N , in the system [39, 55, 63, 64], that is, (1.75)

E ∝ N.

Any terms which scale non-linearly are unphysical and, therefore, should be discarded. Equally, any theory which contains such unphysical terms is not acceptable as a valid many-body method. Either the theory is abandoned or corrections are made in an attempt to restore linear scaling, such as that of Davidson [65], which is used in limited configuration interaction studies. With the development of many-body theories in the 1950s [16, 17], the shortcomings of the Brillouin–Wigner perturbation expansion were widely recognized. The presence of the exact energy in the Brillouin–Wigner denominators in the expressions for the energy components ensures that the unphysical terms arise which scale non linearly with the number of electrons. Theories which lead to a linear scaling with the number of electrons are said to be ‘size-consistent’, ‘size-extensive’, or simply ‘extensive’ [2]. Brillouin–Wigner expansions are not ‘extensive’. They do not scale linearly with the number of electrons in the system.9 It was also recognized that Brillouin–Wigner perturbation theory is not a simple power series in the perturbation parameter, λ. Brillouin–Wigner perturbation theory was, however, used as a step in the development of an acceptable many-body perturbation theory most notably by Brandow [67] in his pioneering work on multi-reference formalisms for the many-body problem. In a review entitled Linked-Cluster Expansions for the Nuclear Many-Body Problem and published in 1967, B.H. Brandow writes: The Goldstone expansion is re-derived by elementary time-independent methods, starting from Brillouin–Wigner (BW) perturbation theory. Interaction energy terms ΔE are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless).

9

We would prefer to avoid the use of the terms ‘size-consistency’ and ‘size-extensivity’ and simply describe a theory as exhibiting correct scaling with the number of particles considered. This point of view is also adopted by Nooijen, Shamasundar and Mukherjee [66] who write: “The notions of size-extensivity and size consistency are used very broadly in the literature and we prefer to speak more speciﬁcally of a physical quantity that is to scale properly in the context of a particular type of physical system or state (e.g. open or closed shell).”

26

I. Hubaˇc and S. Wilson

As we have seen, in their 1967 text on many-body methods for quantum systems, March, Young and Sampanthar [55] dismiss Brillouin–Wigner perturbation theory as a valid many-body technique. They write (p. 71): . . . it will be completely misleading to apply many-body perturbation theory in the Brillouin–Wigner form, short of considering an infinite number of terms in the limit of large N. In a volume entitled Atomic Many-Body Theory published in 1982, Lindgren and Morrison [63] state that: The Brillouin–Wigner form of perturbation theory is formally very simple. It has the disadvantage, however, that the operators depend on the exact energy of the state considered. This requires a self-consistency procedure and limits its application to one energy level at a time. There are also more fundamental difficulties with the Brillouin–Wigner theory . . . The Rayleigh–Schr¨odinger perturbation theory . . . does not have these shortcomings, and it is therefore a more suitable basis for many-body calculations than the Brillouin–Wigner form of the theory. In his monograph Electron correlation in molecules, Wilson [39] writes (p. 65): The perturbation expansion of Lennard-Jones, Brillouin and Wigner does not lead to expressions which are directly proportional to the number of electrons in the system being studied. As recently as 1992, in a book entitled Algebraic and Diagrammatic Methods in Many-Fermion Theory, Harris, Monkhorst and Freeman [64] write (p. 224): A . . . fundamental difficulty with the Wigner–Brillouin expansion is its lack of size consistency. The above extracts serve to demonstrate what had until recently been the ‘standard view’ of Brillouin–Wigner perturbation theory and its applicability to many-body systems. 1.4.2. The re-emergence of Brillouin–Wigner methods The use of Brillouin–Wigner perturbation theory in describing many-body systems has been critically re-examined in recent years [69–96]. It has been shown that under certain well-defined circumstances, it can be regarded as a valid many-body theory. The primary purpose of this volume is to provide a detailed and coherent account of the Brillouin–Wigner methods in the study of the ‘many-body’ problem in atomic and molecular quantum mechanics. The renewal of interest in Brillouin–Wigner perturbation theory for many-body systems seen in recent years, is driven by the need to develop a robust multi-reference theory. As was mentioned in the first section of the present chapter, multi-reference

Brillouin–Wigner Methods for Many-Body Systems

27

formalisms are an important prerequisite for theoretical descriptions of dissociative phenomena. Brillouin–Wigner perturbation theory is seen as a remedy to a problem which plagues multi-reference Rayleigh–Schr¨odinger perturbation theory: the so-called ‘intruder state’ problem. Multi-reference Rayleigh–Schr¨odinger perturbation theory is designed to describe a manifold of states. However, as the perturbation is ‘switched on’, the relative disposition of these states and states outside the reference space, may change in such a way that convergence of the perturbation series is impaired or even destroyed. States from outside the reference space, which assume an energy below that of any of the states among the reference set when the perturbation is switched on, are termed ‘intruder states’. The situation is illustrated schematically in Figure 1.1, which provides a schematic representation of the intruder state problem. In this figure, the reference space P consists of three states with energies E0 , E1 and E2 , which are represented on the left-hand side. The exact energies E0 , E1 , . . . obtained when the perturbation is turned on, are represented on the right-hand side of the figure. But the exact energy E2 corresponds to E3 in the reference space. This is an intruder state. unperturbed system

perturbed system

E6

E6

E5

E5

E4 E4 E3

6

E2

E3

1

intruder state

E2

E1

P

?

E0

λ=0

E1 E0 λ=1

Figure 1.1. In multi-reference Rayleigh–Schr¨odinger perturbation theory, states from outside the reference space, P, which assume an energy below that of any state among the reference set when the perturbation is switched on, are termed ‘intruder states’.

28

I. Hubaˇc and S. Wilson

Intruder states arising when 0 < λ ≤ +1 often have a physical origin. The so-called ‘backdoor’ intruder states, which arise for −1 ≤ λ < 0, are frequently unphysical. The occurrence of ‘backdoor’ intruder states is illustrated schematically in Figure 1.2. In this figure, the reference space P again consists of three states with energies E0 , E1 and E2 , which are represented in the central column. The perturbed system, with energies E0 , E1 , . . ., is represented on the right-hand side and corresponds to λ = +1. The ‘backdoor’ spectrum corresponding to λ = −1 is shown on the left-hand side. ‘Backdoor’ intruder states arise in this ‘backdoor’ spectrum. Multi-reference Brillouin–Wigner theory overcomes the intruder state problem because the energy is contained in the denominator factors. Calculations are therefore ‘state-specific’, that is, they are performed for one state at a time. This is in contrast to multi-reference Rayleigh–Schr¨odinger perturbation theory, which is applied to a manifold of states simultaneously. Multi-reference Brillouin–Wigner perturbation theory is applied to a single state. Wenzel and Steiner [77] write: . . . the reference energy in Brillouin–Wigner perturbation theory is the fully dressed energy . . . This feature guarantees the existence of a natural gap and thereby rapid convergence of the perturbation series. Multi-reference Brillouin–Wigner perturbation theory overcomes the intruder state problem which has plagued multi-reference Rayleigh–Schr¨odinger perturbation theory for many years. However, in general, Brillouin–Wigner expansions do not scale linearly with the number of electrons; they are not extensive. These insights have led to the critical re-examination of Brillouin–Wigner perturbation theory in describing many-body systems in recent years. When we consider the application of multi-reference Brillouin–Wigner methods to many-body systems, two distinct approaches can be taken which we consider now in turn: (i) The Brillouin–Wigner perturbation theory can be employed to solve the equations associated with an explicit ‘many-body’ method. For example, the full configuration interaction problem can be solved by making a Brillouin–Wigner perturbation theory expansion through infinite order, as can the equations of the coupled cluster theory. In the case of the full configuration interaction, it is obvious that provided the Brillouin–Wigner perturbation series is summed through all orders, then a result will be obtained which is entirely equivalent, at least as far as the results are concerned,10 to any other technique for solving the full configuration interaction problem. We might designate the Brillouin–Wigner-based approach to the Full Configuration Interaction (FCI) model “Brillouin–Wigner-Full Configuration Interaction” or (BW- FCI). Hubaˇc and Neogr´ady [68] have explored the use of Brillouin–Wigner perturbation theory in solving the equations of coupled cluster theory. In a paper published in The Physical Review in 1994, entitled Size-consistent Brillouin–Wigner perturbation 10

We are not concerned here with questions of computational eﬃciency.

λ=0

E0

E1

E2

E3

E4

E5

E6

?

P

6

unperturbed system

λ=1

E0

E1

E2

E3

E4

E5

E6

perturbed system

Figure 1.2. In multi-reference Rayleigh–Schr¨odinger perturbation theory, states from outside the reference space, P, which assume an energy below that of any state among the reference set for −1 ≤ λ < 0 are termed ‘backdoor’ intruder states. Unlike the intruder states corresponding to 0 < λ ≤ +1, which often have a physical origin, ‘backdoor’ intruder states are frequently unphysical.

λ = −1

‘backdoor’ intruder state

‘backdoor’ spectrum

Brillouin–Wigner Methods for Many-Body Systems 29

30

I. Hubaˇc and S. Wilson

theory with an exponentially parametrized wave function: Brillouin–Wigner coupled cluster theory, they write Size-consistency of the Brillouin–Wigner perturbation theory is studied using the Lippmann–Schwinger equation and an exponential ansatz for the wave function. Relation of this theory to the coupled cluster method is studied and a comparison through the effective Hamiltonian method is also provided. By adopting an exponential expansion for the wave operator, they ensure that their method is extensive. Hubaˇc and his co-workers, Neogr´ady and M´asˇik, obtained the “Brillouin–Wigner” coupled cluster theory [68–71] which is entirely equivalent to other many-body formulations of coupled cluster theory for the case of a singlereference function [104–107], since the Brillouin–Wigner perturbation expansion is summed through all orders. We designate the Brillouin–Wigner-based approach to the Coupled Cluster (CC) model “Brillouin–Wigner Coupled Cluster” or (BW- CC). If, for example, the equations of a limited coupled cluster expansion, such as that usually designated CCSD (in which all single and double excitation cluster operators with respect to a single-reference function are considered), are solved by means of Brillouin–Wigner perturbation theory, then a method, designated ‘Brillouin–WignerCoupled Cluster Singles Doubles’ or ( BW- CCSD) theory, is obtained which is entirely equivalent to ‘standard’ CCSD theory. (ii) A posteriori corrections can be developed for calculations performed by using the Brillouin–Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin–Wigner perturbation theory itself and, more importantly, for methods such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin–Wigner perturbation expansion. These a posteriori corrections are based on a very simple idea which is suggested by the work of Brandow mentioned in the previous section, Section 1.4.1. Brandow used the Brillouin–Wigner perturbation theory as a starting point for a derivation of the Goldstone “linked diagram” expansion “by elementary time-independent methods”. At a NATO Advanced Study Institute held in 1991, Wilson wrote [107]: The Rayleigh–Schr¨odinger perturbation theory can be derived from the Lennard-Jones–Brillouin–Wigner perturbation theory by expanding the energy-dependent denominators which occur in the latter. In the work of Brandow [67], Brillouin–Wigner perturbation theory is used as a step in the theoretical development of first Rayleigh–Schr¨odinger perturbation theory and then the many-body perturbation theory. In the a posteriori correction developed by the present authors in a paper [62] entitled On the use of Brillouin–Wigner perturbation theory for many-body systems and published in the Journal of Physics B: Atomic, Molecular and Optical Physics in 2000, they write: The use of Brillouin–Wigner perturbation theory in describing many-body systems is critically re-examined.

Brillouin–Wigner Methods for Many-Body Systems

31

Brillouin–Wigner perturbation theory is employed as a computational technique – a technique which avoids the intruder state problem – and then the relation between the Brillouin–Wigner and Rayleigh–Schr¨odinger propagators is used to correct the calculation for lack of extensivity. If we compare the Brillouin–Wigner resolvent for the ground state given in eq. (1.15) (1.15)

B=

|Φm

m=0

1 Φm | (E0 − Em )

with the corresponding Rayleigh–Schr¨odinger resolvent given in eq. (1.47) (1.47)

R=

m=0

|Φm

1 Φm | , E0 − Em

we then see that they differ only in the denominator factors. Using identity relation [80] (1.76)

(E − Ek )−1 = (E0 − Ek )−1 + (E0 − Ek )−1 (−ΔE) (E − Ek )−1

where the exact ground state energy is written as (1.77)

E0 = E0 + ΔE0 ,

where E0 is the ground state eigenvalue of H0 and ΔE0 is termed the level shift, we can relate the Brillouin–Wigner resolvent to the Rayleigh–Schr¨odinger resolvent. In this way, we can find a posteriori extensivity corrections to any Brillouin–Wigner perturbation series. We know that the Rayleigh–Schr¨odinger perturbation theory series leads directly to the many-body perturbation theory by employing the linked diagram theorem. This theory uses factors of the form (E0 − Ek )−1 as denominators. Furthermore, this theory is fully extensive; it scales linearly with electron number. The second term on the right-hand side of eq. (1.76) can be viewed as an ‘extensivity correction term’ for the Brillouin–Wigner series: a correction term which recovers the Rayleigh–Schr¨odinger and many-body perturbation theoretic formulations. This simple idea has been used to find a posteriori correction for the limited configuration interaction method [83], as well as for state-specific multi-reference Brillouin–Wigner coupled cluster theory [81,82]. Indeed, a posteriori corrections for a lack of extensivity can be obtained for any ab initio quantum chemical method, provided that the method can be formulated within Brillouin–Wigner perturbation theory. Whereas the multi-reference Rayleigh–Schr¨odinger perturbation theory approximates a manifold of states simultaneously, the multi-reference Brillouin–Wigner perturbation theory approach is applied to a single state – it is said to be ‘state-specific’. The multi-reference Brillouin–Wigner perturbation theory avoids the intruder state problem. If a particular Brillouin–Wigner-based formulation is not a valid many-body method, then a posteriori correction can be applied. This correction is designed to restore the extensivity of the method. This extensivity may be restored approximately

32

I. Hubaˇc and S. Wilson Brillouin-Wigner

Rayleigh-Schr¨ odinger

λ = 0 −−−−−−−−−−−−−−−−−−−−−→λ = 1

λ = 1 ←−−−−−−−−−−λ = 0

E2 (λ = 0)

E2 (λ = 0) Ebw 2

(λ = 1) E2 (λ = 1)

E2 (λ = 1)

E1 (λ = 0)

E1 (λ = 0)

Ebw 1 (λ = 1)

E1 (λ = 1)

E1 (λ = 1)

E0 (λ = 0)

E0 (λ = 0)

Ebw 0 (λ = 1)

E0 (λ = 1)

E0 (λ = 1)

Figure 1.3. This figure compares schematically the application of Rayleigh–Schr¨odinger perturbation theory and Brillouin–Wigner perturbation theory to the multi-reference electron correlation problem. We indicate states which are considered in a single calculation by enclosing them in a box. The Rayleigh– Schr¨odinger perturbation theory approach approximates the energy expectation values for a manifold of states in a single calculation. In the multi-reference Rayleigh–Schr¨odinger perturbation theory, the states with energies E0 , E1 , E2 , . . . are considered in a single calculation as represented on the right-hand side of the figure. The Brillouin–Wigner perturbation theory approach is ‘state-specific’; that is, we consider a single state in a given calculation and, if the resulting theory is not a valid many-body theory, then we apply a suitable a posteriori correction based on the relation between the Brillouin–Wigner and the Rayleigh– Schr¨odinger denominators. In principle, this a posteriori correction can be rendered exact. In the multireference Brillouin–Wigner perturbation theory, the states with energies E0 , E1 , E2 , . . . are considered in separate calculations as represented on the left-hand side of the figure. The resulting energies are denoted by E0BW , E1BW , E2BW , . . . To each of these energies an a posteriori correction to restore linear scaling can be introduced, if necessary, to yield the energies E0 , E1 , E2 , . . ..

or exactly, depending on the formulation. The situation is illustrated schematically in Figure 1.3. 1.4.3. An overview This monograph is concerned with the application of Brillouin–Wigner methods to the many-body problem in atoms and molecules.

Brillouin–Wigner Methods for Many-Body Systems

33

The outline of this volume is as follows: Chapter 2 uses the partitioning technique to develop both single-reference and multi-reference Brillouin–Wigner methods in a systematic fashion. The Rayleigh–Schr¨odinger perturbation theory is also considered, as is the relation between Brillouin–Wigner and Rayleigh–Schr¨odinger perturbation expansions. The many-body problem in atoms and molecules is discussed in Chapter 3. The properties of a valid many-body theory are elaborated and common approaches to the electron correlation problem are considered in some detail. In Chapter 4, the application of Brillouin–Wigner methods to many-body theories is described, including configuration interaction and cluster expansions, as well as perturbation theory itself. The use of Brillouin–Wigner methods in obtaining a posteriori many-body corrections to theoretical approaches which are not valid manybody theories, such as limited configuration interaction, is also addressed in Chapter 4. Finally, Chapter 5 contains a summary of and considers the prospects for, future progress in the application of Brillouin–Wigner methods to the correlation problem in atoms and molecules and other problems in many-body quantum mechanics. References 1. R. McWeeny, in Handbook of Molecular Physics and Quantum Chemistry, Volume 2, Molecular Electronic Structure, edited by S. Wilson, P.F. Bernath and R. McWeeny, John Wiley & Sons, Chichester, 2003 2. J. Karwowski and I. Shavitt, in Handbook of Molecular Physics and Quantum Chemistry, Volume 2, Molecular Electronic Structure, edited by S. Wilson, P.F. Bernath and R. McWeeny, John Wiley & Sons, Chichester, 2003 3. J. Paldus, in Handbook of Molecular Physics and Quantum Chemistry, Volume 2, Molecular Electronic Structure, edited by S. Wilson, P.F. Bernath and R. McWeeny, John Wiley & Sons, Chichester, 2003 4. S. Wilson, in Handbook of Molecular Physics and Quantum Chemistry, Volume 2, Molecular Electronic Structure, edited by S. Wilson, P.F. Bernath and R. McWeeny, John Wiley & Sons, Chichester, 2003 5. P.A.M. Dirac, The Principles of Quantum Mechanics, 1st edition, Clarendon Press, Oxford, 1930. 6. G.G. Hall, Proc. Roy. Soc. (London) A205, 541, 1951 7. C.C.J. Roothaan, Rev. Mod. Phys. 23, 69, 1951 8. R. McWeeny, Nature 166, 21, 1950 9. S.F. Boys, Proc. Roy. Soc. (London) A200, 542, 1950 10. I. Shavitt, Israel J. Chem. 34, 357, 1993 11. J. Goldstone, Proc. Roy. Soc. (London) A 239, 267, 1957 12. R.P. Feynman, Phys. Rev. 76, 749, 1949 13. R.P. Feynman, Phys. Rev. 76, 769, 1949 14. F. Dyson, Phys. Rev. 75, 486, 1949 15. F. Dyson, Phys. Rev. 75, 1736, 1949 16. K.A. Brueckner, Phys. Rev. 100, 36, 1955 17. N. Hugenholtz, Physica 23, 481, 1957 18. J. Hubbard, Proc. Roy. Soc. (London) A240, 539, 1957 19. F. Coester, Nucl. Phys. 7, 421, 1958

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20. F. Coester and H. K¨ummel, Nucl. Phys. 17. 4777, 1960 21. J. Paldus, in Methods in Computational Molecular Physics, NATO ASI Series, edited by S. Wilson and G.H.F. Diercksen, p. 99ff, Plenum Press, New York, 1992 22. H.P. Kelly, Phys. Rev. 131, 684, 1963 23. H.P. Kelly, Phys. Rev. A134, 1450, 1964 24. H.P. Kelly, Phys. Rev. B136, 896, 1964 25. H.P. Kelly, Phys. Rev. 144, 39, 1966 26. H.P. Kelly, Adv. Theor. Phys. 2, 75, 1968 27. H.P. Kelly, Adv. Chem. Phys. 14, 129, 1969 28. H.P. Kelly, Phys. Rev. Lett. 23, 255, 1969 29. H.P. Kelly, Intern. J. Quantum Chem. Symp. 3, 349, 1970 30. H.P. Kelly, Phys. Rev. A1, 274, 1970 31. H.P. Kelly and A. Ron, Phys. Rev. A4, 11, 1971 ˇ ızˇ ek, J. Chem. Phys. 45, 4256, 1966 32. J. C´ ˇ ızˇ ek and I. Shavitt, Phys. Rev. A 5, 50, 1972 33. J. Paldus, J. C´ 34. U. Kaldor, Phys. Rev. A 7, 427 (1973) 35. S. Wilson and D.M. Silver, Phys. Rev. A 14, 1949, 1976 36. J.M. Roberts, Twentieth Century: The History of the World - 1901 to present, Allen Lane, The Penguin Press, London, 1999 37. J.E. Lennard-Jones, Proc. Roy. Soc. (London) A 129, 598, 1930 38. A. Dalgarno, in Quantum Theory. I. Elements, edited by D.R. Bates, Vol. 1, Chap. 5, Academic Press, New York, 1961 39. S. Wilson, Electron correlation in molecules, Clarendon Press, Oxford, 1984; reprinted, Dover Publications, New York 2007 40. L. Brillouin, J. Physique 7, 373, 1932 41. E.P. Wigner, Math. naturw. Anz. ungar. Akad. Wiss. 53, 475, 1935 42. P.-O. L¨owdin, J. Molec. Spectros. 10, 12, 1963 43. P.-O. L¨owdin, J. Molec. Spectros. 13, 326, 1964 44. P.-O. L¨owdin, J. Math. Phys. 3, 969, 1962 45. P.-O. L¨owdin, J. Math. Phys. 3, 1171, 1962 46. P.-O. L¨owdin, J. Molec. Spectros. 14, 112, 1964 47. P.-O. L¨owdin, J. Molec. Spectros. 14, 119, 1964 48. P.-O. L¨owdin, J. Molec. Spectros. 14, 131, 1964 49. P.-O. L¨owdin, J. Math. Phys. 6, 1341, 1965 50. P.-O. L¨owdin, Phys. Rev. A 139, 357, 1965 51. P.-O. L¨owdin, in Perturbation Theory and its Application to Quantum Mechanics, ed. C.H. Wilcox, John Wiley and Sons, New York, 1966 52. P.-O. L¨owdin, Intern. J. Quantum Chem. 2, 867, 1968 53. P.-O. L¨owdin, and O. Goscinski, Intern. J. Quantum Chem. 5, 685, 1971 54. S. Wilson, in Methods in Computational Molecular Physics, NATO ASI Series B, Vol. 293, p. 203, Plenum Press, New York, 1992 55. N.H. March, W.H. Young and S. Sampanthar, The many-body problem in quantum mechanics, Cambridge University Press, 1967; Reprinted by Dover Publications, New York 56. C. Møller and M.S. Plesset, Phys. Rev., 46, 618, 1934 57. S. Wilson, in Theoretical Chemistry, Senior Reporter: C. Thomson, Specialist Periodical Reports 4, 1, The Royal Society of Chemistry, London, 1981

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58. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 1, 364, The Royal Society of Chemistry, London, 2000 59. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 2, 329, The Royal Society of Chemistry, London, 2002 60. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 3, 379, The Royal Society of Chemistry, London, 2004 61. S. Wilson, in Chemical Modelling: Applications and Theory, ed. A. Hinchliffe, Specialist Periodical Reports 4, 470, The Royal Society of Chemistry, London, 2006 62. S. Wilson, in Chemical Modelling: Applications and Theory, ed. A. Hinchliffe, Specialist Periodical Reports 5, 208, The Royal Society of Chemistry, London, 2008 63. I. Lindgren and J. Morrison, Atomic Many-Body Theory, Springer-Verlag, Berlin, 1982; 2nd edition, 1986 64. F.E. Harris, H.J. Monkhorst and D.L. Freeman, Algebraic and Diagrammatic Methods in Many-Fermion Theory, Oxford University Press, 1992 65. E.R. Davidson, in The world of quantum chemistry, Proceedings of the First International Congress on Quantum Chemistry, ed. R. Daudel and B. Pullman, Reidel, Dordrecht, 1974 66. M. Nooijen, K.R. Shamasundar and D. Mukherjee, Molec. Phys. 103, 2277, 2005 67. B.H. Brandow, Rev. Mod. Phys. 39, 771, 1967 68. I. Hubaˇc and P. Neogr´ady, Phys. Rev. A50, 4558, 1994 69. I. Hubaˇc, in New Methods in Quantum Theory, NATO ASI Series, ed. C.A. Tsipis, V.S. Popov, D.R. Herschbach and J.S. Avery, pp. 183, Kluwer, Dordrecht, 1996 70. J. M´asˇik and I. Hubaˇc, Coll. Czech. Chem. Commun. 62, 829, 1997 71. J. M´asˇik and I. Hubaˇc, in Quantum Systems in Chemistry and Physics: Trends in Methods and Applications, ed. R. McWeeny, J. Maruani, Y.G. Smeyers and S. Wilson, pp. 283, Kluwer Academic Publishers, Dordrecht, 1997 72. J. M´asˇik and I. Hubaˇc, Adv. Quantum Chem. 31, 75, 1998 73. J. M´asˇik, P. Mach and I. Hubaˇc, J. Chem. Phys. 108, 6571, 1998 74. P. Mach, J. M´asˇik, J. Urban and I. Hubaˇc, Molec. Phys. 94, 173, 1998 75. J. M´asˇik, P. Mach, J. Urban, M. Polasek, P. Babinec and I. Hubaˇc, Collect. Czech. Chem. Comm. 63, 1213, 1998 ˇ arsky, V. Hrouda, V. Sychrovsky, I. Hubaˇc, P. Babinec, P. Mach, J. Urban and J. 76. P. C´ M´asˇik, Collect. Czech. Chem. Comm. 60, 1419, 1995 77. W. Wenzel and M.M. Steiner, J. Chem. Phys. 108, 4714, 1998 78. W. Wenzel, Intern. J. Quantum Chem. 70, 613, 1998 ˇ arsky, J. M´asˇik and I. Hubaˇc, J. Chem. Phys. 110, 10275 79. J. Pittner, P. Nechtigall, P. C´ 1999 80. I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 33, 365, 2000 ˇ arsky, J. Chem. Phys. 112, 8779 2000 81. I. Hubaˇc, J. Pittner, and P. C´ ˇ arsky, and I. Hubaˇc, J. Chem. Phys. 112, 8785, 2000 82. J. Sancho-Garc´ıa, J. Pittner, P. C´ 83. I. Hubaˇc, P. Mach and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 33, 4735, 2000 84. I. Hubaˇc, P. Mach and S. Wilson, Adv. Quantum Chem. 39, 225, 2001 85. I. Hubaˇc and S. Wilson, Adv. Quantum Chem. 39, 209, 2001 86. I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 34, 4259, 2001 87. H.M. Quiney, I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 34, 4323, 2001 88. S. Wilson and I. Hubaˇc, Molec. Phys. 99, 1813, 2001

36 89. 90. 91. 92. 93.

94. 95.

96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107.

I. Hubaˇc and S. Wilson I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 34, 4259, 2001 I. Hubaˇc, P. Mach and S. Wilson, Molec. Phys. 100, 859, 2002 I. Hubaˇc and S. Wilson, Int. J. Molec. Science 3, 570, 2002 I. Hubaˇc, P. Mach and S. Wilson, Int. J. Quantum Chem. 89, 198, 2002 I. Hubaˇc and S. Wilson, in Fundamental World of Quantum Chemistry - A Tribute to the Memory of Per-Olov L¨owdin, volume 1, edited by E.J. Br¨andas and E.S. Kryachko, p. 407, Kluwer Academic Publisher, Dordrecht, 2003 I. Hubaˇc, P. Mach and S. Wilson, Molec. Phys. 101, 3493, 2003 ˇ arsky, in Advanced Topics in Theoretical S. Wilson, I. Hubaˇc, P. Mach, J. Pittner and P. C´ Chemical Physics, Progress in Theoretical Chemistry and Physics, edited by J. Maruani, R. Lefebvre and E. Br¨andas, p. 71, Kluwer Academic Publishers, Dordrecht, 2003 I. Hubaˇc and S. Wilson, in Encyclopedia of Computational Chemistry - electronic edition, 2004 DOI: 10.1002/0470845015.cu0032 I. Hubaˇc, P. Mach, P. Papp and S. Wilson, Molec. Phys. 102, 701, 2004 I. Hubaˇc, P. Mach and S. Wilson, Intern. J. Quantum Chem. 104, 387, 2005 P.Papp, P. Mach, J. Pittner, I. Hubaˇc and S. Wilson, Molec. Phys. 104, 2367, 2006 P. Papp, P. Mach, I. Hubaˇc and S. Wilson, Intern. J. Quantum Chem. 107, 2622, 2007 P. Papp, P. Neogr´ady, P. Mach, J. Pittner, I. Hubaˇc and S. Wilson, Molec. Phys. 106, 57, 2008 S. Wilson, in Methods in Computational Molecular Physics, NATO ASI Series B, Vol. 293, p. 99, Plenum Press, New York, 1992 J. Paldus, in Relativistic and Correlation Effects in Molecules and Solids, NATO ASI Series, edited by G.L. Malli, p. 207, Plenum Press, New York (1994) Recent Advances in Coupled Cluster Methods, Recent Advances in Computational Chemistry, Vol. 3, edited by R.J. Bartlett, World Scientific, Singapore (1997) J. Paldus, in Handbook of Molecular Physics and Quantum Chemistry, 2, ed. S. Wilson, P.F. Bernath and R. McWeeny, John Wiley, Chichester, 2002 B.P. Lippmann and J. Schwinger, Phys. Rev. 79, 469, 1950 S. Wilson, in Methods in Computational Molecular Physics, NATO ASI Series B, Vol. 293, p. 222, Plenum Press, New York, 1992

2 BRILLOUIN–WIGNER PERTURBATION THEORY

Abstract

The partitioning technique is described in some detail. The concept of a model space is introduced and the wave operator and the reaction operator are defined. Using these ideas, both the Rayleigh–Schr¨odinger and the Brillouin–Wigner expansions are developed, first for the case of a single-reference function and then for the multi-reference case.

2.1. THE PARTITIONING TECHNIQUE An important feature of modern perturbation theory is the so-called partitioning technique. In this procedure, the functional space for the wave function is separated into two parts, a model space and an orthogonal space. The basic idea here is to find “effective operators” which act only within the limited model space but which generate the same result as do the original operators acting on the entire functional space. These words were written by Lindgren and Morrison in their book Atomic ManyBody Theory [1] which was first published in 1982. The partitioning technique, introduced by Feshbach [2–4] and independently by L¨owdin [5, 6] in the late 1950s and early 1960s, facilitates the systematic development of perturbation expansions for the approximation of the solutions of the timeindependent Schr¨odinger equation. The partitioning technique is the pivotal concept of L¨owdin’s series of papers under the general title “Studies in perturbation theory”, which were reviewed in Chapter 1 and Appendix A. In the partitioning technique, the functional space for the wavefunction is divided into two parts, a model space and its orthogonal complement. Effective operators are derived which act only in the model space, but which yield results equivalent to the original operators acting on the full function space. In this section, we first consider the partitioning technique for the case of a singlereference function. We develop the basic apparatus of the partitioning technique. We define the model function and the associated projection operators. We formulate an effective Hamiltonian whose eigenfunctions lie in the model space, but whose eigenvalue is equal to that of the original Hamiltonian. We define the wave operator, which when applied to the model function yields the exact wave function, and the reaction 37

38

Ivan Hubaˇc and Stephen Wilson

operator, which describes the reaction of the system to a perturbation, yet acts only in the model space. We then develop the extension of the partitioning technique to the multi-reference function case. In the second section of this chapter, we shall employ the partitioning technique to develop various types of perturbation theory, including Rayleigh–Schr¨odinger perturbation theory and Brillouin–Wigner perturbation theory. This involves the expansion of the inverse operators which occur in the effective Hamiltonian operator and other operators obtained by the partitioning technique. Different expansions lead to different types of perturbation theory. 2.1.1. The partitioning technique for a single-reference function Consider the time-independent Schr¨odinger equation: (2.1)

HΨ = EΨ

with Hamiltonian operator, H, having eigenfunctions, Ψ , and eigenvalues E. We wish to develop approximations for the eigenfunction Ψ and the associated eigenvalue E. We restrict our attention to a single non-degenerate eigenstate in this section. In the first subsection, we define the model function and various projection operators. For the case of a single-reference function, the term ‘model function’ is synonymous with ‘zero-order function’. This is followed by a derivative of the effective Hamiltonian operator, and then definitions of the wave operator and the reaction operator. 2.1.1.1. Model function and projection operators We begin by defining the model function, the projector onto the model function (or model space) and its orthogonal complement. Let Φ be some model function, which is an eigenfunction of some model Hamiltonian operator, H0 , that is: (2.2)

H0 Φ = EΦ

H0 is sometimes termed the reference Hamiltonian or the zero-order Hamiltonian operator. We define the projection operator on to the model function, Φ, as (2.3)

P = |Φ Φ| .

If Φ is normalized to unity, i.e. Φ|Φ = 1, then the projection operator is idempotent, i.e. (2.4)

P 2 = P.

Let the projector Q be the orthogonal complement of P so that (2.5)

P +Q=I

and (2.6)

P =I −Q

where I is the identity operator. The projection operator Q is also idempotent, that is

Brillouin–Wigner Methods for Many-Body Systems (2.7)

39

Q2 = Q.

The projectors P and Q also satisfy the condition (2.8)

P Q = P (I − P ) = P − P 2 = 0,

that is, the projection operators are orthogonal. Since (2.9)

|Φ = P |Φ ,

the model eigenproblem (2.2) may be written (2.10)

(H0 − E) P Φ = 0

and thus we can write (2.11)

(H0 − E) P ≡ 0.

The projection operator P is an eigenoperator of the model Hamiltonian operator, H0 . 2.1.1.2. An effective Hamiltonian We can now proceed to define an effective Hamiltonian, which operates only within the model space, but which has the exact energy as its eigenvalue. By introducing the resolution of the identity (2.5), the Schr¨odinger equation (2.1) can be written (2.12)

H(P + Q)Ψ = E(P + Q)Ψ.

Operating on eq. (2.12) from the left with the projector P and using the idempotency of P , i.e. eq. (2.4), and the orthogonality of P and Q, i.e. eq. (2.8), gives (2.13)

P H(P + Q)Ψ = EP Ψ

(2.14)

P HP Ψ + P HQΨ = EP Ψ.

or

Operating on eq. (2.12) from the left with the projector Q and using the idempotency of Q, i.e. (2.7), and the orthogonality relation (2.8) gives (2.15)

QH(P + Q)Ψ = EQΨ

(2.16)

QHP Ψ + QHQΨ = EQΨ.

or

Equation (2.16) may be rearranged as follows: (2.17)

(E−QHQ) QΨ = QHP Ψ

so that, operating from the left by the inverse operator (E−QHQ) (2.18)

−1

QΨ = (E−QHQ)

QHP Ψ

−1

, we have

40

Ivan Hubaˇc and Stephen Wilson

where the inverse is assumed to exist and is defined by (2.19)

−1

= I.

(E−QHQ) (E−QHQ)

Substituting (2.18) into the second term on the left-hand side of (2.14) gives (2.20)

−1

P HP Ψ + P H (E−QHQ)

QHP Ψ = EP Ψ.

Now we let (2.21)

Φ = P Ψ,

that is, applying the projection operator P to the exact wavefunction, yields the model function. It can easily be verified that this will be the case if the intermediate normalization condition, (2.22)

Φ|Ψ = 1,

is assumed. Equation (2.20) can then be written as an eigenvalue equation (2.23)

Heﬀective Φ = EΦ

with an effective Hamiltonian operator given by (2.24) Heﬀective = P HP + P HQ (E − QHQ)−1 QHP . The effective eigenvalue equation (2.23) has an eigenvalue corresponding to the exact eigenvalue of the original Schr¨odinger equation (2.1), but has the model function as an eigenfunction. However, the effective Hamiltonian operator Heﬀective (2.24) depends on the exact energy E that we seek. 2.1.1.3. The wave operator Let us continue by defining the wave operator, Ω. We write (2.25)

Ψ = ΩΦ

where Ψ is the exact wave function and Φ is the model function. The wave operator when applied to the model function Φ, yields the exact wave function Ψ . The action of the wave operator should be contrasted with that of the projection operator P , which when applied to the exact wave function Ψ , yields the model function Φ. Figure 2.1 provides a schematic illustration of the action of the projection operator P and of the wave operator Ω. Equation (2.1), the original Schr¨odinger equation, can then be written (2.26)

HΩΦ = EΩΦ.

Equations (2.1) and (2.26) can also be written in the form (2.27)

(H − E) Ψ = (H − E) ΩΦ = 0.

This equation hold true for any model function, Φ. Thus we can write (2.28)

(H − E) Ω ≡ 0.

Brillouin–Wigner Methods for Many-Body Systems

41

(a) action of the projection operator P orthogonal space (Q)

6

>

Ψ

P

-?

model space (P)

Φ

(b) action of the wave operator Ω orthogonal space (Q)

6

>6

Ψ

Ω

Φ

model space (P)

Figure 2.1. A schematic illustration of the action of the projection operator, P , and the wave operator, Ω. In both diagrams, the horizontal axis represents the model space, P, and the vertical axis represents the orthogonal space, Q. Φ is the model function and Ψ is the exact wave function.

42

Ivan Hubaˇc and Stephen Wilson

The wave operator, Ω is seen to be characterized by three relations: (a) Equation (2.28) can be written as HΩ = EΩ,

(2.29)

so that Ω is an eigenoperator of H associated with the eigenvalue E; (b) We have (2.30)

Ω2 = Ω

that is, Ω is idempotent; (c) It can be shown that (2.31)

tr(Ω) = 1,

that is, Ω has unit trace. The wave operator also satisfies the following relations: (2.32)

ΩP = Ω

and (2.33)

P Ω = P.

Operating on the effective eigenequation (2.23) from the left by the wave operator, Ω, gives (2.34)

ΩHeﬀective Φ = EΩΦ.

Noting that the right-hand side of eq. (2.26) is identical to the right-hand side of eq. (2.34), we can equate the left-hand sides of these two equations to get (2.35)

HΩ = ΩHeﬀective .

Substituting the effective Hamiltonian operator (2.24) into the effective eigenproblem (2.23), gives −1 (2.36) P H P + Q (E − QHQ) QHP Φ = EΦ, Operating on eq. (2.26) from the left with P gives (2.37)

P HΩΦ = EP ΩΦ,

which upon using (2.33) becomes (2.38)

P HΩΦ = EΦ.

Brillouin–Wigner Methods for Many-Body Systems

43

Comparing (2.36) and (2.38) immediately gives the following expression for the wave operator: −1 (2.39) Ω = I + Q (E − QHQ) QH P. Like the effective Hamiltonian, Heﬀective , the wave operator Ω depends on the exact energy E that we are seeking. 2.1.1.4. The reaction operator Multiplying eq. (2.23) from the left by Φ∗ and integrating, gives an expression for the exact energy, E: (2.40)

E = Φ| Heﬀective |Φ ,

where Heﬀective is the effective Hamiltonian operator defined in eq. (2.24). In writing eq. (2.40), we have made use of the fact that Φ is normalized. Multiplying eq. (2.38) from the left by Φ∗ and integrating, gives an alternative expression for the energy, E: (2.41)

E = Φ| HΩ |Φ ,

where we have again used the fact that Φ is normalized. Now let us write the exact Hamiltonian operator, H, as a sum of the model Hamiltonian operator, H0 , and a perturbation (2.42)

H = H0 + (H − H0 )

(2.43)

H = H0 + H1

or where the perturbation operator, H1 , is defined by (2.44)

H1 = H − H0 .

Introducing the exact Hamiltonian operator in the form (2.43) into the expression for the effective Hamiltonian operator given in eq. (2.24), leads immediately to (2.45) Heﬀective = P H0 P + P H1 P + P H1 Q (E − QHQ)−1 QH1 P , which can then be rewritten in the form (2.46)

Heﬀective = P [H0 + V] P

where, by inspection, the operator V is defined by (2.47)

−1

V = H1 + H1 Q (E − QHQ)

QH1 .

The operator V is called the reaction operator associated with the perturbation H1 . Comparing expression (2.47) for the reaction operator, with expression (2.39) for the wave operator, it can be seen that the two operators are related by (2.48)

V = H1 Ω.

44

Ivan Hubaˇc and Stephen Wilson

Substituting the relation (2.46) into expression (2.40) for the exact energy eigenvalue, gives immediately (2.49)

E = Φ| H0 + V |Φ

(2.50)

E = E + Φ| V |Φ

or

where we have used the fact that (2.51)

E = Φ| H0 |Φ .

Substituting the relation (2.43) into expression (2.41) for the exact energy eigenvalue, gives (2.52)

E = Φ| H0 |Φ + Φ| H1 Ω |Φ

(2.53)

E = E + Φ| H1 Ω |Φ .

or

This expression for the exact energy also follows immediately upon substitution of (2.48) into (2.50). 2.1.2. The partitioning technique for a multi-reference function The formalism presented in the preceding section for the case of a single-reference function can be readily generalized to the multi-reference case. The structure of the present section mirrors that of the previous one, in that in the first subsection we consider the choice of model function in the multi-reference case and define various projection operators. Remember that for the case of a single-reference function, the term ‘model function’ is synonymous with ‘zero-order function’. For the multireference case, we shall see that the term ‘model function’ is employed in a somewhat different manner. This is followed by two subsections, the first defining the wave operator and the second defining the reaction operator in the case of a multi-reference function. 2.1.2.1. Model functions and projection operators We divide the zero-order functions into the p functions defining the reference space and the complementary space as follows: ⎞ ⎛ Φ0 ⎜ Φ1 ⎟ ⎟ ⎜ ⎜... ⎟ ⎟ ⎜ Φm Φp−1 ⎟ (2.54) =⎜ ⎟ ⎜ Φn ⎜ Φp ⎟ ⎟ ⎜ ⎝... ⎠ Φn

Brillouin–Wigner Methods for Many-Body Systems

45

where the model space is defined by ⎛ ⎞ Φ0 ⎜ Φ1 ⎟ ⎟ (2.55) Φm = ⎜ ⎝... ⎠ Φp−1 and the complementary space is given by ⎛ ⎞ Φp ⎜ Φp+1 ⎟ ⎟ (2.56) Φn = ⎜ ⎝... ⎠. Φn The projection operator onto the model space is given by (2.57)

P =

p−1

|Φq Φq |

q=0

and its orthogonal complement is (2.58)

Q=

n

|Φq Φq | .

q=p

The projection of the exact wave function, Ψα , for the state α is given by (2.59)

ΨαP = P Ψα .

ΨαP is called the model function for the state α. It can be written as a linear combination of the functions {Φq ; q = 0, 1, . . . , p − 1} spanning the model space: (2.60)

ΨαP

=

p−1

Cq,α Φq

q=0

although, at this stage in the development, the coefficients Cq,α , are unknown. 2.1.2.2. An effective Hamiltonian In the basis (2.54), the Hamiltonian matrix takes the form ⎛ ⎞ H0,0 H0,1 . . . H0,p−1 H0,p . . . H0,n ⎜ H1,0 H1,1 . . . H1,p−1 H1,p . . . H1,n ⎟ ⎜ ⎟ ⎜... ⎟ ... ... ... ... ... ... ⎜ ⎟ ⎜ (2.61) ⎜ Hp−1,0 Hp−1,1 . . . Hp−1,p−1 Hp−1,p . . . Hp−1,n ⎟ ⎟, ⎜ Hp,0 Hp,1 . . . Hp,p−1 Hp,p . . . Hp,n ⎟ ⎜ ⎟ ⎝... ⎠ ... ... ... ... ... ... Hn,0 Hn,1 . . . Hn,p−1 Hn,p . . . Hn,n where (2.62)

Hq,r = Φq |H| Φr .

46

Ivan Hubaˇc and Stephen Wilson

The square sub-matrix with rows and columns labelled by the elements of the reference space is then defined as follows: ⎛ ⎞ H0,0 H0,1 . . . H0,p−1 ⎜ H1,0 H1,1 . . . H1,p−1 ⎟ ⎟. (2.63) Hmm = ⎜ ⎝... ⎠ ... ... ... Hp−1,0 Hp−1,1 . . . Hp−1,p−1 The non-square sub-matrix, with rows labelled by the elements of the reference space and columns labelled by the elements of the complementary space, is defined as follows: ⎛ ⎞ H0,p . . . H0,n ⎜ H1,p . . . H1,n ⎟ ⎟, (2.64) Hmn = ⎜ ⎝... ⎠ ... ... Hp−1,p . . . Hp−1,n whilst the matrix with rows labelled by the elements of the complementary space and columns labelled by the elements of the reference space is defined as follows: ⎞ ⎛ Hp,0 Hp,1 . . . Hp,p−1 ⎠. (2.65) Hnm = ⎝ . . . . . . . . . . . . Hn,0 Hn,1 . . . Hn,p−1 The square sub-matrix with rows and column labelled by the elements of the complementary space is then defined as follows: ⎛ ⎞ Hp,p . . . Hp,n (2.66) Hnn = ⎝ . . . . . . . . . ⎠ . Hn,p . . . Hn,n The matrix (2.61) may then be written in the form Hmm Hmn (2.67) . Hnm Hnn The exact wave function for the μth state can be written as a linear combination of the zero-order functions (2.54). The vector of expansion coefficients can be written in the form Cm (2.68) , Cn where Cm corresponds to the reference space and Cn to the complementary space. The matrix Schr¨odinger equation for a single state, the μth state, then takes the form Hmm Hmn Cm Cm (2.69) = Eμ . Hnm Hnn Cn Cn This matrix eigenvalue equation can be written as two simultaneous matrix equations: Hmm Cm + Hmn Cn = Eμ Cm Hnm Cm + Hnn Cn = Eμ Cn .

Brillouin–Wigner Methods for Many-Body Systems

47

The second of these equations gives (2.70)

Hnm Cm = (Eμ − Hnn ) Cn ,

which can then be rearranged to give (2.71)

−1

Cn = (Eμ − Hnn )

Hnm Cm ,

where we have assumed that the inverse matrix exists. Substituting the expression for Cn given in eq. (2.71), into the first of the simultaneous matrix equations above, gives −1 (2.72) Hmm + Hmn (Eμ − Hnn ) Hnm Cm = Eμ Cm . Defining the matrix (2.73)

−1

= Hmm + Hmn (Eμ − Hnn ) H(eﬀective) mm

Hnm ,

we can write (2.72) in the form (2.74)

Cm = Eμ Cm , H(eﬀective) mm (eﬀective)

with rows and columns labelled so that the effective Hamiltonian matrix Hmm by the elements of the reference space, nevertheless has the exact energy for the μth state, Eμ , as an eigenvalue. This is the fundamental result of the partitioning (eﬀective) technique. The matrix Hmm has p eigensolutions. The solution with the lowest eigenvalue is taken to correspond to Eμ , the exact energy for the μth state. It should be mentioned that the above application of the partitioning technique in matrix form is not unique. However, the approach followed here is by far the most common one adapted in the atomic and molecular physics literature. Alternative approaches are described in the review by Killingbeck and Jolicard [7]. 2.1.2.3. The wave operator In the multi-reference partitioning technique, the exact wave function for state α is given by (2.75)

Ψα = ΩΨαP

where ΨαP is the model function for state α and Ω is the wave operator. ΨαP is now written as a linear combination of the reference functions spanning the model space Φq , q = 0, 1, . . . , p − 1: (2.76)

ΨαP =

p−1

Cq,α Φq ,

q=0

where the coefficients Cq,α are given by solution of the eigenproblem associated with (eﬀective) the effective Hamiltonian matrix Hmm .

48

Ivan Hubaˇc and Stephen Wilson

2.1.2.4. The reaction operator As in the single-reference formalism, the reaction operator for the multi-reference case is given by the product of the perturbation operator and the wave operator. It can be written as (2.77)

V = H1 Ω.

2.2. PERTURBATION EXPANSIONS We are now in a position to obtain perturbation expansions by expanding the inverse operator in the effective Hamiltonian, the wave operator and the reaction operator. We begin, as we did in our discussion of the partitioning technique, by considering the case of a single-reference function and then turn our attention to the multi-reference function case. 2.2.1. Single-reference function perturbation expansions In this subsection, we shall consider first the Rayleigh–Schr¨odinger perturbation theory for a single-reference function and then the Brillouin–Wigner perturbation expansion for a single-reference function. We shall also consider the generalized Brillouin–Wigner perturbation expansion. Finally, we consider the derivation of the single-reference Rayleigh–Schr¨odinger perturbation theory from the Brillouin– Wigner theory, by expanding the denominators. This relation will prove crucial in the methods developed in Chapter 4. In particular, it will facilitate the recovery of a valid many-body theory from Brillouin–Wigner-based methods which do not scale linearly with particle number, i.e. method which are not extensive. However, we begin by discussing the expansion of inverse operators. 2.2.1.1. Expansions of the inverse operator Recall the effective eigenvalue equation (2.23) for the case of a single model function (2.78)

Heﬀective Φ = EΦ

and the corresponding effective Hamiltonian (2.24) −1 (2.79) Heﬀective = P HP + P HQ (E − QHQ) QHP . The key operator to be determined in order to construct the effective Hamiltonian Heﬀective , the wave operator Ω or the reaction operator V, is the resolvent which can be loosely written as (2.80)

−1

Q (E − QHQ)

.

The resolvent, G, is more correctly defined by (2.81)

G (E − QHQ) = I

where I is the identity operator.

Brillouin–Wigner Methods for Many-Body Systems

49

Different forms of perturbation theory can be obtained by expanding the inverse −1 operator in eq. (2.79), i.e. (E − QHQ) . This operator is assumed to exist. It can be written as an infinite expansion using the operator identity (2.82)

−1

(X − Y )

=

∞

n X −1 Y X −1

n=0

or (2.83)

−1

(X − Y )

= X −1 + X −1 Y X −1 + X −1 Y X −1 Y X −1 + · · · .

Equation (2.82) is equivalent to the recursion (2.84)

−1

(X − Y )

−1

= X −1 + X −1 Y (X − Y )

which can be readily proved by operating from the right-hand side by (X − Y ). Different forms of perturbation theory can be obtained according to the partition of E − QHQ into the operator X and Y . More generally, we can write the recursion (2.84) in the form (2.85)

(X − Y )−1 = X −1 + X −1 Y (X − Y )

−1

where X and Y are chosen such that (2.86)

(X − Y ) = (X − Y )

and, of course, we have (2.87)

−1

(X − Y ) = (E − QHQ)

.

We recognize that the more general recursion defined in eq. (2.85), facilitates the development of hybrid approaches. Substituting the expansion (2.82) into the effective Hamiltonian operator (2.79) gives ∞ n (2.88) Heﬀective = P HP + P HQ X −1 Y X −1 QHP . n=0

Similarly, substituting the expansion (2.82) into the wave operator (2.39) gives (2.89)

Ω=I+

∞ n Q X −1 Y X −1 QHP , n=0

whilst substituting (2.82) into the reaction operator (2.47) gives ∞ n (2.90) V = H1 + H1 Q X −1 Y X −1 QH1 . n=0

50

Ivan Hubaˇc and Stephen Wilson

2.2.1.2. Rayleigh–Schr¨odinger perturbation theory The Rayleigh–Schr¨odinger perturbation theory is obtained by putting (2.91)

X = E − H0

and (2.92)

Y = (H − H0 ) − (E − E)

(2.93)

Y = H1 − ΔE,

or

where (2.94)

ΔE = E − E,

in the expansion (2.82) (or (2.83)) for the inverse (X − Y )−1 . ΔE is sometimes termed the ‘level shift’. Specifically, expansion (2.83) becomes ((E − H0 ) − (H1 − ΔE))−1 = (E − H0 )−1 + (E − H0 )−1 (H1 − ΔE)(E − H0 )−1 + (E − H0 )−1 (H1 − ΔE) × (E − H0 )−1 (H1 − ΔE)(E − H0 )−1 + · · · .

(2.95)

To make further progress, we introduce the Rayleigh–Schr¨odinger resolvent. The Rayleigh–Schr¨odinger resolvent In Rayleigh–Schr¨odinger perturbation theory the resolvent is defined by (2.96)

R0 (E − QH0 Q) = I

which can be loosely written (2.97)

−1

R0 = Q (E − QH0 Q)

.

In terms of the resolvent R0 , the expansion given in eq. (2.95) may be written ((E − H0 ) − (H1 − ΔE))−1 = R0 + R0 (H1 − ΔE) R0 + R0 (H1 − ΔE) R0 (H1 − ΔE) R0 + · · · .

(2.98)

The effective Hamiltonian (2.88) can be written in terms of the resolvent R0 for Rayleigh–Schr¨odinger perturbation theory as (2.99)

Heﬀective = P HP +

∞

n

[P HQ (R0 ((H1 − ΔE) R0 ) ) QHP ] .

n=0

The wave operator (2.89) in Rayleigh–Schr¨odinger perturbation theory can be written in terms of the resolvent defined in eq. (2.96) in the form: (2.100) Ω = I +

∞ n=0

n

[Q (R0 ((H1 − ΔE) R0 ) ) QHP ] .

Brillouin–Wigner Methods for Many-Body Systems

51

Similarly, the reaction operator (2.90) takes the following form in the Rayleigh– Schr¨odinger perturbation theory: (2.101) V = H1 +

∞

n

[H1 Q (R0 ((H1 − ΔE) R0 ) ) QH1 ] .

n=0

The level shift ΔE may be expanded in a power series (2.102) ΔE =

∞

λm εm ,

m=1

where λ is the usual perturbation parameter and the coefficients εm depend on the mth power of the perturbation. Substituting (2.102) into eq. (2.99) for the effective Hamiltonian and collecting terms depending on the same power of λ, gives the Rayleigh–Schr¨odinger expansion for the effective Hamiltonian: (0)

(1)

(2)

(3)

(2.103) Heﬀective = Heﬀective + Heﬀective λ + Heﬀectiveλ2 + Heﬀective λ3 + · · · . Likewise, the expansion for the wave operator, eq. (2.100), has the form (2.104) Ω = Ω (0) + Ω (1) λ + Ω (2) λ2 + Ω (3) λ3 + · · · , whilst the reaction operator, eq. (2.101), is then (2.105) V = V(0) + V(1) λ + V(2) λ2 + V(3) λ3 + · · · . Expanding the projection operator Q as a sum-over-states (2.106) Q =

∞

|Φq Φq | ,

q=0

we can write the Rayleigh–Schr¨odinger resolvent as (2.107) R0 =

∞ |Φq Φq | , E − Eq q=0

where the Eq are eigenvalues of the zero-order Hamiltonian, H0 , corresponding to the state Φq . 2.2.1.3. Brillouin–Wigner perturbation theory The Brillouin–Wigner perturbation theory is obtained by taking the following defini−1 tions of X and Y in the expansion (2.82) (or (2.83)) for the inverse (X − Y ) : (2.108) X = E − QH0 Q and (2.109) Y = QH1 Q.

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With this choice of X and Y , the expansion (2.83) takes the form −1

((E − QH0 Q) − (QH1 Q))

= (E − QH0 Q)

−1 −1

(QH1 Q) (E − QH0 Q)

−1

(QH1 Q)

−1

(QH1 Q) (E − QH0 Q)

+ (E − QH0 Q) + (E − QH0 Q) × (E − QH0 Q)

−1

−1

+ ··· .

(2.110) To make further progress, we must now introduce the Brillouin–Wigner resolvent. The Brillouin–Wigner resolvent In Brillouin–Wigner perturbation theory the resolvent is defined as follows: (2.111) B0 (E − QH0 Q) = I which can be loosely written as (2.112) B0 = Q (E − QH0 Q)−1 In terms of the resolvent B0 , the expansion given in (2.110) may be written −1

((E − QH0 Q) − (QH1 Q))

= B0 + B0 (QH1 Q) B0 + B0 (QH1 Q) B0 (QH1 Q) B0 + · · · .

(2.113)

The effective Hamiltonian operator (2.88) in Brillouin–Wigner perturbation theory can be written in terms of the resolvent defined in eq. (2.111) in the form (2.114) Heﬀective = P HP +

∞

n

[P HQ (B0 (H1 B0 ) ) QHP ] .

n=0

The wave operator (2.89) in Brillouin–Wigner perturbation theory can be written in terms of the resolvent defined in eq. (2.111) in the form (2.115) Ω = I +

∞

n

[Q (R0 ((H1 − ΔE) R0 ) ) QHP ] .

n=0

Similarly, the reaction operator (2.90) takes the following form in the Brillouin– Wigner perturbation theory: (2.116) V = H1 +

∞

n

[H1 Q (R0 ((H1 − ΔE) R0 ) ) QH1 ] .

n=0

Expanding the projection operator Q in sum-over-states form (2.117) Q =

∞ k=0

|Φk Φk | ,

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53

we can write the Brillouin–Wigner resolvent as (2.118) B =

∞ |Φk Φk | E − Ek

k=0

where the Ek are eigenvalues of the zero-order Hamiltonian, H0 . 2.2.1.4. Generalized Brillouin–Wigner perturbation theory In the preceding section we have developed the Brillouin–Wigner perturbation theory for a single-reference function. One of the disadvantages of this method when compared with the Rayleigh–Schrodinger perturbation expansion is that the unknown exact energy is contained in the denominator terms in each order of the perturbation expansion. Application of the Brillouin–Wigner expansion therefore demands an iterative approach in which a suitable initial guess for the exact energy is assumed, the perturbation series evaluated and the resulting estimate of the energy used to repeat this calculation. This procedure is repeated until self-consistency is achieved. As we have indicated above, various types of perturbation expansion can be obtained by expanding the inverse operator in the effective Hamiltonian, the wave operator and the reaction operator in different ways. Using the operator identity −1

(X − Y )

=

∞

n X −1 Y X −1

n=0

= X −1 + X −1 Y X −1 + X −1 Y X −1 Y X −1 + · · ·

(2.119) or −1

(2.120) (X − Y )

−1

= X −1 + X −1 Y (X − Y )

,

we have seen that the widely used Rayleigh–Schr¨odinger perturbation theory and the Brillouin–Wigner perturbation theory are obtained by making different choices of X and Y . Here we consider an alternative choice for these operators, namely: X≡E (2.121) Y ≡ QHQ. This leads to what L¨owdin [6] has called generalized Brillouin–Wigner perturbation theory. We can write the resolvent for this approach in the form ∞

1 1 (2.122) Bg = Q = |Φk Φk | , E E k=0

Using (2.88), (2.119) and (2.121) together with (2.236), gives the following expansion for the energy: (2.123) E = Φ0 |H| Φ0 + Φ0 |HBg H| Φ0 + Φ0 |HBg HBg H| Φ0 + · · · .

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We can then introduce the reaction operator (2.124) V = HBg H + HBg HBg H + · · · so that the energy may be written (2.125) E = Φ0 |H| Φ0 + Φ0 |V| Φ0 . Equation (2.124) may be written (2.126) V = HBg H + HBg V. Equation (2.126) is the Lippmann–Schwinger equation [8]. Defining the wave operator [9, 10] by (2.127) |Ψ = Ω |Φ , we write (2.128) E = Φ0 |H| Ψ with (2.129) Φ0 | Φ0 = 1 and (2.130) Φ0 | Ψ = 1 in the form (2.131) E = Φ0 |HΩ| Φ0 . Comparing (2.125) and (2.131) gives (2.132) V = HΩ − H, which upon substituting in (2.126) and rearranging, yields (2.133) Ω = I + Bg HΩ. Equation (2.133) is the Bloch equation [11]. Together with eq. (2.131), it provides the fundamental equation of the generalized Brillouin–Wigner perturbation theory. Iteration of eq. (2.133) and substitution in eq. (2.131) generates the generalized Brillouin–Wigner perturbation expansion for the energy (2.134) E = E + ε1 + ε2 + ε3 + ε4 + · · · , where the energy coefficients, εk , have the general form k−1 1 (2.135) εk = Φ0 H (QH)k−1 Φ0 . E The denominator is simply the exact energy raised to the power k − 1 in order k.

Brillouin–Wigner Methods for Many-Body Systems The energy through second-order, for example, is ∞

(2.136) E = Φ0 |H| Φ0 +

1 Φ0 | H |Φk Φk | H |Φ0 , E k=0

which, if we define (2.137) W1 = Φ0 |H| Φ0 , and (2.138) W2 =

∞

Φ0 | H |Φk Φk | H |Φ0 ,

k=0

can be written (2.139) E = W1 + or

1 W2 E

(2.140) E2 − EW1 − W2 = 0, so that (2.141) E± =

1 2 W1 ± (W1 ) + 4W2 . 2

Through third-order, we have (2.142) E = W1 +

1 1 W2 + 2 W3 , E E

where ∞

(2.143) W3 =

Φ0 | H |Φk Φk | H |Φ Φ | H |Φ0 ,

k=0,=0

so that (2.144) E3 − E2 W1 − EW2 − W3 = 0. In general, the energy through order k may be written (2.145) E =

k

1 Wp , p−1 E p=1

where ∞ ∞

Wp =

k1 =0 k2 =0

(2.146)

···

∞ kp−1 =0

Φ0 | H |Φk1 Φk1 | H |Φk2 · · · Φkp−1 H |Φ0 .

55

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The evaluation of the Wp is non-iterative. The determination of the generalized Brillouin–Wigner perturbation theory energy in order k requires the determination of the lowest root of the k order polynomial: (2.147) E − k

k

Ek−p Wp = 0.

p=1

2.2.1.5. Derivation of Rayleigh–Schr¨odinger perturbation theory from the Brillouin–Wigner perturbation expansion The Rayleigh–Schr¨odinger perturbation expansion can be derived from the Brillouin–Wigner perturbation theory by expanding the denominator factors in the latter. The expressions for the energy coefficient in the Brillouin–Wigner perturbation theory expansion have the form (2.148)

1 Eμ − H0

where Eμ is the exact energy for the state μ, and H0 is the chosen zero-order Hamiltonian operator. Eμ can be written as (2.149) Eμ = Eμ + ΔEμ , where Eμ is the zero-order energy (or reference energy) for the state μ and ΔEμ is the corresponding level shift (2.150) ΔEμ = Eμ − Eμ . ΔEμ can be expanded in a power series (2.151) ΔEμ =

∞

(μ) λm εm ,

m=1

where we have explicitly included the perturbation parameter λ. Writing the Brillouin–Wigner denominator in the form (2.152)

1 , Eμ − H0 + ΔEμ

we can use the identity (2.153)

1 1 1 1 = + Y X −Y X X X −Y

with (2.154) X = Eμ − H0 and (2.155) Y = ΔEμ

Brillouin–Wigner Methods for Many-Body Systems

57

to write (2.156)

1 1 1 1 = + (ΔEμ ) . Eμ − H0 Eμ − H0 Eμ − H0 Eμ − H0

The term on the left-hand side of this equation is the Brillouin–Wigner denominator. The first term on the right-hand-side of this equation is the Rayleigh–Schr¨odinger denominator. Iteration gives 1 1 1 1 = + (−ΔEμ ) + Eμ − H0 Eμ − H0 Eμ − H0 Eμ − H0 1 1 1 (2.157) (−ΔEμ ) (−ΔEμ ) + ··· , Eμ − H0 Eμ − H0 Eμ − H0 so that the left-hand-side is written entirely in terms of Brillouin–Wigner denominators and the right-hand-side contains only Rayleigh–Schr¨odinger denominators. The second-order Brillouin–Wigner energy has the form BW (μ)

(2.158) ε2

=

Φμ | H1 |Φp Φp | H1 |Φμ . Eμ − Ep

p=μ

Substituting 1 1 1 1 = + (−ΔEμ ) + Eμ − Ep Eμ − Ep Eμ − Ep Eμ − Ep 1 1 1 (2.159) (−ΔEμ ) (−ΔEμ ) + ··· , Eμ − Ep Eμ − Ep Eμ − Ep gives a leading term which is just the second-order Rayleigh–Schr¨odinger perturbation theory energy coefficient RS(μ)

(2.160) ε2

=

Φμ | H1 |Φp Φp | H1 |Φμ . Eμ − Ep

p=μ

The second term has the form Φμ | H1 |Φp Φp | H1 |Φμ (2.161) (−ΔEμ ) 2 (Eμ − Ep ) p=μ and, taking the leading term in the power series for ΔEμ , gives (2.162) − ε1

Φμ | H1 |Φp Φp | H1 |Φμ p=μ

2

(Eμ − Ep )

,

which is a third-order term in the Rayleigh–Schr¨odinger perturbation expansion.

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2.2.2. Multi-reference function perturbation expansions 2.2.2.1. Multi-reference Rayleigh–Schr¨odinger perturbation theory Multi-reference Rayleigh–Schr¨odinger perturbation theory can be applied to several states simultaneously. Although the Rayleigh–Schr¨odinger theory can be derived from the Brillouin–Wigner theory by expanding the denominators in the latter as was done by Brandow [12], here we follow the direct approach due to Lindgren [1]. The zero-order eigenproblem is written (2.163) H0 Φp = Ep Φp . In this section, we shall follow the convention that Φi , Φj , . . . are eigenfunctions of the model Hamiltonian, H0 , in the model space, P; that Φa , Φb , . . . are eigenfunctions lying in the orthogonal space, Q, and that Φp , Φq , . . . are arbitrary eigenfunctions. The projection operator onto the model space, P, is written (2.164) P = |Φi Φi | i∈P

and its orthogonal complement is (2.165) Q = |Φa Φa | , a∈P /

so that (2.166) P + Q = I is a resolution of the identity. These projectors satisfy the idempotency conditions (2.167) P = P 2 and (2.168) Q = Q2 , and they are orthogonal (2.169) P Q = P (I − P ) = P − P 2 = 0. Let m be the dimension of the model space. Furthermore, let there be m welldefined eigenfunctions of the full Hamiltonian, H, (2.170) HΨk = EΨk , k = 1, 2, . . . , m, which most closely resemble the corresponding projections onto the model space (2.171) ΨkP = P Ψk , k = 1, 2, . . . , m. The functions ΨkP are termed the model functions. It is clear that (2.172) ΨkP = P ΨkP , k = 1, 2, . . . , m.

Brillouin–Wigner Methods for Many-Body Systems

59

It is assumed that these model functions are linearly independent and that they span the entire model space. There is, therefore, a one-to-one correspondence between the m model functions and m exact solutions of the Schr¨odinger equation. Lindgren defines a state-independent wave operator, Ω, which transforms all the model functions into the corresponding exact functions (2.173) Ψk = ΩΨkP , k = 1, 2, . . . , m. We shall show that it is possible to derive a recursive formula for Ω, but let us begin by examining some of its properties. Operating on eq. (2.173) from the left with the projector P gives (2.174) P Ψk = P ΩΨkP , k = 1, 2, . . . , m so that, using (2.171), we have (2.175) ΨkP = P ΩΨkP ,

k = 1, 2, . . . , m.

Multiplying again from the left by P and using (2.167) and (2.172) gives (2.176) P ΨkP = P ΩP ΨkP , k = 1, 2, . . . , m and thus we have the operator equation (2.177) P = P ΩP. It is sometimes convenient to define the correlation operator, χ, which is defined by (2.178) Ω = 1 + χ so that (2.179) χ = Ω − 1. Application of the correlation operator to a model function generates the projection of the exact wave function in the complementary space Q χΨkP = (Ω − 1) ΨkP = Ψk − ΨkP (2.180)

= Ψk − P Ψk = QΨk .

From (2.177) and (2.178), we have (2.181) P = P (1 + χ)P or (2.182) P = P 2 + P χP and so, using (2.167), (2.183) P χP = 0.

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Ivan Hubaˇc and Stephen Wilson

Using (2.169) and (2.178), it can be shown that QΩP = QχP (2.184) = χP. Let us write the Schr¨odinger equation for the problem of interest (2.185) HΨk = Ek Ψk where (2.186) H = H0 + H1 in the form (2.187) (Ek − H0 ) Ψk = H1 Ψk . H0 is the model Hamiltonian. Operating on eq. (2.187) from the left by P , yields (2.188) (Ek − H0 ) ΨkP = P H1 Ψk where we have used the relation (2.171). Then, operating on eq. (2.188) from the left by the wave operator Ω, gives (2.189) Ek ΩΨkP − ΩH0 ΨkP = ΩP H1 Ψk , which, using eq. (2.173), can be written (2.190) Ek Ψk − ΩH0 ΨkP = ΩP H1 ΩΨkP . Subtracting eq. (2.190) from the Schr¨odinger equation in the form (2.187) (2.191) [Ek Ψk − H0 Ψk ] − Ek Ψk − ΩH0 ΨkP = [H1 Ψk ] − ΩP H1 ΩΨkP the terms involving the energy Ek cancel to give (2.192) ΩH0 ΨkP − H0 Ψk = H1 Ψk − ΩP H1 ΩΨkP . Using eq. (2.173), eq. (2.192) can be written (2.193) ΩH0 ΨkP − H0 ΩΨkP = H1 ΩΨkP − ΩP H1 ΩΨkP , which can then be rearranged to give (2.194) [ΩH0 − H0 Ω] ΨkP = [H1 Ω − ΩP H1 Ω] ΨkP . Equation (2.194) holds for all model functions and can thus be written in operator form as (2.195) [Ω, H0 ] P = H1 ΩP − ΩP H1 ΩP. This equation is termed the generalized Bloch equation. Using the resolution of the identity (2.166), the first term on the right-hand side of eq. (2.195) can be rewritten as (2.196) (P + Q) H1 ΩP = P H1 ΩP + QH1 ΩP.

Brillouin–Wigner Methods for Many-Body Systems

61

Then, using the definition of the correlation operator, χ, given in (2.178), the second term in (2.195) can be written in the form (2.197) − ΩP H1 ΩP = −P H1 ΩP − χP H1 ΩP. Introducing (2.196) and (2.197) into the right-hand side of the generalized Bloch equation (2.195) gives (2.198) [Ω, H0 ] P = P H1 ΩP + QH1 ΩP − P H1 ΩP − χP H1 ΩP so that (2.199) [Ω, H0 ] P = QH1 ΩP − χP H1 ΩP which is the generalized Bloch equation in an alternative form. The generalized Bloch equation in both (2.195) and (2.199) is completely equivalent to the original Schr¨odinger equation for the states in the model space P. It can be employed to generate a Rayleigh–Schr¨odinger perturbation expansion as we shall now show. The correlation operator, χ, generates the components of the exact wave function in the orthogonal space, Q. It can be expanded as follows (2.200) χ = Ω (1) + Ω (2) + · · · , where Ω (n) contains n interactions with the perturbation operator, H1 . The wave operator can, therefore, be written in the form Ω = 1 + Ω (1) + Ω (2) + · · · ∞ = 1+ (2.201) Ω (n) . n=1

This follows immediately from (2.178) and (2.200). The Rayleigh–Schr¨odinger perturbation expansion is obtained by substituting eq. (2.201) into the generalized Bloch equation, eq. (2.199) ∞ ∞ (n) (n) 1+ , H0 P = QH1 1 + P− Ω Ω n=1

(2.202)

χP H1

n=1 ∞

1+

Ω (n)

P

n=1

and then identifying terms which are of the same order in the perturbation, H1 . In this way the following sequence of equations is obtained: (2.203) Ω (1) , H0 P = QH1 P (2.204) Ω (2) , H0 P = QH1 Ω (1) P − Ω (1) P H1 P

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Ivan Hubaˇc and Stephen Wilson

(2.205) Ω (3) , H0 P = QH1 Ω (2) P − Ω (2) P H1 P − Ω (1) P H1 Ω (1) P Ω (4) , H0 P = QH1 Ω (3) P − Ω (3) P H1 P − Ω (2) P H1 Ω (1) P − Ω (1) P H1 Ω (2) P.

(2.206)

... In general, these equations have the form (2.207)

Ω

(n)

, H0 P = QH1 Ω

(n−1)

P−

n−1

Ω (n−m) P H1 Ω (m−1) P.

m=1

Returning to the exact Schr¨odinger equation (2.170), we are now in a position to construct an effective Hamiltonian. Recall eq. (2.170) (2.208) HΨk = EΨk , k = 1, 2, . . . , m and use (2.173) to write this as (2.209) HΩΨkP = EΨk , k = 1, 2, . . . , m. Operating on eq. (2.209) from the left with the projector P on to the model space P, then gives (2.210) P HΩΨkP = EΨkP ,

k = 1, 2, . . . , m.

By inspection, we see that the operator (2.211) Heﬀective = P HΩP operates entirely within the model space, P, and satisfies the eigenequation (2.212) Heﬀective ΨkP = EΨkP , k = 1, 2, . . . , m. Heﬀective is termed the effective Hamiltonian. The eigenfunctions of this effective Hamiltonian lie in the model space P, but the eigenvalues are the exact energies of the states corresponding to the model functions. The effective Hamiltonian (2.211) may be written in the form Heﬀective = P (H0 + H1 ) ΩP (2.213) = P H0 P + P H1 ΩP, where we have used the relation (2.177). Alternatively, using (2.178) and (2.211), we can write (2.214) Heﬀective = P HP + P HχP. To develop the perturbation equations (2.203)–(2.206) and (2.207) into a practical scheme we make use of the eigenfunctions of the zero-order Hamiltonian given as solutions of the eigenproblem (2.163) (2.215) HΦp = Ep Φp .

Brillouin–Wigner Methods for Many-Body Systems

63

In terms of these eigenfunctions, the commutator on the left-hand side of the generalized Bloch equation (2.199) becomes (2.216) Φp | ΩH0 − H0 Ω |Φq = (Eq − Ep ) Φp | Ω |Φq . Similarly, the commutators in (2.203)–(2.206) can be written (2.217) Φp | Ω (1) H0 − H0 Ω (1) |Φq = (Eq − Ep ) Φp | Ω (1) |Φq (2.218) Φp | Ω (2) H0 − H0 Ω (2) |Φq = (Eq − Ep ) Φp | Ω (2) |Φq ... (2.219) Φp | Ω (n) H0 − H0 Ω (n) |Φq = (Eq − Ep ) Φp | Ω (n) |Φq . Assuming that Φi ∈ P and Φa ∈ Q, then the matrix elements of the right-hand side of the generalized Bloch equation (2.203) take the form (2.220) Φa | QH1 P |Φi = Φa | H1 |Φi and so eq. (2.203) becomes (2.221) (Ei − Ea ) Φa | Ω (1) |Φi = Φa | H1 |Φi or (2.222) Φa | Ω (1) |Φi =

Φa | H1 |Φi . Ei − Ea

Similarly, the matrix elements of the right-hand side of the second-order equation, eq. (2.204), have the form (2.223) Φa | H1 Ω (1) |Φi − Φa | Ω (1) P H1 |Φi and thus, using (2.218), eq. (2.204)becomes (2.224) (Ei − Ea ) Φa | Ω (2) |Φi = Φa | H1 Ω (1) |Φi − Φa | Ω (1) P H1 |Φi or (2.225) Φa | Ω (2) |Φi =

Φa | H1 Ω (1) |Φi Φa | Ω (1) P H1 |Φi − . Ei − Ea Ei − Ea

Introducing a resolution of the identity (2.226) I = |Φp Φp | p

or I =P +Q (2.227) |Φi Φi | + |Φa Φa | = i∈P

a∈Q

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Ivan Hubaˇc and Stephen Wilson

gives Φa | Ω (2) |Φi =

Φa | H1 |Φb Φb | Ω (1) |Φi Ei − Ea

b

−

(2.228)

Φa | Ω (1) |Φj Φj | H1 |Φi j

Ei − Ea

.

Substituting for the matrix elements of the first-order wave operator, Ω (1) , given in (2.222) yields Φa | H1 |Φb Φb | H1 |Φi Φa | Ω (2) |Φi = (Ei − Ea ) (Ei − Eb ) b Φa | H1 |Φj Φj | H1 |Φi (2.229) . − (Ei − Ea ) (Ej − Ea ) j Equations (2.222) and (2.229) are the first two terms in the general Rayleigh– Schrodinger perturbation expansion for the multi-reference case. 2.2.2.2. Multi-reference Brillouin–Wigner perturbation theory In this section we briefly survey the basic formalism of the multi-reference Brillouin–Wigner perturbation theory. This will serve to introduce our notation. We seek solutions of the time-independent Schr¨odinger equation (2.230) HΨα = Eα Ψα using a Brillouin–Wigner perturbation expansion based on the Hamiltonian (2.231) H = H0 + H1 , where H0 is the zero-order Hamiltonian and H1 is the perturbation. In particular, we wish to develop a Brillouin–Wigner perturbation theory employing a multi-reference function. We assume that some of the low lying solutions of the zero-order eigenproblem (2.232) H0 Φμ = Eμ Φμ ,

μ = 0, 1, 2, . . .

are known. Let {Φμ ; μ = 0, 1, 2, . . . , p − 1} be a set of linearly independent functions which constitute the reference space, which we label P. Furthermore, let P be the projection operator onto this chosen reference space (2.233) P = |Φμ Φμ | μ∈P

and let Q be its orthogonal complement (2.234) Q = |Φμ Φμ | μ∈P /

Brillouin–Wigner Methods for Many-Body Systems

65

so that P2 = P Q2 = Q P + Q = I. Let us consider the projection of the exact wave function, Ψα , onto the reference space, i.e. (2.235) ΨαP = P Ψα , α = 0, 1, 2, . . . , p − 1. ΨαP is sometimes called the model function. Obviously, ΨαP can be written as a linear combination of the set {Φμ ; μ = 0, 1, 2, . . . , p − 1} (2.236) ΨαP = Cμ,α Φμ . μ∈P

ΨαP

The are, in general, non-orthogonal but are assumed to be linearly independent, and satisfy the equation (2.237) HΨαP = Eα ΨαP , α = 0, 1, 2, . . . , p − 1. In multi-reference Brillouin–Wigner perturbation theory, the exact wavefunctions, Ψα , are expanded as follows (2.238) Ψα = (1 + Bα H1 + Bα H1 Bα H1 + · · · ) ΨαP where Bα is the Brillouin–Wigner type propagator |Φi Φi | (2.239) Bα = . Eα − Ei i∈P /

The exact wavefunction, Ψα , and the model function, ΨαP , satisfy the following intermediate normalization conditions

P Ψα | Ψ α = 1

(2.240) ΨαP | ΨαP = 1. The wave operator may be written (2.241) Ψα = Ωα ΨαP , so that, comparing with (2.238), we have (2.242) Ωα = 1 + Bα H1 + Bα H1 Bα H1 + · · · or (2.243) Ωα = 1 + Bα H1 Ωα , which may be seen as the Bloch equation in Brillouin–Wigner form. We now introduce the ‘effective’ Hamiltonian which acts in the reference space (2.244) Heﬀective = P HΩα P (2.245) Heﬀective ΨαP = Eα ΨαP .

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Using eq. (2.231), we have (2.246) Heﬀective = P H0 P + P H1 Ωα P, which can be written as (2.247) Heﬀective = P H0 P + P Vα P,

α = 0, 1, 2, . . . , p − 1,

where the reaction operator, Vα , is given by (2.248) Vα = H1 Ωα . Combining eqs. (2.243) and (2.248) gives the Lippman–Schwinger equation in Brillouin–Wigner form (2.249) Vα = H1 + H1 Bα Vα . Ω=

p−1

Ωα Pα .

α=0

2.2.2.3. An n-state system For simplicity, let us begin by considering a two-state system. In this case, we have (2.250) P = |Φ0 Φ0 | + |Φ1 Φ1 | . The corresponding reaction operators are V0 = H1 Ω0 (2.251) V1 = H1 Ω1 . The effective Schr¨odinger equation has the form (2.252) Heﬀective ΨαP = Eα ΨαP , α = 0, 1. For the state α = 0, we have (2.253) Ψ0P = C0 Φ0 + C1 Φ1 and the secular equation for the ground state takes the form Φ0 | V0 |Φ0 − E0 Φ0 | V0 |Φ1 C0 (2.254) = 0, Φ1 | V0 |Φ1 − E0 C1 Φ1 | V0 |Φ0 where the matrix elements of the effective Hamiltonian are Φ0 | H1 |Φj Φj | V0 |Φ0 (2.255) Φ0 | V0 |Φ0 = Φ0 | H1 |Φ0 + , E0 − Ej j ∈P /

(2.256) Φ1 | V0 |Φ0 = Φ1 | H1 |Φ0 +

Φ1 | H1 |Φj Φj | V0 |Φ0 , E0 − Ej

j ∈P /

(2.257) Φ0 | V0 |Φ1 = Φ0 | H1 |Φ1 +

Φ0 | H1 |Φj Φj | V0 |Φ1 , E0 − Ej

j ∈P /

Brillouin–Wigner Methods for Many-Body Systems (2.258) Φ1 | V0 |Φ1 = Φ1 | H1 |Φ1 +

67

Φ1 | H1 |Φj Φj | V0 |Φ1 . E0 − Ej

j ∈P /

Equation (2.254) has two roots, of which we take the lowest one. The secular equation for the excited state takes the form C0 Φ0 | V1 |Φ0 − E1 Φ0 | V1 |Φ1 (2.259) = 0, Φ1 | V1 |Φ0 Φ1 | V1 |Φ1 − E1 C1 where the matrix elements are (2.260) Φ0 | V1 |Φ0 = Φ0 | H1 |Φ0 +

Φ0 | H1 |Φj Φj | V1 |Φ0 , E1 − Ej

j ∈P /

(2.261) Φ1 | V1 |Φ0 = Φ1 | H1 |Φ0 +

Φ1 | H1 |Φj Φj | V1 |Φ0 , E1 − Ej

j ∈P /

(2.262) Φ0 | V1 |Φ1 = Φ0 | H1 |Φ1 +

Φ0 | H1 |Φj Φj | V1 |Φ1 , E1 − Ej

j ∈P /

(2.263) Φ1 | V1 |Φ1 = Φ1 | H1 |Φ1 +

Φ1 | H1 |Φj Φj | V1 |Φ1 . E1 − Ej

j ∈P /

Equation (2.259) has two roots, of which we take the lowest one. The reaction operators are given by (2.264) V0 ∼ H1 + H1 B0 H1 + · · · (2.265) V1 ∼ H1 + H1 B1 H1 + · · · We now consider an n-state system and generalize the expression given above for the 2-state case. In this case, we have (2.266) P = |Φ0 Φ0 | + |Φ1 Φ1 | + · · · + |Φn−1 Φn−1 | . The corresponding reaction operators are V0 = H1 Ω0 V1 = H1 Ω1 .... Vn−1 = H1 Ωn−1 For the state α = 0, we have (2.267) Ψ0P = C0 Φ0 + C1 Φ1 + · · · + Cn−1 Φn−1

68

Ivan Hubaˇc and Stephen Wilson

and the secular equation for the ground state takes the form ⎛ ⎞ ⎞⎛ Φ0 | V0 |Φ0 − E0 Φ0 | V0 |Φ1 . . . Φ0 | V0 |Φn−1 C0 ⎜ Φ1 | V0 |Φ0 ⎟ ⎜ Φ1 | V0 |Φ1 − E0 . . . Φ1 | V0 |Φn−1 ⎟ ⎜ ⎟ ⎜ C1 ⎟ = 0, ⎝... ⎠ ⎝ ... ... ... ... ⎠ Φn−1 | V0 |Φ0 Φn−1 | V0 |Φ1 . . . Φn−1 | V0 |Φn−1 Cn−1 (2.268) where the matrix elements of the effective Hamiltonian are Φp | H1 |Φj Φj | V0 |Φq Φp | V0 |Φq = Φp | H1 |Φq + E0 − Ej j ∈P /

(2.269)

p, q = 0, 1, . . . , n − 1.

Equation (2.268) has n roots, of which we take the lowest one. The secular equation for the excited state takes the form ⎛ ⎞ ⎞⎛ Φ0 | V1 |Φ0 − E1 Φ0 | V1 |Φ1 . . . Φ0 | V1 |Φn−1 C0 ⎜ Φ1 | V1 |Φ0 ⎟ ⎜ Φ1 | V1 |Φ1 − E1 . . . Φ1 | V1 |Φn−1 ⎟ ⎜ ⎟ ⎜ C1 ⎟ = 0, ⎝ .. ⎠⎝... ⎠ ... ... ... Φn−1 | V1 |Φ0 Φn−1 | V1 |Φ1 . . . Φn−1 | V1 |Φn−1 Cn−1 (2.270) where the matrix elements are Φp | V1 |Φq = Φp | H1 |Φq +

Φp | H1 |Φj Φj | V1 |Φq E1 − Ej

j ∈P /

(2.271)

p, q = 0, 1, . . . , n − 1.

Equation (2.270) has n − 1 roots of which we take the lowest one. References 1. I. Lindgren and J. Morrison, Atomic Many-Body Theory, Springer-Verlag, Berlin, 1982; 2nd edition, 1986 2. H. Feshbach, Ann. Phys. N.Y. 5, 357, 1958 3. H. Feshbach, Ann. Rev. Nucl. Sci. 8, 49, 1958 4. H. Feshbach, Ann. Phys. N.Y. 19, 287, 1962 5. P.-O. L¨owdin, Phys. Rev. A 139, 357, 1965 6. P.-O. L¨owdin, in Perturbation Theory and its Applications in Quantum Mechanics, edited by C.H. Wilcox, Wiley, New York, 1966 7. J.P. Killingbeck and G. Jolicard, J. Phys. A: Math. & General 36, R105, 2003 8. B.P. Lippmann and J. Schwinger, Phys. Rev. 79, 469, 1950 9. R.J. Eden and N.C. Francis, Phys. Rev. 97, 1366, 1955 10. I. Lindgren, Intern. J. Quantum Chem. 12, 33, 1978 11. C. Bloch, Nucl. Phys. 6, 329, 1958 12. B.H. Brandow, Rev. Mod. Phys. 39, 771, 1967

3 THE MANY-BODY PROBLEM IN ATOMS AND MOLECULES

Abstract

The many-body problem in atomic and molecular physics and in quantum chemistry is described. The second-quantization formulation is introduced together with the diagrammatic techniques which form an essential ingredient of many-body methodology. The use of coupled cluster expansions, mixing of configurations (or configuration interaction) and perturbation theory series for the description of electron correlation effects is considered.

3.1. LINEAR SCALING IN MANY-BODY SYSTEMS Chemistry is primarily concerned not with the properties of single molecules but with periodic trends, homologous series, functional groups, and the like. In chemistry, the systematization of the properties of a series of molecules is just as important as the determination of all of the properties of one particular species. Theoretically, therefore, it is important that any method which is applied to the problem of molecular electronic structure leads to expressions for expectation values which are directly proportional to the number of electrons in the system being studied. Meaningful comparisons of atoms and molecules of different sizes are then possible. [1, 2] In his Nobel Lecture, the late Professor Sir John Pople, FRS, discussed sizeconsistency in limited configuration interaction studies (CID and CISD): Although CID and CISD are well-defined models, given a standard basis set, they suffer some serious disadvantages. These have to do with sizeconsistency. If a method such as CID is applied to a pair of completely separated systems, the resulting energy is not the sum of the energies obtained by applying the same theory to the systems separately. If CID is applied to two separate helium atoms, for example, the wave function does not allow for simultaneous excitation of pairs in each atom, this being strictly a quadruple excitation. This failure of CID and CISD models is likely to lead to poor descriptions of large molecules and interacting systems. [3] In this chapter, we consider the linear scaling of the energy and other expectation values for many-body systems in atomic and molecular physics and quantum 69

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chemistry with the number of particles in the system under study (and the closely related concepts of extensivity and size-consistency). The essential property of any true many-body theory of electronic structure is a linear scaling of the energy components, E, with the number of electrons, N , in the system. A model which is most often used in qualitative discussions of linear scaling with particle number is an array of well-separated identical subsystems. In particular, the model which is most often employed in such discussions is that of a linear array of well-separated, that is, non-interacting, helium atoms. In Section 3.1.1, we consider the exact Schr¨odinger equation for this model system. In Section 3.1.2, we turn our attention to independent particle models. We make some general remarks about many-body methods in Section 3.1.3. 3.1.1. The exact electronic Schr¨odinger equation The time-independent Schr¨odinger equation for a molecular system of arbitrary complexity can be easily written in the form (3.1)

HΨμ = Eμ Ψ μ,

where H is the Hamiltonian operator, Ψμ is the wave function and Eμ is the energy eigenvalue for the μth state. The Hamiltonian operator for a system of n nonrelativistic particles interacting via two-body forces is (3.2)

H=

n k=1

tk +

n

v (rk , rl )

k>l

where the first term is the kinetic energy of the kth particle (3.3)

tk = −

1 ∇2 2mk k

and the second term describes the Coulomb interaction between the particles labelled k and l c k × cl v (rk , rl ) = (3.4) . rkl mk is the mass of particle k and ck is its charge. rkl is the distance between particles k and l, and (3.5)

∇2k =

∂2 ∂2 ∂2 + 2+ 2 2 ∂xk ∂yk ∂zk

where xk , yk and zk are the cartesian coordinates of particle k. We assume the Born–Oppenheimer approximation so as to decouple the motion of the electrons from that of the nuclei. For a molecular system containing N electrons moving in the field of M fixed nuclei, the non-relativistic electronic Hamiltonian has

Brillouin–Wigner Methods for Many-Body Systems

71

the form (3.6)

He =

N M 1 Zp 1 − ∇2i − + 2 r r p=1 i=1 ip i=1 i>j ij

N

where − 12 ∇2i is the kinetic energy of the ith electron and −Zp /rip is the Coulomb interaction between the ith electron and the p nucleus. The latter has a charge of Zp , and 1/rij is the Coulomb repulsion between electrons i and j. The electronic Hamiltonian (3.6) can be written in the form (3.7)

He =

N

1 g (ri , rj ) 2 i=1 j=1 N

h (ri ) +

i=1

N

where (3.8)

M 1 Zp h (ri ) = − ∇2i − 2 r p=1 ip

and (3.9)

g (ri , rj ) =

1 . rij

The prime in (3.7) indicates that terms with i = j are omitted. To simplify our notation, we shall drop the subscript ‘e’ from the electronic Hamiltonian operator since we are only concerned with the electronic structure problem in this volume. In order to gain some understanding of the nature of the many-body problem in atoms and molecules, let us consider an array of well-separated systems, a linear array of helium atoms, for example. By ‘well-separated’ we mean that the systems are not interacting. For simplicity, let us begin by considering just two well-separated systems. The total Hamiltonian operator for the supersystem may be written (3.10)

H = HA + HB

where the Hamiltonian operator for system A is (3.11)

HA =

NA

− 21 ∇2i −

i=1

NA MA NA Zp 1 + r r p=1 i=1 ip i>j ij

whilst that for system B has a similar form (3.12)

HB =

NB i=1

− 12 ∇2i −

NB MB NB Zp 1 + . r r p=1 i=1 ip i>j ij

Here MP is the number of nuclei associated with system P , whilst NP is the number of electrons associated with system P . Because the subsystems are well separated, there are no terms in the Hamiltonian operator (3.10) associated with interactions between them. We conjecture that the exact wave function for the supersystem may

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be written as the product (3.13)

Ψ = ΨA ΨB

where ΨA describes system A and ΨB system B. Substituting (3.10) and (3.13) into the Schr¨odinger equation for the supersystem gives (3.14)

(HA + HB ) ΨA ΨB = EΨA ΨB .

Since HA and ΨA depend only on the coordinates of the electrons and nuclei in system A and likewise HB and ΨB depend only on the coordinates of the particles in system B, eq. (3.14) can be written in the form (3.15)

ΨB HA ΨA + ΨA HB ΨB = EΨA ΨB .

Dividing this equation by ΨA ΨB gives (3.16)

HB ΨB HA ΨA + = E. ΨA ΨB

The first term on the left-hand side of eq. (3.16) depends on the coordinates of the particles in system A only. Similarly, the second term on the left-hand side of this equation depends on the coordinates of the particles in system B only. The righthand side is a constant, the total energy E. But it must be possible to write this total energy as a sum of the energy of system A and that of system B, that is (3.17)

E = EA + EB .

Then eq. (3.16) can be written HB ΨB HA ΨA (3.18) − EA + − EB = 0. ΨA ΨB However, the first parenthesis contains terms which depend only on the coordinates of the particles in system A, whilst the second parenthesis contains terms depending on the coordinates of the electrons and nuclei in system B. Therefore each of these terms can be equated to zero separately, that is HA ΨA (3.19) − EA = 0 ΨA and (3.20)

HB ΨB − EB ΨB

= 0.

Equation (3.19) can then be written as (3.21)

HA ΨA = EA ΨA

and eq. (3.20) as (3.22)

HB ΨB = EB ΨB .

Brillouin–Wigner Methods for Many-Body Systems

73

For the Hamiltonian operator (3.10) which is additively separable, the energy eigenvalue is also additively separable according to (3.17). The above argument can obviously be generalized to an array of n well-separated systems. If the systems are identical, then the energy of the supersystem is just n times that of a single system: (3.23)

E = nEA .

The exact total energy of the system therefore scales linearly with the number of electrons in the supersystem. Additive separability is of fundamental importance in the description of many-body systems. Linear scaling ensures that the energy of n well-separated subsystems is equal to n times the energy of a single subsystem. 3.1.2. Independent particle models In the above discussion we have been concerned with the exact electronic Hamiltonian, energies and wave functions of a supersystem consisting of an array of wellseparated subsystems. We now turn our attention to the description afforded by some independent particle model, in which the electrons move in some mean field. The most commonly used approximation of this type is the Hartree–Fock model, but the discussion presented in this section is not restricted to this particular method. In particular, we write the total electronic Hamiltonian operator in the form (3.24)

H = H0 + H1

where H0 is a model electronic Hamiltonian operator associated with some independent particle model and H1 is the perturbation which recovers the full electronic Hamiltonian operator. The model electronic Hamiltonian operator has the form (3.25)

H0 =

N

(h (ri ) + u (ri ))

i=1

where h (ri ) is the one-electron component of the electronic Hamiltonian, that is (3.26)

h (ri ) = − 12 ∇2i −

M Zp r p=1 ip

and u (ri ) is some mean field potential which describes the averaged interactions of the electron labelled i with the remaining electrons in the system. We shall not concern ourselves with a detailed definition of the potential u (ri ) at this point. We can put (3.27)

H0 =

N

f (ri )

i=1

where (3.28)

f (ri ) =

− 12 ∇2i

M Zp − + u (ri ) r p=1 ip

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I. Hubaˇc and S. Wilson

or (3.29)

f (ri ) = h (ri ) + u (ri ) .

The model electronic eigenproblem takes the form (3.30)

H0 Φ = EΦ.

The perturbation operator is given by (3.31)

H1 =

N N 1 − u (ri ) r i>j ij i=1

so that the full electronic Hamiltonian is recovered on adding the model Hamiltonian and the perturbation. The operator (3.31) is sometimes termed the fluctuation potential. Now let us consider the description of an array of well-separated systems afforded by the model Hamiltonian. For simplicity, we again restrict our attention to a supersystem consisting of two systems labelled A and B initially and then generalize the results. The model Hamiltonian for the supersystem can be written (3.32)

(A)

H0 = H0

(B)

+ H0

since terms describing the interactions between the nuclei in one system with the (A) (B) electrons in the other vanish. The model Hamiltonian operators H0 and H0 have the form (3.33)

(A)

H0

=

NA

f (riA )

iA =1

and (3.34)

(B) H0

=

NB

f (riB ) .

iB =1

We now follow the procedure that we adopted above for the exact electronic Schr¨odinger equation and conjecture that the model electronic wave function for the supersystem can be written (3.35)

Φ = ΦA ΦB .

Substituting (3.32) and (3.35) into the model eigenequation (3.30) gives (A) (B) ΦA ΦB = EΦA ΦB . (3.36) H0 + H0 (A)

Using the fact that H0 depends only on the coordinates of the particles associated (B) with system A and H0 depends only on the coordinates of the particles associated with system B, this equation can be rewritten as (3.37)

(A)

(B)

ΦB H0 ΦA + ΦA H0 ΦB = EΦA ΦB .

Brillouin–Wigner Methods for Many-Body Systems

75

Dividing by ΦA ΦB gives (A)

(3.38)

(B)

H ΦB H0 ΦA + 0 = E, ΦA ΦB

which upon putting (3.39) gives (3.40)

E = EA + EB

(A)

H0 ΦA − EA ΦA

+

(B)

H0 ΦB − EB ΦB

=0

and so (A)

(3.41)

H0 ΦA − EA = 0 ΦA

and (B)

(3.42)

H0 ΦB − EB = 0 ΦB

(3.43)

H0 ΦA = EA ΦA

or (A)

and (3.44)

(B)

H0 ΦB = EB ΦB .

The above argument can be generalized to an array of n well-separated systems. Just as in the case of the exact electronic energy, the energy associated with the model Hamiltonian is n times that of a single system (3.45)

E = nEA .

The total model electronic energy is thus seen to scale linearly with the number of electrons in the system. We have asserted in Section 1.4 that the essential property of any true many-body theory of electronic structure is a linear scaling of the energy components, E, with the number of electrons, N , in the system [1, 4–6], i.e. (3.46)

E ∝ N.

In this section, we have demonstrated that this requirement is satisfied by the solutions of the exact electronic Schr¨odinger equation and by the solutions of the eigenproblem for some approximate electronic Hamiltonian operator associated with an arbitrary independent particle picture.

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3.1.3. Many-body theories of electron correlation Unfortunately, the determination of exact solutions of the Schr¨odinger equation is intractable for almost all systems of practical interest. On the other hand, independent particle models are not sufficiently accurate for most studies of molecular structure. In particular, the Hartree–Fock model, which is the ‘best’ independent particle model in the variational sense, does not support sufficient accuracy for many applications. Some account of electron correlation effects has to be included in the theoretical apparatus which underpins practical computational methods. Although the energy associated with electron correlation is a small fraction of the total energy of an atom or molecule, it is of the same order as most energies of chemical interest. However, such theories may not be true many-body theories. They may contain terms which scale non-linearly with electron number and are therefore unphysical and should be discarded. Any theory which contains such unphysical terms is not acceptable as a true many-body method. Either the theory is abandoned or corrections, such as that of Davidson [7] which is used in limited configuration interaction studies, are made in an attempt to restore linear scaling. The development of many-body theories in the 1950s by Brueckner, Goldstone and others [8–10], revealed the shortcomings of the Brillouin–Wigner perturbation expansion [11–13]. It was at this time that it first became widely recognized that the presence of the exact energy in the denominators of the expressions for the energy components in the Brillouin–Wigner expansion ensure that unphysical terms which scale non-linearly with the number of electrons arise. Brillouin–Wigner perturbation theory was merely used as a step in the development of an acceptable many-body perturbation theory, particularly by Brandow [14] in studies of multi-reference formalisms. In chapter 1, we noted that March et al. [4], in their 1967 text on many-body techniques for quantum systems, dismiss Brillouin–Wigner perturbation theory as a valid many-body method. In particular, they write (p. 71) . . . it will be completely misleading to apply many-body perturbation theory in the Brillouin-Wigner form, short of considering an infinite number of terms in the limit of large N. Lindgren and Morrison in their treatise on Atomic many-body theory [5] recognize that: The Brillouin-Wigner form of perturbation theory is formally very simple. It has the disadvantage, however, that the operators depend on the exact energy of the state considered. This requires a self-consistency procedure and limits its application to one energy level at a time. There are also more fundamental difficulties with the Brillouin-Wigner theory . . . The Rayleigh–Schr¨odinger perturbation theory . . . does not have these shortcomings, and it is therefore a more suitable basis for many-body calculations than the Brillouin-Wigner form of the theory.

Brillouin–Wigner Methods for Many-Body Systems

77

One of the present authors [1] makes a similar point in a text on Electron correlation in molecules published in 1984 (p. 65) The perturbation expansion of Lennard-Jones, Brillouin and Wigner does not lead to expressions which are directly proportional to the number of electrons in the system being studied. Likewise, Harris, Monkhorst and Freeman in their monograph Algebraic and Diagrammatic Methods in Many-Fermion Theory [6] wrote (p. 224) A . . . fundamental difficulty with the Wigner–Brillouin expansion is its lack of size consistency. In contrast to the Brillouin–Wigner expansion, the Rayleigh–Schr¨odinger perturbation series does support a many-body theory. Although unphysical terms arise in each order of the perturbation expansion beyond second-order, these terms can be shown to mutually cancel within a given order. Once this cancellation has been explicitly carried out, we obtain the many-body perturbation theory, a theory which is addressed in detail in the next section. The surviving terms which scale linearly with electron number may be associated with ‘linked connected diagrams’. The ‘linked connected diagram’ theorem is the central concept of the many-body perturbation theory, which we consider in some detail in the following section. 3.2. MANY-BODY PERTURBATION THEORY It has been shown in the previous section that the exact total energy for a many-body system scales linearly with the number of electrons in the system. It has also been shown that a suitably chosen independent particle model leads to an approximation to the energy of a many-body system, which also scales linearly with the number of electrons in the system. In this section, we turn our attention to the development of approximations which are more accurate than the independent particle model and can take account of electron correlation effects. The many-body perturbation theory plays a pivotal role in the development of approximate treatments of correlation effects which scale linearly with the number of electrons. Indeed, many-body perturbation theory provides the foundation upon which almost all modern theories of electron correlation in molecules are constructed. A key feature of the many-body perturbation theory is the use of the method of second-quantization. We therefore open this section by introducing the second quantization formalism. We then discuss the Rayleigh–Schr¨odinger perturbation theory in its many-body form, that is, many-body Rayleigh–Schr¨odinger perturbation theory. We close this section by presenting the many-body perturbation theory with an emphasis on its diagrammatic formulation.

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3.2.1. Second-quantization formalism The antisymmetry of electronic fermion wave functions can lead to a cumbersome notation, especially when describing many-electron systems. In the method of second-quantization, only the occupied single particle levels are specified together with their occupation numbers. This is the only information which is compatible with the indistinguishability of electrons. 3.2.1.1. Second-quantization and the many-body problem Second-quantization forms the basis of a very powerful technique for developing theoretical descriptions of many-body systems. It allows attention to be focussed on specific correlation processes, rather than on the behaviour of the given many-body system as a whole. Furthermore, it has given rise to a diagrammatic representation which is precisely defined and, therefore, wholly equivalent to ‘conventional’ algebraic formulations. However, as well as providing a well-defined representation of components of the correlation energy, the diagrams also afford a picture of the associated correlation processes. In the second-quantization formalism, theoretical expressions are written in terms of matrix elements of operators in a given basis and are manipulated using the algebra of creation and annihilation operators. In the original, first quantized formalism we work with the analytical form of the operators. There are many excellent text books on second-quantization techniques to which we refer the interested reader for a more thorough discussion than that given here. We mention, in particular: • • • • • •

A. Messiah (1961) Quantum Mechanics [15] P. Roman (1969) Introduction to Quantum Field Theory [16] S.S. Schweber (1964) Relativistic Quantum Field Theory [17] J. Ziman (1969) Elements of Advanced Quantum Theory [18] S. Weinberg (1995) The Quantum Theory of Fields [19] P. Jorgensen and J. Simons (1982) Second-Quantization-based Methods in Quantum Chemistry [20]

In this section, we briefly describe the second quantized formalism giving sufficient details for the application which we describe in this monograph. This also serves to establish our notation. We begin by recording something of the historical origins of the second quantization method which should help the reader understand its importance in the study of many-body systems.1 Let us observe that the Schr¨odinger equation can be easily written down for an atom or, more particularly, a molecule of arbitrary complexity. (We also note that there is no known limit to the complexity of possible molecular species). The difficulty is usually said to lie not in writing down the appropriate eigenvalue problem 1 For an interesting account of the history of quantum electrodynamics, see Schweber’s QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga [21].

Brillouin–Wigner Methods for Many-Body Systems

79

but in the development of accurate approximations to the solutions of this molecular Schr¨odinger equation. However, the Schr¨odinger equation for a system of arbitrary complexity has another problem associated with it, namely, it applies to a fixed number of electrons. The (‘first quantized’) Schr¨odinger equation is obtained by applying the correspondence principle to the equations of non-relativistic mechanics. The Schr¨odinger equation applies to systems in which the number of electrons is conserved. The second-quantization formulation of quantum mechanics is equivalent to the Schr¨odinger quantum mechanics. This equivalence was demonstrated by Jordan and Klein [22] for bosons and by Jordan and Wigner [23] for fermions. However, as we shall demonstrate below, the second-quantization formulation provides a notation which has powerful advantages both in terms of ease of use and of generality. Let us digress by briefly turning our attention to the use of the second-quantization formulation in relativistic quantum mechanics. In non-relativistic quantum mechanics, the use of the second quantization formalism has advantages. In relativistic quantum mechanics, the use of the second-quantization formalism is mandatory. Since the methods described in this volume can be applied to the relativistic electronic structure problem, we give here a very short description of the use of second-quantization methods in relativistic quantum mechanics. A much more detailed and thorough discussion can be found in the monograph by Grant [24], Quiney’s contributions to the Handbook of Molecular Physics and Quantum Chemistry [25], or in the review by Quiney, Grant and Wilson [26]. If, following Dirac, we write down the eigenproblem for, say, the hydrogen atom, we find a very different set of solutions to those found in the non-relativistic (Schr¨odinger) case. Solution of the Dirac equation for the hydrogen atom leads to a spectrum which is divided into two branches – a positive energy branch and a negative energy branch. Thus, whilst in the non-relativistic formalism, the ground state of the hydrogen atom consists of a single electron occupying the lowest energy level in the spectrum, in the relativistic formalism, the ground state of the hydrogen atom consists of a single electron occupying the lowest energy level in the positive energy branch of the Dirac spectrum and, according to Dirac’s famous conjecture, this electron is prevented from decaying into one of the negative energy states because these states are themselves filled with electrons. Dirac writes [27]: We assume that nearly all the negative-energy states are occupied, with one electron in each state in accordance with the exclusion principle of Pauli. An unoccupied negative-energy state will now appear as something with a positive energy, since to make it disappear, i.e. to fill it up, we should have to add to it an electron with negative energy. We assume that these unoccupied negative-energy states are the positrons. A consequence of this conjecture, which survives in more modern formulations of relativistic quantum mechanics, is that even the simple hydrogenic atom is a ‘many-body problem’.2 The electrons filling the negative energy branch of the Dirac spectrum are 2

The modern formalism of relativistic quantum mechanics, quantum ﬁeld theory, employs Feynman’s picture in which the negative energy states moving backwards in

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not directly observable. However, when an electron is excited from a negative energy state into a state lying in the positive energy branch of the Dirac spectrum, a ‘hole’ is created in the negative energy ‘sea’. This ‘hole’ is observable; it is a positron. A direct consequence of this Dirac picture is that the number of electrons in a relativistic system is not conserved. Electron–positron pairs can be created. In the Dirac picture, it is the total charge of the system which is conserved and not the number of electrons. Therefore, the use of second-quantization is mandatory in relativistic quantum mechanics and in quantum electrodynamics. This concludes our digression. The development of quantum electrodynamics saw the introduction of diagrammatic techniques. In particular, Feynman [28], in a paper entitled Space-Time Approach to Quantum Electrodynamics, introduced diagrams which provide not only a pictorial representation of microscopic processes, but also a precise graphical algebra which is entirely equivalent to other formulations. They have a simplicity and elegance which is not shared by, for example, purely algebraic methods. With the benefit of hindsight, it is perhaps not surprising that second-quantization and diagrammatic formulations emerged as a powerful approach to the quantum many-body problem in non-relativistic quantum mechanics. In condensed matter physics, the systems studied usually contain an infinite number of electrons. Intuitively, one might expect that correlation processes will involve just a few electrons at a time. Second-quantization and diagrammatic methods provide the necessary theoretical apparatus for doing exactly this. Second-quantization allows the reformulation of the N -body problem which we display in Figure 3.1 in what is usually termed the ‘particle picture’. This is the picture corresponding to the Schr¨odinger equation describing the motion of all N electrons in the studied system. In Figure 3.1, we provide a schematic illustration of a ground state, a singly excited state, a doubly excited state, and a triply excited state in the particle picture. The corresponding states are displayed in the ‘particle–hole’ picture in Figure 3.2(a) which shows the ground state. This is termed the Fermi vacuum or simply the vacuum state. The Fermi level is the energy value shows separates the single particle states, which are occupied in the ground or reference state, from those which are unoccupied. In the Fermi vacuum, there are no holes below the Fermi level, which would be created were an electron to be excited, and there are no corresponding particles above the Fermi level. In Figure 3.2(b), we show a singly excited state with one hole created below the Fermi level and one corresponding particle above it. Figure 3.2(c) and (d) provide a schematic representation of doubly and triply excited states. We emphasize again that the particle–hole picture allows us to concentrate on the essential physics of the electron correlation problem. Later in this chapter, we shall demonstrate how diagrams can encapsulate this physics and, at the same time, provide the basis of a rigorous computational scheme

time, corresponds to positive energy states of antiparticles moving forward in time. The advantage of the Feynman picture is that it is applicable to both fermions and bosons.

Brillouin–Wigner Methods for Many-Body Systems a)

81

b) ↓

↑ ↓

↑ ↓

↑ ↓

↑

↑ ↓

↑ ↓

c)

d) ↓

↑

↑ ↓

↑

↑ ↓ ↑

↓ ↑

↓

↓

Figure 3.1. Simple illustration of the particle formalism. (a) Shows the reference configuration with all particles in their ground states. (b) Shows a configuration resulting from a single excitation. (c) Is a doubly excited configuration and (d) is a triply excited configuration.

for the study of electron correlation effects. Now we turn our attention to the method of second-quantization. 3.2.1.2. Creation and annihilation operators and the occupation number representation The Hamiltonian operator, H, for an N -electron system is invariant with respect to the exchange of any two electrons. So, if Pij is an operator which permutes the electron indices i and j, then (3.47)

[H, Pij ] = 0.

The exact wave function, Ψ, is a solution of the Schr¨odinger equation (3.48)

HΨ = EΨ

82

I. Hubaˇc and S. Wilson a)

b) ↓

↓

c)

d) ↓

↑ ↓

↑

↑

↑

↑ ↓

↓ ↑

Figure 3.2. Simple illustration of the particle-hole formalism. This figure shows the states given in the particle formalism in Figure 3.1 when depicted in the particle-hole picture. (a) shows the reference configuration or vacuum state with no particles above the Fermi level and no holes below it. (b) corresponds to a single excitation which creates a hole below the Fermi level and a particle above it. (c) is a doubly excited state with two holes and two particles and (d) is associated with a triply excited state with three holes and three particles. The particle-hole formalism focusses attention on the excitation process; the particles and holes created during an excitation. The other electrons in the studied many-body system are merely spectators to the excitation process.

and Pij Ψ is also a solution since (3.49)

HPij Ψ = Pij HPij Ψ = EPij Ψ.

If E is non-degenerate Pij Ψ must be a multiple of Ψ (3.50)

Pij Ψ = λΨ

where λ is a constant. Applying Pij to this equation from the left gives (3.51)

(Pij )2 Ψ = λ2 Ψ = Ψ

so that (3.52)

λ2 = 1

Brillouin–Wigner Methods for Many-Body Systems

83

or (3.53)

λ = ±1.

For fermions, and in particular, for electrons (3.54)

λ = −1

and the wave function is antisymmetric (3.55)

Pij Ψ = −Ψ.

The Pauli exclusion principle follows automatically from this. Slater introduced a convenient method for constructing many-electron wave functions that are antisymmetric with respect to the exchange of any two electrons from sets of one-electron spin-orbitals. A Slater determinant |Φ is written as ϕA1 (xA1 ) ϕA2 (xA1 ) . . . ϕAN (xA1 ) 1 − 2 ϕA1 (xA2 ) ϕA2 (xA2 ) . . . ϕAN (xA2 ) (3.56) |Φ = (N !) ... ... ... ϕA1 (xAN ) ϕA2 (xAN ) ϕAN (xAN ) or, more compactly, as (3.57)

|Φ =

N 1 p (−1) P ϕAi (xAi ) (N !)1/2 i=1 P

where ϕAi (xAi ) represents spin-orbitals and x represents the electronic space and spin-coordinates. Let us now arrange the one-electron functions, ϕAi (xAi ), so that corresponding orbital energies, εAi are not decreasing. So the orbital energies are ordered as follows: (3.58)

εA1 ≤ εA2 ≤ · · · ≤ εAi ≤ εAi+1 ≤ · · · ≤ εAm−1 ≤ εAm ,

where m is the total number of spin-orbitals in the given set, and the corresponding orbitals are ordered as follows: (3.59)

ϕA1 ϕA2 . . . ϕAi ϕAi+1 . . . ϕAm−1 ϕAm

In a definition which is entirely equivalent to that given above, we can completely specify a Slater determinant by recording which of the spin-orbitals ϕAi from a given set occur in the Slater determinant and which do not. This may be expressed by the state vector (3.60)

|n1 n2 . . . ni . . . nm ,

where the indices ni can have the value 0 or 1 depending on whether the spin-orbital ϕAi is occupied or unoccupied; that is, whether it occurs in the Slater determinant or not. The numbers ni are called occupation numbers and this representation of Slater determinants is accordingly called the occupation number representation. |n1 n2 . . . ni . . . nm is a vector in Fock space.

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We shall define the creation (Xi+ ) and annihilation (Xi ) operators on the basis of vectors |n1 n2 . . .. The annihilation operator is defined as follows: (3.61)

Xi |n1 n2 . . . ni . . . = (−1)Si ni |n1 n2 . . . (ni − 1) . . .

where (3.62)

Si =

i−1

nk .

k=1

The quantity (−1)Si gives the correct sign of the determinant. In quantum chemical studies, it is usually much more convenient to specify the state vectors so that only the occupied spin-orbitals are listed in the vector. Thus, instead of |n1 n2 . . ., we write |A1 A2 . . .. With this more compact notation, we define the annihilation operator XAi in the following manner: (3.63)

XAi |Ai A2 . . . AN = |A2 . . . AN

(3.64)

XAi |A1 . . . AN = 0

where i = 1, . . . , N . We emphasize that the annihilation operator is defined in such a way that it always acts on the first component of the state vector. The commutation relations, which are considered below, have to be employed to ensure that the state Ai commutes with the first component of state vector |A1 A2 . . . Ai . . .. This guarantees the correct sign. We can now define the vacuum state vector |0 as follows: (3.65)

XAi |Ai = |0 .

The vacuum state |0 is the so-called true physical vacuum state. It is the state with no particles. Let us suppose that our vacuum state |0 and the spin-orbitals |A1 , |A2 . . . satisfy the following relations (3.66)

0|0 = 1

and (3.67)

Ai |Aj = δij

where δij is the Kronecker delta ! 1 when i = j (3.68) δij = . 0 when i = j The state which is Hermitian conjugate to the state |0 can be written as (3.69)

+ . 0| = Ai | XA i

Let us multiply eq. (3.69) from the right by |0 which gives (3.70)

+ |0 . 0|0 = Ai | XA i

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85

According to eq. (3.67), we can write (3.71)

+ Ai | XA |0 = 1 = Ai |Ai i

from which immediately follows that (3.72)

+ |0 = |Ai . XA i

+ is termed a creation operator. We can define relations analogous The operator XA i to those given in eqs. (3.63) and (3.64) for the annihilation operator. In particular, we can write

(3.73)

+ XA |A1 . . . AN = |Ai A1 . . . AN i

and (3.74)

+ |. . . Ai . . . = 0 XA i

where i = 1, . . . , N . In the occupation number representation, eq. (3.74) becomes (3.75)

+ |n1 n2 . . . ni . . . = (−1)Si (1 − ni )|n1 n2 . . . (ni + 1) . . . XA i

where Si is defined in the same way as in eq. (3.62). From eqs. (3.72) and (3.73) it can be seen that any Slater determinant can be written as a product of creation operators acting on the vacuum state |0. Explicitly, we can write N

(3.76)

+ XA |0 . i

|A1 . . . AN = i=1

+ , instead of a Thus we see that we can consider a product of creation operators, XA i Slater determinant. The principal reason for using the second-quantization formulation of the manybody problem is that any one-particle operator, v, and any two-particle operator, g, + can be written in terms of the creation and annihilation operators, XA and XA . The Slater determinants for an N -electron system afford a particular realization of a set of basis vectors in a linear space L (N ). When this space is enlarged so as to include basis vectors corresponding to all possible numbers of electrons as follows:

(3.77)

L (1) ⊕ L (2) ⊕ . . . ⊕ L (N − 1) ⊕ L (N ) ⊕ L (N + 1) ⊕ . . . ≡ F

we obtain the Fock space F. There is an isomorphism between the representation in Fock space and that in Schr¨odinger space. Thus operations in Fock space are in one-to-one correspondence with those in Schr¨odinger space. There are combinations of creation and annihilation operator products together with appropriate numerical coefficients that have matrix elements between Fock-space kets which are identical to those between the corresponding Slater determinants in the Schr¨odinger representation. An arbitrary one-electron operator v S (i) in the Schr¨odinger representation can be written (3.78) v S (i) φk (xi ) = φk (xi ) k | v S |k , k

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I. Hubaˇc and S. Wilson

where k | v S |k is a matrix element of the operator. We note that for a many-electron system there is always a sum of the form (3.79)

vS =

N

v S (i) .

i=1

In the Schr¨odinger representation we can write v S det |φk1 (x1 ) φk2 (x2 ) . . . φki (xi ) . . .| = det φk1 (x1 ) φk2 (x2 ) . . . φk (xi ) . . . ki | v S |ki , i

ki ,ki

where the determinant on the right-hand side of this equation is obtained from that on the left-hand side by making the substitution φki ← φki and ki | v S |ki denotes the matrix element. In Fock space equation (3.80) becomes (3.80) v F |k1 k2 . . . ki . . . = |k1 k2 . . . ki . . . vkFi ,ki . ki ,ki

The coefficient vkF ,ki is equal to the matrix element ki | v S |ki . We note that the ket i on the right-hand side of eq. (3.80) is obtained by making the substitution ki ← ki and can be written as (3.81)

Xk† Xki |k1 k2 . . . kN . i

Furthermore, the restriction that ki be occupied, can be removed, since if ki is not in the list, then the ket is destroyed. Renaming the spin orbitals and noting that the results apply for any ket, we can write (3.82) v F = vr,s Xr† Xs r,s

- a result which is independent of the number of electrons in the system. The operator isomorphism for one-electron operator h (i) which arises in the Schr¨odinger Hamiltonian as the sum (3.83)

N

h (i)

i=1

can be written as N (3.84) h (i) ⇐⇒ φr | h |φs . i=1

r,s

For the two-electron operator g (i, j) we have the isomorphism N (3.85) g (i, j) ⇐⇒ φr φs | g |φt φu Xr† Xs† Xu Xt , i,j=1

r,s,t,u

where the order of the operators Xt and Xu should be noted.

Brillouin–Wigner Methods for Many-Body Systems

87

A one-particle operator, v, can be written as + (3.86) v = A| v |B XA XB , AB

whereas an arbitrary two-particle operator, g, can be written 1 + + (3.87) g = AB| g |CD XA XB XD XC , 2 ABCD

+ XA i

and XB are creation and annihilation operators respectively, defined on where the one-particle basis set |A, |B, . . . . The reader should note the order of the spinorbital indices on the right-hand-side of eq. (3.87). The spin-orbital |A can be written in terms of a spatial orbital |a and a one-electron spin function |σ (3.88)

|A = |a |σ .

|σ, the one-electron spin-function, is given by ! |α (3.89) |σ = . |β In the second-quantization formalism, a Hamiltonian operator H which can be written in the Schr¨odinger formalism for an N -electron system as (3.90)

H=

N i

hi +

N

gij ,

i>j

where hi is a one-electron operator and gij is a two-electron operator, can be written 1 + + + (3.91) H = A| h |B XA XB + AB| g |CD XA XB XD XC . 2 AB

ABCD

In eq. (3.91), the notation is such that A and C (B and D) have the same space and spin coordinates. We emphasize again that the sequence of the creation and annihilation operators in the second term of eq. (3.91) is very important. We note that in the second quantized Hamiltonian, H, defined in eq. (3.91), we have products of creation and annihilation operators. However, note that the Hamiltonian operator in this form does not depend on the number of particles. It is completely defined by the one-electron functions |A, |B . . . . 3.2.1.3. Normal products, contractions and Wick’s theorem To make further progress, we must understand how to handle products of creation and annihilation operators of the type arising in eq. (3.91). We shall introduce Wick’s theorem and also some necessary definitions and relations. The creation and annihilation operators satisfy the anticommutation relation + + + (3.92) XB + XB XA = δAB . XA , X B + = X A

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It is this anticommutation property which ensures that the N -particle system obeys Fermi statistics. The anticommutator allows the order of creation and annihilation operators in a product to be changed by writing it in the form + + + XA XB = XA , XB + − XB XA (3.93)

+ = δAB − XB XA .

Antisymmetry of electrons implies that annihilation operators anticommute XA , XB + = XA XB + XB XA (3.94)

= 0.

and that creation operators also anticommute + + + + + + XA , XB + = XA XB + XB XA (3.95)

= 0.

Next, we have to define the normal product or n-product, and contraction or pairing. The simplification of matrix elements requires that we move creation operators to the left of annihilation operators. A normal product is a reordered operator string which satisfies this requirement. A contraction or pairing of creation and/or annihilation operators is their vacuum expectation value. The normal product is defined as follows: (3.96)

n[Mi1 . . . Mik . . . MiN ] = (−1)p M1 . . . MN

where Mi stands either for a creation or annihilation operator. On the left-hand side of eq. (3.96) the creation and annihilation operators Mik are in arbitrary order. But all creation operators on the right-hand side of eq. (3.96) are to the left of all annihilation operators, with p being the parity of the permutation required to bring the product into this form. So, for example, (3.97)

n[Xi1 Xi+2 Xi3 . . . Xik Xi+k+1 . . . Xi+N ] = (−1)p X1+ X2+ X3+ . . . Xk+ Xk+1 . . . XN ,

where the sign (−1)p is determined by the parity of the associated permutation operation. Specific examples are (3.98)

+ + ] = −XB XA n[XA XB

and + + + + XC XD ] = − n[XA XB XD XC ] n[XA XB + + XA XD XC ] = n[XB + + = − n[XB XD XA XC ]

(3.99)

+ + XD XA XC . = − XB

The contraction or pairing of two creation and/or annihilation operators M1 and M2 is defined as

Brillouin–Wigner Methods for Many-Body Systems

89

(3.100) M 1 M2 = M1 M2 − n[M1 M2 ] that is, the contraction is the product of creation and annihilation operators minus the normal product of the operators. The contraction of operators is designated by a line joining the two operators placed above the operator product. The contraction or pairing of two operators M1 and M2 is equivalent to (3.101) M 1 M2 = 0| M1 M2 |0 . It can be easily shown that the only non-zero contraction is + + X A XB = 0| XA XB |0 (3.102) = δAB .

Note that this contraction is a number and that the creation operator is positioned on the right of the annihilation operator. All other contractions are zero, that is (3.103) X A XB = 0 (3.104) X + A XB = 0 + (3.105) X + A XB = 0.

The only non-vanishing contraction is between an annihilation operator and a creation operator read from left to right. Having defined both a normal product of creation and annihilation operators and the contraction of a pair of these operators, we are now in a position to define a normal product with contractions. Consider, for example, the following simple case: + + (3.106) n[X A XB XC ].

We can permute the operators until the contracted operators are adjacent to one another + + + XC ] = − n[X A XC+ XB ] n[X A XB

(3.107)

+ ] = − n[δAC XB + = − δAC XB .

In general, a normal product with contraction is given by n[M1 M2 . . . M i . . . M j . . . M k . . . Ml . . . MN ] (3.108)

= (−1)p M i M j M k Ml . . . n[M1 M2 . . . MN ],

where p, the parity of the corresponding permutation, is defined in eq. (3.96). Contracted pairs can be ‘removed’ from the normal product, provided account is taken of the sign changes resulting from the permutation required to bring the contracted operators into adjacent positions.

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Now we are ready to formulate the time-independent Wick’s theorem. Wick’s theorem provides a formal mechanism for the simplification of matrix elements and, in particular, of Fermi vacuum expectation values. The time-independent Wick’s theorem may be stated as follows: M1 M2 . . . MN = n[M1 M2 . . . MN ] + n[M 1 M2 . . . MN ] + · · · . . . n[M 1 M2 . . .MN ] + n[M 1 M 2 M 3 M4 . . . MN ]. + ···

(3.109) This equation states that

a product of creation and annihilation operators is equal to a normal product of these operators plus a sum of normal products with one contraction plus the sum of normal products with two contractions etc., up to the normal product where all operators are contracted. The proof of Wick’s theorem proceeds by induction. The theorem is assumed to be true for a product of n creation and annihilation operators and is then shown to be true for a product of n + 1 such operators. Demonstration of the theorem for n = 2 completes the proof. Details of the proof of the time-independent Wick’s theorem are given in Appendix B. The theorem (3.109) can be extended to the so-called generalized Wick’s theorem. This extension allows products of the following form to be handled: (3.110) n[M1 . . . Mi ]n[Mi+1 . . . Mk ] . . . n[Mk+1 . . . Ml ]. This can be expressed in a manner similar to eq. (3.109), but with contractions between operators within the same normal product omitted. Equations (3.65) and (3.69) have an important consequence which should be emphasized here. The vacuum expectation value of a normal product is zero, that is (3.111) 0|n[M1 . . . MN ]|0 = 0, if at least one operator in the normal product n[M1 . . . MN ] is not contracted. The Fermi vacuum expectation value of a normal product is always zero if it contains any uncontracted operators. Non-zero matrix elements arise only from the fully contracted terms of Wick’s theorem. Furthermore, Wick’s theorem requires that contractions are carried out in all possible ways. 3.2.1.4. Particle–hole formalism In eq. (3.76) N

(3.76)

+ XA |0 , i

|A1 . . . AN = i=1

Brillouin–Wigner Methods for Many-Body Systems

91

the number of creation operators is equal to the number of electrons. For manyelectron systems, the number of creation and annihilation operators is large. The handling of products of large numbers of second-quantization operators can become unwieldy. For example, we note the large number of terms which can occur when applying Wick’s theorem. But, in the Slater determinant for a doubly excited configuration, which we write as Φab ij , for example, the only important orbitals are those labelled i and j, which are occupied in the ground state, and those labelled a and b, which are unoccupied in the ground state. The remaining occupied orbitals are essentially ‘spectators’. For both theoretical and practical reasons, it is very con venient to describe the determinant Φab ij in terms of operators associated with the pertinent orbitals, that is, the single particle states labelled i, j, a, and b. This goal can be achieved using the particle–hole formalism, which we described qualitatively above. We are then able to concentrate on the microscopic details of a particular correlation term. This freedom to re-cast the many-body problem in the language of the particle–hole formalism constitutes the main reason for adopting the method of second-quantization in the study of atomic and molecular electronic structure. The essence of the particle–hole formalism lies in the redefinition of the vacuum state that it permits. Consider now the closed-shell ground state Slater determinant |Φ0 which can be expressed, using eq. (3.75), in the form N + XA |0 . i

(3.112) |Φ0 = i=1

In using the particle–hole formalism, we adopt |Φ0 as a new vacuum state. We call this reference state the Fermi vacuum state. With respect to this Fermi vacuum state |Φ0 , we can now define new creation and annihilation operators. To distinguish them from the operators X + and X described above, the creation and annihilation operators in the particle–hole formalism shall be designated Y + and Y , respectively. If we label the occupied spin-orbitals as |A and the virtual spin-orbitals as |A , then the creation (Y + ) and annihilation (Y ) operators in the particle–hole formalism are then defined in the following manner: (3.113) YA+ = XA ;

+ YA = XA

and + (3.114) YA+ = XA ;

YA = XA .

From this definition, it is evident that application of creation operator YA+ to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in |Φ0 . The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in |Φ0 . The effect of YA+ on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA on |Φ0 is the annihilation of a particle in virtual spin-orbitals. Thus, for example, a singly excited Slater determinant |Φai can be described as + + + (3.115) XA X |Φ0 = Y Y |Φ0 . A A A a

i

a

i

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I. Hubaˇc and S. Wilson

The reference state |Φ0 corresponds to the state in which both the number of particles and the number of holes equal zero. The determinant |Φai corresponds to a single particle and a single hole. Similarly, a doubly excited Slater determinant Φab ij can be described in terms of two particles and two holes: + + + + + + (3.116) XA X X X |Φ0 = Y Y Y Y |Φ0 . A A A A A A A a

i

b

j

a

i

b

j

For particle–hole creation and annihilation the Y operators satisfy the same anticommutation relations as those given above for the X operators. Specifically, we have [YA+ , YB ]+ = YA+ YB + YB YA+ = A |B

(3.117)

= δ A B

and [YA+ , YB ]+ = YA+ YB + YB YA+ = A |B

(3.118)

= δA B .

For the Y operators, we can now define a normal product analogous to that defined for the X operators. We shall designate the normal product for the Y operators as (3.119) N [Mi1 . . . Mik . . . MiN ] = (−1)p M1 . . . MN where Mi is now taken to stand for a creation or annihilation operator in the particle– hole formalism. The contraction or pairing of two creation and/or annihilation operators in the particle–hole formalism M1 and M2 is defined as (3.120) M1 M2 = M1 M2 − N [M1 M2 ], that is, the contraction is the product of creation and annihilation operators minus the normal product of the operators. The contraction of operators in the particle–hole formalism is designated by a line joining the two operators in question placed below the product. The contraction or pairing of two operators M1 and M2 is equivalent to (3.121) M1 M2 = 0| M1 M2 |0 . It can be easily shown that the only non-zero contraction is Y A YB+ = 0| YA YB+ |0 (3.122)

= δAB .

Note that this contraction is a number and that the creation operator is positioned on the right of the annihilation operator. All other contractions are zero. We can define a normal product with contraction in the particle–hole formulation as:

Brillouin–Wigner Methods for Many-Body Systems

93

N [M1 M2 . . . Mi . . .Mj . . .Mk . . .Ml . . . MN ] (3.123)

= (−1)p Mi Mj Mk Ml . . . N [M1 M2 . . . MN ],

where p is the parity of the corresponding permutation. Similarly, we can state Wick’s theorem in the particle–hole formalism M1 M2 . . . MN = N [M1 M2 . . . MN ] + N [M1 M2 . . . MN ] + · · · . . . N [M1 M2 . . .MN ] + N [M1 M2 M3 M4 . . . MN ] + · · ·

(3.124)

and the generalized Wick’s theorem for products of the form (3.125) N [M1 . . . Mi ]N [Mi+1 . . . Mk ] . . . N [Mk+1 . . . Ml ] with contractions between operators within the same normal product. Now eqs. (3.117) and (3.118) show that we have two sets of Y operators; one of these sets operates on the holes and satisfies the anticommutation relation (eq. (3.117)) (3.126) [YA+ , YB ]+ = δA B and the other set operate on the particles and obey the anticommutation relation (eq. (3.118)) (3.127) [YA+ , YB ]+ = δA B . To obtain a more convenient form of these anticommutation relations, we shall find it useful to define the following functions

(3.128) τ (A ) = 0;

τ (A ) = 1

and

(3.129) υ(A ) = 1;

υ(A ) = 0

so that τ (A) is 0 if the state A is occupied and 1 if it is not, whereas υ(A) is 1 if the state is occupied and otherwise 1. The contractions of Y operators may now be expressed in terms of the X operators as follows + (3.130) X + A XB = 0;

X+ A XB = υ(A)δAB

and (3.131) X A XB = 0;

+ X A XB = τ (A)δAB .

Hereafter, we shall use the X operators in the sense of (3.130) and (3.131), i.e. we are working in the particle–hole formalism and contractions are indicated below an operator product to emphasize this.

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3.2.1.5. The Hamiltonian operator in normal form Wick’s theorem (3.109), which gives the prescription for handling products of creation and annihilation operators may, of course, be applied to the Hamiltonian operator, H, when it is expressed in the second-quantization formalism, eq. (3.91): 1 + + + (3.132) H = A| h |B XA XB + AB| g |CD XA XB XD XC . 2 AB

ABCD

This leads immediately to the Hamiltonian operator in a form which is of primary importance for perturbation treatments and many-body formalism. This form of the Hamiltonian operator is termed the normal product form. It is usually written as follows: + H = Φ0 | H |Φ0 + A| f |B N [XA XB ] AB

(3.133)

1 + + + AB| v |CD N [XA XB XD XC ], 2 ABCD

where Φ0 | H |Φ0 is the reference energy – most usually the Hartree–Fock ground state energy for a closed shell system – and f is the operator defining the independent electron model used as a reference, which again, is usually the Hartree–Fock operator. v is the so-called fluctuation potential, which when added to the reference operator recovers the full many-body Hamiltonian. 3.2.2. Many-body Rayleigh–Schr¨odinger perturbation theory 3.2.2.1. The many-body problem and quasiparticles Many-Body Perturbation Theory, MBPT, or Many-Body Rayleigh–Schr¨odinger Perturbation Theory, MB - RSPT, is obtained by employing second-quantization and Wick’s theorem to develop a diagrammatic formulation of the ordinary timeindependent Rayleigh–Schr¨odinger perturbation theory. The use of the term ‘manybody’, which originates from the nuclear physics literature, is intended to emphasize that the explicit expressions for the energy coefficients (and other properties) in many-body perturbation theory, are written in terms of matrix elements of the spinorbitals, which reflect the many-electron interaction. This should be contrasted with ordinary Rayleigh–Schr¨odinger perturbation theory for which the matrix elements are expressed over Slater determinants, i.e. N -electron functions. The Feynman-like diagrams which arise in this approach have the advantage that they give a ‘microscopic’ view of the electron interactions in atoms and molecules. Furthermore, the diagrammatic description of individual terms of the perturbation expansion facilitates the use of the ‘linked diagram’ theorem, which has no analogy in the ordinary Rayleigh–Schr¨odinger perturbation theory. It is the linked diagram theorem, which establishes the linear scaling of the energy (and other properties) with the number of electrons in the system.

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95

We have not, so far, discussed the reference model with respect to which a manybody theory may be developed. For a non-interacting many-body system, a complete description can be given using one-body wave functions. An N -electron ground state can be described by a single determinant constructed from a set of one-electron states. The ground state energy is given by the sum of the orbital energies of the occupied orbitals. The non-interacting many-body system is described by the Hamiltonian operator (3.134) H0 =

N

h (ri )

i=1

where the one-electron Hamiltonian, h (ri ), is given by eq. (3.26) (3.135) h (ri ) = − 12 ∇2i −

M Zp . r p=1 ip

We are, of course, interested in describing a many-body system of interacting particles. We can obtain an improved independent particle model which corresponds to the electrons moving in the unscreened field of the nuclei by describing the averaged interactions among the electrons. We write the Hamiltonian operator, H0 , as (3.136) H0 =

N

f (ri )

i=1

where (3.137) f (ri ) = h (ri ) + u (ri ) and u (ri ) is some mean field potential, which describes the averaged interactions of the electron labelled i, with the remaining electrons in the system. If the potential u (ri ) is chosen by invoking the variation theorem and iterated until self-consistency is achieved, then we have the well known Hartree–Fock independent electron model. As we have shown in eq. (3.31), the perturbation is chosen so that the full Hamiltonian is recovered on adding H0 and H1 , i.e. (3.138) H1 =

N N 1 − u (ri ) , r i>j ij i=1

which is sometimes termed the fluctuation potential. The independent ‘particles’ described by the Hartree–Fock Hamiltonian, or indeed by the Hamiltonian associated with any independent particle model other than the non-interacting model (or ‘bare’ particle model) are different from the original particles, i.e. the electrons. These ‘particles’ are termed quasiparticles. The interactions between these quasiparticles are weaker than the interaction between the original ‘bare’ particles. There are important differences between the properties of the quasiparticles and those of the ‘bare’ particles. In particular, we have seen that

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low-lying excitations are specified by the quasiparticles which are present. However, we also note that our simple picture is destroyed when the number of quasiparticles becomes large. 3.2.2.2. The algebraic approximation The algebraic approximation, i.e. the use of approximations based on finite basis set expansions, is ubiquitous in practical quantum chemistry. Gaussian basis sets, in particular, are almost universally employed in contemporary calculations of molecular electronic structure because of the ease and accuracy with which the associated molecular integrals can be evaluated. We have seen in eq. (1.1) that the (cartesian) Gaussian basis functions have the form χi = χi (ζi , i , mi , ni ) (3.139) = xi y mi z ni exp ζi r2 r = (x, y, z) where x, y and z are the cartesian coordinates. ζi is a screening constant. The integers i , mi and ni determine the nodal structure of the Gaussian basis functions. We note that basis sets of spherical harmonic Gaussian-type functions are more useful than cartesian Gaussian-type functions for high precision work because of the computational linear dependence, which can arise from the cartesian functions for higher values of the integers i , mi and ni . Within the algebraic approximation, the single-particle state functions or orbitals, φμ , of the independent particle model are approximated as (3.140) φμ =

m

χi cμi ,

i=1

where m is the size of the basis set and the coefficients cμi are determined by iterative solution of the matrix Hartree–Fock equations – the well-known self-consistent field method. The matrix Hartree–Fock approximation is realized within a basis set of Gaussiantype functions in terms of the following molecular integrals: 1. The overlap integrals Sij = χi |χj " ∞ (3.141) χi (r) χj (r) dr = 0

2. The one-electron integrals hij = χi | h |χj " ∞ (3.142) χi (r) hχj (r) dr = 0

in which the operator h is the sum of the electronic kinetic energy and the nucleus– electron attraction energy.

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97

3. The two-electron integrals 1 gijkl = χi χj | |χk χl r12 " ∞" ∞ 1 (3.143) χi (r1 ) χj (r2 ) χk (r1 ) χl (r2 ) dr1 dr2 = r12 0 0 Implementation of the algebraic approximation requires the determination of the orbital expansion coefficients, cμi , in eq. (3.140). This is achieved by solving the matrix Hartree–Fock equations which may be written (3.144) FC = SC where S is the overlap matrix defined in eq. (3.141), C is a matrix with the orbital expansion vectors arranged in its columns, is a diagonal matrix of orbital energies, and F is the matrix representation of the Fock operator. The Fock operator, F , is written in terms of the Coulomb and exchange operators which are defined as " φi (r) φi (r) drφj (r) (3.145) Ji φj (r) = |r − r| and

"

(3.146) Ki φj (r) =

φi (r) φj (r) drφi (r) |r − r|

respectively. In terms of these operators, the closed-shell Fock operator has the form (3.147) F = h + G where the two-electron component, G, has the form (3.148) G =

N

(2Ji − Ki ) .

i=1

The matrix representation of this Fock operator can be written (3.149) F = h + G where h is defined in eq. (3.142) and (3.150) G = 2J − K. The Coulomb and exchange matrices are defined in terms of the two-electron integrals (3.143) as (3.151) J = Ji , i

where (3.152) (Ji )p,r =

cip cir gpqrs ,

98 and (3.153) K =

I. Hubaˇc and S. Wilson

Ki ,

i

where (3.154) (Ki )p,r =

cip ciq gpqrs .

Matrix elements over molecular orbitals are expressed in terms of the transformed quantities (3.155) hij = cip cjq hpq p

and (3.156) gijkl =

q

p

q

r

cip cjq cir cjs gpqrs .

s

3.2.2.3. Many-body Rayleigh–Schr¨odinger perturbation theory In this section, we shall follow the main features of the derivation of the timeindependent many-body Rayleigh–Schr¨odinger perturbation theory ( MB - RSPT) given ˇ ızˇ ek in their well-known review, published in 1975 [29]. by Paldus and C´ Let us assume that a perturbed Hamiltonian of an atomic or molecular system, H, may be split into two parts as follows: (3.157) H = H0 + λH1 where H0 is the unperturbed Hamiltonian and H1 is the perturbation. λ is the perturbation parameter which interpolates between the unperturbed problem (λ = 0) and the perturbed problem (λ = 1). In order to obtain a direct expression for the correlation energy, we use the notation H for our Hamiltonian. As will be seen later, H differs from the Hamiltonian (3.91) by a scalar quantity. We assume that the following equations hold for H and H0 operators: (3.158) H|Ψi = Ei |Ψi (3.159) H0 |Φi = Ei |Φi . Furthermore, we assume that the complete solution of eq. (3.159) is known. Our goal is to find the solution of the eq. (3.158) under the assumption that |Φi changes into the state |Ψi when the perturbation H1 is switched on. We shall not go into details of the derivation. These are given in Appendix D. Instead, we state that the following RSPT expansion holds for Ei : ∞ (3.160) Ei = Ei + Φi |H1 [Qi (H1 + Ei − Ei )]n |Φi n=0

where (3.161) Qi =

|Φj Φj | 1 − |Φi Φi | = . Ei − Ej Ei − H0

j,j=i

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99

Equation (3.160) can be solved iteratively. We can collect the terms having the same order of perturbation and therefore can write (3.162) Ei =

∞

(j)

εi

j=0 (j)

where Ei is the jth order contribution. The terms up to the third-order have the following forms: (0)

= Φi |H0 |Φi

(1)

= Φi |H1 |Φi

(2)

= Φi |H1 Qi H1 |Φi

(3)

= Φi |H1 Qi (H1 − εi )Qi H1 |Φi .

(3.163) εi (3.164) εi (3.165) εi (3.166) εi

(1)

The whole problem of calculating Ei (at least up to the third-order) is now reduced to the calculation of the individual terms (3.163)–(3.166). The second-quantization formalism has the advantage that these terms can be calculated easily by making use of diagrammatic techniques. We have already shown that the Hamiltonian (3.167) H = f + v where (3.168) f =

f (i),

v=

i

v(i, j)

i<j

are one-particle and two-particle operators respectively, which can be written in the normal product form: + H = Φ0 | H |Φ0 + A| f |B N [XA XB ] AB

(3.169)

1 + + + AB| v |CD N [XA XB XD XC ] 2 ABCD

where, as we have already said, the quantity Φ0 | H |Φ0 is the Hartree–Fock energy of the closed shell ground state. We assume that this operator satisfies the following Schr¨odinger equation: (3.170) H |Ψi = Ei |Ψi and that the zero-order Schr¨odinger equation is (3.171) H0 |Φi = ei |Φi . Let us write (3.172) H = H − Φ0 | H |Φ0

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and (3.173) H0 = H0 − Φ0 | H0 |Φ0 . The quantity Ei is then exactly the correlation energy of the ith electronic state. The simplest partitioning of the Hamiltonian (3.169) is + (3.174) H0 = A| f |B N [XA XB ] AB

(3.175) H1 =

1 + + AB| v |CD N [XA XB XD XC ]. 2 ABCD

Next, we shall assume that our spin-orbital basis |A , |B , . . . is the Hartree–Fock basis, i.e. (3.176) f |A = A |A where A are the Hartree–Fock orbital energies. Then, H0 in eqs. (3.171) and (3.173) is the Hartree–Fock operator, ei in eq. (3.171) is the sum of Hartree–Fock orbital energies over occupied spin-orbitals and eq. (3.174) can be written in the diagonal form: + (3.177) H0 = A N [XA XA ]. A

We will now examine the expression for the correlation energy of the ground state, i.e. i = 0, in eq. (3.162), using eqs. (3.172), (3.173) and (3.176). The individual terms (3.163)–(3.166) have a very simple form in this case, namely (0)

(3.178) ε0 = 0, (1)

(3.179) ε0 = 0, (2)

(3.180) ε0 = Φ0 |H1 Q0 H1 |Φ0 and (3)

(3.181) ε0 = Φ0 |H1 Q0 H1 Q0 H1 |Φ0 . Here we can see that use of this particular partitioning of the Hamiltonian, the Møller– Plesset partitioning, i.e., H0 given by eq. (3.177) and H1 given by eq. (3.175), is very useful since the number of terms in the perturbation expansion (3.178)–(3.181) is considerably reduced. However, we note that we could use some other partitioning of the Hamiltonian, e.g. the so-called Epstein–Nesbet partitioning, which differs from the Møller–Plesset partitioning in that the diagonal terms ij| v |ij − ij| v |ji are shifted from H1 to H0 . Our problem of evaluating the energy coefficients ε0 up to third-order is now reduced to calculating terms (3.180) and (3.181). Higher order terms can be determined in a similar fashion. However, beyond fourth-order the practical calculations become computationally intractable unless a basis set of small or moderate size is employed when invoking the algebraic approximation.

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101

3.2.2.4. Second-order contribution to the correlation energy We will now consider the second-order contribution to the correlation energy as an example of the elementary diagrammatical approach. Construction of the diagrams We use the second-quantization formalism to obtain an expression for the second-order contribution to the correlation energy of the closed-shell system in its ground state. By substituting the expression (3.175) for H1 into eq. (3.180), we obtain: (2)

ε0 =

1 AB| v |CD EF | v |GH 4 ABCD

EF GH

+ + + + XB XD XC ]Q0 N [XE XF XH XG ]. |Φ0 × Φ0 | N [XA

(3.182)

Let us first make a remark concerning the operator Q0 . By using eqs. (3.159), (3.171) and (3.173), we can obtain the denominator of Q0 in eq. (3.161): (3.183) e0 − H0 = e0 − H0 . Using a binomial expansion, we can obtain H0 1 H20 1 1+ (3.184) = + 2 + ··· e 0 − H0 e0 e0 e0 from which we get (3.185)

1 1 |Φn = |Φn . e 0 − H0 e0 − en

Hence, the operator Q0 only determines a denominator in the final expression. This denominator will contain orbital energies because (3.186) en (k) − e0 =

k i=1

(A − A ), i

i

where the index k refers to a k-fold excitation. We already know that in order to calculate the expansion (3.182) using the generalized Wick’s theorem, we have to perform contractions between the N -products on the right hand side of eq. (3.182). We also know that, according to eq. (3.111), all the operators have to be contracted. Here the diagrammatic approach can be introduced since the contractions can be represented diagrammatically. The simple diagram shown in Figure 3.3 corresponds to the following algebraic expression + + AB| v |CD N [XA XB XD XC ].

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I. Hubaˇc and S. Wilson

# + + Figure 3.3. Diagrammatic representation of 12 {A, B, C, D} AB| v |CD N [XA XB XD XC ] in which the lines directed towards the interaction vertex correspond to annihilation operators, whilst lines directed away from the interaction correspond to creation operators.

In the diagram drawn in Figure 3.3, the lines leaving the vertex (which is marked by a dot) correspond to creation operators, whereas the lines entering the vertex correspond to annihilation operators. The dashed line connecting the vertices is called the interaction line. This line represents the two-electron potential 1/rij . Using these conventions, we can schematically rewrite eq. (3.182) in the diagrammatic form

(2)

(3.187) ε0 = Φ0 |

Q0

|Φ0

Let us ignore the operator Q0 for the moment. The rules which can be used to calculate it will be given later. Thus, we are left with two diagrams of the type shown in Figure 3.3. Diagrammatically, the contractions are performed by connecting the diagrams in eq. (3.187), whilst preserving the orientation of the lines. Since all operators in eq. (3.182) have to be contracted, we have to perform all possible connections using eq. (3.187). In this way, we can construct the diagrams shown in Figure 3.4. These diagrams can be redrawn as shown in Figure 3.5, where their more usual form is displayed. The oriented lines are labelled by a single index, because the contractions (3.130) and (3.131) result in a Kronecker delta. Moreover, from eqs. (3.130) and (3.131), we can specify which spin-orbitals are occupied and which are unoccupied or virtual. The diagrams in Figure 3.5 are of the type introduced by Goldstone. Goldstone diagrams are very useful because they lead to obtain a very compact formula for the pertinent order of the perturbation expansion. What is even more important is the realization that by using Goldstone diagrams, the spin summations which arise can be handled very simply. The disadvantage of the Goldstone diagrammatic formalism is that for higher orders of perturbation theory the number of these diagrams increases rapidly. Care must be taken to ensure that all topologically distinct diagrams

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103

Figure 3.4. Connecting the diagrams in eq. (3.187), whilst preserving the orientation of the lines, leads to the following second-order contributions.

Figure 3.5. Goldstone diagrams for the second-order contribution to the correlation energy in the ground state

are included. Therefore, a more compact diagrammatic formalism is a required. Such a formalism might facilitate a systematic procedure which can guarantee that all necessary diagrams have been included. The so-called Hugenholtz diagrams (or degenerate diagrams) afford a more compact diagrammatic formalism. (The Hugenholtz diagrams are in one-to-one correspondence with the so-called Brandow diagrams which

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Figure 3.6. Hugenholtz diagram for the second-order contribution to the correlation energy in the ground state.

are obtained from the Goldstone diagrams by replacing the vertices by ’antisymmertized vertices’. (Several Goldstone diagrams may correspond to a single Brandow diagram.) The relation between Hugenholtz diagrams and Goldstone diagrams is given by

(3.188)

=

+

If the second-order Goldstone diagrams shown in eq. (3.187) are replaced by Hugenholtz diagrams, then we obtain the diagrams shown in Figure 3.6, instead of those displayed in Figure 3.4. On applying relation (3.188) to the Hugenholtz diagram shown in Figure 3.6, we obtain the four Goldstone diagrams displayed in Figure 3.5. Having obtained a diagrammatic representation for the expression (3.182) for the second-order energy, we are now in a position to replace the diagrams by explicit algebraic expressions. A precise set of rules can be given to facilitate the translation of the different diagrammatic terms into the corresponding algebraic expressions. The different diagrammatic conventions (Goldstone, Brandow, Hugenholtz) give rise to slightly different rules, but in each case the translation from diagram to algebraic expression is precisely defined. The rules for translating the Hugenholtz diagrams and the associated Goldstone diagrams into the corresponding algebraic expressions are summarized in the following section. Rules for obtaining algebraic formulae from diagrams The Hugenholtz diagrams and the associated Goldstone diagrams can be translated into the corresponding algebraic expressions according to the following general and precise rules: Rule 1 Draw all possible distinct Hugenholtz diagrams for the pertinent order of the perturbation expansion and generate from them Goldstone diagrams by (3.188):

Brillouin–Wigner Methods for Many-Body Systems

(3.188)

=

105

+

Rule 2 To each interaction line we assign a matrix element AB| v |CD, where the indices A and B refer to the lines leaving the vertex and C and D refer to the lines entering the vertex. Rule 3 There is a denominator factor for each pair of neighbouring interaction lines. Each denominator is a sum of orbital energies. The orbital energies included in the sum are determined by the directed lines lying between the interaction lines. The sign of the orbital energies in the sum is determined by the direction of the lines. A ‘+’sign is taken for those lines going from left to right (‘hole’ lines) and a ‘−’ sign for those going from right to left (‘particle’ lines). Rule 4 The sign of the total expression corresponding to the pertinent Goldstone diagrams is given by (−1)l+h where l is the number of closed loops and h is the number of hole lines. Rule 5 Sum over all indices appearing in the pertinent Goldstone diagram. Rule 6 Sum the individual contributions of all Goldstone diagrams which are generated from the parent Hugenholtz diagram. Rule 7 Multiply the whole expression, that is, the sum of the Goldstone diagram contributions, with the topological factor 1 2n where n is the number of equivalent pairs of lines. An equivalent pair consists of two lines which both start at the same vertex of the Hugenholtz diagram and both end at the same vertex. By applying the rules 1–7 to the problem of the second-order energy, E02 , defined in eq. (3.182), we obtain the following algebraic expressions for Diagrams I–IV in Figure 3.5: Diagram I

+

A B | v |C D

A B C D

1 C D | v |A B A + B − C − pD

Diagram II −

A B C D

A B | v |C D

A

+

B

1 D C | v |A B − C − D

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I. Hubaˇc and S. Wilson

Diagram III −

A B | v |C D

1 C D | v |B A A + B − C − D

A B | v |C D

1 C D | v |A B A + B − C − D

A B C D

Diagram IV + A B C D

Here we can see that Diagrams I and IV give the same contribution to the secondorder energy. Likewise, the contributions from Diagrams II and III are identical. According to rule 7, the topological factor is equal to 14 . Hence, the final expression for the total second-order energy becomes 1 1 (2) E0 = 2 A B | v |C D 4 A + B − C − D AB C D

×{C D | v |A B − D C | v |A B }.

(3.189)

We recall that |A , |B , |C , |D are spin-orbitals. In order to obtain the corresponding expression for the second-order energy component in terms of orbitals, we add a rule for the spin summation: Rule 8 The spin must be conserved along each directed line. It is therefore necessary to introduce an additional numerical factor of 2l by which we multiply each Goldstone diagram. As in rule 4, l is the number of closed loops. On applying this rule to Diagrams I to IV in Figure 3.5, we obtain the following expressions in terms of (spatial) orbitals Diagram I +4

a b | v |c d

1 c d | v |a b a + b − c − d

a b | v |c d

1 d c | v |a b a + b − c − d

a b | v |c d

1 c d | v |b a a + b − c − d

a b c d

Diagram II −2

a b c d

Diagram III −2

a b c d

Brillouin–Wigner Methods for Many-Body Systems

107

Diagram IV +4

a b | v |c d

a b c d

1 c d | v |a b a + b − c − d

By adding together these contributions and then multiplying them by a topological factor of 14 , we arrive at the final energy expression for the second-order energy in the form (2) E0 = a b | v |c d (2 c d | v |a b a b c d

(3.190)

1 − d c | v |a b ). a + b − c − d

Our derivation of the second-order energy may appear anomalous. We first introduced Goldstone diagrams into eq. (3.187), then replaced these with the corresponding Hugenholtz diagrams, before finally returning again to Goldstone diagrams. Actually, the rules we present here are a combination of rules originally given by Hugenholtz with those of Goldstone. These combined rules were suggested by Brandow [14] in 1967. The original rules of Goldstone, which we do not present here, lead to complications in application, because of the difficulty of ensuring that all topologically distinct diagrams are constructed. This is no easy task for higher orders of the perturbation expansion. Furthermore, in the original Goldstone approach, the topological factor has to be determined for each Goldstone diagram. It should be recognized that the perturbation expansion for the quantity Ei is not restricted to the ground state, i.e. i = 0, and for excited states the enumeration of the diagrams and the determination of the topological factor can make the whole calculation very cumbersome. On the other hand, both the spin summations and the determination of the correct sign factor can be carried out correctly only when Goldstone diagrams are employed. A great advantage of Hugenholtz-type diagrams is that they are much fewer in number, each Hugenholtz diagram representing up to 2n Goldstone diagrams, where n is the number of vertices. The topological factor is given by a very simple rule (rule 7), and we can generate all Goldstone diagrams by a systematic procedure (the double projection scheme defined in eq. (3.188).). For these reasons, it is useful to present these combined Hugenholtz–Goldstone rules even though their advantages are not evident at second-order. We note that these combined Hugenholtz–Goldstone rules lead to a larger number of Goldstone diagrams than would be obtained by strictly following the original Goldstone rules. However, this disadvantage is outweighed by the systematic nature of the diagram generation process and by the ease with which the corresponding topological factors can be determined, especially in the case of excited states (i = 0). Algorithms have been constructed to automate the construction of higher order diagrams (see, for example, the work of Paldus and Wong [30,31] and of Lyons et al. [32]). Algorithms have also been devised to automatically translate these diagrams into the corresponding algebraic expressions and the FORTRAN computer code for their evaluation (see, for example, the work of Lyons et al. [32]).

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I. Hubaˇc and S. Wilson

In the above discussion above, we have not discussed two theoretically important concepts: the linked cluster (linked connected graph) theorem and the exclusion principle violating ( EPV) diagrams. The linked cluster theorem and EPV diagrams are of practical importance in the fourth and higher orders of the perturbation expansion. For a detailed discussion of the linked cluster theorem and also the EPV diagrams the ˇ ızˇ ek [29] or to the monographs by reader is referred or to the review by Paldus and C´ March et al. [4], by Raimes [33] or by Fetter and Walecka [34]. 3.2.2.5. Third-order contributions to the correlation energy On taking the Rayleigh–Schr¨odinger perturbation expansion for the energy beyond second-order, we find that so-called renormalization terms appear. In third-order Rayleigh–Schr¨odinger perturbation theory, the energy coefficient can be written in the form (3.191) Φ0 | H1 RH1 RH1 |Φ0 − Φ0 | H1 |Φ0 Φ0 | H1 R2 H1 |Φ0 . The first term in this expression is called the ‘direct’ term whilst the second term is called the ‘renormalization’ term. If the perturbation expansion is developed with respect to a reference function constructed from canonical Hartree–Fock orbitals, then the renormalization term is equal to zero: (3.192) (Φ0 | H1 |Φ0 = 0 and this term is not considered in practical applications of the third-order theory. However, if non-canonical Hartree–Fock orbitals are used in the reference function, then the renormalization term may need consideration. The expression for the third-order energy coefficient given in eq. (3.181): (3)

(3.193) ε0 = Φ0 | H1 Q0 H1 Q0 H1 |Φ0 can be written schematically in the form

(3)

(3.194) ε0 = Φ0 |

Q0

Q0

|Φ0 .

Diagrammatically, the contractions are performed by connecting the diagrams in eq. (3.194), whilst preserving the orientation of the lines. If we perform all possible connections using eq. (3.194) then we can obtain the three different third-order Hugenholtz-type diagrams shown in Figure 3.7. The diagrams differ in the nature of the central interaction: hole–particle, hole–hole and particle–particle. As in secondorder, only doubly excited intermediate states arise.

Brillouin–Wigner Methods for Many-Body Systems

109

Figure 3.7. Hugenholtz diagrams for the third-order contribution to the correlation energy of the ground state.

3.2.2.6. Fourth-order contributions to the correlation energy When the canonical Hartree–Fock orbitals are used to construct the reference function with respect to which the perturbation expansion is made, only one non-vanishing renormalization term occurs in the fourth-order energy component. The fourth-order energy can be written as (3.195) Φ0 |H1 RH1 RH1 RH1 |Φ0 − Φ0 |H1 R2 H1 |Φ0 Φ0 |H1 RH1 |Φ0 where the first term is the ‘direct’ term and the second is the ‘renormalization’ term. The renormalization term may be written (3.196) − Φ0 |H1 R2 H1 |Φ0 Φ0 |WRW|Φ0 ≡ E2 S where E2 = Φ0 |WRW|Φ0 is the second-order energy and S = Φ0 |H1 R2 H1 |Φ0 is an overlap integral. Using the diagrammatic rules, we can construct both the connected and the disconnected diagrams in the direct term in eq. (3.195) (see, for example, the work of Hubaˇc [35]). The direct term contains two components which correspond to disconnected diagrams. These diagrams are displayed in Figure 3.8 where they are labelled E1 and E2. The disconnected diagrams shown in Figure 3.8 satisfy the equation (3.197) E1 + E2 = −E2 S. The importance of the linked cluster theorem in fourth-order of the energy expansion, lies in the fact that the two disconnected contributions to the direct term in eq. (3.197) are exactly cancelled by renormalization terms. This result allows the total fourth-order energy to be written as (3.198) Φ0 |WRWRWRW|Φ0 LC , where the suffix ‘lc’ indicates that only terms associated with linked clusters are included in the expression. In practical applications, only the linked connected

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Figure 3.8. The two disconnected diagrams in the fourth-order direct term. The sum of these two disconnected contributions is exactly cancelled by the renormalization terms.

direct terms have to be evaluated. This is the many-body formulation of Rayleigh– Schr¨odinger perturbation theory. The total number of diagrams corresponding to linked cluster terms in the fourthorder energy component is 39. A complete set of these diagrams was first listed by Wilson and Silver [36]. All 39 diagrams contain doubly excited intermediate states. Four diagrams also contain singly excited intermediate states. Sixteen diagrams contain doubly and triply excited intermediate states. Seven diagrams contain states which are doubly and singly excited with respect to the reference function. The remaining 12 diagrams contain only intermediate states which are doubly excited. Brueckner [8] first demonstrated the cancellation of ‘unlinked’ terms in fourthorder. This result was generalised to all orders of the Rayleigh–Schrodinger perturbation theory by Goldstone and by Hugenholtz. The number of diagrams increases rapidly with order. For example, in fifth order there are a total of 840 Brandow diagrams. They have been generated automatically by Lyons et al. [32]. 3.2.3. Many-body perturbation theory During the 1960s, Kelly [37–43] pioneered the application of what is today the most widely used approach to the description of correlation effects in atomic and molecular systems; namely, the many-body perturbation theory [1, 2, 43–48]. The secondorder theory using the Hartree–Fock model to provide a reference Hamiltonian is particularly widely used. This Møller–Plesset (MP 2) formalism combines an accuracy, which is adequate for many purposes, with computational efficiency allowing both the use of basis sets of the quality required for correlated studies and applications to larger molecules than higher order methods. Using numerical solutions to the Hartree–Fock equations, Kelly applied the manybody perturbation theory to the correlation problem first in atoms [37–39] and

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subsequently in molecules [42, 43]. Kelly’s molecular applications were limited to simple hydrides for which a one-centre expansion could be employed, treating the hydrogen atom(s) as an additional perturbation. Correlation energy calculations based on numerical solutions of the Hartree–Fock equations involve an integration over the continuum which requires some care in its implementation. The range of applicability of the many-body perturbation theory as a method for describing correlation effects in molecules changed radically in the 1970s when a number of authors [50–53] demonstrated how the introduction of the algebraic approximation [53–56], i.e. the use of finite basis sets, could facilitate calculations for arbitrary polyatomic molecular systems. By employing basis sets consisting of subsets centred on each of the component atoms in the system under study (and perhaps on other centres, such as bond mid-points), applications to any molecular system are, in principle, possible. In the algebraic approximation, the integration over the continuum which arose in Kelly’s approach based on numerical solutions of the Hartree–Fock equations, becomes a summation over the virtual states obtained by solution of the matrix Hartree–Fock equations. Essentially, the integration over the continuum becomes a quadrature [57]. The application of many-body perturbation theory to molecules involves the direct application of the Rayleigh–Schrodinger formalism with specific choices of reference Hamiltonian. The most familiar of these is that first presented by Møller and Plesset in 1934 [58]. In the Møller–Plesset formalism, a single-reference function is employed and the partition of the Hamiltonian into a reference or zero-order operator and a perturbation uses the Hartree–Fock model to define the reference. Third-order theory ( MP 3) and fourth-order theory (MP 4) are computationally tractable. There are other choices of reference Hamiltonian. In fact, any operator A which satisfies the commutation relation (3.199) H0 , A = 0 can be used to develop a perturbation series. Equation (3.199) implies that the operator A has the form (3.200) A = |k k| A |k k| k

where |k is an eigenfunction of the zero-order Hamiltonian, H0 . If we put A = H, so that A is set equal to the full Hamiltonian then we get the Epstein–Nesbet formalism or the ‘shifted denominator’ expansion. This formalism was first considered within the context of many-body perturbation theory by Kelly. It may be given the same diagrammatic representation as the expansion based on the Hartree–Fock model Hamiltonian. The use of the ‘shifted denominator’ formalism may also be interpreted as the summation through infinite order of certain types of terms which occur in the Møller–Plesset expansion. The Epstein–Nesbet scheme corresponds to an infinite order summation of diagonal scattering terms in the Møller–Plesset expansion.

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There are several ways in which the terms in the Rayleigh–Schr¨odinger perturbation expansion can be derived. In Chapter 2, we used the approach based on the partitioning technique and we shall briefly recall this here. We begin with the following expression for an effective Hamiltonian (3.201) Heff = [PHP + PHQ(E − QHQ)−1 QHP] Different perturbative schemes can be obtained by expanding the inverse operator in eq. (3.201) (E − QHQ)−1 (assuming it exists) using (3.202) (X − Y)−1 =

∞

X−1 (YX−1 )n

n=0

or alternatively (3.203) (X − Y)−1 = X−1 + X−1 YX−1 + X−1 YX−1 YX−1 + · · · More generally, we can put (3.204) X = E − H0 and (3.205) Y = (H − H0 ) − (E − E), where we have partitioned the Hamiltonian H onto unperturbed part H0 and perturbation H1 . Let us consider the case of a one dimensional model space with projection operator (3.206) P = |Φ0 Φ0 |. We assume that the state |Φ0 is an eigenfunction of the unperturbed problem (3.207) H0 |Φ0 = E0 |Φ0 which is non-degenerate. Furthermore, we assume that the effective Hamiltonian satisfies the eigenvalue equation (3.208) Heff |ΨαP = Eα |ΨαP ,

α = 1, 2, . . . , d

|ΨαP

is the so-called model function. For a non-degenerate ground state |Ψ0 , where we can write the final Rayleigh–Schr¨odinger expansion for the energy as follows: E0 = Φ0 |H0 |Φ0 + Φ0 |H1 |Φ0 + Φ0 |H1 RH1 |Φ0 (3.209) where (3.210) R =

+ Φ0 |H1 RH1 RH1 |Φ0 − Φ0 |H1 RH1 |Φ0 Φ0 |H1 |Φ0 + · · · |Φi Φi | E0 − Ei i=0

is the Rayleigh–Schr¨odinger resolvent. In third order and higher orders, there is a direct term, e.g. +Φ0 |H1 RH1 RH1 |Φ0 , and a renormalization term, e.g. −Φ0 |H1 RH1 |Φ0 Φ0 |H1 |Φ0 .

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A critical step in the development of any perturbation expansion is the division of the Hamiltonian H into a zero-order part H0 and a perturbation H1 . In second quantized formalism, the Hamiltonian for the Møller-Plesset theory is written as + A |f| BN [XA XB ] + H = Φ0 |H|Φ0 + AB

1 + + + AB|v|CDN [XA XB XD XC ], 2

(3.211)

ABCD

where usually we choose the canonical form (3.212) f|A = eA |A, and therefore H = Φ0 |H|Φ0 +

+ eA N [XA XA ] +

A

(3.213)

1 + + + AB|v|CDN [XA XB XD XC ]. 2 ABCD

+ (XB ) are creation (annihiIn these equations, f is the Hartree–Fock operator and XA lation) operators defined with respect to Fermi vacuum |Φ0 and N [ ] is the normal ordered product of creation and annihilation operators. The main advantage of writing the Møller–Plesset partition in second quantized form rests on the fact that we can then exploit the linked cluster theorem in Rayleigh– Schr¨odinger perturbation theory. As a consequence, there is only one direct term in each order in eq. (3.209). Renormalized terms are cancelled by components associated with disconnected diagrams in the direct terms. Therefore, the Rayleigh– Schrodinger perturbation expansion (3.209) takes the form

E0 = Φ0 |H0 |Φ0 + Φ0 |H1 + H1 RH1 + H1 RH1 RH1 + (3.214) + H1 RH1 RH1 RH1 . . . |Φ0 LC , where the subscript LC indicates that only terms associated with linked connected diagrams contribute to expression (3.211). Efficient computer code for evaluating the numerical values of the energy components associated with these low order perturbation theories is available in a number of quantum chemistry packages. We mentioned above that the number of diagrammatic terms rises to 840 in fifth order. Fifth order studies are usually performed for ‘benchmarking’ and for gaining an understanding of the importance of these terms in theories which involve a partial summation of higher order terms. 3.3. MANY-BODY THEORIES FOR ATOMS AND MOLECULES In the first section of this chapter, we emphasized the importance of linear scaling with the number of electrons in the system when developing approximate descriptions of many-body systems. We showed that both the exact solutions of the many-electron

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Schr¨odinger equation and the solutions supported by an independent electron model, scale linearly with electron number. Approximate treatments of electron correlation effects may not necessarily scale linearly with electron number. Approximations which display such a linear scaling are termed many-body methods. In the previous section, we described the many-body perturbation theory which is the basic ingredient of all many-body theories. It provided a fundamental tool for both the synthesis and the analysis of many-body methods. In this section, we consider some manybody theories of atomic and molecular structure and some theories which contain unphysical terms which scale nonlinearly with electron number. 3.3.1. The full configuration interaction method and limited configuration interaction In a detailed review of the history and evolution of the method of configuration interaction published in 1998 [59], Shavitt begins The configuration interaction (CI) method dates back to the earliest days of quantum mechanics, and is the most straightforward and versatile approach for dealing with electron correlation. We refer the reader interested in the history of the configuration interaction method and the evolution of its computational implementation, to Shavitt’s masterly and authoritative account. Here we shall concentrate on the strengths and the weaknesses of the modern configuration interaction approach. To quote Shavitt again: The conceptual simplicity of the configuration interaction method is very appealing, and its variational nature is an important advantage, but its principal strength lies in its flexibility and generality. It can be applied straightforwardly to any electronic state, and can be spin- and symmetry-adapted relatively easily. This “flexibility and generality” has resulted in configuration interaction often being regarded as the method of choice for multi-reference problems. As Shavitt puts it Its multi-reference formulation is straightforward, and applicable readily to any type of reference space, complete or otherwise. Apart from the increased size, and therefore increased computational requirements, multi-reference CI is not notably more difficult than the single-reference form, and the ability to use incomplete active spaces can be employed to limit the computational requirements substantially. He goes on to compare multi-reference CI with other multi-reference formalisms The multi-reference capabilities of CI contrast with the situation in manybody perturbation theory (MBPT) and coupled cluster (CC) theory, for which multi-reference generalizations are substantially more difficult than their

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single-reference counterparts and for which the use of incomplete reference spaces introduces additional difficulties. And furthermore, he points out that . . . unlike multi-reference many-body methods, no problems of intruder states arise in MRCI. As we can see from the above discussion, configuration interaction is a robust and flexible approach to the electron correlation problem in atoms and molecules. Configuration interaction is not, in general, a “many-body” theory in the postBrueckner sense and this is its main deficiency. Shavitt admits that The principal weakness of truncated configuration interaction is its lack of proper scaling with the size of the system. Full configuration interaction does scale linearly with the size of the system but is only computationally tractable for small systems described by small basis sets. In practice, the configuration interaction expansions must be truncated. Truncation is effected by including only states which are singly and doubly excited with respect to some reference configuration(s). Shavitt continues Although the use of higher than doubly-excited CSFs in CI expansions is easy in principle, the exponential increase in the size of a CI expansion with the level of excitation usually makes such calculations impractical. The intractibility of configuration interaction as a computational technique is the weakest feature of the method, as Shavitt makes clear The difficulty of extending the CI expansion to higher excitations is a serious shortcoming because of its very slow convergence. This contrasts with the situation for many-body methods which are far more efficient in accounting for higher order excitations. Again, Shavitt summarizes the situation as follows: Unlike the situation in the many body methods, the connected and disconnected cluster contributions to each excited CSF are inextricably combined in the CI formalism. Today, many-body methodology underpins the most widely used ab initio approaches to the correlation problem in atoms and molecules. However, configuration interaction remains the most robust approach which can be invoked in difficult cases. In the method of configuration interaction, configuration mixing or superposition of configurations, the total wave function for an N -electron system is written (3.215) ΨFCI = Φμ Cμ , μ

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where Φμ is an N -electron function formed as a product of one-electron functions ϕk (3.216) Φμ = |ϕ1 ϕ2 . . . ϕN | . The expansion coefficients, Cμ , are determined by invoking the variation principle which leads to a secular problem of the form (3.217) HC = EC where the coefficient vector can be written as a vector ⎞ ⎛ C1 ⎜ C2 ⎟ ⎟ ⎜ ⎜... ⎟ ⎟ ⎜ (3.218) C = ⎜ ⎟ ⎜ Cμ ⎟ ⎝... ⎠ Cm whilst the Hamiltonian matrix has elements (3.219) {H}μν = Φμ |H| Φν . In the algebraic approximation, finite basis set expansions are employed to approximate the orbital functions. The orbital ϕk is then written in the form (3.220) ϕk =

M

χp cpk ,

p=1

where {χp ; p = 1, 2, . . . , M } denotes the chosen basis set and cpk is a matrix of orbital expansion coefficients which are to be determined. In contemporary quantum chemistry, the basis set, {χp ; p = 1, 2, . . . , M }, is usually taken to be a set of Gaussian functions. This Gaussian algebraic approximation is ubiquitous in molecular electronic structure studies. The algebraic approximation results in the restriction of the domain of the operator H to a finite dimensional subspace S of the Hilbert space h. In most applications of quantum mechanics to atoms and molecules, the N -electron wave function is expressed in terms of the N th rank direct product space V N generated by a finite dimensional single particle space V 1 – that is, (3.221) V N = V 1 ⊗ V 1 ⊗ . . . ⊗ V 1 . The algebraic approximation may be implemented by defining a suitable orthonormal basis set of M one-electron spin orbitals, where M > N , and constructing all unique N -electron determinants. The number of unique determinants which can be formed in this way is M (3.222) η = . N

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η is then the dimension of the subspace S spanned by the set of determinants. In the algebraic approximation, the domain of H is restricted to the η-dimensional space S. If the one-electron basis set were a complete set, then the corresponding configuration interaction problem, which is termed ‘complete configuration interaction’, yields the exact solution of the Schr¨odinger equation. If the one-electron basis set is not a complete set, then the corresponding configuration interaction problem is termed ‘full configuration interaction’. The solutions of the ‘full configuration interaction’ problem are the projections of the exact wave function onto the subspace S. In general, for basis sets which are capable of supporting the level of accuracy required in chemical studies, the full configuration interaction problem is computationally intractable. Even when configuration state functions are employed in place of determinants, the total number of terms in the full configuration interaction expansion is prohibitive. The number of configuration state functions is given by Weyl’s number 2S + 1 M + 1 M +1 (3.223) Dm,N,S = M − 12 N − S M + 1 12 N − S where N is the number of electrons, S is the total spin and M is the number of basis functions. Full configuration interaction calculations with adequate basis sets will remain prohibitive for the foreseeable future. For example, for case of the ground state of the water molecule (N = 10, S = 0) described by a basis set of 75 functions, we find that (3.224) D75,10,0 = 53, 144, 078, 124, 600. In spite of its computational intractability the full configuration interaction method forms the basis of a ‘many-body’ theory. As stated above, the implementation of the algebraic approximation restricts the domain of the Hamiltonian operator, H, to a finite dimensional subspace S of the Hilbert space h. If P denotes the projector onto the subspace S, then the Hamiltonian operator, H, is replaced by (3.225) HP = PHP. The Schr¨odinger equation (3.226) HΨ = EΨ is replaced by its projection onto the space S (3.227) HP ΦP = EΦP where (3.228) ΦP = PΨ is the full configuration interaction wave function corresponding to the space S supported by the chosen basis set. E is the full configuration interaction energy. Let us now consider the description of an array of well-separated systems afforded by the full configuration interaction model. As in Section 3.1, we shall again restrict

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our attention to a supersystem consisting of two subsystems A and B and then generalize the results. The full configuration interaction Hamiltonian operator can be written (A)

(B)

(3.229) HP = HP + HP where (A)

= PH(A) P

(B)

= PH(B) P.

(3.230) HP and (3.231) HP

We can follow the procedure adopted in Section 1 for the exact Schr¨odinger equation and conjecture that the full configuration interaction wave function for the supersystem can be written P (3.232) ΦP = ΦP A ΦB .

Substituting (3.229) and (3.232) into (3.227) gives (A) (B) P P P ΦP (3.233) HP + HP A ΦB = EΦA ΦB . (A)

Since HP depends only on the coordinates of the particles associated with system (B) A and HP depends only on the coordinates of the particles associated with system B, this equation can be rewritten as (A)

(B)

P P P P P (3.234) ΦP B HP ΦA + ΦA HP ΦB = EΦA ΦB . P Dividing by ΦP A ΦB gives (A)

(3.235)

(B)

HP ΦP H ΦP A + PP B =E P ΦA ΦB

which upon putting (3.236) E = EA + EB gives (3.237)

(B) (A) P Φ HP ΦP H A B P − EA + − EB = 0 ΦP ΦP A B

and so (A)

P (3.238) HP ΦP A = EA ΦA

and (B)

P (3.239) HP ΦP B = EB ΦB .

The above argument can be generalized to an array of n well-separated subsystems. Just as in the case of the exact energy discussed in Section 3.1.1 and that of the

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independent particle model discussed in Section 3.1.2, the energy associated with the full configuration interaction Hamiltonian is n times that of a single system: (3.240) E (n) = nEA . The full configuration interaction energy is seen to scale linearly with the number of electrons in the system. It is therefore a ‘many-body’ method. Practical applications of the method of configuration interaction to atoms and molecules almost invariably employ some limited expansion for the wave function. One of the most commonly used configuration interaction techniques includes all single and double excitations with respect to some single determinantal reference function. The CISD (“Configuration Interaction with Single and Double excitations”) wave function has the form (3.241) ΨCISD = Φ0 C0 + Φμ Cμ , μ∈SSD

where SSD denotes the space of all determinants generated by means for single and double excitations with respect to the chosen reference determinant Φ0 . This method cannot be described as a ‘many-body’ technique. This can easily be seen by considering the application to two well-separated helium atoms. The method of single and double excitation configuration interaction applied to each of the atoms separately gives the exact energy within the chosen basis set. However, application to the supersystem consisting of two helium atoms requires the inclusion of triply and quadruply excited determinants in order to describe the dimer. This problem becomes more and more pronounced as the number of electrons in the system is increased. Shavitt has summed up the situation as follows The fact that in a configuration interaction expansion unlinked cluster contributions can only be accounted for by including quadruple- (and higher-order) excitations is one of the principal drawsbacks of the method. In contrast, such contributions are automatically accounted for without explicitly computing higher-order terms, in some cluster-based methods and in many-body perturbation theory. In this sense the CI expansion is much less compact and less efficient than these approaches and becomes progressively less efficient as the number of electrons increases. 3.3.2. Cluster expansions The introduction of coupled cluster theory into quantum chemistry is attributed to ˇ ızˇ ek3 who, in 1966, published a seminal paper [61] in the Journal of ChemiC´ cal Physics entitled On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using QuantumField Theoretical Methods. The theory uses an exponential ansatz which was originally employed in statistical mechanics to compute the partition function of a non 3

ˇ ıˇzek has published a short historical review [60] of coupled cluster theory. C´

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ideal gas with pairwise interactions between the molecules: an approach associated with the names of Ursell and Mayer.4 The use of the exponential ansatz in formulating a quantum mechanical manybody theory – a theory which is extensive and scales linearly with the number of electrons studied – was first realized in nuclear physics by Coester and K¨ummel [64– 66]. The origins of the cluster approach to many-fermion systems, goes back to the early 1950s, when the first attempts were made to understand the correlation effect in an electron gas [67, 68] and in nuclear matter [69]. For both of these systems, it was absolutely essential that the method employed scaled linearly with the number of particles, i.e. that it is ‘size extensive’. During the 1950s, quantum chemists attempted to simplify the treatment of manyelectron systems by partitioning a given system into weakly interacting groups of electrons. Pioneering work by Fock [70] and by Hurley, Lennard-Jones and Pople [71] introduced the ‘pair function’ or ‘geminal model’. The philosophy of the more general group function approach was succinctly expressed by Parr [72] when he wrote that we should destroy the fallacy that large molecules are forever inaccessible to accurate treatment . . . in a molecule there are never more than a few electrons in one region of space. The group function model5 was developed by Parr, Ellison and Lykos [74], by McWeeny [75, 76], and by McWeeny and Sutcliffe [77, 78]. The group function ansatz . . . clearly leads to difficulties if the electrons do not fall into well separated groups. However, it is possible to decompose an N -electron system into smaller subsystems without dividing it into disjoint subsystems. This may be achieved by the cluster expansion approach . . . [1, 2] A cluster approach to the quantum chemical many-electron problem was developed by Sinano˘glu in his Many-Electron Theory of Atoms and Molecules [79]. What is ˇ ızˇ ek using today called coupled cluster theory was introduced by C´ creation and annihilation operators, hole–particle formalism, Wick’s theorem, and the technique of Feynman-like diagrams. ˇ ızˇ ek established the C´ connection of this method with the configuration interaction method as well as with the perturbation theory in the quantum-field theoretical form. 4

Ursell had published a paper [62] in the Proceedings of the Cambridge Philosophical Society entitled The evaluation of Gibbs phase-integral for imperfect gases in 1927 whilst Mayer published a paper [63] in 1937 entitled The Statistical Mechanics of Condensing Systems. I in the Journal of Chemical Physics. 5 A recent review of the group function model has been given by McWeeny [73].

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In a volume entitled Electron correlation in small molecules published in 1976, Hurley [80] derived the equations of coupled cluster theory without using what he ˇ ızˇ ek. Hurley claims that describes as the elaborate graphical analysis used by C´ The present determinantal derivation is much simpler and brings out clearly the relationship between the CPA [which is equivalent to the CCSD approximation] and the standard secular equation for ΨDE [the double excitation component of the configuration interaction wave function] Undoubtedly, Hurley’s book did much to bring coupled cluster theory to the attention of a wider audience.6 The coupled cluster theory may be derived from the many-body perturbation theory which we have presented above. Each coupled cluster approximation can be obtained by summing certain well-defined types of diagrammatic terms through all orders of the perturbation expansion. We shall not present here the details of the relation between coupled cluster and many-body perturbation theories. For a detailed discussion, the reader is referred to the review by Paldus and Li [81], published in 1999, entitled A critical assessment of coupled cluster method in quantum chemistry and the chapter on coupled cluster theory by Paldus [82] in the Handbook of Molecular Physics and Quantum Chemistry. The coupled cluster methods exploit an exponential ansatz for the wave operator. A chosen reference function |Φ0 , is transformed into the exact wave function |Ψ , as follows: (3.242) |Ψ = expT |Φ0 . Here T is the cluster operator which consists of the sum of independent connect cluster components Ti of various orders of excitation, that is: (3.243) T =

N

Ti .

i

N is the total number of electrons in the system. The advantage of the exponential ansatz for the wave operator can be seen by considering the application to systems consisting of independent subsystems, A, B, C,. . . . The cluster ansatz for the supersystem considered at the same excitation level can be written as the product of exponential cluster ans¨atze truncated at that excitation level, that is (3.244) expTA expTB expTC + . . . = expTA +TB +TC +··· . This result rests on the fact that the cluster operators TA , TB , TC , . . . commute because of the independence of the subsystems. 6

In the preface to the second edition of his Methods of Molecular Quantum Mechanics in 1989, McWeeny comments on the methods of theoretical physics which have been introduced into quantum chemistry. These methods are “still unfamiliar in chemistry” and, “because of their complexity and sophistication, their use has been widely resisted”, but “their power and generality is now beyond doubt”.

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Let us discuss the relation between coupled cluster theories and the many-body perturbation expansion. Recall eq. (3.214), the Rayleigh–Schr¨odinger expansion in ‘linked cluster’ form E0 = Φ0 |H0 |Φ0 + Φ0 |H1 + H1 RH1 + H1 RH1 RH1 + (3.245)

+ H1 RH1 RH1 RH1 . . . |Φ0 LC .

We can rewrite this equation as (3.246) E0 = Φ0 |H0 |Φ0 + Φ0 |VR |Φ0 , where the (Rayleigh–Schr¨odinger) reaction operator VR has the form $ % (3.247) VR = H1 + H1 RH1 + H1 RH1 RH1 + · · · LC or

% $ (3.248) VR = H1 + H1 RVR LC .

This is a Lippmann–Schwinger-like equation. If we introduce the wave operator ΩR as % $ (3.249) VR = H1 ΩR LC , then we have from eq. (3.248) % $ (3.250) ΩR = 1 + RH1 ΩR LC which is the Bloch equation. Substituting eq. (3.249) into eq. (3.246), we obtain the following expression for the exact energy E0 : (3.251) E0 = Φ0 |H0 |Φ0 + Φ0 |H1 ΩR |Φ0 LC . Using eq. (3.250), we get the following expression for the wave operator: $ % (3.252) ΩR |Φ0 = |Φ0 + RH1 ΩR |Φ0 LC . Introducing an exponential ansatz for wave operator ΩR , that is: (3.253) ΩR = eT we are led to the ‘standard’ non-degenerate coupled cluster theory.7 7

Note that the same expressions can be obtained by using the Baker–Hausdorf– Campbell theorem [83, 84]. We can write HeT |Φ0 = E0 eT |Φ0 or

Φ0 |e−T HeT |Φ0 = E0 .

Now, using Baker–Hausdorf–Campbell theorem, we have $ −T % $ % e HeT = H N eT lc ,

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Now we shall use simple arguments based on our previous results for many-body perturbation theory to show how the exponential ansatz of the coupled cluster method arises. Let us recall eq. (3.251) which gives the exact energy in the form (3.254) E0 = Φo |H0 |Φ0 + Φ0 |H1 ΩR |Φo LC where ΩR is the Rayleigh–Schr¨odinger wave operator and the subscript LC indicates that only terms associated with ‘linked connected’ diagrams are included. Let us digress and consider the Brillouin–Wigner formalism. Let M denote the Brillouin– Wigner wave operator. Its precise form need not concern us in the present discussion. Then, eq. (3.254) takes the form (3.255) E0 = Φ0 |H0 |Φ0 + Φ0 |H1 M |Φ0 . It should be emphasised that the Brillouin–Wigner expansion contains only direct terms. Unlike the Rayleigh–Schr¨odinger expansion, it does not contain renormalization terms. In the Brillouin–Wigner case the subscript LC can be omitted. The matrix elements of the operator H1 M are unknown, so we use a resolution of the identity operator to rewrite this equation as (3.256) E0 = Φ0 |H0 |Φ0 + Φ0 |H1 |Φj Φj |M |Φ0 . j

In general, the operator M is an N -particle operator which we can write as (3.257) M =

N

M (i)

i=0 (i)

where M depends on i particles. Inspection of eq. (3.256) reveals that we need to know at most the two-particle operator M (2) . We assume that the canonical Hartree– Fock orbitals are employed. In this way, we arrive at the expression (2) (2) (3.258) E0 = Φ0 |H0 |Φ0 + Φ0 |H1 |Φj Φj |M (2) |Φ0 j (2)

where the configuration state functions |Φj are doubly excited with respect to the reference configuration. We omit the contributions of single excitation so as to make our arguments more transparent, i.e. we neglect the one-particle M (1) operator. (2) The matrix element Φj |M (2) |Φ0 can be obtained by using the Bloch equation in Brillouin–Wigner form (2.243). (3.259)

(2) Φj |M (2) |Φ0

(2)

Φj |H1 M |Φ0 = . E0 − Ej % $ Φ0 | H N eT lc |Φ0 = E0

and for the amplitudes

% $ Φexc | H N eT |Φ0 = 0,

which is equivalent to eq. (3.252).

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Using the resolution of identity again, we can re-write eq. (3.259) as (3.260)

(2) Φj |M (2) |Φ0

=

Φ(2) j |H1 |Φk Φk |M |Φ0 E0 − Ej

k

.

By inspecting eq. (3.260), we see that we have to know the matrix elements for the components of M up to and including those involving fourfold excitations, i.e. M (4) . Thus we can write (3.261)

(2) Φj |M (2) |Φ0

=

(2) 4 Φj |H1 |Φk Φk |M (n) |Φ0

E0 − Ej

n=0 k

or, more explicitly, (2) Φj |M (2) |Φ0

+ (3.262) +

=

&

(2)

Φj |H1 |Φk Φk |M (0) |Φ0 E0 − Ej

k (2) Φj |H1 |Φk Φk |M (1) |Φ0

E0 − Ej (2) Φj |H1 |Φk Φk |M (3) |Φ0

E0 − Ej

(2)

+

Φj |H1 |Φk Φk |M (2) |Φ0 E0 − Ej

' (2) Φj |H1 |Φk Φk |M (4) |Φ0 . + E0 − Ej

We have demonstrated that in order to obtain the matrix elements of M (2) , we need to know the matrix elements M (i) with i taking values up to 4. By repeating these arguments, we can show that in order to obtain the matrix elements of M (4) , we require matrix elements for the components of M up to and including M (6) , and so on in higher orders. To solve eq. (3.261) for the matrix elements of the two particle operator M (2) , we proceed in the following manner: Starting from eq. (3.257), we put the one- and two particle components of M in the form (3.263) M (1) = m1 and 1 (3.264) M (2) = m2 + m1 m1 , 2 where 1 (3.265) m2 = M (2) − m1 m1 . 2 Similarly, the three particle component can be written (3.266) M (3) = m3 +

1 1 m1 m1 m1 + m1 m2 3! 2

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where 1 1 m1 m1 m1 − m1 m2 . 3! 2 The fourth-order component is (3.267) m3 = M (3) −

(3.268) M (4) = m4 +

1 1 1 1 m1 m1 m1 m1 + m1 m1 m2 + m2 m2 + m1 m3 4! 3! 2 2

and so on in higher orders. We can also write eq. (3.257) in the form (3.269) M = em where (3.270) m =

N

m(i) .

i=0

Now, to simplify our arguments, we neglect the one particle operator M (1) = m1 and the three particle operator M (3) operator. We can then make the approximation (3.271) M (2) = m2 and 1 m2 m2 . 2 Equation (3.258) for the energy then takes the form: (3.273) E0 = Φ0 |H0 |Φ0 + Φ0 |H1 |Φj Φj |m2 |Φ0 . (3.272) M (4) =

j

Equation (3.262) for the matrix elements of m2 is & Φ(2) j |H1 |Φk Φk |m2 |Φ0 (2) Φj |m2 |Φ0 = E0 − Ej k

(3.274)

' (2) 1 Φj |H1 |Φk Φk |m2 m2 |Φ0 . + 2 E0 − Ej k

In this way, we have obtained a set of ‘closed’ equations for the m2 operator. Equations (3.132) and (3.274) form the basis of the coupled cluster approximation at the double excitation level. The formalism discussed above can be used in the derivation of the single-reference Brillouin–Wigner coupled cluster theory. Let us now return to the Rayleigh–Schr¨odinger formalism which is used in developing the ‘standard’ coupled cluster formalism. The single-reference coupled cluster expansion can be developed by first of all writing the Schr¨odinger equation for the non-degenerate case as (3.275) H|Ψ0 = E0 |Ψ0 .

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The exact wave function is written (3.276) |Ψ0 = eT |Φ0 where (3.277) T =

∞

ti .

i−0

Substituting eq. (3.276) into eq. (3.275) gives (3.278) HeT |Φ0 = E0 eT |Φ0 . Multiplying from the left by e−T , we obtain (3.279) e−T HeT |Φ0 = E0 |Φ0 . The essence of this formulation is the use of the Baker–Hausdorf–Campbell expansion which is (3.280) e−T HeT |Φ0 = {HN eT }LC |Φ0 = E0 |Φ0 or (3.281) {HN eT }LC |Φ0 = E0 |Φ0 , where HN is the Hamiltonian in normal order form and { }LC indicates that we only take into account terms corresponding to linked connected diagrams. Expressions for the energy are obtained from (3.282) Φ0 |{HN eT }LC |Φ0 = E0 . Since E0 is a scalar quantity, we construct scalar diagrammatic contributions from the left-hand side of eq. (3.282). Diagrammatically, we have the following representation of HN : HN = Φ0 |H|Φ0 + A|f |BN [x+ A xB ] 1 + 2

(3.283)

A,B

A,B,C,D

or

(3.284) HN = Φ0 |H|Φ0 +

where (3.285)

A,B

+ AB|v|CDN [x+ A xB xD xC ]

A|f |BN [x+ A xB ] −→

+

Brillouin–Wigner Methods for Many-Body Systems and

(3.286)

1 2

+ AB|v|CDN [x+ A xB xD xC ] −→

A,B,C,D

127

.

For simplicity, we shall restrict our attention to the approximation T ≈ t2 , so that 1 (3.287) eT = 1 + t2 + t2 t2 2 and diagrammatically

(3.288) t2 =

1 + P Q|t2 |RSN [x+ P xQ xS xR ] −→ 2 P QRS

Now eq. (3.282) for energy E0 can be written diagrammatically as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎫⎪ ⎧ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎨ ⎬⎪ ⎨ 1+ + + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(3.289)

.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

= E0 .

If the one-particle basis is the solution of the Hartree–Fock equations, then we obtain the following expression for energy E0 : A C (3.290)

B D

A +D

C B

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which, in terms of spin orbitals, gives E0 = Φ0 |H|Φ0 +

1 2

A B |v|C D

A ,B ,C ,D

/ 0 C D |t2 |A B − D C |t2 |A B .

(3.291)

From this equation, we observe that the correlation energy is given in terms of the matrix elements of the operator v and t2 only. In order to obtain t2 matrix elements, we proceed by first noting that eq. (3.282) is obtained by the projection of eq. (3.281) onto the ground state Φ0 |. This results in an expression for E0 , which is a scalar quantity. If we now project eq. (3.281) onto states other than |Φ0 , that is, onto |Φi with i = 0, then we obtain expressions for the t2 amplitudes: (3.292) Φi |{HN eT }LC |Φ0 = 0,

i = 0

We will not give the details8 at this point, but just present the final results. ˇ ızˇ ek [29]) Equation (3.292) has the form (taken from the review of Paldus and C´ λ(d1 , d1 , d2 , d2 ) + λ(d2 , d2 , d1 , d1 ) = 0 where 1 d1 d2 |v|d1 d2 + d1 |f|d3 d3 d2 |t2 |d1 d2 2 d3 2d1 d3 |v|d1 d3 d1 |f|d3 d1 d2 |t2 |d3 d2 + −

λ(d1 , d1 ; d2 , d2 ) =

d3

d3 ,d 3

− d1 d3 |v|d1 d3 d2 d3 |t2 |d3 d2 − d2 d3 |v|d3 d1 d1 d3 |t2 |d3 d2 1 d d |v|d3 d4 d3 d4 |t2 |d1 d2 + 2 1 2

−

d1 d3 |v|d3 d1

d3 d2 |t2 |d3 d2

d3 ,d4

1 d d |v|d3 d4 d1 d2 |t2 |d3 d4 2 1 2 d3 ,d4 / 2d3 d4 |v|d3 d4 − d3 d4 |v|d4 d3 +

+

d3 ,d4 ;d 3 ,d4

× d1 d3 |t2 |d1 d3 d2 d4 |t2 |d2 d4 − d1 d3 |t2 |d3 d1

8

These details can be found in a number of review articles. See, for example, the ˇ arsky [85], by Paldus and C´ ˇ ıˇzek [29], by Bartlett and his reviews by Hubaˇc and C´ coworkers (see [86] and references therein), and by Urban et al. [87].

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× d2 d4 |t2 |d2 d4 − d3 d4 |t2 |d3 d1 d1 d2 |t2 |d4 d2 1 − d1 d3 |t2 |d4 d3 d2 d4 |t2 |d2 d1 + d3 d4 |v|d3 d4 2 (3.293)

× d1 d3 |t2 |d3 d1 d2 d4 |t2 |d4 d2 + d2 d3 |t2 |d4 d1 d1 d4 |t2 |d3 d2 0 + d1 d2 |t2 |d3 d4 d3 d4 |t2 |d1 d2 .

Efficient computer code for evaluating the numerical values of the energies associated with various coupled cluster approximations is available in a number of quantum chemistry packages. Coupled cluster calculations are more computationally demanding than those based on low order perturbation theory. However, they can support higher accuracy, particularly in cases where there is some quasi-degeneracy. Actually, the so-called ‘gold standard’ adopted in recent years as a best compromise between computational demands and accuracy, is the hybrid CCSD ( T ) method which combines use of a coupled cluster expansion with perturbation theory for the ‘triple excitation’ component. References 1. S. Wilson, Electron correlation in molecules, Clarendon Press, Oxford, 1984 2. S. Wilson, Electron correlation in molecules, reprinted, Dover Publications, New York, 2007 3. J.A. Pople, Rev. Mod. Phys. 71, 5, 1999 4. N.H. March, W.H. Young and S. Sampanthar, The many-body problem in quantum mechanics, Cambridge University Press, 1967, reprinted, Dover Publications, New York, 1996 5. I. Lindgren and J. Morrison, Atomic Many-Body Theory, Springer-Verlag, Berlin, 1982, 1986 6. F.E. Harris, H.J. Monkhorst and D.L. Freeman, Algebraic and Diagrammatic Methods in Many-Fermion Theory, Oxford University Press, 1992 7. E.R. Davidson, The World of Quantum Chemistry, edited by R. Daudel and B. Pullman, p. 17, Reidel, Dordrecht, 1974; see also S.R. Langhoff, and E.R. Davidson, Int. J. Quantum Chem., 8, 61, 1974 8. K. A. Brueckner, Phys. Rev. 100, 36, 1955 9. J. Goldstone, Proc. Roy. Soc. A239, 267, 1957 10. N. Hugenholtz, Physica 23, 481, 1957 11. J.E. Lennard-Jones, Proc. Roy. Soc. (London) A129, 598, 1930 12. L. Brillouin, J. Physique 7, 373, 1932 13. E.P. Wigner, Math. naturw. Anz. ungar. Akad. Wiss. 53, 475, 1935 14. B.H. Brandow, Rev. Mod. Phys. 39, 771, 1967 15. A. Messiah Quantum Mechanics, North Holland Publishing Company, 1961 16. P. Roman 1969 Introduction to Quantum Field Theory, Wiley, New York 17. S.S. Schweber 1964 Relativistic Quantum Field Theory, Harper and Row, New York 18. J. Ziman 1969 Elements of Advanced Quantum Theory, Cambridge University Press, Cambridge

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19. S. Weinberg 1995 The Quantum Theory of Fields, Cambridge University Press, Cambridge 20. P. Jørgensen and J. Simons 1982 Second-Quantization-based Methods in Quantum Chemistry, Academic Press 21. S.S. Schweber 1994 QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga, Princeton University Press, Princeton 22. P. Jordan and O. Klein, Z. Phys. 45, 755, 1927 23. P. Jordan and E.P. Wigner, Z. Phys. 47, 631, 1928 24. I.P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, Springer, 2006 25. H.M. Quiney, Handbook of Molecular Physics and Quantum Chemistry, vol. 2, Molecular Electronic Structure, ed. S. Wilson, P.F. Bernath and R. McWeeny, Wiley, Chichester, 2003 26. H.M. Quiney, I.P. Grant and S. Wilson, in Many-body methods in Quantum Chemistry, Lecture Notes in Chemistry, Springer Verlag, Berlin, 1989 27. P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edition, p. 274–275, Clarendon Press, Oxford 1958 28. R.P. Feynman, Phys. Rev. 76, 769, 1949 ˇ ızˇ ek, Advan. Quant. Chem. 9, 105, 1975 29. J. Paldus and J. C´ 30. J. Paldus and H.C. Wong Comput. Phys. Commun. 6, 1, 1973 31. H.C. Wong and J. Paldus Comput. Phys. Commun. 6, 9, 1973 32. J. Lyons, D. Moncrieff and S. Wilson, Comput. Phys. Commun. 84, 91, 1994 33. S. Raimes, Many-electron theory, North-Holland Publishing Company, 1972 34. A.L. Fetter, J.D. Walecka, Quantum theory of many-particle systems, New York: McGraw-Hill, 1971 35. I. Hubaˇc, International J. of Quantum Chem. XVII, 195, 1980 36. S. Wilson and D.M. Silver, Int. J. Quantum Chem. 5, 683, 1979 37. H.P. Kelly, Phys. Rev. 131, 684, 1963 38. H.P. Kelly, Phys. Rev. 136, 896, 1964 39. H.P. Kelly, Phys. Rev. 144, 39, 1966 40. H.P. Kelly, Adv. Theor. Phys. 2, 75, 1968 41. H.P. Kelly, Intern. J. Quantum Chem. Symp. 3, 349, 1970 42. H.P. Kelly, Phys. Rev. Lett 23, 455, 1969 43. J.H. Miller and H.P. Kelly, Phys. Rev. Lett. 26, 679, 1971 44. S. Wilson, in Theoretical Chemistry, Senior Reporter: C. Thomson, Specialist Periodical Reports 4, 1, The Royal Society of Chemistry, London, 1981 45. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 1, 364, The Royal Society of Chemistry, London, 2000 46. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 2, 329, The Royal Society of Chemistry, London, 2002 1034 47. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 3, 379, The Royal Society of Chemistry, London, 2004 48. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 4, 470, The Royal Society of Chemistry, London, 2006 49. S. Wilson, in Chemical Modelling: Applications and Theory, Senior Reporter: A. Hinchliffe, Specialist Periodical Reports 5, 208, The Royal Society of Chemistry, London, 2008 50. J.M. Schulman and D.N. Kaufman, J. Chem. Phys. 53, 477, 1970 51. U. Kaldor, Phys. Rev. A7, 427, 1973

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52. R.J. Bartlett and D.M. Silver, J. Chem. Phys. 62, 3258, 1975 53. S. Wilson and D.M. Silver, Phys. Rev. A14, 1969, 1976 54. I.G. Kaplan, Symmetry of Many-Electron Systems, translated by J. Gerratt, Academic Press, New York & London 1975 (p. 269, the term ‘algebraic approximation’ is employed to describe the use of finite analytic basis sets in molecular electronic structure calculations) 55. S. Wilson, in Methods in Computational Molecular Physics, ed. G.H.F. Diercksen and S. Wilson, p. 71, Reidel, Dordrecht 1983 56. S. Wilson, Adv. Chem. Phys. 67, 439, 1987 57. E.J. Heller and H.A. Yamani, Phys. Rev. A9, 1201, 1974 58. Ch. Møller and M.S. Plesset, Phys. Rev. 46, 618, 1934 59. I. Shavitt, Molec. Phys. 94, 3, 1998 ˇ ızˇ ek, Theor. Chem. Acc. 80, 91, 1991 60. J. C´ ˇ ızˇ ek, J. Chem. Phys. 45, 4256, 1966 61. J. C´ 62. H.D. Ursell, Proc. Cambridge Phil. Soc. 23, 685, 1927 63. J.E. Mayer J. Chem. Phys. 5, 67, 1937 64. F. Coester, Nucl. Phys. 7, 421, 1958 65. F. Coester and H. Kummel, Nucl. Phys. 17, 477, 1960 66. H. Kummel, Nucl. Phys. 22, 177, 1961 67. W. Macke, Z. Naturforsch. 5a, 192, 1950 68. M.G.-Mann and K. A. Brueckner, Phys. Rev. 106, 364, 1957 69. H.A. Bethe, Phys. Rev. 103, 1353, 1956 70. V. Fock, Dokl. Akad. Nauk. SSSR 73, 735, 1950 71. A.C. Hurley, J.E. Lennard-Jones and J.A. Pople, Proc. Roy. Soc. (London) A220, 446, 1953 72. R.G. Parr, The quantum theory of molecular electronic structure, Benjamin, New York 1963 73. R. McWeeny, in Handbook of Molecular Physics and Quantum Chemistry, 2, Molecular Electronic Structure, ed. S. Wilson, P.F. Bernath and R. McWeeny, Wiley, Chichester, 2003 74. R.G. Parr, F.O. Ellison and P. Lykos, J. Chem. Phys. 24, 1106, 1956 75. R. McWeeny, Proc. Roy. Soc. (London) A253, 242, 1959 76. R. McWeeny, Rev. Mod. Phys. 32, 335, 1960 77. R. McWeeny and B.T. Sutlciffe, Proc. Roy. Soc. (London) A273, 103, 1963 78. R. McWeeny and B.T. Sutlciffe, Methods of molecular quantum mechanics, Academic Press, New York 1976 79. O. Sinano˘glu J. Chem. Phys. 36, 706, 1962 80. A.C. Hurley, Electron correlation in small molecules, Academic Press, London, 1976 81. J. Paldus, X. Li, Advan. in Chem. Phys. 110, 1, 1999 82. J. Paldus, in Handbook of Molecular Physics and Quantum Chemistry, 2, Molecular Electronic Structure, ed. S. Wilson, P.F. Bernath and R. McWeeny, Wiley, Chichester, 2003 83. G.H. Weiss and A.A. Maradudin, The Baker-Hausdorﬀ formula and a problem in crystal physics, J. Math. Phys. 3, 771, 1962 84. R.M. Wilcox, J. Math. Phys. 8, 962, 1967 ˇ arsky, Topics in Current Chemistry 75, 97, 1978 85. I. Hubaˇc and P. C´ 86. R.J. Bartlett and M. Musial, Rev. Mod. Phys. 79, 291, 2007 ˇ sa´ k, V. Kell¨o and J. Noga, in Electron correlation in atoms and 87. M. Urban, I. Cernuˇ molecules, edited by S. Wilson, Methods in Computational Chemistry 1, 117, 1987

4 BRILLOUIN–WIGNER METHODS FOR MANY-BODY SYSTEMS

Abstract

The Brillouin–Wigner many-body problem in atomic and molecular physics and in quantum chemistry is described. The use of coupled cluster expansions, configuration interaction and perturbation series is considered both for the single-reference function case and for those cases requiring the use of a multi-reference formalism.

4.1. INTRODUCTION In this chapter, we come to the central purpose of this volume – the application of Brillouin–Wigner methods to many-body atomic and molecular systems.1 We have seen in Chapter 2 that Brillouin–Wigner perturbation theory [2–4] leads to energy expressions with denominators which contain the exact energy, Ei . As a consequence of this, the Brillouin–Wigner expansion yields energy components which scale nonlinearly with the number of electrons in the system. The method does not have. . . . extensivity (sometimes redundantly called size extensivity). A computational model is extensive if the computed value of an extensive property (in the thermodynamic sense), particularly the energy, varies correctly with the size of the system. For example, if a system is made up of n identical units, the model is extensive if the computed energy for this system is proportional to n as n → ∞. This concept applies to a system consisting of interacting subsystems, such as a uniform electron gas, or a solid, as well as to noninteracting systems, such as n non-interacting H2 molecules. [5] As we have seen in Chapter 3, that linear scaling of the energy with the number of electrons in a given system is the distinguishing feature of ‘many-body’ theories of electronic structure. It is because of their lack of extensivity that Brillouin–Wigner methods have been largely dismissed as the basis for a viable many-body theory for almost half a century. However, it has been recognized in recent years [6–57] 1

An earlier account of Brillouin–Wigner methods for many-body systems was given by the authors [1] in the Encyclopedia of Computational Chemistry.

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that there are circumstances in which the Brillouin–Wigner perturbation theory can be regarded as a true ‘many-body’ theory and these situations demand more careful examination. Since the seminal work of Brueckner [58], Goldstone [59] and others [60, 61] in the mid-1950s, Rayleigh–Schr¨odinger perturbation theory has been the approximation technique of choice in describing electron correlation effects in atoms and in molecules. In its ‘many body’ form, the many-body Rayleigh–Schr¨odinger perturbation theory has formed the basis of practical schemes for studying the electronic structure problem in molecular physics and quantum chemistry, beginning with the work of Kelly [62–71] in the 1960s on atoms and diatomic hydrides. The many-body Rayleigh–Schr¨odinger perturbation theory provides a powerful approach for systems which can be described by a single-reference function, but for systems requiring the use of a multi-reference formalism, such as bond breaking processes, it is plagued by what has become known as the ‘intruder state problem’. Intruder states manifest themselves when branch point singularities exist within the unit circle of the complex plane for the function E (λ). Because the energy denominators in Brillouin–Wigner perturbation theory contain the exact energy, this creates a natural gap and the intruder state problem is completely avoided. It is immediately apparent that, assuming convergence, the Brillouin–Wigner perturbation expansion taken through infinite order does provide an acceptable manybody theory, when it is used to solve the equations defining some many-body method. Indeed, in their treatise The many-body problem in quantum mechanics, March, Young and Sampanthar [72] admit the possibility “of considering an infinite number of terms in the limit of large N”. Trivially, we can note that infinite order Brillouin– Wigner perturbation theory is a physically acceptable theory when it is used to solve the full configuration interaction (FCI) problem. The Brillouin–Wigner expansion provides an alternative computational strategy for solving the equation defining the full configuration interaction method. When infinite order Brillouin–Wigner perturbation theory is used to solve the equations defining any valid ‘many-body’ technique, then a physically acceptable solution (that is a solution that is extensive) is obtained, provided that the perturbation expansion converges. For example, Brillouin–Wigner perturbation theory has been used by Hubaˇc and Neogr´ady [6] (see also [8]) to solve the equations of coupled cluster theory for a single-reference function. The approximation which they designate BWCCSD, i.e. Brillouin–Wigner Coupled Cluster expansion with Single and Double excitations, yields numerical results which are entirely equivalent to other coupled cluster formalisms, such as the ‘standard’ CCSD theory [73], in the case of a single-reference function [9, 10]. There is a well-known relation between Brillouin–Wigner perturbation theory and the Rayleigh–Schr¨odinger perturbation expansion. The Rayleigh–Schr¨odinger series can be obtained by expanding the denominator factors occurring in the Brillouin– Wigner expansion. The relation was used by Brandow [76], in deriving a multireference ‘many-body’ Rayleigh–Schr¨odinger perturbation theory. In our paper entitled On the use of Brillouin–Wigner perturbation theory for manybody systems [18], we have suggested that a given calculation using

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Brillouin–Wigner perturbation theory can be corrected so as to restore linear scaling with particle number, that is extensivity can be achieved and a many-body theory obtained. This correction may be approximate, but in certain circumstances it may be exact. Whereas Brandow used the relation between Brillouin–Wigner perturbation theory and Rayleigh–Schr¨odinger perturbation theory in his theoretical formalism, the approach that we advocate here exploits this relation as part of the computational strategy. In our approach, extensivity corrections are made during the computational process. In multi-reference formalisms, the Brillouin–Wigner expansion avoids the ‘intruder state problem’ which plagues Rayleigh–Schrodinger methods. We submit that Brillouin–Wigner methods coupled with extensivity corrections offer a robust approach to problems requiring a multi-reference formalism. In this chapter, we shall consider the application of Brillouin–Wigner-based methods to the three most commonly employed ab initio methods of approximation in molecular electronic structure theory: 1. Coupled Cluster (CC) expansions A brief introduction to coupled cluster theory has been given in Chapter 3, Section 3.3.2. Approximations based on CC expansions include, CCD, CCSD CCSDT, etc. and their multi-reference variants MR - CCD, MR - CCSD, MR - CCSDT, etc. The CCSD approximation is also known as the coupled-pair approximation ( CPA) [75] or the coupled pair many electron theory ( CPMET) [76, 77]. 2. Configuration Interaction (CI) expansions A brief introduction to the method of configuration interaction has been given in Chapter 3, Section 3.3.1. Approximations based on CI expansions include CI D, CI SD , etc. and their multi-reference variants MR - CI D , MR - CI SD , etc. 3. Perturbation Theory (PT) expansions A brief introduction to the perturbation theory as a many-body method has been given in Chapter 3, Section 3.2.3. Approximations based on PT expansions include MBPT 2, MBPT 3, MBPT 4, etc.2 and their multi-reference variants MR - MBPT 2, MR - MBPT 3, MR - MBPT 4, etc.3 We shall consider in turn each of these widely used approximation methods in this chapter. We shall provide an account of the Brillouin–Wigner formulation of each of these methods in a self-contained manner so that extensive cross referencing can be avoided. We shall establish the value of the Brillouin–Wigner method in the study of problems requiring a multi-reference formalism for a broad range of theoretical approaches. In this way, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid 2

These methods are often written as mp2, mp3, mp4, etc., that is Møller–Plesset perturbation theory [78] in second, third, fourth-order. 3 These methods are often written as mr-mp2, mr-mp3, mr-mp4, etc., that is multi-reference Møller–Plesset perturbation theory in second, third, fourth-order.

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methods such as the widely used CCSD ( T ) which employs CCSD theory together with a perturbative estimate of the triple excitation component of the correlation energy. In Section 4.2, we described the Brillouin–Wigner coupled cluster theory. We turn our attention to the use of Brillouin–Wigner methodology in configuration interaction studies in Section 4.3. In Section 4.4, the Brillouin–Wigner perturbation theory is considered. We note that, provided the respective expansions are convergent, each of the methods listed above should, when all terms are included, lead to the same result, i.e. the exact solution of the appropriate Schr¨odinger equation within the chosen basis set. Furthermore, if, for any reference function, all terms are included in the respective expansions, then, again subject to the proviso that the expansions converge, they will lead to the same result – the exact solution of the relevant Schr¨odinger equation within the basis set chosen. In the presence of strong quasi-degeneracy effects, an appropriately chosen multi-reference function may support a convergent expansion whereas another multi-reference function or a single-reference function may not. The appropriate choice of multi-reference function for quasi-degenerate problems is a significant problem and one which we do not address here. The use of a multireference formalism is required for problems as simple as the dissociation of the ground state of the hydrogen molecule. The choice of multi-reference function is dictated by the physics and chemistry of the systems under study. For more complicated problems the choice of reference requires considerable care. This choice certainly represents a significant barrier to the development of ‘black box’ quantum chemical software packages for problems demanding the use of a multi-reference formalism. In practice, the different expansions for the correlation energy afforded by coupled cluster theory, configuration interaction and perturbation theory have to be truncated in order to render computations tractable. The different methods differ only in the way in which this truncation is carried out. However, the method of truncation can significantly affect not only the theoretical properties of a particular approach but also, to some extent, its computational feasibility. [79, 80] Furthermore, The fact that one method may include more terms in an expansion than another method does not necessarily imply that it is superior. The terms which are left out of an expansion of the wavefunction or expectation value are, in fact, often just as important as the ones which are actually included. [79, 80] It is well-known, for example, that in a perturbation theory, analysis of the method of configuration interaction when restricted to single- and double-excitations with respect to a single determinant reference function includes many terms, corresponding to unlinked diagrams, which are exactly cancelled by terms involving higher order excitations. We leave it as an exercise for the interested reader to consider the application of Brillouin–Wigner methods, together with an a posteriori extensivity correction

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where appropriate, to other approximation techniques in molecular structure theory. The Brillouin–Wigner approach is a very general technique which has been largely overlooked in many-body studies of molecules and which shows considerable promise for situations in which a single-reference formalism is inadequate. We do not propose to describe here the details of specific applications of Brillouin– Wigner methods to many-body systems in chemistry and physics. Such details can be found in our article in the Encyclopedia of Computational Chemistry [1] and in our review entitled Brillouin–Wigner expansions in quantum chemistry: Bloch-like and Lippmann–Schwinger-like equations [36]. We have established a website at quantumsystems.googlepages.com/Brillouin–Wigner where further details of the applications of Brillouin–Wigner methodology are given. On this website, we plan to give details of further development in Brillouin–Wigner many-body theory.4 4.2. BRILLOUIN–WIGNER COUPLED CLUSTER THEORY The use of the exponential ansatz in formulating a quantum mechanical many-body theory was briefly described in Chapter 3, Section 3.3.2. This approach was first realized in nuclear physics by Coester and K¨ummel [81, 82] and its introduction ˇ ızˇ ek [76]. A recent overview of this into quantum chemistry is usually attributed to C´ method has been given by Paldus [73]. The single-reference coupled cluster approach has been described [83] as the best compromise between high accuracy and relatively low computer cost We shall divide our discussion of the Brillouin–Wigner formulation of coupled cluster theory into three parts. In the first section, Section 4.2.1, we consider the single-reference case for which the Brillouin–Wigner approach is entirely equivalent to the well-established formulations of coupled cluster theory based on the Rayleigh– Schr¨odinger expansion. This is important since it establishes that, under the appropriate conditions, the Brillouin–Wigner expansion forms the basis of a valid many-body theory. In the following section, Section 4.2.2, we turn our attention to the multireference formulation of coupled cluster theory. We present a multi-root formulation of the multi-reference Brillouin–Wigner coupled cluster theory in which all d exact energies are of interest. In the third section, Section 4.2.3, we describe a single-root formulation of the multi-reference Brillouin–Wigner coupled-cluster theory. This approach, which is termed ‘one state’, ‘state selective’ or ‘single root’, avoids the problems that may arise with the multi-root, multi-reference coupled cluster theories. 4.2.1. Single-reference Brillouin–Wigner coupled cluster theory We begin by giving a brief derivation of the basic equations of Brillouin–Wigner series in a form that is suitable for developing the Brillouin–Wigner coupled cluster 4

We will also collect the inevitable errors in the present volume.

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expansion. We use the notation that we have employed in previous chapters, which is similar to that used elsewhere in the literature (e.g. in the monograph by Lindgren and Morrison [84]). Let us divide our Hilbert space onto two orthogonal subspaces P and Q such that (4.1)

P ⊕ Q = 1,

The Hamiltonian operator in the exact Schr¨odinger equation, (4.2)

H|Ψi = Ei |Ψi ,

may be divided into two parts as follows (4.3)

H = H0 + H1

where H0 is the unperturbed Hamiltonian operator and H1 is the perturbation. Furthermore, we assume that we know all of the solutions of the unperturbed Schr¨odinger equation, (4.4)

H0 |Φi = Ei |Φi ,

so that we have the following resolution of the identity: (4.5) |Φi Φi | = 1. i

Now we can specify the projection operators P and Q on to the subspaces P and Q in sum-over-states form as (4.6) |Φi Φi | P = Φi ∈P

and (4.7)

Q=

|Φa Φa |,

Φa ∈Q

respectively. For the case of a non-degenerate (single-reference) ground state, we have (4.8)

P = |Φi Φi | ≡ |Φ0 Φ0 |.

The exact wave function for the ground state is Ψ0 and we assume the following intermediate normalization condition (4.9)

Φ0 |Ψ0 = 1.

Therefore, we can write the exact ground state wavefunction, Ψ0 , as |Ψ0 = (P + Q) |Ψ0 = |Φ0 + Q |Ψ0 (4.10)

= |Φ0 + |χ0

where |χ0 = Q|Ψ0 is the so-called correlation function for the ground state. In order to solve the exact Schr¨odinger equation (4.2) for the ground state, we have to find E0

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and |χ0 . Using eq. (4.3), we can write the exact Schr¨odinger equation, eq. (4.2), in the form (4.11)

(H0 + H1 )|Ψ0 = E0 |Ψ0

or, rearranging terms, (4.12)

(E0 − H0 )|Ψ0 = QH1 |Ψ0 .

Multiplying this equation from the left by Q, we obtain (4.13)

(E0 − H0 )Q|Ψ0 = QH1 |Ψ0 .

Let us introduce the Brillouin–Wigner type resolvent B0 which satisfies the equation (4.14)

B0 (E0 − H0 ) = Q

and which can be written as Q (4.15) B0 = (E0 − H0 ) This B0 satisfies the following equation (4.16)

B0 Q = QB0 = B0

and the relation (4.17)

B0 |Φa =

Q |Φa . (E0 − Ea )

Let us now multiply eq. (4.13) from the left by B0 giving (4.18)

B0 (E0 − H0 )Q |Ψ0 = B0 QH1 |Ψ0 .

Using the equalities (4.16) this immediately gives (4.19)

Q |Ψ0 = B0 H1 |Ψ0

whereupon, by substituting in eq. (4.10), we have the result (4.20)

|Ψ0 = |Φ0 + B0 H1 |Ψ0 .

Equation (4.20) can be used to generate the Brillouin–Wigner perturbation series for the exact wave function by recursion giving (4.21)

|Ψ0 = (1 + B0 H1 + B0 H1 B0 H1 + · · · ) |Φ0 .

Let us introduce the wave operator Ω0 , which we define as (4.22)

Ω0 = 1 + B0 H1 + B0 H1 B0 H1 + · · ·

which allows eq. (4.21) to be written in the form (4.23)

|Ψ0 = Ω0 |Φ0 .

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Using the intermediate normalization (4.9), we multiply eq. (4.11) from the left by Φ0 | and thus obtain the following expression for the exact ground state energy: (4.24)

E0 = Φ0 |H0 |Ψ0 + Φ0 |H1 |Ψ0 .

Using eqs. (4.4) and (4.23) we can re-write (4.24) as (4.25)

E0 = E0 + Φ0 |H1 Ω0 |Φ0 .

One of the possible ways of realizing the wave operator Ω0 is to use the exponential ˇ ızˇ ek ansatz introduced into atomic and molecular electronic structure theory by C´ [76] in coupled pair many-electron theory and coupled cluster theory. Here the wave operator is written as Ω0 = eT (4.26)

= 1 + T + 12 T 2 +

1 3 3! T

+

1 4 4! T

+ O T5

where T is the excitation operator, which is written as the sum (4.27) T = Ti . i

T1 being a single excitation operator, T2 a double excitation operator and so on up to Tn for an n-electron excitation. Analyzing eq. (4.22) we find that the wave operator Ω0 , satisfies the following expression [6] (4.28)

Ω0 = 1 + B0 H1 Ω0

(4.29)

H|Φ0 = |Φ0 + B0 H1 Ω0 |Φ0 .

or

Equation (4.29) can be regarded as the analogue of the Bloch equation [39] in Brillouin–Wigner form. We digress here in order to clarify the situation by returning to the ‘standard’ Rayleigh–Schr¨odinger formalism and writing the expression for the ground state energy using the theory of the effective Hamiltonian [84]: (4.30)

Heff = P HΩ0 P

which is defined through the equation (4.31)

Ψ0 = Ω0 (P Ψ0 )

and which fulfils the so-called Bloch equation [85]. If we partition the Hamiltonian H into an unperturbed part, H0 , and a perturbation, H1 , then the effective Hamiltonian operator can be written in the form (4.32)

Heff = PH0 P + PH1 Ω0 P.

This gives us the following expression for the ground state energy (4.33)

E0 = E0 + Φ0 |H1 Ω0 |Φ0 .

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Now, in order to calculate Ω0 |Φ0 , we use the Bloch equation [85] which has the form (4.34)

(E0 − E0 )Ω0 P = QH1 Ω0 P − χ0 PH1 Ω0 P

where χ0 is defined as follows (4.35)

Ω0 = 1 + χ0 ,

For the non-degenerate (single-reference) case we can write from eq. (4.34) |Φi Φi |H1 Ω0 |Φ0 χ0 |Φ0 Φ0 |H1 Ω0 |Φ0 (4.36) Ω0 |Φ0 = |Φ0 + − , E0 − H0 E0 − H0 i=0

Introducing the Rayleigh–Schr¨odinger type resolvent |Φi Φ0 | Q (4.37) R0 = ≡ , E0 − H0 E0 − H0 i=0

we can re-write eq. (4.36) in the following form: (4.38)

Ω0 |Φ0 = |Φ0 + R0 H1 Ω0 |Φ0 −

χ0 |Φ0 Φ0 |H1 Ω0 |Φ0 , E0 − H0

A great simplification of eq. (4.38) is achieved by exploiting the linked cluster theorem [84]. This implies that in eq. (4.36) or eq. (4.38), we calculate only the terms represented by linked connected (LC) diagrams. Therefore, eq. (4.38) can be written as (4.39)

Ω0 |Φ0 = |Φ0 + R0 H1 Ω0 |Φ0 LC

where the subscript LC indicates that only terms corresponding to linked connected diagrams are included. Now we have completed our digression and return to the Brillouin–Wigner formalism. We see that eq. (4.29) is analogous to Bloch equation (4.39). We have simply two ways of realizing Ω0 |Φ0 . We can either use the Bloch equation (4.39) with the Rayleigh–Schr¨odinger resolvent and the linked cluster theorem or we can use eq. (4.29) with the Brillouin–Wigner resolvent. We describe eq. (4.29) as ‘the Bloch equation in Brillouin–Wigner form’. The expressions (4.29) and (4.39) are entirely equivalent. If we use the exponential ansatz defined in (4.26) for wave operator Ω0 , we can obtain the following equations for the exact ground state energy: (4.40)

E0 = E0 + Φ0 |H1 eT |Φ0

and (4.41)

eT |Φ0 = |Φ0 + B0 H1 eT |Φ0 ,

These two equations together are the basic equations of non-degenerate (singlereference) Brillouin–Wigner coupled cluster (BWCC) theory. We emphasize that these equations are obtained directly from Brillouin–Wigner perturbation expansion. In particular, we have not used the linked cluster theorem and neither have we employed

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a Baker–Campbell–Hausdorff ( BCH) expansion, which is exploited in the ‘standard’ derivation of the coupled cluster equations.5 By adopting various approximations for the excitation operator T , we obtain a hierarchy of different approximate versions of BWCC theory in just the same way as in ‘standard’ coupled cluster theory (e.g. if we put T ∼ T1 + T2 , then we obtain an approximation which may be designated BWCCSD theory, that is, BWCC theory with Single and Double excitations. Putting T ∼ T2 gives BWCCD theory, that is, BWCC theory with Double excitations, the approximation T ∼ T1 + T2 + T3 leads to BWCCSDT theory, again, that is, BWCC theory with Single, Double and Triple excitations, and so on). The solution of eqs. (4.40) and (4.41) can be realized in the same way as in ‘standard’ coupled cluster theory. It has been demonstrated by Hubaˇc and Neogr´ady [6] that BWCC theory is fully equivalent to ‘standard’ coupled cluster theory. Furthermore, BWCC theory scales linearly with particle number. It is fully extensive. The cancellation of disconnected diagrams in BWCC theory is realized through the iterative solution of eqs. (4.40) and (4.41). We shall now demonstrate the cancellation of disconnected diagrams in BWCC theory in the case of BWCCD theory [9]. If we adopt a simple approximation in the exponential ansatz (4.26), such that (4.42)

T = T2

we are lead to the BWCC with Double excitations (BWCCD) theory. In this case, the expression for the correlation energy (4.43)

ΔE0 = E0 − E0 = Φ0 |H1 eT |Φ0

can be simplified so as to assume the form (4.44)

ΔE0 = Φ0 |H1 T2 |Φ0

Equation (4.41) for the two-body cluster operator T2 will then be AB T2 AB (E0 − Ei )ΦAB IJ |T2 |Φ0 = ΦIJ |H1 e |Φ0 + ΦIJ |H1 (1 + T2 +

1 2 T )|Φ0 2! 2

(4.45) where |ΦAB IJ represents a configuration which is doubly excited with respect to the ground state. Since we do not employ the linked cluster theorem, the contributions T2 originating from the matrix element ΦAB IJ |H1 e |Φ0 can be partitioned into connected terms, designated by the subscript (c), and disconnected terms, designated by the subscript DC. Using this partition, we can write (4.46)

AB T2 (E0 − Ei )ΦAB IJ |T2 |Φ0 = ΦIJ |H1 e |Φ0 C +

1 AB Φ |H1 T22 |Φ0 DC 2! IJ

5 The Baker-Campbell-Hausdorﬀ formula is a fundamental expansion in elementary Linear Algebra and Lie group theory (J. E. Campbell, Proc. London Math. Soc. 29, 14 (1898); H. F. Baker, Proc. London Math. Soc. 34, 347 (1902); F. Hausdorﬀ, Ber. Verhandl. Saechs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 58, 19 (1906)).

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The contribution of the disconnected component is given by (4.47)

1 AB Φ |H1 T22 |Φ0 DC = tAB IJ ΔE0 , 2! IJ

where tAB IJ denotes the antisymmetrized bi-excited cluster amplitudes. This component can be computed easily by using diagrammatic techniques. If we now substitute for the disconnected component into eq. (4.36), we obtain (4.48)

AB T2 AB (E0 − Ei )tAB IJ = ΦIJ |H1 e |Φ0 C + ΔE0 tIJ

which can be simplified to the following: (4.49)

AB T2 (E0 − Ei )tAB IJ = ΦIJ |H1 e |Φ0 C .

Here we have to note that eq. (4.48) only reduces to eq. (4.49) in the case when the cluster amplitudes are fully converged. We see that the cancellation of disconnected contributions in the BWCCD theory is achieved iteratively and, furthermore, exact cancellation is achieved by the full convergence of cluster amplitudes. It can therefore be concluded that BWCCD theory is fully equivalent to the ‘standard’ CCD approximation, which exploits the linked cluster theorem. Similar arguments have been used to demonstrate the equivalency of BWCCSD and CCSD theories [9]. The extension of these arguments to other coupled cluster approximations, such as BWCCSDT and CCSDT theories, is straightforward. 4.2.2. Multi-reference Brillouin–Wigner coupled cluster theory Paldus [73] has emphasised that, when exploiting a multi-reference formalism in coupled cluster theory, additional complications arise because of the fact that the extension of the standard SR [single-reference] exponential cluster ansatz to the MR [multireference] case is far from being unambiguous Here we shall discuss two versions of the multi-reference Brillouin–Wigner coupled cluster (BWCC) theory which are based on the use of effective Hamiltonians. To derive a multi-reference version of the BWCC theory, we shall follow a procedure which is similar to that used in the case of single-reference function. We begin again with eqs. (4.1)–(4.7): (4.1)

P⊕Q=1

(4.2)

H|Ψi = Ei |Ψi

(4.3)

H = H0 + H1

(4.4)

H0 |Φi = Ei |Φi |Φi Φi | = 1

(4.5)

i

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I. Hubaˇc and S. Wilson

P =

|Φi Φi |

Φi ∈P

(4.7)

Q=

|Φa Φa |.

Φa ∈Q

In contrast to the non-degenerate case, we now consider the case in which the P subspace is d-dimensional. In general, projections of the exact wave functions, Ψα , onto the model space P (4.50)

ΨαP = P Ψα ,

α = 1, . . . , d

are non-orthogonal. But the functions ΨαP , which are sometimes termed the model functions, are assumed to be linearly independent. Therefore, they can be written as a linear combination of configurations spanning the model space: (4.51) ΨαP = Cν,α Φν ν∈P

and, in turn, the model space configurations can be expressed using the inverse transformation (4.52)

Φμ =

d

−1 P Cα,μ Φα .

α=1

The expansion coefficients cαμ obey the following identity relation: (4.53)

d

−1 cνα Cα,μ = δνμ .

α=1

By following procedures similar to those employed in the case of a single-reference function, the exact wave function Ψα , for α = 1, 2, . . . , d, in the Brillouin–Wigner perturbation theory can be written as the expansion (4.54)

Ψα = (1 + Bα H1 + Bα H1 Bα H1 + · · · )ΨαP ,

where ΨαP is the projection of the exact wave function onto the model space P and Bα is the Brillouin–Wigner type propagator: |Φq Φq | (4.55) Bα = . Eα − Eq q∈Q

It should be mentioned that, in developing the Brillouin–Wigner expansion (4.54), we have used the intermediate normalization condition, i.e. (4.56)

ΨαP |Ψα = ΨαP |ΨαP = 1.

We now introduce a wave operator Ωα to denote the operator expansion in parenthesis in eq. (4.54) (4.57)

Ωα = 1 + Bα H1 + Bα H1 Bα H1 + · · ·

α = 1, . . . , d.

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We can then write eq. (4.54) as follows (4.58)

Ψα = Ωα ΨαP

α = 1, . . . , d.

As we can see the wave operator Ωα has the property that it transforms the model function, ΨαP , into the corresponding exact wave function, Ψα . We should emphasise that the wave operator, Ωα , maps just one projected wave function into the corresponding exact wave function. This is a special feature of the Brillouin–Wigner perturbation formulation. We note that there is no expression analogous to (4.58) in the Rayleigh–Schr¨odinger formulation. This feature of the Brillouin–Wigner formulation is a consequence of the fact that the exact energy for just one state Eα , occurs in the resolvent, eq. (4.55). To avoid any confusion, we shall term Ωα as the state-specific (or single root) wave operator. From eq. (4.57), it follows that the state-specific wave operators obey the following system of equations (4.59)

Ωα = 1 + Bα H1 Ωα

α = 1, . . . , d.

Each equation may be regarded as a Bloch equation in the Brillouin–Wigner form for the state α. The formulation of a multi-reference BWCC theory can now proceed in two distinct ways. In the first option, we can formulate a multi-root version of the multi-reference BWCC theory which yields all roots of the d-dimensional P space simultaneously. This is the approach employed in most multi-reference coupled cluster formulations which are based on the Rayleigh–Schr¨odinger expansion. In the second option, we can use the state-specific wave operator (4.59) and formulate a state-specific (or single root) version of multi-reference BWCC theory [10]. These two possible formulations of multi-reference Brillouin–Wigner coupled cluster theory are discussed further below. In Section 4.2.2.1, we present a multiroot formulation of Brillouin–Wigner theory. This formalism is employed in Section 4.2.2.2 to develop a multi-root, multi-reference Brillouin–Wigner coupled cluster theory, using a Hilbert space approach. In Section 4.2.2.3, we discuss the basic approximations employed in the multi-reference Brillouin–Wigner coupled cluster method. 4.2.2.1. Multi-root formulation of multi-reference Brillouin–Wigner coupled cluster theory For cases in which all d exact energies are of interest, we can construct a ‘multi-root’ wave operator, Ω, acting on states from the model space. The operator Ω can be obtained by summing all of the state-specific wave operators, Ωα , as follows: (4.60)

Ω=

d

Ωα Pα

α=1

where Pα is the projection operator (4.61)

Pα =

d β=1

−1 |ΨαP Sαβ ΨαP |.

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−1 Here Sαβ is the inverse of the overlap matrix Sαβ

(4.62)

P Sαβ = ΦP β |Φα .

The multi-root wave operator Ω has the property (4.63)

Ψα = ΩΨαP = Ωα ΨαP ,

for α = 1, . . . , d

which implies that it represents a ‘true’ wave operator. But we should recall that the projected wave functions ΨαP are, in general, non-orthogonal. As a result, the wave operator is a non-Hermitian operator and should be described more precisely as the Bloch wave operator [85] (this result is in contrast with Kato’s approach [86] which leads to a Hermitian wave operator). Using the multi-root wave operator (4.60), the system of eqs. (4.58) can be replaced by a single equation, eq. (4.63), and, likewise, the intermediate normalization condition, eq. (4.56), can be replaced by the equation in the operator form (4.64)

P Ω = P.

However, similar substitutions cannot be made in the case of the system of equations for state-specific wave operators, eqs. (4.59). Specifically, we are not able to replace this system of equations by a single equation in operator form, whilst simultaneously retaining its Brillouin–Wigner character. To some extent, this replacement can be achieved within a matrix formulation. By using the multi-root wave operator (4.60), the system of eqs. (4.59) can be shown to be equivalent to (4.65)

ΩΨαP = (1 + Bα H1 Ω)ΨαP .

Projecting this equation onto configurations from the Q space, we obtain a system of equations for the multi-root wave operator (4.66)

(E0 − Eq )Φq |Ω|ΨαP = Φq |H1 Ω|ΨαP ,

which may be viewed as a matrix formulation of the Bloch equation in the Brillouin– Wigner form. The determination of the exact energies can be achieved by substituting eq. (4.63) into the Schr¨odinger equation (4.2). In this way, we obtain the following equation (4.67)

H(ΩΨαP ) = Eα Ψα .

Then, by projecting this equation onto the model space, P, we get (4.68)

P HΩΨαP = Eα P Ψα = Eα ΨαP ,

so that the exact energies can be obtained as eigenvalues of the effective Hamiltonian operator (4.69)

Heff = P HΩP.

This operator acts within the model space and will be referred to as a ‘multi-root’ effective Hamiltonian operator. The eigenfunctions of the effective Hamiltonian operator are projected wave functions ΨαP . Since the functions ΨαP are not necessarily orthogonal, the effective Hamiltonian operator, Heff , is, in general, non-Hermitian. Clearly, the eigenvalues of Heff are always real since they correspond to true energies.

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In order to obtain the wave operator Ω in a form that is suitable for practical calculations, we substitute the expansion for the model function, eq. (4.51), into the Schr¨odinger equation in the Brillouin–Wigner form, eq. (4.66). This substitution results in the following system of equations (4.70) (E0 − Eq ) Φq |Ω|ΦP Φq |Ω|ΦP ν cνα = ν cνα ν∈P

ν∈P

where the coefficients cνα are obtained by diagonalization of the effective Hamiltonian. Now, we can proceed in two different ways. We can eliminate the summation on either the left-hand side (LHS) of eq. (4.70) or on the right-hand side (RHS) of eq. (4.70). We consider each of these two possibilities in turn. (a) In the first approach, we move the denominator factors back to the RHS of eq. (4.70) and then multiply both sides of the resulting equation by c−1 αμ . We then carry out a summation over all values of α. These manipulations lead to the following set of equations (4.71) Φq |Ω|Φμ = Φq |H1 Ω|Φμ dνμ ν∈P

where (4.72)

dνμ =

d cνμ c−1 αμ . E − Eq α=1 α

Equations (4.71) and (4.72) can be used to calculate of the wave operator Ω. The coefficients dνμ depend explicitly on the exact energies as well as on the expansion coefficients within the model space. Both the wave operator and the effective Hamiltonian are therefore energy dependent. Consequently, the system of eqs. (4.69) and (4.71) must be solved simultaneously in an iterative fashion. (b) The second possible approach in developing eq. (4.70) is to ‘glue’ together the operators H0 and H1 . In this approach, terms containing the zero-order energy Eq are added to the RHS of eq. (4.70) giving: (4.73) Φq |Ω|Φν cνα Eα = Φq |HΩ|Φν cνα . ν∈P

ν∈P

Multiplying both sides of this equation by c−1 αμ and then summing over all α, results in the following set of equations 1 ! d −1 (4.74) Φq |Ω|Φν cνα Eα cαμ = Φq |HΩ|Φμ . ν∈P

α=1

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If we now recognize that the term in brackets {. . . } is the matrix element of the effective Hamiltonian [9], then we obtain the Bloch equation in the matrix form eff (4.75) Φq |HΩ|Φμ = Φq |Ω|Φν Hνμ . ν∈P

The corresponding operator form of this equation is the well-known Bloch equation [85, 88–91] which can be written (4.76)

HΩP = ΩP HΩP

or the so-called generalized Bloch equation [85, 88–92, 130] (4.77)

[Ω, H0 ] P = H1 ΩP − ΩP H1 ΩP,

which determines the wave operator in the multi-root case. It should be noted that the wave operator Ω no longer depends on the exact energies and therefore represents a much more suitable formulation for practical calculations. Within the multi-reference Brillouin–Wigner perturbation theory, we have been able to construct a multi-root wave operator together with an effective Hamiltonian operator, Heff , which formally possess the same properties as those employed in the multi-reference theories based on the Bloch equation. For this reason, the adjective ‘multi-root’ is clearly not necessary here. Second, we derived the Bloch equation starting from the Brillouin–Wigner perturbation expansion, but, in contrast to the approach described by Brandow in his review [74] on Linked-Cluster Expansions for the Nuclear Many-Body Problem, we did not expand the denominator factors in order to remove the exact energy dependence. Finally, the two sets of equations given above for the wave operator (4.71) and (4.75), are entirely equivalent. Our first approach represented by the set of eqs. (4.71) may be regarded as a Bloch equation [85] in Brillouin–Wigner form. Similarly, in terms of perturbation theory, the generalized Bloch equation (4.77) may be viewed as a Bloch equation in the Rayleigh–Schr¨odinger form. 4.2.2.2. Multi-root multi-reference Brillouin–Wigner coupled cluster: Hilbert space approach In the discussion given in the previous subsection, we have specified the wave operator Ω by means of the multi-reference Brillouin–Wigner perturbation expansion defined by eqs. (4.59) and (4.60): (4.59)

Ωα = 1 + Bα H1 Ωα

(4.60)

Ω=

d

α = 1, . . . , d

Ωα Pα .

α=1

If we now adopt an exponential ansatz for the wave operator Ω, then we are lead to the multi-reference Brillouin–Wigner coupled-cluster (MR BWCC) theory.

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The simplest way to realize the idea of an exponential expansion for the wave operator is to exploit the so-called state-universal or Hilbert space exponential ans¨atz originally presented by Jeziorski and Monkhorst [87] in 1981, namely: μ (4.78) Ω = eT |Φμ Φμ |. μ∈P

In this equation, T μ is a cluster operator defined with respect to the μ-th configuration involving, in general, one-body (T1μ ), two-body (T2μ ), and so on up to the N -body (TNμ ) cluster components: (4.79)

T μ = T1μ + T2μ + · · · + TNμ ,

with N being the total number of electrons in the system. The Jeziorski–Monkhorst ansatz, eq. (4.78), employs different particle-conserving cluster operators for different reference determinants. It is well suited to cases involving significant quasidegeneracy. Comparing the two forms for the wave operator given in eqs. (4.60) and (4.78), the Hilbert space approach appears as a more straightforward way of employing an exponential ansatz for the wave operator in the multi-reference case, when compared with the so-called valence-universal or Fock space approach [84, 91, 93–99] which employs a single (valence universal) cluster operator. The valence universal/Fock space exponential wave operator generates ground and excited states of the system under investigation, together with ions obtained by removing active electrons one at a time. The valence universal/Fock space approaches are particularly useful for valence systems and for various differential properties, such as ionization energies and electron affinities. Substituting the Jeziorski–Monkhorst cluster ansatz (4.78) into eqn. (4.58), leads to the following cluster expansion for the exact wave function μ (4.80) Ψα = cμα eT |Φμ , μ∈P

where the coefficients cμα are obtained by diagonalization of the effective Hamiltonian. The resulting wave function represents a generalization of the cluster expansion previously introduced by Silverstone and Sinanoglu [100] for open-shell wave functions with an empty set of core orbitals. Although the past 20 years have witnessed a great progress in the Hilbert space multi-reference coupled cluster methods (see, for example, the work of Mukherjee and Pal [99], Paldus [101], Jeziorski and Paldus [102], Jankowski et al. [103], Paldus et al. [104], Paldus et al. [105], Meissner et al. [106], Kucharski and Bartlett [107], Balkov´a et al. [108], Balkov´a and Bartlett [109], Balkov´a et al. [110], Balkov´a et al. [111], Berkovic and Kaldor [112]) only a few applications of this approach have been reported, mostly oriented to the simple model systems exploiting a lowdimensional model space. Among the reasons for this paucity of applications are the choice of an appropriate model space, convergence difficulties arising from intruder state problems and from multiple solutions of non-linear coupled cluster equations,

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and also spin-contamination when handling open-shell systems. From the computational point of view, excessively large model spaces require a large number of cluster amplitudes, since each configuration in the model space is associated with its own set of cluster amplitudes. Furthermore, calculations making use of large model spaces are more likely to be plagued by intruder states. On the other hand, in order to guarantee linear scaling with particle number, the extensivity property central to manybody formalisms, most of the Hilbert space coupled cluster developments employ a complete model space. The extensivity of calculations employing incomplete model spaces is often destroyed by diagonalization of the effective hamiltonian. Although incomplete model space formulations [111] can also yield theories having extensivity, the use of a complete model space is still conceptually simpler. In the following discussion, we shall limit our discussion to the complete model space case, which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. Now let us begin the determination of exact energies within the multi-root multireference BWCC theory. In the following analysis, we will work with the Hamiltonian in the normal-ordered product form, i.e. (4.81)

H = Φμ |H|Φμ + HN (μ) = Hμμ + HN (μ)

where the μth configuration plays the role of a Fermi vacuum. When calculating matrix elements containing the wave operator Ω, we can, in general, work with different Fermi vacua for different reference configurations. In our derivation, the role of the Fermi vacuum will be played by the configuration with the μth subscript. Using the exponential ansatz (4.78), the off-diagonal matrix elements of the effective Hamiltonian (4.69) can be written as (4.82)

μ

eff Hνμ = Φν |HΩ|Φμ = Φν |HN (μ)eT |Φμ .

In a similar fashion, the diagonal elements can be written as μ

eff = Φμ |HΩ|Φμ = Hμμ + Φμ |HN (μ)eT |Φμ Hμμ

(4.83)

μ

= Hμμ + HN (μ)eT μ .

We recall that all matrix elements of the effective Hamiltonian are expressed in terms of connected diagrams only. In the case of diagonal matrix elements, only the connected vacuum diagrams may arise, whilst in the case of off-diagonal matrix elements, at least one part of a disconnected diagram will correspond to an internal excitation. We turn now to the determination of the unknown cluster amplitudes. There are at least these in turn for the implementation of the Hilbert space exponential ansatz (4.78). We shall now discuss three possible schemes: (i) In the first scheme, we keep the original Brillouin–Wigner character of the calculation; i.e. the formal dependence of the wave operator on exact energies. We start from the Bloch equation in the Brillouin–Wigner form, eq. (4.71). By using the exponential ansatz (4.78) we get the equation

Brillouin–Wigner Methods for Many-Body Systems (4.84)

μ

Φq (μ)|eT |Φμ =

151

μ

Φq (μ)|H1 eT |Φν dνμ ,

ν∈P

where Φq (μ) is an excited configuration with respect to the μ-th configuration. This approach may be regarded as a ‘pure’ multi-reference BWCC formalism, as is manifested by the presence of typical Brillouin–Wigner-like denominators in eq. (4.72). Examination of the summation on the right-hand side of eq. (4.84) reveals that, besides the direct term ν = μ with a structure the same as in the non-degenerate case, there are so-called coupling terms with ν = μ, which differ in structure from the coupling terms used in the multi-reference coupled cluster Hilbert space approach by the presence of a perturbation. Hence, the coupling among the various cluster operators T μ , is provided in two ways: directly through the so-called coupling terms and indirectly through the presence of exact energies in denominators. From a computational point of view, this approach is not advantageous because of the complicated strucμ ture of coupling terms. Although the exponential expansion eT terminates at some level of excitation, this level is too high for practical applications. For instance, if we confine our attention to a two-dimensional model space spanned by two closed-shelltype configurations (inter-related as bi-excited configurations), the level of excitation is 6 in the case of the CCSD approximation, because the Hamiltonian is at most a twoparticle operator. Moreover, the explicit splitting of the Hamiltonian into a zero-order Hamiltonian and a perturbation, introduces another formal complication when using various vacuum states. (ii) The second possible scheme for implementing the Hilbert space exponential ans¨atz, is to employ the standard Bloch equation (4.75) in its matrix form. Substituting the exponential ansatz (4.78) into eq. (4.75), we obtain the following set of equations for calculating the cluster amplitudes: μ

(4.85)

μ

eff (Hμμ − Hμμ )Φq (μ)|eT |Φμ = Φq (μ)|HN (μ)eT |Φμ μ eff Φq (μ)|H1 eT |Φν Hνμ . − ν=μ

Now, in contrast to the first scheme, in this second scheme the explicit dependence on the exact energies is removed, so that the description of this scheme as a ‘Brillouin– Wigner’ scheme is no longer appropriate. Therefore, we will refer to this second scheme as the multi-reference coupled cluster (MR CC) method. The coupling among the various cluster operators is provided only through the terms on the RHS of the above equation. It can be seen that their structure is very simple, because the exponenμ tial expansion eT terminates at some level of excitation. Moreover, the matrix eleμ ments Φq (μ)|eT |Φμ do not contain internal summations. For instance, in the case of a model space spanned by two closed-shell-type configurations, the level of excitation is 4 for the CCSD approximation. However, it should be emphasized that such an approach is manifested by the presence of disconnected diagrams as well, so care must be taken when making lower approximations in order to avoid destroying the extensivity property. This approach is very suitable for practical calculations and it has been intensively developed and tested by Meissner, Kucharski, Bartlett and their

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collaborators [106–111], especially at the level of the CCSD approximation using a Slater determinant-based spin-orbital formulation. Nevertheless, there is a formal difference between the present derivation and that employed by Meissner et al., in that we do not partition the Hamiltonian, whereas Meissner et al. started from the Bloch equation in the ‘Rayleigh–Schr¨odinger’ form (4.77). (iii) The third possible scheme for implementing the Hilbert space exponential ansatz takes advantage of the Baker–Campbell–Hausdorff formula. As we mentioned above, this approach was initiated by Jeziorski and Monkhorst [87] in 1981. It has been intensively developed and tested by Jankowski, Jeziorski, Paldus and their collaborators [101–105], especially within the CCSD approximation using the spinadapted formulation. We start from the standard Bloch equation in the operator form, eq. (4.76). However, before projecting this equation against configurations spanning μ the Q and P spaces, we multiply both sides from the left by e−T which gives μ μ μ ν eff (4.86) Φq (μ)|e−T HeT |Φμ = Hνμ Φq (μ)|e−T eT |Φν . ν

Using the Baker–Campbell–Hausdorff formula leads to the result that only terms corresponding to connected diagrams survive μ μ ν eff (4.87) Φq (μ)|HN (μ)eT |Φμ C = Hνμ Φq (μ)|e−T eT |Φν C ν=μ

and this is the set of equations that is used to calculate cluster amplitudes6 . The apparent advantage of this approach is the freedom it affords to introduce various approximations without the loss of extensivity. However, this freedom is balanced by μ μ the complicated structure of coupling terms. The product of exponentials e−T eT leads to the presence of internal summations which result in an increasing number of diagrams, which has been described as “cumbersome” and “unwieldy” [108]. 4.2.2.3. Basic approximations employed in the multi-reference Brillouin–Wigner coupled cluster method In the previous section, we have presented three different computational schemes for the determination of the ‘multi-root’ wave operator within the multi-reference Brillouin–Wigner coupled cluster MR BWCC method. Whilst the first scheme, which is characterized by the set of eqs. (4.84), may be regarded as a ‘pure’ MR BWCC approach (it must be solved simultaneously with the equation for the effective Hamiltonian), the other two schemes, which are characterized by the set of eqs. (4.85) and (4.87), respectively, represent ‘standard’ MR BWCC approaches and, in these cases, the adjective ‘Brillouin–Wigner’ is inappropriate (the wave operators do not depend on exact energies). There is no doubt that of the three schemes the most computationally tractable approach is the second. This scheme is employed, for example, 6

To be more precise, the Baker–Campbell–Hausdorﬀ lemma only guarantees that the direct term corresponds to connected diagrams. The connectedness of diagrams associated with the coupling terms has to be proven a posteriori.

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by Bartlett and his co-workers. The third scheme is the next most computationally tractable. Of course, all three schemes are equivalent when the full expansion of exponentials is carried out. However, a full expansion can only be considered in a very limited number of cases for systems containing a small number of electrons. For this reason, some ‘reasonable’ approximation is made by truncating the expansion for the wave operator. The most commonly used approximation is the so-called CCSD method in which the operators T μ in the cluster expansion are approximated by their singly and doubly excited components, i.e. (4.88)

T μ = T1μ + T2μ .

To date, a number of the Hilbert space MR CCSD approximation schemes have been published in the literature. They are known by a variety of names. We should mention the linear MR CCSD approximation introduced by Jeziorski and Paldus [102], which is designated MR L - CCSD. This method was first applied to the H4 model system by Paldus et al. [104] in 1989, who exploited a two-dimensional reference space. This approximation neglects all terms which are non-linear in T μ in the calculation of the effective Hamiltonian defined in eqs. (4.81)–(4.83), as well as in the equations for the cluster amplitudes, eq. (4.87). Although such a linear theory is extensive and performs well in the highly quasi-degenerate region, it undergoes singular behaviour in the non-degenerate region. Singularities arise whenever the energies of low-lying excited configurations become close to energies of the reference configurations. It is remarkable, that while the lowest root of the effective Hamiltonian always describes the ground state, the second root successively approximates higher and higher excited states. Furthermore, the MR L - CCSD method fails to produce real energies in some regions near singularities. It is worth noting that such a breakdown does not have an analogue in the linear single-reference theory which may become singular but never yields complex energies. The next step in the development and implementation of the MR CCSD method is to include the quadratic terms and, in general, non-linear terms. Here, we should mention the orthogonally spin-adapted MR CCSD -1, MR CCSD -2 and MR CCSD -3 approximations developed by Paldus et al. [105] and tested for the H4 model system. The first two approximations were designed just for testing purposes in order to better assess the importance of various non-linear terms. All three approximations are extensive. They differ by the presence of quadratic and bi-linear terms in the direct component, as well as in the coupling terms in the equation for cluster amplitudes (4.87). To be more precise, in addition to absolute and linear terms, the MR CCSD -1 method contains the quadratic (T2μ )2 term involved in the direct term; the MR CCSD -2 method contains the quadratic (T2μ )2 term involved in both the direct component, as well as in the coupling terms, and, finally, the MR CCSD -3 method represents a fully quadratic MR CCSD approximation which considers all bi-linear terms. The main conclusions to be drawn from these studies are that the inclusion of quadratic terms eliminates the singular behaviour of the linear MR CCSD approximation, MR L - CCSD, (even at the MR CCSD -1 level) and that the inclusion of bi-linear components usually further improves the results.

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Meissner, Kucharski and others [106, 107] developed the so-called ‘quadratic’ method. These researchers start from eq. (4.85) and retain only the linear and quadratic (bi-linear) terms, in coupling terms whilst using the full expansion of the direct term. The ‘quadratic’ MR CCSD method is extensive. Its application to molecular systems, as well as the corresponding equations for mono-excited and biexcited amplitudes, is described in the work of Meissner, Kucharski, Bartlett and others [106–110]. The quadratic MR CCSD method is very similar to the MR CCSD 3 approximation and one may expect somewhat better results due to the presence of cubic and quartic terms involved in the direct term. Moreover, the spin-orbital formulation also enables us to obtain some open-shell (although spin-contaminated) energies in the special case of a model space spanned by two open-shell Slater determinants that combine into the singlet and triplet. This method was first proposed by Balkov´a and Bartlett [109] and applied to a study of open-shell singlet states of ozone, singlet-triplet separation in methylene, etc., see [110]. Needless to say, all non-linear approximations presented so far employ the full expansion of the effective Hamiltonian (4.81)–(4.85). In spite of great progress achieved in the development of non-linear approximations, some problems still persist. Although the description of the ground state is very accurate in the quasi-degenerate region, and remains satisfactory in the nondegenerate region, the description of excited states is often qualitatively poorer and it is probable that this may also affect the ground state energy through the effective Hamiltonian. Another problem is the fact that non-linear MR CCSD equations possess multiple solutions capable of describing not only the d states of interest, but also various other states which contain a significant contribution from the model space. Some of these other states may not represent physically meaningful eigenstates. The Newton–Raphson iterative procedure may converge to different solutions depending on the starting approximation. The initial approximation of cluster amplitudes becomes a non-trivial task. The choice of initial approximations may be a particular problem when highly accurate CI results are not available. For this reason, a very useful approximation is the linearized version of the second approach represented by eq. (4.87). We designate this as a linear MR BWCCSD (MR L - BWCCSD) method. The effective Hamiltonian (4.81)–(4.85) in this case has the form: MR CCSD

(4.89)

ef f Hνμ = Hμμ δνμ + Φν |HN (μ)(1 + T2μ + T2μ )|Φμ

and the mono-excited and bi-excited cluster amplitudes are given by A(B) A(B) ef f − Hμμ tI(J) (μ) = ΦI(J) |HN (μ)(1 + T2μ + T2μ )|Φμ Hμμ A(B) μ (4.90) ΦI(J) |T1 + T2μ |Φμ Hνμ . − ν=μ

It is apparent that such an approach is not extensive, because the left-hand side is diagrammatically represented by unlinked diagrams which have no counterpart on the right-hand side (a product of a closed vacuum diagram and the cluster amplitude). Nevertheless it may serve as an initial guess when convergence difficulties with the extensive formulations may be expected. This approximation is very similar to

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MR CISD (multi-reference configuration interaction limited to singles and doubles), but is not equivalent, because the so-called semi-internal triples and quadruples are missing (i.e. those configurations from the Q space that are, at most, doubly excited with respect to one reference configuration, but triply and quadruply excited with respect to another one).

4.2.3. Single-root formulation of the multi-reference Brillouin–Wigner coupled-cluster theory We have shown in Section 4.2.2 that the multi-root multi-reference coupled cluster MR CC approach, involving the simultaneous calculation of all d states in correspondence with the model space, may be very demanding computationally. Indeed, the multi-root approach may even be impossible in some cases, because of problems associated with intruder states, or because of the existence of multiple solutions or convergence difficulties. To a lesser extent, there is the problem of the choice of the basis set, in that the description of different electronic states may often have different orbital requirements. The lack of balance in the description of different states may have adverse effects of convergence. A relatively poor description of one state (due, for example, to the omission of significant configurations from the reference space or truncation of the cluster operator) may have a negative effect on the convergence of the description of other states. Because of these difficulties, it is highly desirable to develop a theory which handles one state at a time, while retaining a multi-configurational reference function. Approaches which consider one state at a time are often referred to as ‘one-state’ or ‘state-selective’ or ‘single-root’. They were first proposed in the late 1970s. A paper published by Harris [113] in 1977, entitled Coupled cluster methods for excited states, first introduced the ‘state-selective’ approach. Four papers which were published in 1978 and 1979 advancing the ‘state-selective’ approach: parts 6 and 7 of a series of papers entitled Correlation problems in atomic and molecular systems: part 6 entitled Coupled cluster approach to open-shell systems by Paldus et al. [114] and part 7 with the title Application of the open-shell coupled cluster approach to simˇ ızˇ ek [115], and two papers by ple π-electron model systems by Saute, Paldus and C´ Nakatsuji and Hirao on the Cluster expansion of wavefunction, the first paper [116] having the subtitle Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell theory and the second paper [117] having the subtitle Pseudo-orbital theory based on SAC expansion and its application to spin-density of open-shell systems. The ‘state-selective’ approach to the multi-reference problem was further developed by Banerjee and Simons [118], by Laidig and Bartlett [119], by Hoffmann and Simons [120], by Li and Paldus [121, 122] and by Jeziorski, Paldus and Jankowski [123] who formulated extensive open-shell CC theory, based on the unitary group approach (UGA) formalism. The Rayleigh–Schr¨odinger formulation of a ‘stateselective’ approach to the multi-reference correlation problem has been developed more recently by Mukherjee and his collaborators [124–130] and also by Schaefer and his colleagues [131–133].

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This section is organized as follows: In Section 4.2.3.1, the single-root formulation of multi-reference Brillouin–Wigner perturbation theory is presented in a form suitable for the development of single-root multi-reference Brillouin–Wigner coupled cluster theory. A Hilbert space approach is adopted in Section 4.2.3.2 to obtain the basic formalism of single-root multi-reference Brillouin–Wigner coupled cluster theory. In Section 4.2.3.3, we present the approximation designated single root MR BW CCSD , single-root multi-reference Brillouin–Wigner coupled cluster theory including single- and double-excitations. Linear-scaling corrections in Brillouin–Wigner coupled cluster theory are introduced and discussed in Section 4.2.3.4. 4.2.3.1. Single-root formulation of multi-reference Brillouin–Wigner perturbation theory Here we formulate a single-root version of the multi-reference Brillouin–Wigner perturbation theory BWPT, which can be used to develop a single-root multi-reference Brillouin–Wigner coupled cluster theory and various approximate coupled cluster theories. We shall be interested in one exact state and, for simplicity, we shall take this to be the ground state Ψ0 . In the multi-reference BWPT, we can introduce a state2 Ω 2 acts on states from specific or single-root wave operator which we designate Ω. the model space as follows: (4.91)

2 = 1 + B0 H1 + B0 H1 B0 H1 + · · · Ω

or, equivalently (4.92)

2 2 = 1 + B0 H1 Ω. Ω

In eqs. (4.91) and (4.92), B0 is the Brillouin–Wigner type of propagator |Φq Φq | (4.93) B0 = E0 − Eq q∈Q

in which E0 is the exact energy of the ground state. However, it should be emphasized that eqs. (4.91) and (4.92) are only equivalent when the Brillouin–Wigner perturbation expansion converges. Then, according to eq. (4.54), the state-specific wave 2 has the following property: operator, Ω (4.94)

2 2 P Ψ0 = Ω(PΨ 0 ) = ΩΨ0 .

2 transforms the projection of the ground state wave Equation (4.94) shows that Ω function on to the model space, Ψ0P , back into the exact ground state wave function. In contrast to the multi-root wave operator, Ω, which was introduced in the previ2 is a state-specific wave operator, (see ous section, our single-root wave operator, Ω, 2 eq. (4.57)). Ω does not transform projections of other exact wave functions PΨα on to the model space into the wave functions for the corresponding exact states. To avoid any confusion we shall use a tilde to distinguish the single-root wave operator. 2 on the model space configurations If we denote the ‘action’ of the wave operator Ω 3 as Ψμ i.e.

Brillouin–Wigner Methods for Many-Body Systems (4.95)

157

2 μ, Ψ2μ = ΩΦ

2 is a bijection (a one-to-one mapping). then it is very easy to show that the operator Ω Using the Brillouin–Wigner perturbation expansion, eq. (4.91), gives us (4.96)

Ψ2μ = Φμ + B0 (H1 + H1 B0 H1 + · · · )Φμ ,

which implies that the resulting states Ψ2μ are linearly independent and the operator 2 represents a bijection (it transforms d linear independent states). Since the states Ω 2 is, in general, Ψ2μ are not necessarily orthogonal, the single-root wave operator Ω non-Hermitian (as was the case in the multi-root theory). The only drawback of such 2 and this depends on the conan approach is the existence of the wave operator Ω, vergence of the Brillouin–Wigner perturbation expansion (4.91). In the case of the ground state, it is quite reasonable to assume that the convergence can be achieved in view of the expected larger differences in the denominator factors (E0 − Eq ). But, in general, we are not assured convergence for excited states, when the exact energy Eα of interest may become close to some zero-order energy Eq . In such a case, one can 2 as the inverse start from eq. (4.92) which prescribes an explicit form of the operator Ω of (1 − B0 H1 ); i.e. (4.97)

2 = (1 − B0 H1 )−1 . Ω

The inverse operator may exist even in the case when the corresponding power series in terms of B0 H1 does not converge. In the following, we will, therefore, start from eq. (4.92) instead of eq. (4.91). 2 eff acting Now, we can introduce the state-specific ‘effective’ Hamiltonian, H within the model space, in the same way as in the multi-root theory, i.e. (4.98)

2 eff = PHΩP. 2 H

Using eq. (4.94), we can write (4.99)

2 eff Ψ P = PHΨ0 = E0 Ψ P , H o 0

which implies that the exact energy of the ground state can be obtained as an 2 eff . Likewise, the projection of the exact eigenvalue of the effective Hamiltonian, H ground state wave function onto the model space, Ψ0P , can be found as an eigen2 eff . As for the remainvector of the state-specific effective Hamiltonian operator, H 2 eff , ing eigenvalues and eigenvectors of the state-specific effective Hamiltonian, H they are uniquely determined by definition of the single-root wave operator (4.91), even though they do not represent solutions which have any physical meaning. As 2 eff , is a nonin the multi-root approach, the state-specific effective Hamiltonian, H Hermitian operator because of the non-Hermicity of the state-specific wave operator, 2 We should emphasize that in both ‘multi-root’ and the ‘single-root’ approaches, Ω. the effective Hamiltonian acts within the same d-dimensional reference space. However, whereas in the ‘multi-root’ approach, all of the roots (eigenvalues) of the effective Hamiltonian are physically meaningful, in the ‘single-root’ approach, just one

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root (eigenvalue) has physical meaning. Hence the terms ‘multi-root’ and ‘singleroot’ are appropriate to distinguish these two approaches. 2 in a form suitable for practical calculations, In order to obtain the wave operator, Ω, we project eq. (4.92) onto configurations from the Q and P subspaces: 2 μ = (E0 − Eq )−1 Φq |H1 Ω|Φ 2 μ (4.100) Φq |Ω|Φ which gives us the expression 2 μ = Φq |H1 Ω|Φ 2 μ . (4.101) (E0 − Eq )Φq |Ω|Φ If we use the Schr¨odinger equation for the zero-order Hamiltonian, eq. (4.2), then we obtain the system of coupled equations for μ = 1, 2, . . . , d: 2 μ = Φq |HΩ|Φ 2 μ , (4.102) E0 Φq |Ω|Φ 2 in the single-root case. The splitting of the which determine the wave operator, Ω, exact Hamiltonian, H, is now eliminated (which can be advantageous from the com2 still putational point of view in coupled cluster theory), but the wave operator, Ω, depends on the exact energy. 2 can We conclude that in the multi-reference BWPT a single-root wave operator, Ω, 2 eff , which be constructed together with the corresponding effective Hamiltonian, H determine the exact wave function, as well as energy for the one state of interest using a multi-configurational reference function. Secondly, we have derived an analogue of the Bloch equation for the single-root wave operator, eq. (4.91), but, in contrast to the multi-root theory, this equation is dependent on the exact energy of the state of interest. It must, therefore, be solved simultaneously with the eigenvalue problem for 2 eff . the effective Hamiltonian, H 4.2.3.2. Single-root multi-reference Brillouin–Wigner coupled cluster theory: Hilbert space approach 2 in the Brillouin–Wigner In Section 4.2.3.1, we have defined the wave operator, Ω, 2 we can form (4.92). If we adopt an exponential ansatz for the wave operator, Ω, develop the single-root (state-specific) multi-reference Brillouin–Wigner coupledcluster (MR BWCC) theory. This is the purpose of the present section. The simplest way to realize an exponential expansion is to employ the exponential ans¨atz of Jeziorski and Monkhorst [87] which exploits a complete model space. This is the approach that we followed in Section 4.2.2.2 in developing a multi-root multireference Brillouin–Wigner coupled cluster theory. The Jeziorski and Monkhorst exponential ansatz may be written μ 2= (4.103) Ω eT |Φμ Φμ | μ∈P

where the summation runs over all functions in the model space. If we employ the exact Hamiltonian in the normal-ordered-product form (4.81)

H = Φμ |H|Φμ + HN (μ) = Hμμ + HN (μ),

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with the μ − th configuration being a Fermi vacuum, then the basic equation for the single-root wave operator (4.102) takes the form 2 μ = Φq |HN (μ)Ω|Φ 2 μ . (4.104) (E0 − Hμμ )Φq |Ω|Φ If we now substitute the Hilbert space exponential ansatz of Jeziorski and Monkhorst, 2 then we obtain the following system of expression (4.103), for the wave operator, Ω, equations μ

μ

(4.105) (E0 − Hμμ )Φq |eT |Φμ = Φq |HN (μ)eT |Φμ . These equations can be used to calculate the cluster amplitudes in the single-root MR BWCC theory. These equations do not mix the various sets of cluster amplitudes, (i.e. amplitudes belonging to various reference configurations). Coupling among them is provided indirectly through the exact energy E0 ; which means that they are coupled through the off-diagonal matrix elements of the effective Hamiltonian. This fact dramatically simplifies the theory as a whole, because, in contrast to the situation for the multi-root approach, there is no necessity to calculate the coupling terms. Furthermore, the structure of matrix elements in eq. (4.105) is precisely the same as those which occur in the non-degenerate coupled cluster theory (the only distinguishing feature being that the cluster amplitudes corresponding to internal excitations are equal to zero). We turn now to the calculation of the effective Hamiltonian (4.98) for single-root multi-reference Brillouin–Wigner coupled cluster theory. Using the Hilbert space exponential ansatz of Jeziorski and Monkhorst, expression (4.103), the off-diagonal elements can be expressed in the form 2 eff = Φν |HN (μ)|eT μ |Φμ , (4.106) H νμ whilst the diagonal elements are given by 2 eff = Hμμ + Φμ |HN (μ)eT μ |Φμ = Hμμ + HN (μ)eT μ μ . (4.107) H μμ Obviously, we obtain precisely the same matrix elements as in the multi-root theory (see, eqs. (4.82)–(4.82) and also the work of Meissner et al. [106] and Kucharski and Bartlett [107]), because in the present single-root strategy we postulated the same expression for the effective Hamiltonian as was employed in the multi-root theory. Like those in the multi-root theory, all matrix elements of the state-specific effective 2 eff , can be written in terms of algebraic expressions which can be Hamiltonian, H represented by connected diagrams only. 4.2.3.3. Single-root multi-reference Brillouin–Wigner coupled cluster single- and double-excitations approximation In Section 4.2.3.2, we presented the basic equations of single-root (state-specific) multi-reference Brillouin–Wigner coupled cluster theory. We derived these equations from the single-root (state-specific) multi-reference Brillouin–Wigner perturbation theory presented in Section 4.2.3.1. In this section, we turn our attention to the coupled cluster single- and double-excitations approximation, CCSD. We present

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the theoretical apparatus of the single-root (state-specific) multi-reference Brillouin– Wigner coupled cluster approximation designated MR BWCCSD. Specifically, we derive the basic equations for the mono-excited and bi-excited cluster amplitudes in a spin-orbital form at the CCSD level of approximation, i.e. with the cluster operators T μ being approximated by their singly and doubly excited cluster components (4.88). To make progress, let us assume that the q-th configuration is a mono-excited configuration with respect to the μ-th configuration, i.e. (4.108) Φq (μ) = ΦA I (μ). In this case, eq. (4.105) reduces to μ

A T (4.109) (E0 − Hμμ )tA |Φμ I (μ) = ΦI (μ)|HN (μ)e

where we used the well-known fact that μ

T (4.110) ΦA |Φμ = tA I (μ)|e I (μ).

According to the work of M´asˇik and Hubaˇc [9] and of Paldus et al. [134], the direct term on the RHS of eq. (4.109) can be split into connected (subscript C) and disconnected parts as follows: μ

μ

μ

T T T |Φμ = ΦA |Φμ C + tA μ . ΦA I (μ)|HN (μ)e I (μ)|HN (μ)e I (μ)HN (μ)e (4.111)

If we take into account the expression for diagonal elements of the effective Hamiltonian given in eq. (4.107), then we get a system of non-linear equations of the form 2 eff tA (μ) = ΦA (μ)|HN (μ)eT μ |Φμ C . (4.112) E0 − H μμ I I This equation can be used to calculate the singly excited cluster amplitudes within the single-root MR BWCCSD approximation. Turning to the case of the bi-excited amplitudes, we can proceed in a similar fashion. Taking the q-th configuration to be a bi-excited configuration with respect to μ-th configuration, i.e. (4.113) Φq (μ) = ΦAB IJ (μ), equation (4.105) can be written in the form A B A B AB Tμ (4.114) (E0 − Hμμ ) tAB |Φμ , IJ + tI tJ − tJ tI μ = ΦIJ (μ)|HN (μ)e where we omit the subscript μ at individual amplitudes for the sake of simplicity. In this equation, we have used the well-known fact that Tμ A B A B (4.115) ΦAB |Φμ = tAB IJ (μ)|e IJ + tI tJ − tJ tI μ . Then, according to the work of M´asˇik and Hubaˇc [10] and of Li and Paldus [122], the direct term on the RHS of eq. (4.114) can be split into connected and disconnected parts in the following way:

Brillouin–Wigner Methods for Many-Body Systems μ

161 μ

T T ΦAB |Φμ = ΦAB |Φμ C IJ (μ)|HN (μ)e IJ (μ)|HN (μ)e AB μ A B A B + tIJ + tI tJ − tJ tI μ HN (μ)eT μ B Tμ (4.116) , + PIJ PAB tA Φ (μ)|H (μ)e |Φ N μ C I J

where PIJ and PAB are antisymmetrizers of the form (4.117) PIJ = 1 − P (I ↔ J). If we take account of expression (4.107) for diagonal elements of the effective Hamiltonian, then we obtain the following system of non-linear equations 2 ef f tAB + tA tB − tA tB = ΦAB (μ)|HN (μ)eT μ |Φμ C E0 − H μμ IJ I J J I μ IJ B Tμ (4.118) + PIJ PAB tA |Φμ C , I ΦJ (μ)|HN (μ)e which can be used for the calculation of doubly excited cluster amplitudes within the single-root MR BWCCSD approximation. No additional calculation is required for Tμ |Φμ C . The configuration ΦB the matrix element ΦB J (μ)|HN (μ)e J (μ) either represents an external excitation, in which case the matrix element can be replaced by eq. (4.112), or an internal excitation which implies that the matrix element is an offdiagonal matrix element of the effective Hamiltonian (4.106). In the special case of a two-determinant model space corresponding to two active orbitals of different symmetry, the internal mono-excitations cannot contribute for symmetry reasons, so one can exploit eq. (4.112) even in the case of internal excitations, which leads to a system of equations: 2 ef f tAB (μ) = ΦAB (μ)|HN (μ)eT μ |Φμ C E0 − H μμ IJ IJ 2 ef f tA tB − tA tB . (4.119) + E0 − H μμ I J J I μ Equations (4.112) and (4.118) can be used for the calculation of mono-excited and bi-excited cluster amplitudes in the single-root MR BWCCSD approximation in a general case. In the case of two-determinant model space, eq. (4.118) for the bi-excited cluster amplitudes can be replaced by its simpler, two-determinant version (4.119). As we have already stated, these equations do not mix various sets of cluster amplitudes and the coupling among amplitudes is provided through the off-diagonal elements of the effective Hamiltonian (if all of the off-diagonal elements were zero then the coupling would vanish). On the other hand, the equations explicitly depend on the exact energy of interest and they must be solved simultaneously with the eigenvalue problem for the effective Hamiltonian. Hence, the set of eqs. (4.112) and (4.118) must be solved iteratively and after each iteration we compute the effective Hamiltonian (4.98)–(4.99). Diagonalization of the effective Hamiltonian yields several eigenvalues and the lowest of these is taken as a new exact energy, E0 , for use in the next iteration. The remaining eigenvalues are not physically meaningful and are therefore discarded. This completes our presentation of the formalism of the single-root Hilbert space MR BWCCSD approximation.

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Let us now consider the convergence properties of the single-root Hilbert space approximation that we have presented above. It can be seen that the denominators appearing in eqs. (4.112) and (4.118) above will always be negative, so when calculating the ground state energy, no problems should arise with vanishing denominators or intruder states. Furthermore, since the cluster amplitudes are coupled through the off-diagonal elements of the state-specific effective Hamiltonian, 2 eff , we may expect good convergence over the whole potential energy curve of the H νμ system studied, because the off-diagonal elements enable us to ‘switch’ in a continuous manner between the single-reference and multi-reference approaches. In the non-degenerate case, for which the coupling between the reference configurations is 2 eff , are negligible (in weak, the off-diagonal elements of the effective Hamiltonian, H νμ fact, in this case the off-diagonal elements would correspond to amplitudes of internal excitations divided by pertinent denominators). This implies that the difference 2 eff ) approaches zero for the ground state configuration, Φμ , and eqs. (4.112) (E0 − H μμ and (4.118) reduce to standard non-degenerate CCSD equations for the ground state. In the quasi-degenerate case, for which the coupling between the reference configurations is becoming large, the coupling between eqs. (4.112) and (4.118) becomes stronger. These equations then correspond to a multi-reference case. From the above discussion we may expect that MR BWCCSD approximation will be well suited to the calculation of the ground state energy. As far as calculations for excited states are concerned, because of the unknown behaviour of the denominator factors, the possibility of convergence problems cannot be excluded. Moreover, for excited states, the problems of choosing the appropriate eigenvalue of the effective Hamiltonian from several eigenvalues may not be an unambiguous matter as it is in the case of the ground state. MR BWCCSD

4.2.3.4. Linear scaling corrections in Brillouin–Wigner coupled cluster theory In Section 4.2.1, it was shown that, for example, the BWCCD approximation is fully equivalent to the ‘standard’ (Rayleigh–Schr¨odinger) CCD approximation, which is based on the linked cluster theorem. Similar arguments have been advanced to demonstrate the equivalency of BWCCSD and CCSD theories [9]. The extension of these arguments to other coupled cluster approximations, such as BWCCSDT and CCSDT theories, is straightforward. The single-reference Brillouin–Wigner coupled cluster theory and the various approximations which can be based on it ( CCD, CCSD, CCSDT and so on) leads to energies which scale linearly with the number of particles in the system under investigation. They are extensive methods. The application of the Brillouin–Wigner coupled cluster theory to the multireference function electron correlation problem yields two distinct approaches: (i) the multi-root formalism which was discussed in Section 4.2.2 and (ii) the single-root formalism described in the previous subsections of this section, Section 4.2.3. The multiroot multi-reference Brillouin–Wigner coupled cluster formalism reveals insights into other formulations of the multi-reference coupled cluster problem which often suffer from the intruder state problem which, and in practice, may lead to spurious

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singularities in potential energy curves and surfaces. The single root or state-specific multi-reference Brillouin–Wigner coupled cluster theory avoids the problem of intruder states. However, the state-specific multi-reference Brillouin–Wigner coupled cluster formalism does not lead directly to energies which scale linearly with the number of particles in the system. The single-reference BW- CC formalism is not extensive. We have critically re-examined Brillouin–Wigner perturbation theory and its use in describing many-body systems [18], i.e. the use of Brillouin–Wigner formalism in developing expressions for energies and other expectation values which scale linearly with the number of particles, N , in the system studied. At any order of the Brillouin– Wigner perturbation expansion for the energy, the energy coefficients can be divided into a part which scales linearly with N and a part which does not. Writing the exact energy, E, as (4.120) E = E0 + ΔE, where E0 is the ground state energy eigenvalue of the Hamiltonian H0 and ΔE is the level shift, we can expand the denominator factors which arise in the Brillouin– Wigner perturbation expansion as −1

(4.121) (E − Ek )

= (E0 − Ek )

−1

−1

+ (E0 − Ek )

−1

(−ΔE) (E − Ek )

.

Now, if we re-write this identity in the form −1

(4.122) (E − Ek )

−1

+ (E0 − Ek )

(ΔE) (E − Ek )

−1

−1

= (E0 − Ek )

,

we see that the first term on the left-hand side of this equation is the Brillouin–Wigner denominator factor containing the exact energy, E. The term on the right-hand side is the denominator which arises in the Rayleigh–Schr¨odinger formalism. We know that the Rayleigh–Schr¨odinger perturbation theory series leads directly to the many-body perturbation theory by employing the linked diagram theorem. This theory uses factors of the form (E0 −Ek )−1 as denominators. Furthermore, this theory is fully extensive; it scales linearly with electron number. The second term on the left-hand side of eq. (1.76) can be taken as an ‘extensivity correction term’ for the Brillouin–Wigner series; a correction term which recovers the Rayleigh–Schr¨odinger and many-body perturbation theoretic formulations. This simple idea has been used to find a posteriori corrections for Brillouin–Wigner many-body formalisms [18]. In this section, we are concerned with the use of a posteriori corrections for the state-specific multireference Brillouin–Wigner coupled cluster theory [19, 20]. We recall that the state-specific Brillouin–Wigner analogue of the Bloch equation has the form: 2 = (1 − B0 H1 )−1 . (4.97) Ω For the ground state, this takes the form (4.123) Ω0 = (1 − B0 H1 )−1 , where we have dropped the tilde to simplify the notation. By substituting eq. (4.121) into the Bloch equation (4.92), we obtain the following expression for the wave operator:

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(4.124) Ω0 = 1 +

|Φk Φk | −ΔE |Φk Φk | H1 Ω0 . H1 Ω0 + E0 − Ek (E0 − Ek ) (E0 − Ek ) k>1

k>1

The first two terms on the right-hand side of this equation scale linearly with particle number. They are a component of a true many-body theory. The first two terms on the right-hand side of eq. (4.124) may be viewed as the analogue of the Rayleigh– Schrodinger Bloch equation when it is cast in Brillouin–Wigner form. The third term in eq. (4.124) gives, on iterating the state-specific multi-reference BW CCSD equations, terms which do not scale linearly with particle number. Calculations carried out using the state-specific multi-reference BW CCSD theory may therefore be corrected a posteriori by dropping these ‘unphysical’ terms in the converged amplitudes. This is achieved by first iterating the amplitudes with the Bloch equation, i.e. (4.123) |Φk Φk | (4.125) Ω0 = 1 + H1 Ω0 , E0 − Ek k>1

and then including an additional term on the right-hand side in the final iteration, i.e. |Φk Φk | ΔE |Φk Φk | H1 Ω0 . (4.126) Ω0 = 1 + H1 Ω0 + E0 − Ek (E0 − Ek ) (E0 − Ek ) k>1

k>1

From the corrected amplitudes, the effective Hamiltonian matrix can be constructed which, upon diagonalization, gives the final energy. In Brillouin–Wigner coupled cluster theory, the simple a posteriori correction described above is exact in the case of the single-reference formalism. In the statespecific multi-reference Brillouin–Wigner coupled cluster theory, the simple a posteriori correction is approximate. An iterative correction for lack of extensivity has been studied by Pittner [38], but this reintroduces the intruder state problem. 4.3. BRILLOUIN–WIGNER CONFIGURATION INTERACTION THEORY The method of configuration interaction, otherwise known as ‘configuration mixing’ or ‘superposition of configurations’, is certainly the most easily understood approach to the non-relativistic electron correlation problem in atoms and molecules [5]. The configuration interaction method has a long history (see, for example, [5, 135–138] in atomic and molecular electronic structure theory). It provides a robust approximation which can be applied in situations where the more modern ‘many-body’ methods, such as diagrammatic many-body perturbation theory and coupled cluster expansions may give rise to difficulties. Indeed, configuration interaction is used to provide standards against which other correlation treatments are assessed (see, for example, reference [139]). In the method of configuration interaction, the total atomic or molecular wave function, Ψ , is written as a linear combination of some known N -electron determinantal functions, Φi , (4.127) Ψ = Φi Ci , i

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where the expansion coefficients, Ci , are to be determined by invoking the variation principle. This requirement leads to a secular problem of the form (4.128) HCi = Ci where H is the hamiltonian matrix with elements given by (4.129) [H]i,j = Φi |H| Φj , in which the non-relativistic electronic hamiltonian, H, is a semi-bounded, selfadjoint operator in Hilbert space h. The N -electron determinantal functions, Φi , are Slater determinants constructed from a set of M spin orbitals. These orbitals are approximated by invoking the algebraic approximation, i.e. they are written as a linear combination of some chosen basis functions. Most often, the basis functions are taken to be Gaussian-type functions [140]. The algebraic approximation results in the domain of the operator being restricted to a finite-dimensional subspace S of Hilbert space. For an N -electron system, the algebraic approximation may be implemented by defining a suitable orthonormal basis set of M (> N ) one-electron spin orbitals (most often solutions of the matrix Hartree–Fock equations) and then constructing all unique N -electron determinants. The number of unique determinants that can be formed is given by M (4.130) η = . N η is the dimension of the subspace S and the algebraic approximation restricts the domain of the hamiltonian to this η-dimensional subspace. Solution of the eigenproblem (4.128) in a given basis set constitutes the full configuration interaction problem. Solution of the eigenproblem (4.128) in a complete basis set constitutes the complete configuration interaction problem, which the exact solution of the Schr¨odinger equation. Both full configuration interaction and complete configuration interaction are true many-body theories. In practice, difficulties arise in setting up and solving secular equations of high order, except in the case of small basis sets. For most adequate basis sets, only a small subset of the determinantal functions Φi can be used in the expansion. It is then usual to write the expansion (4.127) in the form (4.131) Ψ = Φi Ci + Φi Ci + Φi Ci . i∈P

i∈Q

i∈R

where P defines the set of reference configurations with respect to which the set of configurations Q is generated by replacement of orbitals which are occupied in the set P. In the limited configuration interaction method the eigenproblem (4.128) is solved in the space P ⊕ QR and the set consists of those configurations which are neglected. In describing the historical development of methods for describing electron correlation in molecules, Paldus writes [141] “ The initial hope that the configuration interaction approach limited to doubly excited configuration, originating from a single-reference state, will provide a satisfactory description of correlation effects soon had to be given up”.

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Limited configuration interaction, when developed with respect to a singlereference function, is particularly widely used in its single and double excitation form, designated CI SD. The so-called ‘extensivity’ problem [5] with the limited, single and double excitation configuration interaction expansion with respect to a single determinantal reference function is well documented [5]. The popularity of the CI SD method has fostered the development of corrections which attempt to minimize the error arising from terms which scale non-linearly with the number of electrons in the system. The first use of such a posteriori corrections was made by Davidson [142] and by Langhoff and Davidson [143] in 1974. However, the first analysis of the problem was given by Brueckner [58, 144] in 1955 using perturbation theoretic arguments. As is well known, Brueckner traced the problem to terms which scale non-linearly with the number of electrons in the system. These considerations led to the development of the many-body perturbation theory. Analysis of the CI SD problem in terms of the many-body perturbation theory shows that configuration interaction calculations which involve only double excitations have truncation errors of fourth-order. The correction obtained by Davidson is now often termed the renormalized Davidson correction [145–150]. The Davidson correction and its variations [145–149] attempt to estimate the contribution of the quadruple excitations that are required to cancel fourth-order ‘unlinked diagram’ components. The Davidson correction takes no account of the linked energy diagrams containing triply excited intermediate states which also arise in fourth-order [148,149]. Other forms of correction were derived from consideration of a system consisting of non-interacting identical subsystems [151–154]. A useful summary of the various a posteriori corrections that have appeared in the literature can be found in the work of Meissner [150]. Multi-Reference configuration interaction is robust and thus, for example, Meissner et al. wrote [155]: The multi-reference configuration interaction (MRCI) method with single and doubles (MR - CISD) is one of the few quantum chemical methods which are used in routine calculations for systems requiring a multi-reference description. The main reason for that is its formal and computational simplicity and resistance to the intruder-state problem which frequently occurs in other multi-reference-type calculations. However, these authors also caution An important drawback of the MR - CISD scheme is, however, a relatively poor description of the dynamic electron correlation provided by the linear expansion. This is a fundamental difficulty with the MR - CISD formalism. A generalization of the Davidson correction to the multi-reference case was given by Buenker and his co-workers [156,157] but, as Shavitt [158] has pointed out, “without formal justification”. Jankowski, Meissner and Wasilewski [159] have proposed a more systematic generalization of the Davidson correction for multi-reference configuration interaction.

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In this section, we present the Brillouin–Wigner configuration interaction theory. In Section 4.3.1, we consider the single-reference Brillouin–Wigner configuration interaction theory. An analysis of the single-reference limited configuration interaction method CISD using Brillouin–Wigner perturbation theory and a Lippmann– Schwinger-like equation [160] leads to a posteriori corrections for the lack of extensivity of the method. Multi-Reference configuration interaction theory is considered in Section 4.3.2. Again, by using Brillouin–Wigner methodology an a posteriori correction is derived for the lack of extensivity of the method. 4.3.1. Single-reference Brillouin–Wigner configuration interaction theory In this section, we use Brillouin–Wigner perturbation theory, and, in particular, an infinite order summation employing a Lippmann–Schwinger-like equation [160], to investigate a posteriori corrections to the limited configuration interaction method. For simplicity, we restrict our attention to limited configuration interaction developed with respect to a single-reference function. However, our approach is general and may be developed for any limited configuration interaction method, as will be shown in Section 4.3.2. This section is divided into three subsections. In the first of these, Section 4.3.1.1, we use Brillouin–Wigner perturbation theory to analyse the single-reference, singleand double-excitation configuration interaction method, CI SD. Section 4.3.1.2 is a digression in which we compare the Rayleigh–Schrodinger formulation of the limited configuration interaction problem with the Brillouin–Wigner formalism. In Section 4.3.1.3, we describe ‘many-body’ corrections for the Brillouin–Wigner limited configuration interaction. 4.3.1.1. Brillouin–Wigner perturbation theory and limited configuration interaction Let us write the exact Schr¨odinger equation as (4.132) H |Ψi = Ei |Ψi , where H is the hamiltonian, Ei is the exact energy of the i-th state and Ψi is the corresponding wave function. Let us separate the hamiltonian, H, into two parts as follows: (4.133) H = H0 + H1 , where H0 is the unperturbed part and H1 is the perturbation. Furthermore, we assume that we know the solutions of the unperturbed Schr¨odinger equation which has the form: (4.134) H0 |Φi = Ei |Φi . The lowest (ground state) non-degenerate energy is given by the expression [8, 17, 161] E0 = Φ0 |H0 | Φ0 (4.135)

+ Φ0 |H1 + H1 B0 H1 + H1 B0 H1 B0 H1 + · · · | Φ0 ,

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where B0 is the Brillouin–Wigner resolvent |Φi Φi | Q0 (4.136) B0 = = . E0 − H0 E0 − H0 i=0

Now let us introduce two operators, namely, the reaction operator (4.137) VB = H1 + H1 B0 H1 + H1 B0 H1 B0 H1 + · · · and the wave operator (4.138) Ψ0 = Ω0 Φ0 , where (4.139) Ω0 = 1 + B0 H1 + B0 H1 B0 H1 + · · · . Expressions (4.137) and (4.139) may be re-written in the forms (4.140) VB = H1 + H1 B0 VB and (4.141) Ω0 = 1 + B0 H1 Ω0 , respectively. Equation (4.140) is a Lippmann–Schwinger-like equation [160], which is known from scattering theory, and eq. (4.141) is the Bloch equation in Brillouin– Wigner form. Using eqs. (4.140) and (4.141), we can write two expressions for the energy (4.142) E0 = Φ0 |H0 | Φ0 + Φ0 |VB | Φ0 and (4.143) E0 = Φ0 |H0 | Φ0 + Φ0 |H1 Ω0 | Φ0 . Expression (4.143) was used by Hubaˇc and Neogr´ady [6, 8] in their development of Brillouin–Wigner coupled cluster theory. It is important to recognize that expression (4.143) represents an infinite order Brillouin–Wigner perturbation theory and unlike finite order Brillouin–Wigner perturbative expansions, this does not give rise to an unphysical terms. We shall now investigate eq. (4.142). This is also an infinite order scheme. Using eqs. (4.140) and (4.136), we can write eq. (4.142) in the form [8]: (4.144) E0 = Φ0 |H0 | Φ0 + Φ0 |H1 + H1 B0 VB | Φ0 or E0 = Φ0 |H| Φ0 Φ0 |H1 | Φi Φi |VB | Φ0 (4.145) . + E0 − Ei i=0

Since H1 is a two-particle operator, the configuration |Φi involves, at most, a double replacement with respect to |Φ0 . Thus, given the matrix elements Φi |VB | Φ0 ,

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we could then calculate the exact energy, E0 . The required matrix elements can be obtained by using (4.140) Φi |VB | Φ0 = Φi |H1 | Φ0 Φi |H1 | Φj Φj |VB | Φ0 (4.146) . + E0 − Ej j=0

Now we can see that |Φi involves, at most, quadruple replacements with respect to the reference configuration, |Φ0 . Repeated application of eq. (4.140) leads to higher order replacements in the configuration |Φi . If we restrict our attention to double replacements in eq. (4.146), then eq. (4.145) and (4.146) provide a computational scheme which realizes the limited configuration interaction method in its CI SD form [7, 8]. 4.3.1.2. Rayleigh–Schr¨odinger perturbation theory and limited configuration interaction Let us now turn our attention to the Rayleigh–Schr¨odinger perturbation theory for the problem defined by eq. (4.132), (4.133) and (4.134). The expansion for the exact ground state energy given by Rayleigh–Schr¨odinger perturbation theory may be written E0 = Φ0 |H0 | Φ0 + Φ0 |H1 | Φ0 + Φ0 |H1 R0 H1 | Φ0 + Φ0 |H1 R0 H1 R0 H1 | Φ0 − Φ0 |H1 | Φ0 Φ0 |H1 R0 H1 | Φ0 + Φ0 |H1 R0 H1 R0 H1 R0 H1 | Φ0

− Φ0 |H1 R0 H1 | Φ0 Φ0 H1 R20 H1 Φ0 + · · ·

(4.147) where (4.148) R0 =

|Φi Φi | i=0

Ei − E0

is the Rayleigh–Schr¨odinger resolvent. It can be seen that in addition to the direct terms, there are also renormalization terms. These terms prevent the introduction of a reaction-like operator directly into eq. (4.147). But we know that Rayleigh– Schr¨odinger perturbation theory and/or many-body perturbation theory in its diagrammatic formulation encompasses the linked diagram theorem. This ensures that only connected diagrams are included in the expansion (4.147), which in turn guarantees that the theory scales linearly with the number of electrons in the system. However, expansion (4.147) may be re-written in the form: E0 = Φ0 |H0 | Φ0 + Φ0 |H1 + H1 R0 H1 (4.149)

H1 R0 H1 R0 H1 + H1 R0 H1 R0 H1 R0 H1 + · · · | Φ0 linked diagrams

where the subscript ‘linked diagrams’ ensures that only terms corresponding to linked diagrams are included in the expansion. In a similar fashion, the reaction operator can be written as

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(4.150) VR = {H1 + H1 R0 H1 + H1 R0 H1 R0 H1 + · · · }linked diagrams or (4.151) VR = {H1 + H1 R0 VR }linked diagrams . Using eq. (4.151), we can write eq. (4.149) in the form: (4.152) E0 = Φ0 |H0 | Φ0 + Φ0 |H1 + H1 R0 VR | Φ0 linked diagrams or E0 = Φ0 |H| Φ0 (4.153)

Φ0 |H1 | Φi Φi |VR | Φ0 + E0 − Ei i=0

. linked diagrams

Recalling our discussion of eq. (4.145), we note that, since H1 is a two-particle operator, the configuration |Φi involves, at most, a double replacement with respect to |Φ0 . The matrix elements Φi |VR | Φ0 can be calculated using eq. (4.151) Φi |VR | Φ0 = Φi |H1 | Φ0 (4.154)

Φi |H1 | Φj Φj |VR | Φ0 + E0 − Ej j=0

.

linked diagrams

If we limit our attention to the case in which |Φi involves double replacements, then by solving (4.153) and (4.154), we obtain the CI SD method. To obtain a many-body CI SD method we should solve eqs. (4.153) and (4.154) diagrammatically and include only connected diagrammatic terms. 4.3.1.3. ‘Many-body’ corrections for Brillouin–Wigner limited configuration interaction An alternative approach, which we shall follow in this work, is to solve eqs. (4.145) and (4.146), which define the much simpler Brillouin–Wigner scheme, and introduce a correction term for the non-linear effects. The following identity relates the Brillouin–Wigner and the Rayleigh–Schr¨odinger resolvent denominators [17, 18, 74] (4.155) (E0 − Ek )−1 = (E0 − Ek )−1 − (E0 − Ek )−1 ΔE (E0 − Ek )−1 where (4.156) E0 = E0 + ΔE0 . The second term on the right-hand-side of eq. (4.155) can be regarded as a correction term. Equations (4.155) and (4.156) can be used to correct eqs. (4.145) and (4.146) for the unphysical components that they contain. Using (4.155), the correction terms may be written:

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∞ Φ| H1 |Φk Φk | VB |Φ E − Ek k=0 ⎧ ⎫ ∞ ⎨ E − E0 Φ| H1 |Φk Φk | VB |Φ ⎬ + ⎩ ⎭ E − Ek E0 − Ek

E = Φ |H0 | Φ + Φ |H1 | Φ +

(4.157)

k=0

and Φk |H1 | Φ = Φk |H1 | Φ +

∞ Φk |H1 | Φp Φp |VB | Φ E − Ep p=0

(4.158)

⎧ ⎫ ∞ ⎨ E − E0 Φk |H1 | Φp Φp |VB | Φ ⎬ + , ∀k. ⎩ ⎭ E − Ek (E0 − Ep ) p=0

where the correction terms given in brackets {. . .} are only added in the final iteration. Equations (4.157) and (4.158) constitute the main results of the present section. 4.3.2. Multi-reference Brillouin–Wigner configuration interaction theory In this section, we turn our attention to the multi-reference Brillouin–Wigner configuration interaction theory. We use Brillouin–Wigner perturbation theory, and, in particular, an infinite order summation employing a Lippmann–Schwinger-like equation [160], to investigate a posteriori corrections to the limited multi-reference configuration interaction method. In Section 4.3.2.1, we briefly survey the basic formalism of the Brillouin–Wigner perturbation theory [2–4, 18, 25] in its multireference form. This will serve both to provide the necessary background and to introduce our notation. We then, in Section 4.3.2.2, present an explicit formulation of the multi-reference Brillouin–Wigner perturbation theory for a p-state system. We use this formalism in Section 4.3.2.3 to present the multi-reference configuration interaction problem in Brillouin–Wigner form. Finally, in Section 4.3.2.4 we consider the introduction of a posteriori corrections in multi-reference Brillouin–Wigner configuration interaction. 4.3.2.1. Multi-reference Brillouin–Wigner perturbation theory for limited configuration interaction We wish to approximate the solutions of the time-independent Schr¨odinger equation (4.159) HΨα = Eα Ψα with (4.160) H = H0 + H1 , where H0 is the zero-order hamiltonian and H1 is the perturbation. We assume that the solutions of the zero-order eigenproblem (4.161) H0 Φμ = Eμ Φμ ,

μ = 0, 1, 2, . . . .

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are known. Let (4.162) {Φμ ; μ = 0, 1, 2, . . . , p − 1} be a set of linearly independent functions which constitute the reference space, which we label P. Let P be the projection operator onto this reference space (4.163) P = |Φμ Φμ | μ∈P

and let Q be its orthogonal complement Q=I −P = (4.164) |Φμ Φμ | . μ∈P /

The projectors P and Q satisfy the relations P2 = P Q2 = Q (4.165) P Q = 0. We can write the subspace S as (4.166) S = P ⊕ Q. Let us consider the projection of the exact wave function, Ψα , onto the reference space P, i.e. (4.167) ΨαP = P Ψα , α = 0, 1, 2, . . . , p − 1. ΨαP is sometimes called the model function. Obviously, ΨαP can be written as a linear combination of the set (4.162) (4.168) ΨαP = Φμ Cμα . μ∈P

ΨαP

are, in general, non-orthogonal but are assumed to be linearly independent. The The exact wavefunction, Ψα , is expanded as follows: (4.169) Ψα = (1 + Bα H1 + Bα H1 Bα H1 + · · · ) ΨαP , where Bα is the Brillouin–Wigner type propagator |Φi Φi | (4.170) Bα = . Eα − Ei i∈P /

which depends on the unknown exact energy eigenvalue Eα . The exact wavefunction, Ψα , and the model function, ΨαP , satisfy the following intermediate normalization conditions:

P Ψα | Ψ α = 1

(4.171) ΨαP | ΨαP = 1

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The Brillouin–Wigner wave operator, Ωα , is defined by (4.172) Ψα = Ωα ΨαP so that application of the wave operator to the model function yields the exact wave function. Comparing eq. (4.172) with eq. (4.169), we have (4.173) Ωα = 1 + Bα H1 + Bα H1 Bα H1 + · · · or, re-writing (4.173) as a recursion, (4.174) Ωα = 1 + Bα H1 Ωα , which may be seen to be the Bloch-like equation [39] in Brillouin–Wigner form. We now introduce the ‘effective’ hamiltonian which acts in the reference space P (4.175) Hα = P HΩα P, but has the exact energy, Eα , as an eigenvalue, i.e. (4.176) Hα ΨαP = Eα ΨαP . Using (4.160), we can write the effective hamiltonian operator in the form: (4.177) Hα = P H0 P + P H1 Ωα P, which can then be re-written in the form (4.178) Hα = P H0 P + P Vα P,

α = 0, 1, 2, . . . , p − 1,

where we have introduced the reaction operator, Vα , defined as follows: (4.179) Vα = H1 Ωα Combining the Bloch-like equation (4.174) and the definition of the reaction operator, eq. (4.179), gives a Lippmann–Schwinger-like equation [160] in Brillouin–Wigner form: (4.180) Vα = H1 + H1 Bα Vα . It should be emphasized that Vα is a state-specific reaction operator corresponding to only one eigenenergy Eα . Note that the wave operator arising in the Rayleigh–Schr¨odinger formalism, Ω, can be related to the wave operators, Ωα , in the Brillouin–Wigner method through the relation (4.181) Ω =

p−1

Ωα Pα .

α=0

The exact energies, Eα , are eigenvalues of different effective hamiltonian operators (4.182) Hα ΨαP = Eα ΨαP , α = 0, 1, 2, . . . , p − 1.

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4.3.2.2. A p-state system Let us consider a p-state system and obtain an explicit formulation of the multireference Brillouin–Wigner perturbation theory for this case. In the p-state case, we have a reference space spanned by p functions, Φ0 , Φ1 , . . . , Φp−1 . The projector onto this space is (4.183) P =

p−1

|Φμ Φμ | .

μ=0

The corresponding wave operators are (4.184) Ω0 , Ω1 , . . . , Ωp−1 and the reaction operators are V0 = H1 Ω0 V1 = H1 Ω1 ... (4.185) Vp−1 = H1 Ωp−1 . The effective Schr¨odinger equation has the form: (4.186) Hα ΨαP = Eα ΨαP , α = 0, 1, . . . , p − 1. The effective hamiltonian operator takes the form: Hα = P HΩα P (4.187)

= P H0 P + P H1 Ωα P = P H0 P + P Vα P,

so that Hα =

! p−1

1 ! p−1 1 |Φμ Φμ | H0 |Φμ Φμ |

μ=0

(4.188)

+

! p−1

μ=0

|Φμ Φμ |}Vα

! p−1

μ=0

1 |Φμ Φμ | .

μ=0

For the state α = 0, we have a model function given by (4.189) Ψ0P =

p−1

Φμ Cμ .

μ=0

The secular equation for the lowest state takes the form: ⎛

Φ0 | H α |Φ0 − E0 ⎜ Φ1 | V 0 |Φ0 ⎜ ⎝...

Φp−1 V 0 |Φ0 (4.190)

Φ0 | V 0 |Φ1 Φ1 | H α |Φ1 − E0 . . . Φp−1 V 0 |Φ1

... ... ... ...

⎞⎛ ⎞ C0 Φ0 | V 0 Φp−1 ⎟ ⎜ C1 ⎟ Φ1 | V 0 Φp−1 ⎟⎜ ⎟ ⎠ ⎝ . . . ⎠ = 0, . . . Cp−1 Φp−1 H α Φp−1 − E0

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where the matrix elements of the effective hamiltonian are Φμ | H1 |Φj Φj | V0 |Φν (4.191) Φμ | V0 |Φν = Φμ | H1 |Φ0 + E0 − Ej j ∈P /

and (4.192) Φμ | Hα |Φμ = Φμ | H0 |Φμ + Φμ | V0 |Φμ , with μ, ν = 0, 1, . . . , p − 1. Since H1 is a two-particle operator, the configuration |Φj is, at most, a double replacement with respect to |Φμ in (4.191). The matrix elements of the reaction operator are obtained from the Lippmann–Schwinger-like equation [160], eq. (4.180). Specifically, we have Φj | H1 |Φk Φk | V0 |Φμ (4.193) Φj | V0 |Φμ = Φj | H1 |Φμ + , ∀j ∈ /. E0 − Ek j ∈P /

4.3.2.3. Multi-reference configuration interaction in Brillouin–Wigner form If |Φj is a determinant related to one of the reference determinants by a double replacement, then |Φk involves, at most, quadruple replacements with respect to |Φμ in eq. (4.193). Repeated application of the Lippmann–Schwinger-like equation [160] leads to higher order replacements. If we restrict the degree of replacement admitted in (4.193) then we realize a limited multi-reference configuration interaction method. It is this realization of the multi-reference limited configuration interaction method that we use to obtain an a posteriori correction based on Brillouin–Wigner perturbation theory. Equation (4.190) has p roots of which we take only one. The exact energy, E0 , occurs in the denominator factors in eqs. (4.191) and (4.193). The eq. (4.190) must, therefore, be solved iteratively until self-consistency is achieved. These are the basic equations of the multi-reference configuration interaction method in a Brillouin– Wigner formulation in the case of a p state reference. If we restrict our attention to single and double replacements, then our equations are entirely equivalent to multi-reference configuration interaction in the truncated form - usually designated ‘MR - CISD’ . We can write: (4.194) S = S1 ⊕ S2 ⊕ . . . ⊕ SN , where Sm is the subspace of S defined by those determinants obtained by m-fold replacements with respect to determinants in S0 . We can then define (4.195) S1+2 = S1 ⊕ S2 , so that the MR - CISD method can be written in Brillouin–Wigner form as Φμ | H1 |Φj Φj | V0 |Φν (4.196) Φμ | V0 |Φν = Φμ | H1 |Φ0 + E0 − Ej j∈S1+2

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and Φj | V0 |Φμ = Φj | H1 |Φμ +

Φj | H1 |Φk Φk | V0 |Φμ , ∀j ∈ S12 . E0 − Ek

k∈S1+2

(4.197) Equations (4.196) and (4.197) are solved together with (4.190) and (4.192) to obtained the MR - CISD model. They are the working equations of the Brillouin–Wigner MR - CISD method. 4.3.2.4. A posteriori Brillouin–Wigner correction to limited multi-reference configuration interaction We are now in a position to develop an a posteriori correction to state-specific limited multi-reference configuration interaction in the case of a p-state reference function. The following identity relates the Brillouin–Wigner and the Rayleigh– Schr¨odinger denominators [18, 76, 77] for the ground state (α = 0): 1 1 1 1 (4.198) = − ΔE (E0 − Ej ) (E0 − Ej ) (E0 − Ej ) (E0 − Ej ) and (4.199) E0 = E0 + ΔE0 , so that ΔE0 is the usual ‘level shift’. Equation (4.198) may be rewritten in the form: 1 1 1 1 (4.200) = + ΔE . (E0 − Ej ) (E0 − Ej ) (E0 − Ej ) (E0 − Ej ) The left-hand-side of (4.200) is the Rayleigh–Schr¨odinger denominator. The first term on the right-hand-side of (4.198) is the Brillouin–Wigner denominator, whilst the second term can be regarded as a correction term. Equations (4.199) and (4.200) can be used to correct the matrix elements of the effective hamiltonian: (4.191)– (4.193) in the case of the lowest state. For the lowest state the corrected matrix elements for the p-state case may be written: Φμ | H1 |Φj Φj | V0 |Φν Φμ | V0 |Φν = Φμ | H1 |Φν + E0 − Ej j∈S12 ⎧ ⎫ ⎨ ΔE Φ | H |Φ Φ | V |Φ ⎬ μ 1 j j 0 ν + (4.201) , ⎩ ⎭ (E0 − Ej ) (E0 − Ej ) j∈S12

where the correction terms are given in brackets {. . .}. The matrix elements of the reaction operator V0 are determined recursively using eqs. (4.197) and (4.201). Recursion is continued until convergence to some tolerance is achieved for each matrix element. Only in the final iteration are the correction terms added to the matrix elements. The ground state secular equation (4.190) is then solved to obtain an estimate of the exact ground state energy, E0 , which can then be used repeatedly to construct and solve the secular equation until self-consistency is achieved.

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4.4. BRILLOUIN–WIGNER PERTURBATION THEORY Møller–Plesset second-order perturbation theory [78, 162] is the most widely used approach to the electron correlation problem in contemporary ab initio molecular electronic structure studies [163–168]. For systems which are well described by a single determinantal reference function, this theory - based on the use of Rayleigh–Schr¨odinger perturbation theory to describe electron correlation corrections to the Hartree–Fock independent electron model - affords an approach which combines accuracy with computational efficiency. The method, which is often designated ‘MP 2’, is based on the lowest order of the many-body perturbation theory expansion to take account of correlation effects. Multi-Reference second-order Rayleigh–Schr¨odinger perturbation theory might be expected to provide the theoretical apparatus for the description of the electronic structure of molecular systems, for which a single determinantal functions does not form an appropriate zero-order approximation. Multi-Reference functions are required, for example, in the description of many dissociative molecular processes. In practice, the multi-reference Rayleigh–Schr¨odinger perturbation theory is not a robust technique. It cannot, therefore, be regarded as a general purpose ab initio electronic structure method. The problems arising in the multi-reference theory can be traced to the occurrence of ‘intruder states’. Eigenstates may be re-ordered when the perturbation parameter λ is varied. In particular, eigenstates with energies lying above those in the reference space at λ = 0 may have energies below the higher levels in the reference space for some |λ| > 0. For 1 ≥ λ > 0, these states are termed ‘intruder states’. For −1 ≤ λ < 0, these states are termed ‘back door intruder states’. Brillouin–Wigner perturbation theory [2–4, 18, 25] is seen as a solution to the demand for a robust multi-reference formalism. Figure 4.1 compares the Brillouin-Winger and RayleighSchr¨odinger approaches. The relationship between single-reference Brillouin–Wigner perturbation theory and its Rayleigh–Schrodinger counterpart is well known, but for completeness we include a brief account of the single-reference case in Section 4.4.1 before turning to the multi-reference case in Section 4.4.2. 4.4.1. Single-reference Brillouin–Wigner perturbation theory We begin by writing the exact Schr¨odinger equation as (4.202) H |Ψi = Ei |Ψi , where H is the hamiltonian, Ei is the exact energy of the i-th state and Ψi is the corresponding wave function. We continue by separating the hamiltonian, H, into two parts as follows: (4.203) H = H0 + H1 , where H0 is the unperturbed part and H1 is the perturbation. Furthermore, we assume that we know the solutions of the unperturbed Schr¨odinger equation which has the form:

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Brillouin-Wigner perturbation theory

Rayleigh-Schr¨ odinger perturbation theory

λ = 0 −−−−−−−−−−−−−−−−−−−−−→λ = 1

λ = 1 ←−−−−−−−−−−λ = 0

E2 (λ = 0)

E2 (λ = 0) Ebw 2

(λ = 1) E2 (λ = 1)

E2 (λ = 1)

E1 (λ = 0)

E1 (λ = 0)

Ebw 1 (λ = 1)

E1 (λ = 1)

E1 (λ = 1)

E0 (λ = 0)

E0 (λ = 0)

Ebw 0 (λ = 1)

E0 (λ = 1)

E0 (λ = 1)

Figure 4.1. This figure compares schematically the application of Rayleigh–Schr¨odinger perturbation theory and Brillouin–Wigner perturbation theory to the multi-reference electron correlation problem. We indicate states which are considered in a single calculation by enclosing them in a box. The Rayleigh– Schr¨odinger perturbation theory approach approximates the energy expectation values for a manifold of states in a single calculation. In the multi-reference Rayleigh–Schr¨odinger perturbation theory, the states with energies E0 , E1 , E2 , . . . are considered in a single calculation, as represented on the right-hand side of the figure. The Brillouin–Wigner perturbation theory approach is ‘state-specific’; that is, we consider a single state in a given calculation and, if the resulting theory is not a valid many-body theory, then we apply a suitable a posteriori correction based on the relation between the Brillouin–Wigner and the Rayleigh–Schr¨odinger denominators. In principle, this a posteriori correction can be rendered exact. In the multi-reference Brillouin–Wigner perturbation theory, the states with energies E0 , E1 , E2 , . . . are considered in separate calculations, as represented on the left-hand side of the figure. The resulting energies are denoted by E0BW , E1BW , E2BW , . . . To each of these energies an a posteriori correction to restore linear scaling can be introduced if necessary to yield the energies E0 , E1 , E2 , . . . .

(4.204) H0 |Φi = Ei |Φi . The lowest (ground state) non-degenerate energy is given by the expression [8, 17, 161]: E0 = Φ0 |H0 | Φ0 (4.205) + Φ0 |H1 + H1 B0 H1 + H1 B0 H1 B0 H1 + · · · | Φ0 ,

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where B0 is the Brillouin–Wigner resolvent |Φi Φi | Q0 (4.206) B0 = = . E0 − H0 E0 − H0 i=0

4.4.2. Multi-reference Brillouin–Wigner perturbation theory We turn, in this section, to the multi-reference Brillouin–Wigner perturbation theory. We divide our discussion into two parts. In Section 4.4.2.1, we survey the basic theoretical apparatus of multi-reference second-order Brillouin–Wigner perturbation theory. In Section 4.4.3, we describe an a posteriori correction to multi-reference Brillouin–Wigner perturbation theory. 4.4.2.1. Multi-reference second-order Brillouin–Wigner perturbation theory In this section, we briefly survey the basic theoretical apparatus of multi-reference second-order Brillouin–Wigner perturbation theory. This will serve both to provide the necessary background and to introduce our notation. Our problem is to approximate the solutions of the time-independent Schr¨odinger equation associated with the Hamiltonian H which we write as (4.207) HΨα = Eα Ψα , where Eα is the exact eigenvalue for the state α and Ψα is the corresponding eigenfunction. To develop a perturbation theory, the Hamiltonian is written as a sum of two parts, i.e. (4.208) H = H0 + λH1 , where H0 is the zero-order Hamiltonian and H1 is the perturbating operator. λ is the perturbation parameter which interpolates between the reference model when λ = 0 and the perturbed problem when λ = 1. It is assumed that the solutions of the zeroorder eigenproblem (4.209) H0 Φμ = Eμ Φμ ,

μ = 0, 1, 2, . . . .

are known. We let S be the set of all Φμ arising from the solution of (4.209). Let (4.210) {Φμ ; μ = 0, 1, 2, . . . , p − 1} be a subset of linearly independent functions which constitute the reference space, which we label P. The choice of the functions included in P is dictated by the nature of the problem under investigation. Let P be the projection operator onto this reference subspace (4.211) P = |Φμ Φμ | , μ∈P

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and let Q be its orthogonal complement: Q=I −P = (4.212) |Φμ Φμ | , μ∈P /

so that the projectors P and Q satisfy the idempotency and orthogonality relations P2 Q2 (4.213) PQ P +Q

=P =Q =0 = I.

We can write the space S as (4.214) S = P ⊕ Q. Let us consider the projection of the exact wave function, Ψα , onto the reference space P, i.e. (4.215) ΨαP = P Ψα , α = 0, 1, 2, . . . , p − 1 ΨαP is sometimes called the model function. Obviously, ΨαP can be written as a linear combination of the subset (4.210) (4.216) ΨαP = Cμα Φμ , μ∈P

where Cμα is a coefficient which, at this stage, is undetermined. The functions ΨαP are, in general, non-orthogonal, but are assumed to be linearly independent. The exact wavefunction, Ψα , is expanded as follows: (4.217) Ψα = (1 + Bα H1 + Bα H1 Bα H1 + · · · ) ΨαP , where Bα is the Brillouin–Wigner type propagator: |Φμ Φμ | (4.218) Bα = , Eα − Eμ μ∈P /

which depends on the unknown exact energy eigenvalue Eα . The exact wavefunction, Ψα , and the model function, ΨαP , satisfy the following intermediate normalization conditions:

P Ψα | Ψ α = 1

(4.219) ΨαP | ΨαP = 1. The wave operator, Ωα , is defined by (4.220) Ψα = Ωα ΨαP .

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Application of the wave operator to the model function yields the exact wave function. Comparing (4.220) with (4.217), we see immediately that: (4.221) Ωα = 1 + Bα H1 + Bα H1 Bα H1 + · · · or, re-writing (4.221) as a recursion, (4.222) Ωα = 1 + Bα H1 Ωα , which may be seen to be the Bloch equation [39] in Brillouin–Wigner form. We now introduce the ‘effective’ Hamiltonian which acts in the reference subspace ˆ α = P HΩα P (4.223) H ˆ α , operates only in the reference space P but has the The effective Hamiltonian, H exact energy, Eα , as an eigenvalue, i.e. ˆ α Ψ P = Eα Ψ P (4.224) H α α Using (4.208) we can write the effective Hamiltonian operator (4.223) in the form ˆ α = P H0 P + λP H1 Ωα P (4.225) H which can then be re-written in the form ˆ α = P H0 P + λP Vα P, α = 0, 1, 2, . . . , p − 1 (4.226) H where we have introduced the reaction operator, Vα , defined as follows: (4.227) Vα = H1 Ωα Combining (4.222) and (4.227) gives a Lippmann–Schwinger-like equation [117] in Brillouin–Wigner form: (4.228) Vα = H1 + H1 Bα Vα It should be emphasized that Vα is a state-specific reaction operator corresponding to only one eigenenergy Eα . Note that the wave operator arising in the Rayleigh–Schr¨odinger formalism, Ω can be related to the wave operators Ωα in the Brillouin–Wigner method through the relation (4.229) Ω =

p−1

Ωα Pα

α=0

The exact energies, Eα , are eigenvalues of different effective hamiltonian operators ˆ α Ψ P = Eα Ψ P , α = 0, 1, 2, . . . , p − 1 (4.230) H α α Let us consider a p-state system and obtain an explicit formulation of the multireference Brillouin–Wigner second-order perturbation theory for this case. In the pstate case, we have a reference space spanned by p functions, Φ0 , Φ1 , . . . , Φp−1 . The projector onto this space is

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(4.231) P =

p−1

|Φμ Φμ | .

μ=0

The corresponding wave operators are (4.232) Ω0 , Ω1 , . . . , Ωp−1 and the reaction operators are V0 = H1 Ω0 V1 = H1 Ω1 ... (4.233) Vp−1 = H1 Ωp−1 . The effective Schr¨odinger equation has the form for the state α: ˆ α Ψ P = Eα Ψ P , α = 0, 1, . . . , p − 1. (4.234) H α

α

The effective Hamiltonian operator for this state-specific formalism takes the form: ˆ α = P HΩα P H = P H0 P + P H1 Ωα P (4.235)

= P H0 P + P Vα P,

which can be written more explicitly as &p−1 ' p−1 ˆ |Φμ Φμ |}H0 { |Φμ Φμ | Hα = μ=0

(4.236)

+

& p−1

μ=0

|Φμ Φμ |}Vα {

μ=0

p−1

' |Φμ Φμ | .

μ=0

For the state α = 0, we have a model function given by (4.237) Ψ0P =

p−1

Cμ Φμ .

μ=0

The secular equation for the lowest state takes the form: ⎛

ˆ α |Φ0 − E0 Φ0 | H ⎜ Φ1 | V 0 |Φ0 ⎜ ⎝...

Φp−1 V 0 |Φ0

Φ0 | V 0 |Φ1 ˆ α |Φ1 − E0 Φ1 | H ...

Φp−1 V 0 |Φ1

... ... ... ...

⎞⎛ ⎞ Φ0 | V 0 Φp−1 C0 ⎟ ⎜ C1 ⎟ Φ1 | V 0 Φp−1 ⎟⎜ ⎟ ⎠ ⎝ . . . ⎠ = 0, ...

ˆ α Φp−1 − E0 Cp−1 Φp−1 H

(4.238)

where the matrix elements of the effective Hamiltonian are Φμ | H1 |Φj Φj | V0 |Φν (4.239) Φμ | V0 |Φν = Φμ | H1 |Φ0 + E0 − Ej j ∈P /

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and ˆ α |Φμ = Φμ | H0 |Φμ + Φμ | V0 |Φμ , (4.240) Φμ | H with μ, ν = 0, 1, . . . , p − 1. Since H1 is a two-particle operator, the configuration |Φj is, at most, a double replacement with respect to |Φμ in (4.239). The matrix elements of the reaction operator are obtained from the Lippmann–Schwinger-like equation [160], eq. (4.228). Specifically, we have Φj | H1 |Φk Φk | V0 |Φμ , ∀j ∈ / P. Φj | V0 |Φμ = Φj | H1 |Φμ + E0 − Ek k∈P /

(4.241) Equation (4.238) has p roots of which we take only one. The exact energy, E0 , occurs in the denominator factors in eqs. (4.239) and (4.241) and the eq. (4.238) must, therefore, be solved iteratively until self-consistency is achieved. If we restrict the order of perturbation admitted in (4.241) then we realize a finite order multi-reference Brillouin–Wigner perturbation theory. Specifically, if we neglect terms of order λ3 are higher, we are led immediately to the second-order theory for which the matrix elements of the effective Hamiltonian (4.239) take the form: Φμ | H1 |Φj Φj | H1 |Φν (4.242) Φμ | V0 |Φν = Φμ | H1 |Φ0 + . E0 − Ej j ∈P /

4.4.3. A posteriori correction to multi-reference Brillouin–Wigner perturbation theory Now it is well known that Brillouin–Wigner perturbation theory is not, in general, a many-body theory, in that it contains terms which scale non-linearly with the number of electrons in the system. However, it has been shown that a posteriori corrections to Brillouin–Wigner perturbation theory can be made based on the identity (4.243) (E0 − Ek )−1 = (E0 − Ek )−1 + (E0 − Ek )−1 (−ΔE0 ) (E0 − Ek )−1 , where ΔE0 = E0 −E0 , can be applied to the matrix elements in (4.238). Rearranging (4.243) we have −1

(4.244) (E0 − Ek )

+ (E0 − Ek )

−1

(ΔE0 ) (E0 − Ek )

−1

−1

= (E0 − Ek )

.

The first term on the left hand side of (4.244) is a Brillouin–Wigner denominator and the term on the right hand side is a Rayleigh–Schr¨odinger denominator. The second term on the left hand side can be regarded as an a posteriori removal of unlinked diagram terms from a Brillouin–Wigner expansion. In general, this identity relation is applied a posteriori, so the state-specific multi-reference Brillouin–Wigner theory avoids convergence problems associated with intruder states. However, this introduces terms which are not extensive, i.e. do not scale linearly with particle and the a posteriori modification removes these terms and restores extensivity so that a true many-body theory is recovered.

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For the state k = α the ‘modified’ matrix elements obtained after using the identity relation for the p-state case may be written Φμ | H1 |Φj Φj | V0 |Φν Φμ | V0 |Φν = Φμ | H1 |Φν + E0 − Ej j ∈P / ⎧ ⎫ ⎨ ΔE Φ | H |Φ Φ | V |Φ ⎬ μ 1 j j 0 ν + (4.245) . ⎩ ⎭ (E0 − Ej ) (E0 − Ej ) j ∈P /

Matrix elements of the reaction operator, Vα , could be determined by using eq. (4.245) including the terms contained in the braces {. . .} after convergence has been achieved. However, it can be shown that in the case p = 1, i.e. the singlereference case, eq. (4.245) reduces to the familiar and widely used ‘MP2’ expression. In the multi-reference case, eq. (4.245) is independent of the exact ground state energy, E0 , and becomes Φi | H1 |Φm Φm | H1 |Φj (4.246) Φi | Vα |Φj = Φi | H1 |Φj + , Eα − Ej m∈P /

which could be regarded as a Rayleigh–Schr¨odinger-like expression in that the denominator depends only on the unperturbed energies. However, according to Lindgren and Morrison (p. 207) [84] one of the fundamental differences between the Brillouin–Wigner and the Rayleigh–Schr¨odinger formalisms [is that in] the former case there is one effective Hamiltonian for each energy, while in the latter case a single operator yields all the model states and corresponding energies. and expression (4.246) should be viewed, therefore, as in Brillouin–Wigner from. Certainly, the Brillouin–Wigner expansion described in Section 4.4.2.1 is of central importance to the derivation of expression (4.246). Equation (4.246) is our working equation and is obviously more efficient in computational implementations than eq. (4.245), which contains the a posteriori modification explicitly. In the multi-reference case, application of the a posteriori modification to second-order Brillouin–Wigner perturbation theory leads to state-specific second-order Rayleigh– Schr¨odinger-like perturbation theory. (Note that other formulations of multi-reference Rayleigh–Schr¨odinger perturbation theory7 are not state-specific in that the wave operator employed does not depend on the model function on which it operates.) References 1. I. Hubaˇc and S. Wilson, Encyclopedia of Computational Chemistry, electronic edition, P. von Ragu´e Schleyer, N.L. Allinger, H.F. Schaefer III, T. Clark, J. Gasteiger, P. Kollman and P. Schreiner, Wiley, Chichester, 2003 7

A pedagogical presentation of multi-reference many-body perturbation theory is given by Kucharski and Bartlett [169].

Brillouin–Wigner Methods for Many-Body Systems 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

35. 36.

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J.E. Lennard-Jones, Proc. Roy. Soc. (London) A 129, 598, 1930 L. Brillouin, J. Physique 7, 373, 1932 E.P. Wigner, Math. naturw. Anz. ungar. Akad. Wiss. 53, 475, 1935 J. Karwowski and I. Shavitt, in Handbook of Molecular Physics and Quantum Chemistry, Volume 2, Molecular Electronic Structure, edited by S. Wilson, P.F. Bernath and R. McWeeny, John Wiley & Sons, Chichester, 2003 I. Hubaˇc and P. Neogr´ady, Phys. Rev A50, 4558, 1994 ˇ arsky, V. Hrouda, V. Sychrovsk´y, I. Hubaˇc, P. Babinec, P. Mach, J. Urban and J. P. C´ M´asˇik, Collect. Czech. Chem. Comm. 60, 1419, 1995 I. Hubaˇc, in New Methods in Quantum Theory, NATO ASI Series, ed. C.A. Tsipis, V.S. Popov, D.R. Herschback and J.S. Avery, pp. 183, Kluwer, Dordrecht, 1996 J. M´asˇik and I. Hubaˇc, Coll. Czech. Chem. Commun. 62, 829, 1997 J. M´asˇik and I. Hubaˇc, in Quantum Systems in Chemistry and Physics: Trends in Methods and Applications, ed. R. McWeeny, J. Maruani, Y.G. Smeyers and S. Wilson, pp. 283, Kluwer Academic Publishers, Dordrecht, 1997 W. Wenzel, Int. J. Quant. Chem. 70, 613, 1998 W. Wenzel and M.M Steiner, J. Chem. Phys. 108, 4714, 1998 J. M´asˇik, P. Mach and I. Hubaˇc, J. Chem. Phys. 108, 6571, 1998 P. Mach, J. M´asˇik, J. Urban and I. Hubaˇc, Molec. Phys. 94, 173, 1998 J. M´asˇik, P. Mach, J. Urban, M. Polasek, P. Babinec and I. Hubaˇc, Collect. Czech. Chem. Comm. 63, 1213, 1998 J. M´asˇik and I. Hubaˇc, Adv. Quantum Chem. 31, 75, 1998 ˇ arsky, J. M´asˇik and I. Hubaˇc, J. Chem. Phys. 110, 10275, J. Pittner, P. Nechtigall, P. C´ 1999 I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 33, 365, 2000 ˇ arsky, J. Chem. Phys. 112, 8779, 2000 I. Hubaˇc, J. Pittner and P. C´ ˇ arsky and I. Hubaˇc, J. Chem. Phys. 112, 8785, 2000 J. Sancho-Garc´ıa, J. Pittner, P. C´ I. Hubaˇc, P. Mach and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 33, 4735, 2000 H.M. Quiney, I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 34, 4323, 2001 ˇ arsky and I. Hubaˇc, J. Phys. Chem. A 105, 1354, 2001 J. Pittner, P. Nachtigall, P. C´ ˇ ˇ J. Pittner, J. Smydke, P. C´arsky and I. Hubaˇc, J. Mol. Struct.-THEOCHEM 547, 239, 2001 I. Hubaˇc, P. Mach and S. Wilson, Adv. Quantum Chem. 39, 225, 2001 S. Wilson and Hubaˇc, Molec. Phys. 99, 1813, 2001 I. Hubaˇc and S. Wilson, Adv. Quant. Chem. 39, 209, 2001 I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 34, 4259, 2001 I. Hubaˇc, P. Mach and S. Wilson, Molec. Phys. 100, 859, 2002 I. Hubaˇc and S. Wilson, Int. J. Molec. Sci. 3, 570, 2002 I. Hubaˇc, P. Mach and S. Wilson, Intern. J. Quantum Chem. 89, 198, 2002 N.D.K. Petraco, L. Horn´y, H.F. Schaefer and I. Hubaˇc, J. Chem. Phys. 117, 9580, 2002 ˇ arsky, A. Mavridis and I. Hubaˇc, J. Chem. Phys. 117, I.S.K. Kerkines, J. Pittner, P. C´ 9733, 2002 I. Hubac and S. Wilson, in Fundamental World of Quantum Chemistry - A Tribute to the Memory of Per-Olov L¨ owdin, volume 1, edited by E.J. Br¨andas and E.S. Kryachko, p. 407, Kluwer Academic Publisher, Dordrecht, 2003 I. Hubac, P. Mach and S. Wilson, Molec. Phys. 101, 3493, 2003 ˇ arsky, in Advanced Topics in TheoretS. Wilson, I. Hubac, P. Mach, J. Pittner and P. C´ ical Chemical Physics, Progress in Theoretical Chemistry and Physics, edited by J. Maruani, R. Lefebvre and E. Br¨andas, p. 71, Kluwer Academic Publishers, Dordrecht, 2003

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5 SUMMARY AND PROSPECTS

Abstract

A summary of the present status of Brillouin–Wigner many-body methodology is given. Future prospects are assessed.

We have been concerned in this volume with the application of Brillouin–Wigner theory to the many-body problem in atoms and molecules and, in particular, to the description of electron correlation effects. Because of their linear scaling with particle number, many-body methods are now pre-eminent in quantum chemistry and in molecular physics. The exponential growth in the power of computing machines over the past half century has facilitated theoretical studies of larger and increasingly complex molecular systems. The theoretical apparatus underpinning these studies exploits many-body methodology. Over recent years, it has emerged that the manybody Brillouin–Wigner formalism is an important theoretical tool which offers a powerful alternative perspective to the ‘standard’ Rayleigh–Schr¨odinger approach for a wide range of problems. The seminal paper entitled “Many-Body Problem for Strongly Interacting Particles. II. Linked Cluster Expansion” by K.A. Brueckner [1], published more than 50 years ago in The Physical Review, established the validity of Rayleigh–Schr¨odinger perturbation theory in underpinning the theoretical description of many-body systems. Many-body perturbation theory, which is widely used in quantum chemical applications using the ‘Møller–Plesset’ or ‘Hartree–Fock model’1 choice of zeroorder hamiltonian, is the many-body formulation of the Rayleigh–Schr¨odinger perturbation theory. The key property of many-body methods in general and many-body perturbation theory in particular is recognized as their extensivity or linear scaling with the number of particles in the studied system. Rayleigh–Schr¨odinger perturbation theory contains terms in each order of the series which scale non-linearly with particle 1

The use of ‘Møller–Plesset’ or ‘Hartree–Fock model’ to label particular choices of zero-order Hamiltonian in many-body perturbation theory dates from the work of Pople et al. [2] and of Wilson and Silver [3]. In their original publication of 1934, Møller and Plesset [4] did not recognize the “many-body” character of the theory in the modern (post-Brueckner) sense.

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number, but these terms are known to cancel mutually in a given order. The surviving terms are associated with linked diagrams in a diagrammatic representation of the series. The terms which cancel in each order are associated with unlinked diagrams. Rayleigh–Schr¨odinger perturbation theory can be used to derive and analyze different approaches to the many-body problem in atoms and molecules [5, 6]. Approximate theories of electron correlation effects based on configuration interaction or on various cluster expansions can be derived and analyzed in terms of finite-order or infinite-order Rayleigh–Schr¨odinger perturbation expansions. Brillouin–Wigner theory has been largely neglected in the quantum chemistry and molecular physics literature, because it is found to lack extensivity. Brillouin–Wigner perturbation theory contains terms in each order which do not scale linearly with particle number and furthermore, these terms are not cancelled in a given order. This lack of extensivity is associated with the presence of the exact energy in the denominator factors. The use of a Hartree–Fock reference function is ubiquitous in molecular electronic structure theory because of the beneficial computational consequences of the orthogonality of the Hartree–Fock molecular orbitals. However, many quantum chemical studies require the use of a multi-reference formalism. For example, studies of systems involving bond breaking processes almost invariably require the use of a reference function constructed as a linear combination of a number of reference functions. For cases where electron correlation effects are large and, in particular, when the Hartree–Fock model gives qualitatively incorrect results, the system is said to be strongly correlated. In spite of the success of single-reference many-body methods and, in particular (single-reference) many-body perturbation theory, multi-reference formulations of the many-body problem have been beset by problems for more than 20 years.2 Multi-reference Rayleigh–Schr¨odinger perturbation theory is plagued by the socalled ‘intruder state’ problem. Multi-reference Rayleigh–Schr¨odinger perturbation theory is applied to a manifold of states simultaneously and as the perturbation is switched on, states within the reference space may move above some of the lower states in the complementary space. These intruder states can degrade or even destroy the convergence of the perturbation expansion.3 In his original publication, which appeared in 1935, Wigner [7] recognized that what is now called Brillouin–Wigner perturbation theory has certain advantages, which we summarized in Chapter 1, when dealing with systems involving degenerate or quasidegenerate states. In this volume, we have described the application 2

See, for example, the monograph Electron correlation in molecules [5, 6] published by one of us in 1984 for an account of the then current thinking on the problems associated with quasi-degenerate diagrammatic perturbation theory. 3 In the monograph Electron correlation in molecules [5,6] published by one of us in 1984, three solutions to the intruder state problem were oﬀered: 1. The formation of matrix Pad´e approximants to the eﬀective interaction operator.

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of Brillouin–Wigner theory to systems demanding the use of a multi-reference formulation. The approach is “state specific”; it is applied to an individual state rather than the manifold of states considered in the multi-reference Rayleigh–Schr¨odinger perturbation theory and incorporates, if necessary, an a posteriori adjustment for the lack of extensivity associated with Brillouin–Wigner theory. The Brillouin–Wigner-based formalism may yield results which are identical to those obtained by employing the Rayleigh–Schr¨odinger-based many-body formulation for cases where a single-reference is appropriate. For example, single-reference Brillouin–Wigner coupled cluster theory with single and double excitations (BWCCSD ) was shown to be exactly equivalent to other formulations of the CCSD coupled cluster problem by Hubaˇc and Neogr´ady [8] in a paper entitled “Sizeconsistent Brillouin–Wigner perturbation theory with an exponentially parameterized wave function: Brillouin–Wigner coupled-cluster theory” which was published in 1994. Single-reference Brillouin–Wigner perturbation theory through second-order (BWPT 2) yields results which are identical to the ‘standard’ MBPT 2 theory, i.e. MP 2, after making an a posteriori adjustment for its lack of extensivity [9–11]. In the case of multi-reference problems, that is problems which require the use of a multi-determinantal reference function, the Brillouin–Wigner approach has unique characteristics. It completely avoids the intruder state problem because of the natural energy gap which arises when the exact energy is included in denominator factors [17]. It can be applied to a single state. It is ‘state specific’. By introducing a posteriori adjustments, extensivity can be ensured and a many-body theory results [18]. In some cases the a posteriori adjustment is approximate, in other cases it may be exact. For example, multi-reference Brillouin–Wigner coupled cluster theory including single and double excitations ( MR - BWCCSD) using the a posteriori adjustment for its lack of extensivity introduced by Hubaˇc and Wilson [18] in a paper entitled “On the use of Brillouin–Wigner perturbation theory for many-body systems” yields an theory which is “approximately extensive” [19]. On the other hand, using the same a posteriori adjustment for lack of extensivity in second-order multi-reference Brillouin–Wigner perturbation theory (MR - BW 2), results in a theory which can be seen to be ‘exactly extensive’ [9–11]. Other authors [12–16] have developed secondorder Rayleigh–Schr¨odinger methodologies which are exactly extensive, but often suffer from problems caused intruder states. The Brillouin–Wigner approach completely avoids these problems. It is very flexible and, in principle, can deal with a model space of arbitrary dimensions. Pittner has given an interesting perspective on the extensivity of Brillouin–Wigner theory. He proposed [20] the use of an a posteriori adjustment for lack of extensivity based on a ‘continuous transition’ between 2. The inclusion of the intruder states in the reference space. This leads to an incomplete model space which can be much larger that the original reference space. It may also destroy the extensivity of the method. 3. Exclusion of higher energy states from the reference space. Again, this may also destroy the extensivity of the method.

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Brillouin–Wigner expansions and expansions based on Rayleigh–Schr¨odinger perturbation theory. Very recent work by Pittner [21], has provided an alternative perspective on the Rayleigh–Schr¨odinger formulation of the multi-reference correlation problem developed by Mukherjee and his collaborators [22–28], and developed very recently by Schaefer and his colleagues [29–31]. The Brillouin–Wigner approach to the multi-reference correlation problem can be used to simplify [21] certain coupling terms which arise in the Rayleigh–Schr¨odinger formulation developed by Mukherjee and by Schaefer and their respective co-workers. In this monograph, we have concentrated exclusively on the calculation of approximations to energy expectation values for the most widely used techniques for describing correlation effects in atoms and molecules. In particular, we have considered configuration interaction, cluster expansions and perturbation theory. It should be emphasized that the Brillouin–Wigner formalism described here can be applied to any ab initio quantum chemical method, for example, the so-called augmented coupled cluster theories such as CCSD ( T ), as well as higher order single-reference coupled cluster theories such as CCSDT and CCSDTQ. The augmented coupled cluster theory designated CCSD ( T ) [32] is essentially a hybrid coupled cluster/perturbation theory approach which has been dubbed the ‘gold standard’ in contemporary applied electronic structure theory being a good compromise between the accuracy it typically supports and the computational demands of the associated algorithms.4 The singlereference CCSDT and CCSDTQ may offer higher accuracy, but at the cost of severely increased computational demands. A Brillouin–Wigner realization of the CCSD ( T ), CCSDT and CCSDTQ theories has been described in recent work by Demel and Pittner [34] and the interested reader is referred to their original publication for further details. In contemporary molecular structure studies, energy derivatives are often calculated both to determine spectroscopic parameters and constants, and in geometry optimizations during structure determination. The calculation of energy derivatives dates from the pioneering work of Gerratt and Mills [35, 36] and of Pulay [37–39] in the late 1960s. Many of these calculations employ energy derivatives for methods developed with respect to a single determinant Hartree–Fock reference function even though this function may not be appropriate for all geometries considered. We have shown that Brillouin–Wigner-based methods can be successfully applied to multi-reference formulations. Such methods therefore facilitate the determination of energy derivatives with respect to a reference function which provides a useful zero-order approximation for complete potential energy curves. For example, it is known that the single determinant Hartree–Fock function often leads to harmonic force constants which are too high and this behaviour is mirrored in correlated 4 See, for example, C.J. Cramer in his book Essentials of Computational Chemistry [33] writes

“... ccsd(t) ... has come to be the eﬀective gold standard for single-reference calculations.” p. 226.

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treatments based on the single-reference function case. The use of multi-reference functions should obviate this distortion. Equally, although single-reference methods can be successfully employed in studies of the structure of molecules which are well described by a single-reference function, the study of, for example, transition states frequently requires a multi-reference formulation. Energy derivatives can be evaluated within the Brillouin–Wigner-based methods described in this book. For the first example of energy derivatives using Brillouin–Wigner theory see the recent work of ˇ Pittner and Smydke [40] entitled Analytic gradient for the multi-reference Brillouin– Wigner coupled cluster method and for the state-universal multi-reference coupled cluster method. The interested reader is referred to the original publication [40] for further details. The Brillouin–Wigner-based methods described in this book can also be employed in the calculation of molecular properties other than the energy. Because of the central role played by the energy expectation value in studies of molecular structure, we have concentrated exclusively on the calculation of approximations to energy. However, the Brillouin–Wigner-based techniques can be applied to the calculation of first-order properties, such as dipole moments and multipole moments, as well as second-order properties, such as polarizabilities and hyperpolarizabilities. In calculations of properties such as ionization potentials and electron affinities, the use of a Fock space formulation is more appropriate that the Hilbert space formulation that we have followed in this monograph. Some progress has been made [41, 42] in formulating the necessary Fock space Brillouin–Wigner methodology. The interested reader can find further details in the original publications [41, 42]. The theoretical apparatus presented in this monograph is formulated entirely within the framework of non-relativistic quantum mechanics. This may be adequate for studies of most properties of molecules containing light atoms.5 However, it is now well established that a fully relativistic formulation is essential for the treatment of heavy and superheavy elements and molecules containing them [44]. A fully relativistic, four-component Brillouin–Wigner-based treatment of hydrogen-like systems by Quiney and the present authors has demonstrated [43] the feasibility of general four-component Brillouin–Wigner-based theories. There is no problem of principle in extending this approach to arbitrary atomic and molecular systems. The most widely used ab initio approach to the correlation energy in molecular species is the MP 2 theory – second-order many-body perturbation theory. This approach is available in many quantum chemistry packages,6 such as the well-known G AUSSIAN suite of computer programs [50] for which the British theoretical chemist Professor Sir John Pople FRS was awarded a half share of the 1998 Nobel Prize in Chemistry.7 MP 2 is essentially a robust, ‘black-box’ method which is today the most 5

Even for molecules containing light atoms, properties depending on the wave function close to the nuclear region can require the use of a relativistic formalism. 6 There are many quantum chemistry packages available to the contemporary researcher. For a review of some of theses see reference [45]. 7 Pople’s Nobel citation reads “for his development of computational methods in quantum chemistry”.

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widely used ab initio electronic structure method.8 We submit that second-order, multi-reference Brillouin–Wigner perturbation theory, together with an a posteriori adjustment for its lack of extensivity as proposed in our paper [18], could provide a robust ‘black-box’ which may be comparable with MP 2 theory for cases requiring the use of a multi-reference function. This would open up the possibility of routine calculations to the vast range of applications to systems involving bond breaking processes for which existing single-reference formalisms are inadequate. There is one very important aspect of multi-reference quantum chemical studies that we have not considered in this work and that is the choice of reference function. This must be driven by the chemical and physical properties of the system under investigation. The choice of reference function is of crucial importance since the application of the Brillouin–Wigner-based formalism described in this book will not compensate for a poor choice of reference with respect to which the electron correlation treatment is developed. We believe that the development of an automated approach to the choice of reference functions for molecular electronic structure problems requiring a multi-reference formulation is the major barrier to the development of a robust, ‘black-box’ methodology. We submit that valence bond theory and in particular modern valence bond theory has much to offer as a basis for general purpose quantum chemistry computer program packages capable of routine applications to problems involving bond breaking and other processes which demand the use of a multi-reference approach. For example, it is known that a simple picture of the whole potential energy curve for the simplest of molecules, the hydrogen molecule, can be obtained by adopting the valence bond model [51]. This approach can be applied to larger molecular systems, either by imposing the so-called ‘strong orthogonality’ condition (or weaker orthogonality restrictions [52]) between group functions associated with different ‘groups of electrons’ [53]9 or by exploiting the increasing power of modern computing machines to confront the N ! problem arising from wave function ans¨atz based directly on products of non-orthogonal orbitals [55–57].10 In this monograph, we have assumed that the model space is complete. All configurations that can be generated from a given set of active orbitals are included in the model space. This requirement ensures that extensivity errors are not introduced when the effective Hamiltonian operator is diagonalized. Pittner, Li and Paldus have presented [61] a formulation for an incomplete model space in a paper entitled Multireference Brillouin–Wigner coupled-cluster method with a general model space. We refer the interested reader to the original publication for further details. 8 For a brief overview of the many applications of mp2 in contemporary molecular electronic structure studies see the recent report by one of us [45] which continues earlier reports on the subject [46–49]. 9 When the electron group functions are associated with electron pairs this approach becomes the well known generalized valence bond method (gvb) [54]. 10 Some algorithms exploit the relation between the modern valence bond wavefunction and the complete active space self-consistent ﬁeld (cas-scf) functions. This leads to the so-called cas-vb approach [58–60].

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In closing, we note that we have restricted our attention to the use of Brillouin– Wigner-based methods in handling the many-body problem in atomic and molecular systems requiring a multi-reference formulation. We suggest that such methods will have a broad range of applications in studies of the solid state, in condensed matter theory, in material science and in nuclear physics. In each of these fields problems arise requiring the use of a multi-reference formulation. In some fields such problems are termed ‘strongly correlated’. We submit that, after being abandoned for almost half a century in favour of Rayleigh–Schr¨odinger many-body formalism, the Brillouin–Wigner approach has much to offer in studies of the quantum many-body problem. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

K.A. Brueckner, Phys. Rev. 100, 36, 1955 J.A. Pople, J.S. Binkley and R. Seeger, Intern. J. Quantum Chem. Symp. 10, 1, 1976 S. Wilson and D.M. Silver, Phys. Rev. A 14, 1949, 1976 Ch. Møller and M.S. Plesset, Phys. Rev. 46, 618, 1934 S. Wilson, Electron correlation in molecules, Clarendon Press, Oxford, 1984 S. Wilson, Electron correlation in molecules, (reprint, with corrections, of the 1984 edition), Dover Publications, New York, 2007 E.P. Wigner, Math. naturw. Anz. ungar. Akad. Wiss. 53, 475, 1935 I. Hubaˇc and P. Neogr´ady, Phys. Rev. A50, 4558, 1994 P. Papp; P. Mach; J. Pittner; I. Hubaˇc and S. Wilson Molec. Phys. 104 2367, 2006 P. Papp, P. Mach, I. Hubaˇc and S. Wilson, Intern. J. Quantum Chem. 107, 2622, 2007 P. Papp, P. Neogr´ady, P. Mach, J. Pittner, I. Hubaˇc and S. Wilson, Molec. Phys. 106, 57, 2008 S.A. Kucharski and R.J. Bartlett, Int. J. Quantum Chem. Symp. 22, 383, 1988 K. Hirao (ed.), Recent Advances in Multi-reference Methods World Scientific Publishing, Singapore, 1999 Y.-K. Choe, J.P. Finley, H. Nakano, and K. Hirao, J. Chem. Phys. 113, 7773, 2000 H. Nakano, J. Nakatani, and K. Hirao, J. Chem. Phys. 114, 1133, 2001 H.A. Witek, H. Nakano, and K. Hirao, J. Chem. Phys. 118, 8197, 2003 chap5-Hirao2000, chap5-Hirao2001, chap5-Hirao2003 W. Wenzel, Intern. J. Quant. Chem. 70, 613, 1998 I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 33, 365, 2000 ˇ arsky J. Chem. Phys. 112 8779, 2000 I. Hubaˇc, J. Pittner, and P. C´ J. Pittner, J. Chem. Phys. 118, 10876, 2003 J. Pittner, personal communication U.S. Mahapatra, B. Datta, B. Bandyopadhyay, and D. Mukherjee, Adv. Quantum Chem. 30, 163, 1998 U.S. Mahapatra, B. Datta, and D. Mukherjee, J. Phys. Chem. A 103, 1822, 1999 U.S. Mahapatra, B. Datta, and D. Mukherjee, J. Chem. Phys. 110, 6171, 1999 U.S. Mahapatra, B. Datta, and D. Mukherjee, Chem. Phys. Lett. 299, 42, 1999 S. Chattopadhyay, U.S. Mahapatra, and D. Mukherjee, J. Chem. Phys. 112, 7939, 2000 S. Chattopadhyay, D. Pahari, D. Mukherjee, and U.S. Mahapatra, J. Chem. Phys. 120, 5968, 2004

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28. S. Chattopadhyay, P. Ghosh, and U.S. Mahapatra, J. Phys. B: At. Mol. Opt. Phys. 37, 495, 2004 29. F.A. Evangelista, W.D. Allen and H.F. Schaefer III, J. Chem. Phys. 125, 154113, 2006 30. F.A. Evangelista, W.D. Allen and H.F. Schaefer III, J. Chem. Phys. 127, 024102, 2007 31. F.A. Evangelista, A.C. Simmonett, W.D. Allen, H.F. Schaefer III, and J. Gauss, J. Chem. Phys. 128, 124104, 2008 32. K. Raghavachari, G.W. Trucks, J.A. Pople and M. Head-Gordon, Chem. Phys. Lett. 157, 479, 1989 33. C.J. Cramer, Essentials of Computational Chemistry, John Wiley & Sons, 2004 34. O. Demel and J. Pittner, J. Chem. Phys. 128, 104108, 2008 35. J. Gerratt and I.M. Mills, J. Chem. Phys. 49, 1730, 1968 36. J. Gerratt and I.M. Mills, J. Chem. Phys. 49, 1719, 1968 37. P. Pulay Molec. Phys. 17, 197, 1969 38. P. Pulay, Molec. Phys. 18, 473, 1970 39. P. Pulay, G. Fogarasi, F. Pang, and J. E. Boggs, J. Am. Chem. Soc. 101, 2550, 1979 ˇ 40. J. Pittner and J. Smydke, J. Chem. Phys. 127, 114103, 2007 ˇ Horn´y, H.F. Schaefer and I. Hubaˇc, J. Chem. Phys. 117, 9580, 2002 41. N.D.K. Petraco, L. ˇ Horn´y, I. Hubaˇc and S. Pal, Chem. Phys. 315, 240, 2005 42. H.F. Schaefer, III, L. 43. H.M. Quiney, I. Hubaˇc and S. Wilson, J. Phys. B: At. Mol. & Opt. Phys.34, 4323, 2001 44. U. Kaldor and S. Wilson, (eds), Theoretical Chemistry and Physics of Heavy and Superheavy Elements Progress in Theoretical Chemistry and Physics, Kluwer Academic Press (2003) 45. S. Wilson, in Specialist Periodical Reports, Chemical Modelling: Applications and Theory, 5 ed. A. Hinchliffe, Royal Society of Chemistry, London, 2008 46. S. Wilson, in Specialist Periodical Reports, Chemical Modelling: Applications and Theory, 1 ed. A. Hinchliffe, Royal Society of Chemistry, London, 2000 47. S. Wilson, in Specialist Periodical Reports, Chemical Modelling: Applications and Theory, 2 ed. A. Hinchliffe, Royal Society of Chemistry, London, 2002 48. S. Wilson, in Specialist Periodical Reports, Chemical Modelling: Applications and Theory, 3 ed. A. Hinchliffe, Royal Society of Chemistry, London, 2004 49. S. Wilson, in Specialist Periodical Reports, Chemical Modelling: Applications and Theory, 4 ed. A. Hinchliffe, Royal Society of Chemistry, London, 2006 50. M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, Jr., T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, and J.A. Pople, G AUSSIAN 03, Gaussian, Inc., Wallingford CT, 2004. 51. S. Wilson and J. Gerratt, Molec. Phys. 30, 777, 1975 52. S. Wilson, J. Chem. Phys. 64, 1692, 1976 53. S. Wilson and J. Gerratt, Molec. Phys. 30, 765, 1975; ibid. 30, 789, 1975

Brillouin–Wigner Methods for Many-Body Systems 54. 55. 56. 57.

58. 59. 60.

61.

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W.A. Goddard, T.H. Dunning, Jr., W.J. Hunt and P.J. Hay, Acc Chem. Res. 6, 368, 1973 J. Gerratt, Adv. At. Molec. Phys. 7, 141, 1971 N.C. Pyper and J. Gerratt, Proc. Roy. Soc. (London) A 355, 407, 1977 J. Gerratt, D.L. Cooper, P.B. Karadakov and M. Raimondi, in Handbook of Molecular Physics and Quantum Chemistry 2: Molecular Electronic Structure, edited by S. Wilson, P.F. Bernath and R. McWeeny, p. 146, John Wiley & Sons, Chichester, 2003 K. Hirao, H. Nakano, K. Nakayama, and M. Dupuis, J. Chem. Phys. 105, 9227, 1996 K. Hirao, H. Nakano, and K. Nakayama, J. Chem. Phys. 107, 9966, 1997 T. Thorsteinsson and D.L. Cooper, in Quantum Systems in Chemistry and Physics 1, Prog. Theoret. Chem. & Phys., edited by A. Hernandez-Laguna, J. Maruani, R. McWeeny and S. Wilson, p. 303, Kluwer Academic Press, Dordrecht, 2000 J. Pittner, X. Li and J. Paldus, Molec. Phys. 103, 2239, 2005

A ¨ LOWDIN’S STUDIES IN PERTURBATION THEORY

In this appendix, we present the titles and abstracts of L¨owdin’s influential series of papers under the general title “Studies in Perturbation Theory”. There are fourteen titles in the series, which was published between 1963 ˜ and 1971. The fourteenth paper is co-authored by O. Goscinski.

• Journal of Molecular Spectroscopy 10, 12 (1963) Studies in Perturbation Theory. I. An Elementary Iteration-Variation Procedure for Solving the Schr¨odinger Equation by Partitioning Technique P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Abstract The fundamental Schr¨odinger equation in quantum mechanics may be transformed to a discrete representation by introducing a complete set of basis functions in which the eigenfunctions may be developed. The eigenvalues are then determined by solving a secular equation, and this problem is here attacked by a partitioning which leads to an implicit relation for the energy E of the form E = f (E), which corresponds to the Schr¨odinger-Brillouin perturbation formula but has a more condensed form. The first-order iteration process based on the relation E (k+1) = f (E (k) ) is studied, and it is shown that, independent of whether this process is convergent or not, one can go over to a second-order process which appears to be closely connected with the variational expression. This second-order iterative procedure turns out to be very convenient for numerical work, particularly since it treats degenerate eigenvalues just as easily as single eigenvalues. The

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I. Hubaˇc and S. Wilson general behavior of the curve y = E − f (E) is discussed, and the method is illustrated by a few numerical examples. In an appendix, a brief survey of the classification of iteration processes is also given.

• Journal of Molecular Spectroscopy 13, 326 (1964) Studies in Perturbation Theory. II. Generalization of the Brillouin-Wigner Formalism P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Received 11 November, 1963 Abstract The solution of the Schr¨odinger equation by means of the variation principle and a discrete basis leads to a certain secular equation. In a previous paper (Part I), it was shown that, if this equation is handled by a partitioning technique, generalization of the Schr¨odinger-Brillouin formula is obtained. The procedure contains an inverse matrix, and it is shown here that, if in the expression for the wave function this matrix is approximated to the nth order, the corresponding matrix in the expectation value for the energy is approximated to order (2n + 1). This gives a simple derivation and generalization of the Brillouin-Wigner theorem. The connection between the variation principle and perturbation theory is further discussed.

• Journal of Molecular Spectroscopy 13, 326 (1964) Studies in Perturbation Theory. III. Solution of the Schr¨odinger equation under a variation of a parameter P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Received 11 November, 1963 Abstract The solution of he Schr¨odinger equation by means of the variation principle and a discrete but not necessarily orthonormal basis is studied in this case where the Hamiltonian H = H(α) is a function os a parameter α. The coefficients in the power series expansions in α of the wave function and the energy are explicitly determined in the form of expectation values, and the connection with the conventional Schr¨odinger theory is discussed.

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• Journal of Mathematical Physics 3, 969 (1962) Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism P.-O. L¨ owdin Quantum Chemistry Group, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Received December 15, 1961 Abstract The partitioning technique for solving secular equations is briefly reviewed. It is then reformulated in terms of an operator language in order to permit a discussion of the various methods of solving the Schr¨odinger equation. The total space is divided into two parts by means of a self-adjoint projection operator O. Introducing the symbolic inverse T = (1 − O)/(E − H), one can show that there exists an operator Ω = O + T HO, which is an idempotent eigenoperator to H and satisfies the relations HΩ = EΩ and Ω 2 = Ω. This operator is not normal but has a form which directly corresponds to infinite-order perturbation theory. Both the Brillouin- and Schr¨odinger-type formulas may be derived by power series expansion of T , even if other forms are perhaps more natural. The concept of the reaction operator is discussed, and upper and lower bounds for the true eigenvalues are finally derived.

• Journal of Mathematical Physics 3, 1171 (1962) Studies in Perturbation Theory. V. Some Aspects on the Exact Self-Consistent Field Theory P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Received 9 April 1962 Abstract The independent particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely

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I. Hubaˇc and S. Wilson analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.

• Journal of Molecular Spectroscopy 14, 112 (1964) Studies in Perturbation Theory. VI. Contraction of secular equations P.-O. L¨ owdin Quantum Theory Project, University of Florida, Gainesville, Florida and Quantum Chemistry Group, Uppsala University, Sweden Received December 7, 1963 Abstract The partitioning technique for solving secular equations is discussed. It is shown that the original secular equation may be transformed to a contracted secular equation referring to the specific subspace under consideration. The technique may be applied to hetero-atoms in a molecule, to central atoms in a crystal field, to chemical bonds in a molecular environment, and to discuss the question of ‘dressed’ and ‘undressed’ particles. If the system under consideration is not complete, the addition of a complementary subspace leads to a new term in the Hamiltonian corresponding to a ‘dressing’ of the system involved. It is shown that eigenvalue problems associated with contracted Hamiltonians may be solved by iteration procedures and particularly that there exists a secondorder iteration procedure which is a generalization of the convention variational principle.

• Journal of Molecular Spectroscopy 14, 119 (1964) Studies in Perturbation Theory. VII. Localized perturbation P.-O. L¨ owdin Quantum Theory Project, University of Florida, Gainesville, Florida and Quantum Chemistry Group, Uppsala University, Sweden Received December 7, 1963 Abstract A perturbation is said to be localized if only a few elements of the energy matrix are influenced in a specific representation. The corresponding Schr¨odinger equation can be transformed to a form which focuses the interest on those energy values as are changed by the perturbation. The associated secular equation contains the energy as a non-linear parameter, and it may be solved numerically by the

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second-order iteration procedure developed in a previous paper. The cases of a perturbation of a single diagonal element and a pair of non-diagonal elements are treated separately as an illustrative example. The method is applicable to heteroatoms in molecules, to impurities in solids and to molecular structures with ‘extra’ bonds, etc. Numerical applications are carried out elsewhere.

• Journal of Molecular Spectroscopy 14, 131 (1964) Studies in Perturbation Theory. VIII. Separation of the Dirac equation and study of the spin-orbit coupling and Fermi contact terms P.-O. L¨ owdin Quantum Theory Project, University of Florida, Gainesville, Florida and Quantum Chemistry Group, Uppsala University, Sweden Received December 7, 1963 Introduction In the treatment of atomic, molecular, and solid-state theory in the non-relativistic approximation, it is often desirable to study also the relativistic correction including the variation of the mass with velocity, spin-orbit coupling, Fermi contact interactions, and similar effects. In order to treat this problem, one has to start from the relativistic equation for the system under consideration in a more or less complete form and to go over to the nonrelativistic approximation. This procedure was first carried out for the one-electron case by Pauli1 by separating the Dirac equation into two equations involving the large and small components of the wave functions separately. Since then, the connection between the Dirac equation and the nonrelativistic approximation has been studied by a large number of authors and, for a survey, we would like to refer to the handbook article by Bethe and Salpeter2 A very elegant method for deriving the nonrelativistic form by means of contact transformations has been developed by Foldy and Wouthuysen3 Recent studies in the partitioning technique reported in a series of papers, which in the following will be referred to as PT I-VII, indicate that this method is perhaps more powerful than previously expected. In this paper, we will reconsider the separation of the Dirac equation for a single-particle system from this point of view.

1

W. Pauli, Z. Physik 43, 601 (1927). H.A. Bethe and E.E. Salpeter, in Hanbuch der Physik, ed. S. Fl¨ ugge, Vol. 35, p. 88, Springer-Verlag, Berlin, (1957). 3 L.L. Foldy and S.A. Wouthuysen, Phys. Rev. 78, 29 (1950); L.L. Foldy, Phys. Rev. 87, 688 (1952). 2

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• Journal of Mathematical Physics 6, 1341 (1965) Studies in Perturbation Theory. IX. Connection Between Various Approaches in the Recent Development - Evaluation of Upper Bounds to Energy Eigenvalues in Schr¨odinger’s Perturbation Theory P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Abstract The treatment of Schr¨odinger’s perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. lt is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schr¨odinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ‘bracketing theorem’ and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.

• The Physical Review 139, A357 (1965) Studies in Perturbation Theory. X. Lower Bounds to Energy Eigenvalues in Perturbation-Theory Ground State P.-O. L¨ owdin Quantum Theory Project, University of Florida, Gainesville, Florida and Department of Quantum Chemistry, Uppsala University, Sweden Received 7 December 1964 Abstract The eigenvalue problem HΨ = EΨ in quantum theory is conveniently studied by means of the partitioning technique. It is shown that, if E is a real variable, one may construct a function E1 = f (E) such that each pair E and E1 always bracket at least one true eigenvalue E. If E is chosen as an upper bound by means of, e.g., the variation principle, the function E1 is hence going to provide a lower bound. The reaction operator t associated with the perturbation problem H = H0 + V

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for a positive-definite perturbation V is studied in some detail, and it is shown that a lower bound to t may be constructed in a finite number of operations by using the idea of “inner projection” closely associated with the Aronszajn projection previously utilized in the method of intermediate Hamiltonians. By means of truncated basis sets one can now obtain not only upper bounds but also lower bounds which converge towards the correct eigenvalues when the set becomes complete. The method is applied to Brillouin-type perturbation theory, and lower bounds may be obtained either by pure expansion methods, by inner projections, or by a combination of both approaches leading to perturbation expansion with estimated remainders. The application to Schr¨odinger’s perturbation theory is also outlined. The method is numerically illustrated by a study of lower bounds to the ground state energies of the helium-like ions: He, Li+ , B2+ , etc.

• The journal of chemical physics 43, S 175 (1965) Studies in Perturbation Theory. XI. Lower bounds to energy eigenvalues, ground state, and excited state P.-O. L¨ owdin Quantum Theory Project, University of Florida, Gainesville, Florida and Quantum Chemistry Group, Uppsala University, Sweden Received 13 April 1965 Abstract The bracketing theorem in the partitioning technique for solving the Schr¨odinger equation may be used in principle to determine upper and lower bounds to energy eigenvalues. Practical lower bounds of any accuracy desired may be evaluated by utilizing the properties of ‘inner projections’ on finite manifolds in the Hilbert space. The method is here applied to the ground state and excited states of a Hamiltonian H = H0 + V having a positive definite perturbation V . Even if inspiration is derived from the method of intermediate Hamiltonians, the final results are of bracketing type and independent of this approach. The method is numerically illustrated in some accompanying papers.

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• In Perturbation theory and its applications in quantum mechanics, ed. C.H. Wilcox, p.255, Wiley, New York (1966) The Calculation of Upper and Lower Bounds of Energy Eigenvalues in Perturbation Theory by means of Partitioning Technique P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Abstract The partitioning technique for solving the eigenvalue problem HΨ = EΨ is briefly reviewed in matrix and operator form. It is shown that, if E, is a real variable; one may construct a single- or multi-valued function E1 = f (E) such that each pair E and E1 bracket at least one true eigenvalue E. If E is chosen as an upper bound by means of the variation principle, the function E1 is hence going to provide a lower bound. The partioning technique is here used to derive Brillouintype and Schr¨odinger-type perturbation expansions and, in the case of a positive definite perturbation V > 0, upper and lower bounds for the sums are determined by means of operator inequalities. By using the method of ‘inner projection’, it is further shown that the remainders in the Brillouin-type expansion may be evaluated with any accuracy desired, and the corresponding problem for the Schr¨odinger-type expansion is briefly discussed. Numerical applications are given elsewhere.

• International Journal of Quantum Chemistry 2, 867 (1968) Studies in Perturbation Theory. XIII. Treatment of Constants of the Motion in Resolvent Method, Partitioning Technique, and Perturbation Theory P.-O. L¨ owdin Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Abstract After a brief survey of some basic concepts in the theory of linear spaces, the eigenvalue problem is formulated in the resolvent technique based on the introduction of a reference function φ and a complex variable E. This leads to a series of fundamental concepts including the trial wave function, the inhomogeneous equation, and finally the transition and expectation values of the Hamiltonian, of which the former renders a ‘bracketing function’for the energy. In order to avoid the explicit limiting procedures in this approach, the eigenvalue problem is then reformulated in terms of the partitioning technique which, in turn, leads to a closed form of infinite-order perturbation theory. The eigenvalue problem is greatly simplified if the Hamiltonian H has a constant of motion Λ or has symmetry properties characterized by the group G = {g},

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and the question is now how these simplifications can be incorporated into the partitioning technique and into perturbation theory. In both cases, there exists a set of projection operators {Qk } which lead to a splitting of the Hilbert space into subspaces which have virtually nothing to do with each other. It is shown that, in the partitioning technique, it is sufficient to consider one of these subspaces at a time, and the results are then generalized to perturbation theory. It turns out that the finite-order expansions are no longer unique, and the commutation rules connecting the various forms are derived. The infinite-order results are finally presented in such a form that they are later suitable for the evaluation of upper and lower bounds to the energy eigenvalues.

• International Journal of Quantum Chemistry 5, 685 (1971) Studies in Perturbation Theory. XIV. Treatment of Constants of the Motion, Degeneracies and Symmetry Properties by Means of Multidimensional Partitioning P.-O. Lowdin and O. Goscinski Department of Quantum Chemistry, Uppsala University, Sweden and Quantum Theory Project, University of Florida, Gainesville, Florida Abstract The multi-dimensional partitioning technique is reviewed, with particular emphasis on its formal properties. The treatment of constants of motion by this method coincides with a treatment given previously, involving a modified resolvent in the ordinary approach. On the other hand, the present method, upon expansion, coincides with Van Vleck’s perturbation theory for quasidegenerate state. Bounds and estimates of the remainder related to Pad´e approximants obtained via inner projections are discussed. The construction of effective Hamiltonians is considered, in particular the energy-independent non self-adjoint ones.

B PROOF OF THE TIME-INDEPENDENT WICK’S THEOREM

The algebraic manipulation of products of creation and annihilation operators is greatly simplified by the time-independent Wick’s theorem [1, 2]. This theorem states that: If A is a product of creation and annihilation operators then (B.1)

A = {A} + {A}

where {A} represents the normal product form of A (see eq. (3.96)) and {A} represents the sum of the normal-ordered terms obtained by making all possible single, double, triple, . . . contractions within A (see eq. (3.108)). The proof is by induction. We assume that the theorem is valid for an operator A which is a product of N particle-hole creation and annihilation operators (x1 , x2 , . . . , xN ), that is (B.2)

A = {A} + {A}

where (B.3)

A = x1 x2 . . . xN

and show that this implies that the theorem is valid for the operator B, that is (B.4)

B = {B} + {B}

where B is given by (B.5) (B.6)

B = AxN +1 = x1 x2 . . . xN xN +1

and xN +1 is a particle-hole creation or annihilation operator. The general theorem is then proved by induction. In eq. (B.1) the term {A} is the sum of all normal ordered terms with single, double, triple, . . . contractions within A. Explicitly, {A} may be written

211

212

I. Hubac and S. Wilson 1 ! 1 ! 1 ! {A} = x1 x2 x3 . . . xN + x1 x2 x3 . . . xN + x1 x2 x3 . . . xN + · · · ' 1 & ! +

x1 x2 x3 x4 . . . xN & (B.7)

x1 x2 x3 x4 . . . xN

' x1 x2 x3 x4 . . . xN

+

! + · · · + x1 x2 x3 x4 x5 x6 . . . xN

1 + ····

From eq. (B.1) and eq. (B.5) we can write (B.8)

B = {A} xN +1 + {A}xN +1 .

We now consider two cases: (i) xN +1 is a particle-hole annihilation operator, b. b should be placed to the right in normal form. It can be placed inside the braces {} in the two terms on the right-hand side of eq. (B.8) giving (B.9)

{A} b = {Ab}

and (B.10)

{A}b = {Ab}.

All contractions vanish when b appears on the right. {Ab} therefore represents the sum of all contracted terms in B = Ab. Equation (B.4) is proved for case (i). (ii) xN +1 is a particle-hole creation operator, b† . b† should be moved to the left to obtain Ab† in normal form. Using the anticommutation relations (B.11)

xb† = xb† + b† x

where x is a creation or annihilation operator, we have x1 x2 x3 . . . xN −1 xN b† = −x1 x2 x3 . . . xN −1 b† xN + x1 x2 x3 . . . xN −1 xN b† (B.12) From the first term on the right-hand side of eq. (B.12) we obtained x1 x2 x3 . . . xN −1 b† xN = −x1 x2 x3 . . . b† xN −1 xN + x1 x2 x3 . . . xN −1 b† xN in which the second term on the right-hand side can be manipulated to yield x1 x2 x3 . . . xN −1 b† xN = −x1 x2 x3 . . . b† xN −1 xN − −x1 x2 x3 . . . xN −1 xN b† (B.13)

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Repeating this process, we are led to the equation x1 x2 x3 . . . xN −1 xN b† = (−1)N b† x1 x2 x3 . . . xN −1 xN + x1 x2 x3 . . . xN −1 xN b† + x1 x2 x3 . . . xN −1 xN b† + · · · + x1 x2 x3 . . . xN −1 xN b† (B.14) It can be assumed without any loss of generality that x1 x2 x3 . . . xN −1 xN is the normal form of A. We therefore have (B.15)

{A} b† = (−1) b† {A} + {Ab† } N

where the second term on the right-hand side represents the sum of all normal-ordered terms with contractions between b† and the operators that make up A. In the first term on the right-hand side of eq. (B.15), b† can be moved inside the braces {}, since in normal form creation operators appear on the left $ % (B.16) b† {A} = b† A . Moving the creation operator b† to the right inside the braces {} introduces a phase factor depending on the parity of the permutation required: $ % (B.17) b† {A} = (−1)N Ab† Equation (B.15) can then be written in the form % $ (B.18) {A} b† = Ab† + {Ab† } Following a similar procedure, it can be shown that $ % † $ †% / †0 (B.19) A b = Ab + Ab where the second term on the right-hand side is the sum of all normal-ordered terms with contractions between b† and the uncontracted operators of A. By combining eqs. (B.8), (B.18) and (B.19), we have % $ % $ $ Ab† = Ab† + Ab† + Ab† + {· · · We have now shown that if Wick’s theorem (B.1) is true for an arbitrary product of N creation/annihilation operators then it is also true for an arbitrary product of N + 1 such operators. Our proof is completed by noting that the theorem holds for the case N = 1. References 1. G.C. Wick, Phys. Rev. 80, 268, 1950 2. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson & Co., New York, 1961 Dover Publications, 2005; F.J. Dyson, Phys. Rev. 82, 428, 1951; N.H. March, W.H. Young and S. Sampanthar, The Many-Body Problem in Quantum Mechanics, Cambridge University Press, 1967 Dover Publications, 1996;

214

I. Hubac and S. Wilson S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson & Co., New York, 1961 Dover Publications, 2005; J. Paldus and J. Cizek, Adv. Quantum Chem. 9, 105, 1975; B.T. Pickup, in Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson, P.F. Bernath and S. Wilson, Volume 1: Fundamentals, Chapter 27: The Occupation Number Representation and the Many-Body Problem, pp. 435, John Wiley & Sons, Chichester, 2003

C DIAGRAMMATIC CONVENTIONS

There are a number of diagrammatic conventions commonly in use in the study of many-body systems. Each of these conventions employs diagrams which contain the same essential physical information. The different diagrammatic conventions differ in the economy they offer in representing the many-body perturbation expansion and in the complexity of the rules for translating a given diagram into the corresponding algebraic expression. In this appendix, we briefly describe each of the diagrammatic conventions in common usage. The rules for translating diagrams into the corresponding algebraic expressions are given in detail in Chapter 3 and will not be repeated here. The use of Feynman-like diagrams in describing electron correlation effects in many-body systems was introduced in the seminal paper by Goldstone. In Goldstone diagrams, holes are represented by downward directed lines

and particles are represented by upward directed lines

A one-body interaction is represented by a dashed horizontal line terminated at one end by a cross or some other suitable chosen symbol. Thus, for example, a Goldstone diagram component containing a one-electron interaction might typically have the form

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A two-body interaction in the Goldstone diagram consists of a horizontal dashed line, so a typical component of a Goldstone diagram containing a two-body interaction might have the form

Goldstone’s original treatment of the many-body problem employed time-dependent perturbation theory for what is essentially a time-independent problem. In the Goldstone picture, time is taken to progress from the bottom to the top of the diagram. For example, the diagram:

t2

t1

represents a hole-particle interaction at time t1 followed by a subsequent hole-particle interaction at time t2 .

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Now an entirely time-independent derivation of the linked diagram theorem and the many-body perturbation theory might seem more natural for a time-independent problem. Diagrammatic methods remain invaluable in the time-independent approach. However, in order to emphasize their time-independent interpretation, some authors turn the Goldstone diagrams clockwise through 90◦ . So, for example, the diagram shown above is displayed in the following manner:-

Particle lines are directed from left to right. Hole lines go from right to left. In the Goldstone diagrammatic convention, the diagrams

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and

are distinct. However, they differ only by exchange of the two particle lines involved. Brandow recognized that by including exchange in the two-body interaction a significant economy could be achieved. In the Brandow diagrams the vertices are said to be antisymmetrized. The two Goldstone diagrams shown above correspond to the single Brandow diagram:

The Brandow diagrams can be given a time-dependent or time-independent interpretation. The time-independent interpretation may be emphasized by employing time-independent Brandow diagrams such as:

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219

The third and final type of diagram in common usage is associated with Hugenholtz. The Hugenholtz diagrams are in one-to-one correspondence with those of Brandow. For example, the ‘time-dependent’ and ‘time-independent’ Brandow diagrams given immediately above correspond to the following Hugenholtz diagrams:

and

The two-body interaction is represented by a dot. This dot is identified as a two-body interaction by having a valency of four. A one-body interaction is represented in the Hugenholtz diagrammatic convention by a dot with a valency of two, for example

Different one-body operators are distinguished by employing different styles of dot, for example,

or by labelling the dot μ

D ¨ RAYLEIGH–SCHRODINGER PERTURBATION THEORY

Rayleigh–Schr¨odinger perturbation theory is usually developed by writing the exact Schr¨odinger equation in the form: (D.1)

(H0 + H1 ) Ψi (λ) = Ei (λ) Ψi (λ)

where H0 and H1 are respectively the reference Hamiltonian operator and the perturbation. λ is the perturbation parameter which will be set to unity. The eigenvalue, Ei (λ), and eigenfunction, Ψi (λ), are assumed to be continuous functions of λ, that is (D.2)

Ei (λ) =

∞

(p)

Ei λp

p=0

and (D.3)

Ψi (λ) =

∞

(p)

χi λp

p=0

The equations of Rayleigh–Schr¨odinger perturbation theory are obtained by substituting the expansions (D.2) and (D.3) in the exact Schr¨odinger equation (D.1) and equating powers of λ. Details are given in Section 1.3.2 (See also [1]). The Rayleigh–Schr¨odinger perturbation theory can also be derived from the Brillouin–Wigner perturbation theory by expanding the energy dependent denominators which occur in the latter expansion. This is the approach that we follow here since it emphasises the relation between the two expansions. An elementary derivation of the Brillouin–Wigner perturbation theory is given in Section 1.3.1. If we define the Rayleigh–Schr¨odinger perturbation theory resolvent operator as follows (see eq. (1.47)): (D.4) Ri = (Ei − Ek )−1 |Φk Φk | k=i

and the Brillouin–Wigner perturbation theory resolvent operator as follows (see eq. (1.15)): 221

222

I. Hubac and S. Wilson Bi =

(D.5)

(Ei − Ek )−1 |Φk Φk | ,

k=i

where Ei is the exact energy of the ith state and Ei is the energy of the ith state in the reference spectrum, then it is clear that by writing Ei = Ei + ΔEi ,

(D.6)

where ΔEi , the level shift, is given by the Taylor expansion (D.7)

ΔEi =

∞

(p)

Ei ,

p=1

or, explicitly including the perturbation parameter, ΔEi =

(D.8)

∞

(p)

λp Ei ,

p=1

one can obtain the relation Bi = Ri + Ri (−ΔEi ) Bi

(D.9) or

Bi = Ri + Ri (−ΔEi ) Ri + Ri (−ΔEi ) Ri (−ΔEi ) Ri + Ri (−ΔEi ) Ri (−ΔEi ) Ri (−ΔEi ) Ri + · · · .

(D.10)

which follows from the identity (D.11) (Ei − Ek )−1 = (Ei − Ek )−1 + (Ei − Ek )−1 (−ΔEi )(Ei − Ek )−1 . The Brillouin–Wigner perturbation expansion for the energy of the ith state takes the form (see eq. (1.20)): Ei = Ei + Φi | H1 |Φi + Φi | H1 Bi H1 |Φi + Φi | H1 Bi H1 Bi H1 |Φi + Φi | H1 Bi H1 Bi H1 Bi H1 |Φi + · · ·

(D.12)

The Brillouin–Wigner energy expansion has the form of a simple geometric series in H1 . Explicitly, the coefficients in the Brillouin–Wigner energy expansion are therefore (1)

Ei

= Φi | H1 |Φi

(2) Ei (3) Ei (4) Ei

= Φi | H1 Bi H1 |Φi

(D.13)

= Φi | H1 Bi H1 Bi H1 |Φi = Φi | H1 Bi H1 Bi H1 Bi H1 |Φi ...

(n) Ei

= Φi | H1 (Bi H1 )n−1 |Φi .

Brillouin–Wigner Methods for Many-Body Systems

223

(p)

The energy coefficients Ei in the Brillouin–Wigner expansion can be written in sum-over-states form as follows: (2)

Ei

(3)

Ei

=

Φi | H 1 |Φk Φk | H 1 |Φi (Ei − Ek ) k=i

Φi | H 1 |Φk Φk | H 1 |Φk Φk | H 1 |Φi (Ei − Ek ) (Ei − Ek )

=

k,k =i (4) Ei

=

k,k ,k =i

Φi | H 1 |Φk Φk | H 1 |Φk Φk | H 1 |Φk Φk | H 1 |Φi (Ei − Ek ) (Ei − Ek ) (Ei − Ek )

... (n)

Ei

= k,k ,...,kn−1 =i

Φi | H 1 |Φk Φk | H 1 |Φk Φk | H 1 . . . H 1 Φkn−1 Φkn−1 H 1 |Φi (Ei − Ek ) (Ei − Ek ) . . . Ei − Ekn−1

(D.14)

Using (D.9) in (D.13) (2)

Ei

= Φi | H1 Ri H1 |Φi + (−ΔEi ) Φi | H1 Ri2 H1 |Φi 2

+ (−ΔEi ) Φi | H1 Ri3 H1 |Φi + . . . (3)

Ei

= Φi | H1 Ri H1 Ri H1 |Φi + (−ΔEi ) Φi | H1 Ri H1 Ri2 H1 |Φi 2

+ (−ΔEi ) Φi | H1 Ri2 H1 Ri H1 |Φi + (−ΔEi ) Φi | H1 Ri2 H1 Ri2 H1 |Φi + (−ΔEi )2 Φi | H1 Ri H1 Ri3 H1 |Φi + (−ΔEi )2 Φi | H1 Ri3 H1 Ri H1 |Φi + ... (4) Ei

= Φi | H1 Ri H1 Ri H1 Ri H1 |Φi + (−ΔEi ) Φi | H1 Ri H1 Ri H1 Ri2 H1 |Φi + (−ΔEi ) Φi | H1 Ri H1 Ri2 H1 Ri H1 |Φi + (−ΔEi ) Φi | H1 Ri2 H1 Ri H1 Ri H1 |Φi 2

+ (−ΔEi ) Φi | H1 Ri H1 Ri2 H1 Ri2 H1 |Φi 2

+ (−ΔEi ) Φi | H1 Ri2 H1 Ri H1 Ri2 H1 |Φi + (−ΔEi )2 Φi | H1 Ri2 H1 Ri2 H1 Ri H1 |Φi + . . . ... (n) Ei

n−1

= Φi | H1 (Ri H1 )

|Φi + · · ·

(D.15) Introducing the expansion (D.7) (or (D.8)) and collecting terms of the same order in λ we can obtain the Rayleigh–Schr¨odinger perturbation expansion for the ith state.

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I. Hubac and S. Wilson (2)

Ei

= Φi | H1 Ri H1 |Φi

(3) Ei (4) Ei

= Φi | H1 Ri H1 Ri H1 |Φi − Ei

(1)

Φi | H1 Ri2 H1 |Φi (1)

= Φi | H1 Ri H1 Ri H1 Ri H1 |Φi − Ei Φi | H1 Ri2 H1 Ri H |Φi 2 (1) (1) Φi | H1 Ri3 H1 |Φi − Ei Φi | H1 Ri H1 Ri2 H |Φi + Ei (2)

− Ei

(D.16)

Φi | H1 Ri2 H1 |Φi .

In each order, the Rayleigh–Schr¨odinger energy coefficient consists of a principal term of the form (D.17)

n

Φi | H1 (Ri H1 ) |Φi . (p)

The energy coefficients Ei in the Rayleigh–Schr¨odinger expansion can be written in sum-over-states form as follows (2)

Ei

=

Φi | H 1 |Φ Φ | H 1 |Φi k k (Ei − Ek ) k=i

(3)

Ei

Φi | H 1 |Φ Φ | H 1 |Φ Φ | H 1 |Φi k k k k (Ei − Ek ) (Ei − Ek )

=

k,k =i

(1)

− Ei (4)

Ei

=

Φi | H 1 |Φ Φ | H 1 |Φi k k k=i

k,k ,k =i (1)

− 2Ei

(Ei − Ek )2

Φi | H 1 |Φk Φk | H 1 |Φk Φk | H 1 |Φk Φk | H 1 |Φi (Ei − Ek ) (Ei − Ek ) (Ei − Ek ) Φi | H 1 |Φk Φk | H 1 |Φk Φk | H 1 |Φi

k,k =i

(Ei − Ek )2 (Ei − Ek )

Φ | H |Φ Φ | H |Φ (1) 2 1 1 i i k k + Ei 3 (E − E ) i k k=i (2)

− Ei

Φi | H 1 |Φk Φk | H 1 |Φi

k=i

(D.18)

(Ei − Ek )2

...

with a principal term of the form (n) Ei = k,k ,...,kn−1 =i

Φi | H1 |Φk Φk | H1 |Φk Φk | H1 . . . H1 Φkn−1 Φkn−1 H1 |Φi (Ei − Ek ) (Ei − Ek ) . . . Ei − Ekn−1 (D.19)

+ ...

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References 1. S. Wilson, in Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson, P.F. Bernath and Roy McWeeny, volume 2: Molecular Electronic Structure, Chapter 8: Perturbation Theory, John Wiley & Sons, Chichester, 2003

INDEX

n-product, 88 n-state system, 66 The Physical Review, 191 a posteriori Brillouin–Wigner correction to limited multi-reference conﬁguration interaction, 176 a posteriori correction, 135, 136, 164, 166, 167, 171, 175, 176, 179, 183, 184, 193, 196 cas-scf, 196 cas-vb, 196 ccsd(t), 194 gvb, 196 mp2, 135 mp3, 135 mp4, 135 mr-mp2, 135 mr-mp3, 135 mr-mp4, 135 bch, 142 bwccdt, 162 bwccd, 162 bwccsd, 162 bwcc, 137 bwcc, linear scaling corrections, 162 bwcc, one-state formulation of the multi-reference, 155 bwcc, single-reference, 137 bwcc, single-root formulation of the multi-reference, 155 bwcc, state-selective formulation of the multi-reference, 155

bwci, ‘many-body’ corrections for, 170 bwpt2, 193 bwpt, 177 bwpt and limited ci, 167 bwpt, multi-reference, 179 bwpt, multi-reference second order, 179 bwpt, one-state formulation of the multi-reference, 156 bwpt, single-reference, 177 bwpt, single-root formulation of the multi-reference, 156 bwpt, state-selective formulation of the multi-reference, 156 ccd, 135, 143, 162 ccsd(t), 136, 194 ccsdtq, 194 ccsdt, 135, 143, 162, 194 cc, 135, 155 ci d, 135 ci sd, 135, 166, 167, 169, 170 ci, 135, 154 cpa, 135 cpmet, 135, 140 fci, full conﬁguration interaction, 134, 165 mbpt2, 135 mbpt3, 135 mbpt4, 135 mp2, 177, 195 mr bwccsd, one-state, 159 mr bwccsd, single-root, 159 mr bwccsd, state-speciﬁc, 159

227

228 mr bwcc, 143, 148, 152, 158, 159 mr bwcc, multi-root, 148 mr bwcc, multi-root formulation of, 145 mr bwcc, one-state, 158 mr bwcc, one-state formulation of, 155 mr bwcc, single-root, 158 mr bwcc, single-root formulation of, 155 mr bwcc, state-selective, 158 mr bwcc, state-selective formulation of, 155 mr bwci, 175 mr bwci for limited conﬁguration interaction, 171 mr bwpt, a posteriori correction to, 183 mr bwpt, one-state formulation of, 156 mr bwpt, single-root formulation of, 156 mr bwpt, state-selective formulation of, 156 mr ccsd-1, 153 mr ccsd-2, 153 mr ccsd-3 , 153 mr ccsd, linear, 153 mr l-ccsd, 153 mr-bwccsd, 193 mr-bwci, 171 mr-ccd, 135 mr-ccsdt, 135 mr-ccsd, 135 mr-ci d, 135 mr-ci sd, 135 mr-cisd, 166, 167, 175, 176 mr-mbpt2, 135 mr-mbpt3, 135 mr-mbpt4, 135 pt, 135 rspt, 134, 135, 137, 145, 163, 169, 177, 184 rspt and limited ci, 169 uga, 155 a posteriori correction to mr bwpt, 183 a posteriori correction to multi-reference Brillouin–Wigner perturbation theory, 183 ˇ ıˇzek, J., 4, 137, 140, 155 C´

Index ‘black box’ quantum chemical software package, 136 ‘many-body’ corrections for Brillouin– Wigner limited conﬁguration interaction, 170 a posteriori corrections to Brillouin– Wigner perturbation theory, 30 activation energy, 2 additive separability, 73 advantages of the Brillouin–Wigner theory, 18 algebraic approximation, 3, 4, 96, 165 analytic basis set, 3 annihilation operator, 81, 87 anticommutation relation, 87 approximation, algebraic, 96 augmented coupled cluster theories, 194 backdoor intruder states, 28 Baker–Campbell–Hausdorﬀ expansion, 142, 152 Balkov´ a, A., 149 Banerjee, A., 155 barriers to rotation, 2 Bartlett, R.J., 149, 151, 153–155, 159, 184 basis function, Gaussian, 3, 96 basis set, 3, 4 Berkoviv, S., 149 binding energy, 2 Bloch equation, 140, 141, 148, 151, 152 Bloch-like equation, 137, 140, 141, 145, 146, 148, 150, 163, 164, 168, 173, 181 Born–Oppenheimer approximation, 70 Boys, S.F., 3 Brandow, B.H., 25, 30, 76, 107, 148 Brillouin’s 1932 Paper, 7 Brillouin, L., 5, 7, 19 Brillouin–Wigner conﬁguration interaction theory, 164 Brillouin–Wigner conﬁguration interaction theory, multi-reference, 171 Brillouin–Wigner coupled cluster singleand double-excitations approxi-

Index mation, one-state multi-reference, 159 Brillouin–Wigner coupled cluster singleand double-excitations approximation, single-root multi-reference, 159 Brillouin–Wigner coupled cluster singleand double-excitations approximation, state-speciﬁc multi-reference, 159 Brillouin–Wigner coupled cluster singles and doubles theory, 30 Brillouin–Wigner coupled cluster theory, 30, 137 Brillouin–Wigner coupled cluster theory, linear scaling corrections, 162 Brillouin–Wigner coupled cluster theory, multi-reference, 143, 152 Brillouin–Wigner coupled cluster theory, single-reference, 137 Brillouin–Wigner coupled cluster, multi-root multi-reference, 148 Brillouin–Wigner coupled cluster, one-state multi-reference, 158 Brillouin–Wigner coupled cluster, single-root multi-reference, 158 Brillouin–Wigner coupled cluster, state-selective multi-reference, 158 Brillouin–Wigner coupled-cluster theory, one-state formulation of the multi-reference, 155 Brillouin–Wigner coupled-cluster theory, single-root formulation of the multi-reference, 155 Brillouin–Wigner coupled-cluster theory, state-selective formulation of the multi-reference, 155 Brillouin–Wigner form, multi-reference conﬁguration interaction in, 175 Brillouin–Wigner full conﬁguration interaction, 28 Brillouin–Wigner limited conﬁguration interaction, ‘many-body’ corrections for, 170 Brillouin–Wigner methods, reemergence of , 26 Brillouin–Wigner perturbation expansion, derivation of Rayleigh–

229 Schr¨ odinger perturbation theory from, 56 Brillouin–Wigner perturbation theory, 5, 7, 17, 26, 30, 51, 133, 177 Brillouin–Wigner perturbation theory and limited conﬁguration interaction, 167 Brillouin–Wigner perturbation theory through second order, bwpt2, 193 Brillouin–Wigner perturbation theory, a posteriori corrections to, 30 Brillouin–Wigner perturbation theory, basic structure of, 12 Brillouin–Wigner perturbation theory, comparison with Rayleigh– Schr¨ odinger perturbation theory, 17 Brillouin–Wigner perturbation theory, generalized, 53 Brillouin–Wigner perturbation theory, multi-reference, 28, 64, 179 Brillouin–Wigner perturbation theory, multi-reference second order, 179 Brillouin–Wigner perturbation theory, one-state formulation of the multi-reference, 156 Brillouin–Wigner perturbation theory, single-reference, 177 Brillouin–Wigner perturbation theory, single-root formulation of the multi-reference, 156 Brillouin–Wigner perturbation theory, state-selective formulation of the multi-reference, 156 Brillouin–Wigner perturbation theory, a posteriori corrections to, 30 Brillouin–Wigner resolvent, 17, 52 Brillouin–Wigner theory, advantages of the, 18 Brillouin–Wigner theory, disadvantages of the, 22 Brillouin–Wigner type propagator, 13 Brillouin–Wigner type resolvent, 13 Brueckner, K.A., 3, 4, 25, 76, 134, 166, 191 Buenker, R.J., 166

230 ccsd, 134–136, 143, 151–153, 159, 160, 162 Coester, F., 3, 137 complete conﬁguration interaction, 165 computers, digital, 2 computing, high performance, 4 condensed matter physics, 80 condensed matter theory, 197 conﬁguration interaction, 4 conﬁguration interaction expansion, 134–136, 155, 164–167, 169, 171, 175, 176 conﬁguration interaction theory, Brillouin–Wigner, 164 conﬁguration interaction theory, single-reference Brillouin–Wigner, 167 conﬁguration interaction, complete, 165 conﬁguration interaction, full (fci), 134, 165 construction of diagrams, 101 contraction, 87, 88 correlation eﬀects, 2 correlation energy, second order contribution to the, 101 coupled cluster expansion, 135–137, 140, 142, 143, 145, 149–151, 155, 156, 158, 159, 162, 164 coupled cluster single- and doubleexcitations approximation, one-state multi-reference Brillouin–Wigner, 159 coupled cluster single- and doubleexcitations approximation, single-root multi-reference Brillouin–Wigner, 159 coupled cluster single- and doubleexcitations approximation, state-speciﬁc multi-reference Brillouin–Wigner, 159 coupled cluster singles and doubles theory, Brillouin–Wigner, 30 coupled cluster theory, 4, 193 coupled cluster theory, Brillouin– Wigner, 30 coupled pair approximation, 135 coupled pair many electron theory, 135, 140

Index Cramer, C.J., 194 creation and annihilation operators, products of, 94 creation operator, 81, 87 Dalgarno, A., 5, 7, 18, 19, 21 Davidson, E.R., 76, 166 Demel, O., 194 diagram construction, 101 diagrammatic techniques, 3, 80 diagrams, Goldstone, 104 diagrams, Hugenholtz, 104 digital computers, 2 Dirac spectrum, 79 Dirac, P.A.M., 2, 79 disadvantages of the Brillouin–Wigner theory, 22 dissociative processes, 5 Dyson, F., 3 eﬀective Hamiltonian, 39, 45 electron correlation, many-body theories of, 76 electron–positron pair, 80 electronic Schr¨ odinger equation, 70 extensivity, 25, 70, 133, 135, 136, 150–152, 163, 164, 166, 167, 183 extensivity, size, 133, 135, 136, 150–152, 163, 164, 166, 167, 183 Fermi statistics, 88 Fermi vacuum, 80 Fermi vacuum expectation value, 89 Feshbach, H., 37 Feynman, R.P., 3 ﬁnite analytic basis set, 3 ﬁnite basis set expansion, 4 ﬁrst order wave function in Rayleigh– Schr¨ odinger perturbation theory, 16 ﬂuctuation potential, 74, 94, 95 Fock space approach, 149 Freeman, D.L., 25, 26, 77 full conﬁguration interaction, 28 full conﬁguration interaction (fci), 134, 165 full conﬁguration interaction, Brillouin– Wigner, 28

Index

231

Gaussian basis function, 3, 96 generalized Brillouin–Wigner perturbation theory, 53 generalized Wick’s theorem, 90, 101 Gerratt, J., 194 Goldstone diagrams, 104 Goldstone, J., 3, 4, 24, 76, 134 Goscinski, O., 5, 10

intruder states, backdoor, 28 inverse operator, expansion of, 48

Hall, G.G., 3 Hamiltonian operator in normal form, 94 Harris, F., 25, 26 Harris, F.E., 77, 155 Hartree–Fock approximation, matrix, 96 Hartree–Fock equations, 3 Hartree–Fock model, 191, 192 high performance computing, 4 Hilbert space approach, 148, 149, 153, 158 Hilbert space approach, multi-root multi-reference Brillouin–Wigner coupled cluster, 148 Hilbert space approach, one-state multi-reference Brillouin–Wigner coupled cluster, 158 Hilbert space approach, single-root multi-reference Brillouin–Wigner coupled cluster, 158 Hilbert space approach, state-selective multi-reference Brillouin–Wigner coupled cluster, 158 Hirao, K., 155 Historical background, 5 Hoﬀmann, M., 155 Hubaˇc, I., 28, 30, 134, 142, 160, 168, 193 Hubbard, J., 3 Hugenholtz diagrams, 104 Hugenholtz, N.M., 3, 25

K¨ ummel, H., 3, 137 Kaldor, U., 4, 149 Kato, T., 146 Kelly, H.P., 4, 134 Klein, O., 79 Kucharski, S.A., 149, 151, 154, 159, 184

incomplete model space, 150 independent particle model, 2, 3, 70, 77 Intel, 4 interaction energy, 2 intruder state problem, 27, 134, 135, 149, 150, 155, 162–164, 177, 183, 192 intruder states, 28

Jankowski, K., 149, 152, 155, 166 Jeziorski, B., 149, 152, 153, 155, 158, 159 Jordan, P., 79 Jorgensen, P., 78

L¨ owdin, P.-O., 5, 10, 37 Laidig, W.D., 155 Langhoﬀ, S.R., 166 Lennard-Jones’ 1930 Paper, 5 Lennard-Jones, J.E., 5, 18, 19 Lennard-Jones–Brillouin–Wigner perturbation theory, 5, 7 Li, X., 155, 196 limited ci and rspt, 169 limited ci, bwpt and, 167 limited conﬁguration interaction expansion, 166 limited conﬁguration interaction, Brillouin–Wigner perturbation theory and, 167 limited conﬁguration interaction, multi-reference Brillouin–Wigner perturbation theory, 171 limited multi-reference conﬁguration interaction, a posteriori Brillouin–Wigner correction to, 176 Lindgren, I., 25, 26, 37, 76, 138, 184 linear mr ccsd, 153 linear scaling, 25, 69 linear scaling corrections, 162 linear scaling of the energy, 133 linear scaling with particle number, 135, 150, 162 linked cluster theorem, 141–143, 162 linked diagram theorem, 3, 4 Lippmann–Schwinger method, 13

232 Lippmann–Schwinger-like equation, 137, 167, 168, 171, 173, 175, 181, 183 M´ aˇsik, J., 30 Møller, Ch., 191 Møller–Plesset perturbation theory, 135, 177 Møller–Plesset theory, 191 many-body perturbation theory, 3, 4, 30, 77 many-body problem, 25 many-body problem and quasiparticles, 94 many-body problem, second quantization and, 78 many-body Rayleigh–Schr¨ odinger perturbation theory, 94, 98 many-body theories of electron correlation, 76 many-body theory, 133–135, 137, 150, 164, 165, 183, 191–193, 197 many-electron correlation eﬀects, 2 March, N.H., 23–26, 76, 134 material science, 197 matrix Hartree–Fock approximation, 96 matrix Hartree–Fock equations, 3 mbpt2, 193 McWeeny, R., 3 Meissner, L., 149, 151, 152, 154, 159, 166 Messiah, A, 78 Mills, I.M., 194 model electronic Hamiltonian, 73 model function, 38, 44 molecular quantum mechanics, 2 Monkhorst, H.J., 25, 26, 77, 149, 152, 158, 159 Moore’s Law, 4 Moore, G., 4 Morrison, J., 25, 26, 37, 76, 138, 184 mp2, 193 Mukherjee, D., 149, 155, 194 multi-reference bwpt, 179 multi-reference Brillouin–Wigner conﬁguration interaction theory, 171 multi-reference Brillouin–Wigner coupled cluster single- and

Index double-excitations approximation, one-state, 159 multi-reference Brillouin–Wigner coupled cluster single- and double-excitations approximation, single-root, 159 multi-reference Brillouin–Wigner coupled cluster single- and double-excitations approximation, state-speciﬁc, 159 multi-reference Brillouin–Wigner coupled cluster theory, 143, 152 multi-reference Brillouin–Wigner coupled cluster theory, multi-root formulation of, 145 multi-reference Brillouin–Wigner coupled cluster, multi-root, 148 multi-reference Brillouin–Wigner coupled cluster, one-state, 158 multi-reference Brillouin–Wigner coupled cluster, single-root, 158 multi-reference Brillouin–Wigner coupled cluster, state-selective, 158 multi-reference Brillouin–Wigner coupled-cluster theory, one-state formulation of, 155 multi-reference Brillouin–Wigner coupled-cluster theory, single-root formulation of, 155 multi-reference Brillouin–Wigner coupled-cluster theory, stateselective formulation of, 155 multi-reference Brillouin–Wigner perturbation theory, 28, 64, 179 multi-reference Brillouin–Wigner perturbation theory for limited conﬁguration interaction, 171 multi-reference Brillouin–Wigner perturbation theory, a posteriori correction to, 183 multi-reference Brillouin–Wigner perturbation theory, one-state formulation of, 156 multi-reference Brillouin–Wigner perturbation theory, single-root formulation of, 156

Index multi-reference Brillouin–Wigner perturbation theory, state-selective formulation of, 156 multi-reference conﬁguration, 166 multi-reference conﬁguration interaction, 166 multi-reference conﬁguration interaction in Brillouin–Wigner form, 175 multi-reference conﬁguration interaction with single and double excitations, 166, 175, 176 multi-reference function perturbation expansion, 58 multi-reference functions, 5 multi-reference Rayleigh–Schr¨ odinger perturbation theory, 27, 28, 58 multi-reference second order bwpt, 179 multi-reference second order Brillouin– Wigner perturbation theory, 179 multi-root formulation of mr bwcc, 145 multi-root formulation of multi-reference Brillouin–Wigner coupled cluster theory, 145 multi-root multi-reference Brillouin– Wigner coupled cluster: Hilbert space approach, 148 multiple solutions of the coupled cluster equation, 149 Nakatsuji, H., 155 negative energy branch (of the Dirac spectrum), 80 negative energy sea, 80 Neogr´ ady, 193 Neogr´ ady, P., 30, 134, 142, 168 Neogrady, P., 28 Nobel Prize, 195 normal form, Hamiltonian operator in, 94 normal product, 87, 88 nuclear physics, 3, 197 occupation number representation, 81 one-state mr bwccsd, 159 one-state formulation of mr bwcc, 155 one-state formulation of mr bwpt, 156

233 one-state formulation of the multireference Brillouin–Wigner coupled-cluster theory, 155 one-state formulation of the multireference Brillouin–Wigner perturbation theory, 156 one-state multi-reference Brillouin– Wigner coupled cluster single- and double-excitations approximation, 159 one-state multi-reference Brillouin– Wigner coupled cluster: Hilbert space approach, 158 Pad´e approximants, 192 pairing, 88 Pal, S., 149 Paldus, J., 3, 4, 137, 143, 149, 152, 153, 155, 160, 165, 196 particle picture, 80 particle–hole formalism, 90 particle–hole picture, 80 partitioning technique, 10, 37 partitioning technique for a multireference function, 44 partitioning technique for a singlereference function, 38 Pauli exclusion principle, 83 perturbation expansion, 48 perturbation expansion, multi-reference function, 58 perturbation expansion, single-reference function, 48 perturbation theory, 1, 2, 12 perturbation theory, a posteriori corrections to Brillouin–Wigner, 30 perturbation theory, basic structure of Brillouin–Wigner, 12 perturbation theory, Brillouin–Wigner, 5, 7, 17, 26, 30, 51, 133, 177 perturbation theory, Brillouin–Wigner, comparison with Rayleigh– Schr¨ odinger perturbation theory, 17 perturbation theory, generalized Brillouin–Wigner, 53

234 perturbation theory, Lennard-Jones– Brillouin–Wigner, 7 perturbation theory, many-body, 3, 4, 30, 77 perturbation theory, many-body Rayleigh–Schr¨ odinger, 94 perturbation theory, many-body Rayleigh–Schr¨ odinger perturbation, 98 perturbation theory, multi-reference Brillouin–Wigner, 28, 64 perturbation theory, multi-reference Rayleigh–Schr¨ odinger, 27, 28, 58 perturbation theory, Rayleigh– Schr¨ odinger, 9, 14, 15, 17, 30, 50, 134, 135, 137, 145, 163, 169, 177, 184 perturbation theory, Rayleigh– Schr¨ odinger, and limited conﬁguration interaction, 169 perturbation theory, Rayleigh– Schr¨ odinger, comparison with Brillouin–Wigner perturbation theory, 17 Perturbation Theory, Studies in, 10 perturbation theory, a posteriori corrections to Brillouin–Wigner, 30 Pittner, J., 193–196 Plesset, M.S., 191 Pople, J.A., 191, 195 positive energy branch (of the Dirac spectrum), 80 positron, 80 products of creation and annihilation operators, 94 projection operator, 38, 44 propagator, Brillouin–Wigner type, 13 Pulay, P., 194 quantum chemical software package, ‘black box’, 136 quantum ﬁeld theory, 2, 3 quasiparticles, many-body problem and, 94 Quiney, H.M., 195 Rayleigh, J.W.S., 9

Index Rayleigh–Schr¨ odinger perturbation theory, 9, 14, 15, 17, 30, 50, 134, 135, 137, 145, 163, 169, 177, 184 Rayleigh–Schr¨ odinger perturbation theory and limited conﬁguration interaction, 169 Rayleigh–Schr¨ odinger perturbation theory, comparison with Brillouin– Wigner perturbation theory, 17 Rayleigh–Schr¨ odinger perturbation theory, derivation from the Brillouin–Wigner perturbation expansion, 56 Rayleigh–Schr¨ odinger perturbation theory, ﬁrst order wave function, 16 Rayleigh–Schr¨ odinger perturbation theory, many-body, 94, 98 Rayleigh–Schr¨ odinger perturbation theory, multi-reference, 27, 28, 58 Rayleigh–Schr¨ odinger perturbation theory, second order energy, 16 Rayleigh–Schr¨ odinger resolvent, 50 re-emergence of Brillouin–Wigner methods, 26 reaction operator, 38, 43, 48 relativistic formalism, 2 resolvent, Brillouin–Wigner, 17, 52 resolvent, Brillouin–Wigner type, 13 resolvent, Rayleigh–Schr¨ odinger, 50 Roberts, J.M., 4 Roman, P., 78 Ron, A., 4 Roothaan, C.C.J., 3 Sampanthar, 134 Sampanthar, S., 23–26, 76 Saute, M., 155 scaling, linear, 25, 69 Schaefer, iii, H.F., 155, 194 Schr¨ odinger equation, 1, 15 Schr¨ odinger equation, electronic, 70 Schr¨ odinger, E., 9 Schweber, S.S., 78 second order contribution to the correlation energy, 101

Index second order energy in Rayleigh– Schr¨ odinger perturbation theory, 16 second quantization and the many-body problem, 78 second-quantization formalism, 78 self-consistent ﬁeld method, 3 separability, additive, 73 Shavitt, I., 4, 25, 166 Silver, D.M., 4, 191 Simons, J., 78, 155 single-reference bwcc, 137 single-reference bwpt, 177 single-reference Brillouin–Wigner conﬁguration interaction theory, 167 single-reference Brillouin–Wigner coupled cluster theory, 137 single-reference Brillouin–Wigner perturbation theory, 177 single-reference conﬁguration interaction expansion, 167 single-reference function perturbation expansion, 48 single-root mr bwccsd, 159 single-root formulation of mr bwcc, 155 single-root formulation of mr bwpt, 156 single-root formulation of the multireference Brillouin–Wigner coupled-cluster theory, 155 single-root formulation of the multireference Brillouin–Wigner perturbation theory, 156 single-root multi-reference Brillouin– Wigner coupled cluster single- and double-excitations approximation, 159 single-root multi-reference Brillouin– Wigner coupled cluster: Hilbert space approach, 158 size extensivity, 133, 135, 136, 150–152, 163, 164, 166, 167, 183 size-consistency, 25, 70 size-extensivity, 25, 70 Slater determinant, 83, 91, 92

235 Slater, J.C., 83 ˇ Smydke, J., 195 solid state physics, 3 solid state theory, 197 spectators, 91 state-selective formulation of mr bwcc, 155 state-selective formulation of mr bwpt, 156 state-selective formulation of the multi-reference Brillouin–Wigner coupled-cluster theory, 155 state-selective formulation of the multi-reference Brillouin–Wigner perturbation theory, 156 state-selective multi-reference Brillouin– Wigner coupled cluster: Hilbert space approach, 158 state-speciﬁc mr bwccsd, 159 state-speciﬁc multi-reference Brillouin– Wigner coupled cluster single- and double-excitations approximation, 159 Steiner, M.M., 28 Studies in Perturbation Theory, 10 theoretical models, 1 two-state problem, 19 unitary group approach, 155 valence-universal approach, 149 wave operator, 37, 40, 47 Weinberg, S., 78 well-separated systems, 71 Wenzel, W., 28 Wick’s theorem, 87, 90, 91, 94 Wick’s theorem, generalized, 90, 101 Wick’s theorem, proof of, 90 Wigner’s 1935 Paper, 8 Wigner, E.P., 5, 8, 19, 21, 22, 79, 193 Wilson, S., 4, 5, 7, 19, 25, 26, 77, 191, 193 Young, W.H., 23–26, 76, 134 Ziman, J., 78