FUNDAMENTAL WORLD OF QUANTUM CHEMISTRY Fundamental World of Quantum Chemistry A Tribute to the Memory of Per-Olov Lowdin Volume III Edited by Erkki J. Brandas Uppsala University, Uppsala, Sweden and Eugene S. Kryachko Bogoliubov Institute for Theoretical Physics, Kiev, Ukraine and University of Leuven, Leuven, Belgium SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6687-9 ISBN 978-94-017-0448-9 (eBook) DOI 10.1007/978-94-017-0448-9 Printed on acid-free paper Cover Picture: Lowdin's asymmetrical wave function 'I' in love with Hamiltonian All Rights Reserved © 2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004. Softcover reprint of the hardcover 1st edition 2004 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. Photo: Per-Olov Lowdin (Michael Hehenberger. Property of the Uppsala Quantum Chemistry Group) PREFACE TO VOLUME III It is with respect and devotion that we point out Per-Olov Lowdin's pioneering and limitless efforts to found and establish the science of Quantum Chemistry. With his extraordinary capacity, Per-Olov Lowdin made basic contributions to the development, provided the conceptual framework and introduced fundamental concepts, and added the rigour to the field. Let us remind the reader of Lowdin's definition of a molecule as a Coulombic system of electrons and nuclei whose Hamiltonian with the center of mass translational degrees of freedom removed, has at least a bound state (in Molecules in Physics, Chemistry and Biology II, J. Maruani (Ed.), Kluwer, Dordrecht, 1988). But let us also recall his critique, i. e. of the impossibility to derive the Coulomb Hamiltonian from the first principles and the non-unique existence of molecular structures due to the general incompatibility between the number of nuclear positions and the number of internuclear distances. One of the most urgent problems of modem quantum chemistry is to treat the motions of the atomic nuclei and the electrons on a more or less equivalent basis. Per-Olov Lowdin also established, developed and strengthened the links between physics, chemistry and biology. In addition, by organizing a staggering amount of regular summer and winter schools, institutes and symposia, by founding the International Journal of Quantum Chemistry and the Advances of Quantum Chemistry, he commended undisputable leadership at the highest national and international levels. The clarity of expression, the vision of the possible and the optimism for the future are beautifully reflected in his "Present Situation of Quantum Chemistry" published in J. Phys. Chem. 61, 55-68 (1957), where, with eagerness of reaching a wide audience and capability of attracting many scientists, Per Olov Lowdin introduced and defined the field of Quantum Chemistry by formulating its goals, i.e. (i) to be able to predict accurately the properties of a hypothetical polyatomic molecule before it is synthezised in the laboratories; (li) to obtain such knowledge of the electronic structure of matter that one can construct new substances of particular value to mankind; (iii) to learn to think in terms of electrons and their quantum mechanical behaviour. In his revelations "Quantum Theory of Many Particle Systems I-IIf' (Phys. Rev. 97, 1474-1520 (1955», Per-Olov Lowdin made a rigorous and ground-breaking formulation of the all important correlation problem in Vlll quantum chemical applications, from which emanated fundamental concepts like the correlation energy, the natural spin orbital, the general configuration interaction (CI) method, the reduced density matrix and the charge and bond order matrix. The constraints imposed by symmetry considerations in various independent particle models termed the "symmetry dilemma" (Rev. Mod. Phys. 35, 496 (1963)) have further led to important developments in solid state physics, chemical physics and physical chemistry. Well-known is Lowdin's series of "Studies in Perturbation Theory l XIV", a framework for the analysis of intermolecular forces as well as the technical development of very large CI expansions. The impact of his work on Quantum Biology are to be found in new applications and developments, i.e. in possible conformation work on prions, proton trans correlations in DNA, macromolecular recognitions and specific motifs, in cancer research, etc. where there is a highly relevant quantum mechanical component to be identified. These accomplishments together with close to 70 summer and winter schools and Sanibel Symposia have contributed to the education of several generations of quantum chemists and solid state physicists. It is a great privilege and honor to have participated in this edification. Looking at the picture of Per-Olov Lowdin, climbing in the mountains, we recollect the famous song by the Prophetes Deborah after the victory over the Philistines: "The mountains flowed before the Lord." This is precisely whom he was, is and will be -Per-Olov Lowdin or Pelle for all of us. Indeed, these years, when he was with us, were the most colourful, exciting and fascinating period in the history of quantum chemistry. It was the most enjoyable time for all of us because we all worked in quantum chemistry "just for FUN", as he always maintained. He will always be with us, the whole family of Quantum Chemists -including his own family with the "mother of the Uppsala Quantum Chemistry Group", Karin Lowdin, as prima motor and inner connection, and he is particularly close to all who have contributed to this tribute volume to his memory appearing as Volumes 4 and 5 in the Series "Conceptual Trends in Quantum Chemistry", already included Volume 1, "Conceptual Trends in Quantum Chemistry" (1994), Volume 2, "Structure and Dynamics of Atoms and Molecules: Conceptual Trends" (1995), and Volume 3, "Conceptual Perspectives in Quantum Chemistry" (1997), edited by Jean-Louis Calais and Eugene Kryachko. We deeply thank all authors for their wonderful and profound contributions. Erkki 1. Brandas Eugene S. Kryachko Uppsala University Bogoliubov Institute for Theoretical Physics, Uppsala Kiev, Ukraine and University of Liege, Belgium Contents of Volume III G. G. Hall Per-Olov Lowdin 1 Stockholm-Uppsala Symposium 1 Visiting Uppsala 1957-58 1 Assessment 2 J.-M.Andre In SilicoChemistry: Past, Present and Future 3 Introduction 3 In Silico Chemistry, Past and Present 7 In Silico Chemistry, Future 10 Conclusions 16 Acknowledgements 19 Notes 20 References 20 J. Katriel Weights of Spin and Permutational Symmetry Adapted States for Arbitrary Elementary Spins 23 1. Introduction 23 2. Methodology of the Present Exploration 25 2.1. Generating the data 25 2.2. Analysis of the data 29 3. One and Two Particle States 30 4. Three Particle States 30 5. Four Particle States 32 6. Five Particle States 33 7. Some Six-Particle States 36 8. Attempt at a Generalization 37 8.1. Modularity strings 37 8.1.1. Repetition of modularity strings 37 8.1.2. Composition of modularity strings 38 8.1.3. Absorption of a modularity string 38 8.1.4. Amalgamation of modularity strings 38 8.1.5. Irreducible modularity string 38 x 8.2. Towards a generalizing conjecture 38 9. Conclusions 41 Acknowledgements 41 References 41 B. L. Burrows and M. Cohen Schrodinger's Wave Equation - A Lie Algebra Treatment 43 1. Introduction 44 2. Some One-Dimensional Problems 46 2.1. The Heisenberg algebra 46 2.2. The SO(3) and SO(2,1) algebras 50 3. Problems in Two Space Variables 54 3.1. Single term perturbations 57 3.2. Two-term perturbations 57 3.3. Three-term perturbations 59 3.4. Four-term perturbations 60 Appendix A: Complex Extensions of Some Real Lie Algebras 61 Appendix B: The Algebra 0(5) and Some of Its Subalgebras 62 Acknowledgements 65 References 65 M. R. Kibler and M. Daoud On Supersymmetric Quantum Mechanics 67 1. Introducing Supersymmetry 67 2. A Generalized Weyl-Heisenberg Algebra W 71 k 2.1. The algebra W 71 k 2.2. Projection operators for W 72 k 2.3. Representation ofW 72 k 2.4. A deformed-boson + k-fermion realization of W 73 k 2.4.1. The realization ofW 73 k 2.4.2. Actions on the space F 75 2.5. Particular cases for W 75 k 3. A General Supersymmetric Hamiltonian 77 3.1. Axiomatic of supersymmetry 77 3.2. Supercharges 77 3.3. The general Hamiltonian 79 3.4. Particular cases for the Hamiltonian 80 3.5. A connection between fractional sQM and ordinary sQM 81 4. A Fractional Supersymmetric Oscillator 83 4.1. A special case ofW 83 k 4.2. The resulting fractional supersymmetric oscillator 83 Xl 4.3. Examples 84 4.3.1. Example I 84 4.3.2. Example 2 85 5. Differential Realizations 86 6. Closing Remarks 88 Acknowledgments 89 Appendix A: Connection Between W and Uq(s12) 89 k Appendix B: A Q-uon • Boson + k-Fermion Decomposition 90 References 93 P. W. Langhoff, J. A. Boatz, R. J. Hinde, and J. A. Sheehy Application Of Lowdin's Metric Matrix: Atomic Spectral Methods for Electronic Structure Calculations 97 1. Introduction 98 2. Definition of the Atomic Spectral-Product Basis 99 3. Hamiltonian Matrix in the Spectral-Product Basis 99 4. Convergence in the Spectral-Product Basis 100 4.1. Prior Antisymmetry 100 4.2. Metrically-Defined Hamiltonian Representation 102 4.3. Removal of Linear Deepnndece in the Moffitt Basis 103 4.4. Isolation of the Antisymmetric Spectral-Product Subspace 105 4.5. Equivalence of Prior and Post Antisymmetry 107 5. Illustrative Calculations -The Electron Pair Bond 108 6. Concluding Remarks 113 Acknowledgments 113 References 113 F. E. Harris Integrals for Exponentially Correlated Four-Body Systems of General Angular Symmetry 115 Dedication 115 1. Introduction 116 2. Wavefunctions 117 3. Matrix Elements 119 4. Angular Integration 119 5. Rotational Invariants 122 6. Radial Integration 124 7. Discussion 125 Acknowledgments 126 Appendix: Angular Momentum Identities 126 References 127 xu P. R. Surjan and A. Szabados Appendix to "Studies in Perturbation Theory": The Problem of Partitioning 129 1. Introduction 130 2. The Concept of Partitioning 131 3. Traditional Partitionings in Quantum Chemistry 133 3.1. Epstein-Nesbet partitioning 133 3.2. Adams partitioning 136 3.3. M~ler-Plesset partitioning 136 4. Level Shifts 138 4.1. Basic definition 138 4.2. Connection between MP and EN 139 4.3. Complex level shifts 139 5. Feenberg Scaling 140 6. Optimized Partitioning 140 6.1. General formulation 141 6.2. Properties of the optimized partitioning 142 6.2.1. Vanishing ofthe third order correction 142 6.2.2. Consequence on the higher orders 142 6.2.3. Extensivity 143 6.2.4. Resummation of RS-PT series 143 6.2.5. Derivation by projection operator technique 145 6.3. The example of the anharmonic oscillator 146 7. Optimized Partitioning in Single Reference PT 146 8. Using Noncanonical Orbital Energies in MBPT 149 8.1. Davidson-Kapuy partitioning 149 8.2. Dyson partitioning 151 8.3. Optimized orbital energies in MBPT 152 Optimized orbitals in MBPT: Lindgren's approach 154 9. Zero order Hamiltonians with Two-Body Terms 154 10. Optimized Partitioning with Multi-Configurational Zero Order 155 10.1. Multi-configurational perturbation theory 155 10.1.1. Generalized MP partitioning 156 10.1.2. Generalized DK partitioning 157 10.1.3. Generalized Dyson partitioning 157 10.1.4. Generalized EN partitioning 158 10.2. Optimized partitioning in multi-reference theories 158 10.2.1. Optimized partitioning in MCPT 158 10.2.2. Witek-Nakano-Hirao approach 158 10.2.3. Freed's optimization approach 159 11. Minimizing the Norm of RW 160 11.1. On the convergence of the PT series 160 11.2. The norm of RW 161