Fundamental World of Quantum Chemistry: A Tribute to the Memory of Per-Olov Löwdin Volume III

701 Pages·2004·15.645 MB·English

by  G. G. Hall (auth.), Erkki J. Brändas, Eugene S. Kryachko (eds.)

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
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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