A General SCF Theory

220 Pages·1978·3.99 MB·English

by  Ramon Carbo, Josep M. Riera (auth.)

Lecture Notes in
Chemistry
Edited by G. Berthier, M. J. S. Dewar, H. Fischer
K. Fukui, H. Hartmann, H. H. Jaffe, J. Jortner
W. Kutzelnigg, K. Ruedenberg, E. Scrocco, W. Zeil
5
Ramon Carbo
Josep M. Riera
A General SCF Theory
Springer-Verlag
Berlin Heidelberg New York 1978
Authors
Ramon Carbo
Division of Theoretical Chemistry
Department of Chemistry
University of Alberta
Edmonton, Alberta, Canada T6G 2G2
and
Departamento de Biomatematicas
Facultad de Medicina
Universidad Autonoma de Barcelona
BeliaterralSpain
Josep M. Riera
Centro de Calculo and
Seccion de Ouimica Cuantica
Departamento de Ouimica Organica
Instituto Ouimico de Sarria
Barcelona-17/Spain
Library of Congress Cataloging in PublicatioD Data
Carb6, RamOn.
A general SCF theory.
(Lecture notes in chemistry; 5)·
Bibliography: p.
Includes index.
1. Self consistent field theory. I. Riera,
Josep M., joint author. II. Title.
Q~61.C26 541' .24 77-19004
ISBN-13: 978-3-540-08535-5 e-ISBN-13: 978-3-642-93075-1
001: 10.1007/978-3-642-93075-1
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re
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agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978
Softcover reprint of the hardcover 1s t edition 1978
2152/3140-543210
Assassins de raons i de vides
que mai no tingueu repos en
cap dels vostres dies i que
en la mort us persegueixin
les nostres memories
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Lluis Llac (1976)
Tots sabiem on era la Paret,
pero ignoravem que hi havia
darrera .•.
Tocant a rna.
J.V. Foix (1972)
Introduction
We live in a molecular world, almost closed shell in nature, and for
this reason Chemistry has been a science dealing with closed shell mol
ecules.
However, the high degree of experimental sophistication reached in the
past decade has made more apparent the role of open shell structures in
chemical research.
A parallel phenomenon can be observed in the development of SCF theory,
where closed shell molecular calculations at any level of complexity
compose the main body of references which can be obtained in Quantum
Chemistry today.
Besides the linkage between experimental and theoretical behaviour,
there are, obviously, other reasons which can be attached to a lack of
molecular open shell calculations. Among others, there was no connec
tionbetween closed or open shell theoretical treatments. In this
manner, many computational features used by closed shell connoisseurs
have not been extended to other computational areas. Since the work
of Roothaan in 1960, the open shell molecular landscape has been, the
oretically, a very closed one. Further development of SCF theory, which
has led to an outburst of multiconfigurational procedures, has paid
no, or very faint, attention to the interconnection between these SCF
theory advanced features, the open shell framework and closed shell
common practice. A good theoretical goal, generally speaking, and in
particular inside SCF theory, may consist of a procedure which can be
used to solve a given chemical problem, within the physical and approx
imate limits of the theory.
A restricted SCF formalism has been chosen here. From this point of
view the present schemes should be seen as one of the last steps in
Roothaan's formalism. As a consequence, all theoretical developments
are mainly based on projection operator algebra, which is used in order
to obtain a unique coupling operator and a unique SCF equation. This
choice is not arbitrary, but follows from the already exposed idea of
the connection between closed and open shell techniques. Beside this
primary reason there are others, unavoidable if general convergence,
extrapolation or eigenspace manipulation devices are to be described
once and for all SCF levels.
It shall be noted that each computational scheme has been tested in one
way or another, and for this reason each of them can be used without
further general speculation.
Without pretending to be exhaustive, some assorted examples are shown
in order to give a hint on the usefulness and capability of the
VI
procedures described herein. In this sense, molecular geometries of
excited states, radicals and ions are presented, as well as some rele
vant properties of various open shell systems.
A final statement of completeness will be futile and, perhaps, outra
geous. The authors believe in a neverending development of Science.
From this scope, this work should be finally taken as an attempt to
rationalize and construct the starting point of a more sophisticated
and elegant SCF theory.
Chemical problems, even those attached to open shell states,are mainly
related to closed shell systems. In fact, when an excitation or ioni
zation takes place a completely new chemical structure arises. In a
cornmon closed shell species are hidden a considerable number of new
structures, each one possessing its own particular chemical properties.
If SCF theory has proved to be accurate enough to study closed shell
problems, it hopefully can be used also to study open shell ones, par
ticularly in cases where experimental techniques can hardly be used,
and if so, only at an extraordinary cost, when compared with straight
forward computations.
A theoretical structure capable of handling chemical problems must
contain a set of properties which, until now, have not been reported
altogether in the literature. The scattering of techniques applicable
to the various parts of the SCF computational pathway is considerable,
and furthermore a galaxy of open shell computational methods can be
added to these.
This work should be seen as an attempt to describe a stable, general
and integrated SCF theory framework, in .such a manner that it can be
used as well for teaching purposes, as for research and program con
struction.
In order to enhance this last goal a modular structure has been used
throughout the description of the various SCF computational levels,
although care has also been taken to establish the relationship and
uniqueness of them all.
Edmonton, September, 1977.
Acknowledgements
Many people have made possible the development of a general SCF theory
and the writing of this book. The authors should acknowledge the con
tribution of Dr. R. Caballol and Dr. R. Gallifa, the continuing work of
Mr. J. A. Hernandez and Mr. F. Sanz as well as the efforts of Mr. J.
Huguet, Mr. R. Carulla, Ms. D. Gibbs, Ms. N. Gilbey and Ms. J. Jorgensen.
Without their help this manuscript would never have been written.
One of us (R.C.) wishes to thank Professor S. Huzinaga for the invitation
during 1977, to the Department of Chemistry of the University of Alberta
(Canada), making feasible the preparation of the final version of this
work. The support and facilities given by the Faculty of Medicine of
the Universitat Autonoma de Barcelona are also deeply acknowledged.
The authors also acknowledge the computing facilities kindly provided
by Professors E. Scrocco and S. Fraga.
contents Page
I. Historical Review 1
1. The Open Shell Development
2. The Multiconfigurational Scheme
II. Electronic Energy, Fock Operators and Coupling Operators 5
1. Introduction
2. General Energy Expression
3. General Coulomb and Exchange Operators
4. Energy and Lagrangian Variation: Fock Operators
and Euler Equations
5. Coupling Operator
6. Null Gradient Coupling Operator Part
7. Lagrange multipliers Hermitean Condition Coupling
Operator Part
8. LCAO Form of Coupling Operator
9. Simplified Energy Forms. Monoconfigurational Open Shell
10. Closed Shell
11. Corollary
III. Eigenspace Hanipulations 23
1. Introduction
2. General Formalism
3. Unconditional Convergence in SCF Procedures: Level
Shift Techniques
3.1.The Ordering Principle
3.2. Level Shift Technique in a General SCF Procedure
3.3. Koopman's Theorem and Level Shifted Closed Shell
Fock Operators
3.4. Final Remarks
IV. Multiconfigurational Structure of Monoconfigurational
SCF Procedures 34
1. Monoconfigurational Energy and Euler Equations
2. Results
3. Final Remarks
V. Paired Excitation Multiconfigurational SCF 41
1. Closed Shell MCPESCF Theory
2. Complete MCPE Energy Expression
x
3. Two Electron Systems
4. MCPESCF Theory with an Invariant Closed Shell
5. Open Shell PEMCSCF
6. Special Cases
VI. SCF Perturbation Theory 50
1. Perturbation Theory
1.1. General Scheme
1.2. Orthonormalization Conditions
1.3. Eigenvalue Corrections
1.4. Wigner's Theorem
1. 5. Eigenvector Corrections
1. 6. Alternative Formalism
2. Open Shell SCF Theory
2.1. Energy
2.2. Eigenvector Corrections
2.3. Procedure
2.4. Closed Shell SCF Perturbation Theory
3. Interaction of Two Molecules as an Application Example
3.1. General Background
3.2. Variational Equations
3.3. Perturbational Scheme
3.4. Nature of the Interaction Energy
3.5. Electrostatic Molecular Potential
VII. General Theory for Two and Three Electron Systems 68
1. Two Electron Systems
1.1. Singlet States
1.2. Triplet States
2. Three Electron Systems
2.1. Doublet States. Case A
2.2. Doublet States. Case B
2.3. Quadruplet States
VIII. Approximate SCF Theories 78
1. Atomic Orbital Representation
1.1. Introduction
1.2. Representation of AO's
1.3. Representation of Charge Density: Mulliken's Gross
Atomic Populations as a Natural Way of Charge
Partitioning
XI
1.4. Expectation Values of Monoelectronic Operators
1.5. SCF Theory and Energy Partitioning into Atomic
Contributions
1.6. General Remarks
2. Decomposition of the Electronic Repulsion Matrix
3. Empirical Approximate Methods
4. Model Potentials: Huzinaga's Approach
IX. Miscellaneous Remarks 93
1. A Synthetic Approach
2. The Concept of Shell
3. Symmetry
4. Optimization of Non-Linear Parameters
5. Generalized Brillouin's Theorem and Off-Diagonal
Herrnitean Conditions on Lagrange Multipliers
6. Error Analysis
7. Mathematical Structure of SCF
X. The Problem of the Helium Atom First Excited Singlet 104
1. A Possible Solution State
2. The "Triplet Catastrophe"
3. Further Analysis of the "The Triplet Catastrophe"
4. Some Results on Monoconfigurational He SCF
XI. Applications 114
1. Introduction
2. SCF Study of Water: Ground and Excited States
3. Paired Excitation Calculation on Water
4. Formaldehyde
5. Magnesium Oxyde
6. Nitrogen Dioxyde
7. Methanol
8. Diimine
9. Methylenimine
10. Glycine
11. Excited States of Some Molecules with c=o and C=N
Bonds: INDO Procedure
*Appendix A: Monoconfigurational State Parameters 160
*Appendix B: Slater Rules 173
*Appendix C: Multiconfigurational Fock Operators 176
XII
*Suggested Reading 188
*Bibliographical Survey
A. Open Shell SCF Theory 191
B. Multiconfigurational SCF Theory 201
Subject Index 209