A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory
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A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory
Yukio Yamaguchi John D. Goddard
Yoshihiro Osamura Henry F. Schaefer DI
Oxford OXFORD UNIVERSITY PRESS 1994 New York
Oxford University Press Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland Madrid and associated companies in Berlin Ibadan
Copyright O 1994 by Oxford University Press, Inc. Published by Oxford
University Press, Inc.,
200 Madison Avenue, New York, New York 10016 Oxford is
a registered trademark of Oxford University Press
AU rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data The new dimension to quantum chemistry: analytic derivative methods in ab initio molecular electronic structure theory / Yukio Yamaguchi ... [et al.]. p. cm. — (International series of monographs on chemistry: [#29]) Includes bibliographical references and index. ISBN 0-19-507028-3 1. Quantum chemistry. I. Yamaguchi, Y. (Yukio) H. Series.
QD462.N49 1994 541.2'8'015118—dc20 93-31798
987654321 Printed
in the United States of America on acid-free paper
Preface theoretical chemistry, the importance of the analytic evaluation of energy derivatives from reliable wavefunctions can hardly be overestimated. The first and second energy derivatives of the total energy of a system with respect to geometrical variables are absolutely essential to interrogate reactive potential energy hypersurfaces unambiguously, efficiently, and accurately. The third and fourth energy derivatives are related very directly to the anharmonicity of the potential energy. These higher derivatives are crucial for a true understanding of the internal motions of molecules. The formulation and implementation of analytic energy derivative methods has been one of the "hottest" topics in ab initio quanturn chemistry over the past two decades and remains today an area of great importance and activity. In modern
There are many excellent books that handle the fundamental concepts involved in elementary linear combination of atomic orbital-molecular orbital (LCAO-MO) theory. However, there are few sources that explain in detail recent developments in analytic derivative methods in ab initio quantum chemistry. This book aims to enlighten beginning graduate students and young researchers who are interested in the development and implementation of analytic derivative methods in ab initio quantum chemistry. In this book we limit ourselves to derivative methods for wavefunctions based on the variational principle, namely, restricted Hartree-Fock (RHF), configuration interaction (CI) and multiconfiguration selfconsistent-field (MCSCF) wavefunctions. Thus, we exclude, with some reluctance, methods based on perturbation theory such as Moller-Plesset perturbation theory to second-order, MP2. A recent development in derivative methods, the derivatives for the coupled cluster method, is omitted as well. Some references to these nonvariational derivative schemes are given in the bibliographical survey, and a brief discussion of the future of coupled cluster derivative methods is given in the final chapter on the Future. Even the most casual observer will appreciate that the analytic evaluation of derivatives involves a vast amount of algebra. We have endeavored to put down in a permanent record a systematic presentation of the necessary algebraic formulae. Critical to such a permanent record is that it be error-free, and much effort has been expended to this end. We especially thank Oxford University Press for their good efforts in the computerized transfer of equations to the text. Of course, all human efforts are prone to error, and we would be greatly appreciative if readers would bring any errors to our attention.
PREFACE
vi
It is our pleasure to thank our former and present co-workers for their significant contributions to the subjects discussed in this book, namely, Drs. Russell M. Pitzer, Nicholas C. Handy, Peter Pn lay, Bernard R. Brooks, Michel Dupuis, William D. Laidig, Paul Saxe, Douglas J. Fox, Mark A. Vincent, Michael J. Frisch, Jeffrey F. Gaw, George Fitzgerald, Mark R. Hoffmann, Timothy J. Lee, Michael E. Colvin, Julia E. Rice, Miguel Duran, Peter J. Knowles, Gustavo E. Scuseria, Roger S. Grey, Richard B. Remington, Wesley D. Allen, Andrew C. Scheiner, Tracy P. Hamilton, Ian L. Alberts, Neil A. Burton, Curtis L. Janssen, Edward T. Seidl, and Seung-Joon Kim. The authors also thank Dr. Cynthia Meredith for careful proofreading of the entire manuscript. July, 1993 Yukio Yamaguchi
Yoshihiro Osamura John D. Goddard Henry F. Schaefer III
Contents
i
Preface
List of Abbreviations
I.
xvii
Introduction
3
References
7
Suggested Reading
8
2 Basic Concepts and Definitions
9
9
2.1
The Schrödinger Equation and the Hamiltonian Operator
2.2
The Variation Method
10
2.3
The Linear Combination of Atomic Orbitals-Molecular Orbital (LCAO-MO) Approximation
11
2.4
Basis Sets
12
2.5
The Hartree-Fock and Self-Consistent-Field Wavefunctions
13
2.6
The Configuration Interaction (CI) Wavefunction
16
2.7
The Multiconfiguration (MC) SCF Wavefunction
18
2.8
Energy Expressions in MO and AO Bases
20
2.8.1
One-Electron MO Integral
20
2.8.2
Two-Electron MO Integral
21
CONTENTS
viii The Electronic Energy for a Closed-Shell SCF Wavefunction
2.8.4
The Electronic Energy for a CI Wavefunction
.
22 23
Potential Energy Surface
24
2.10 Wigner's 2n+1 Theorem
25
References
27
Suggested Reading
28
Derivative Expressions for Integrals and Molecular Orbital Coefficients
29
2.9
3
2.8.3
3.1
Expansions of the Hamiltonian Operator and the Atomic Orbital Integrals
29
3.2
Expansion of the Molecular Orbital Coefficients
30
3.3
A Relationship between 11a, U b , and Uab
31
3.4
The Derivative Expressions for Overlap AO Integrals
32
3.5
The Derivative Expressions for One-Electron AO Integrals
33
3.6
The Derivative Expressions for Two-Electron AO Integrals
33
3.7
The Derivative Expressions for Overlap MO Integrals
34
3.7.1
The First and Second Derivatives of Overlap MO Integrals
3.7.2
An Alternative Derivation of the First Derivatives of Overlap MO
34
Integrals 3.7.3
37
A Derivative Expression for the Skeleton (Core) Overlap Derivative
Integrals 3.7.4
38
An Alternative Derivation of the Second Derivatives of Overlap MO
Integrals 3.8
39
The Derivative Expressions for One-Electron MO Integrals
39
3.8.1
The First and Second Derivatives of One-Electron MO Integrals
3.8.2
An Alternative Derivation of the First Derivatives of One-Electron
MO Integrals
.
39 41
CONTENTS 3.8.3
3.8.4
3.9
ix A Derivative Expression for the Skeleton (Core) One-Electron
Derivative Integrals
42
An Alternative Derivation of the Second Derivatives of One-Electron MO Integrals
42
The Derivative Expressions for Two-Electron MO Integrals 3.9.1
3.9.2
3.9.3
3.9.4
The First and Second Derivatives of the Two-Electron MO Integrals
43
An Alternative Derivation of the First Derivatives of Two-Electron MO Integrals
47
The Derivative Expression for the Skeleton (Core) Two-Electron
Derivative Integrals
48
An Alternative Derivation of the Second Derivatives of Two-Electron MO Integrals
49
3.10 The Derivatives of the Nuclear Repulsion Energy
51
3.10.1 The First Derivative of the Nuclear Repulsion Energy
51
3.10.2 The Second Derivative of the Nuclear Repulsion Energy
51
References
4
43
52
Closed Shell Self Consistent Field Wavefunctions -
-
-
53
4.1
The SCF Energy
4.2
The Closed-Shell Fock Operator and the Variational Conditions
4.3
The First Derivative of the Electronic Energy
55
4.4
The Evaluation of the Energy First Derivative
57
4.5
The First Derivative of the Fock Matrix
58
4.6
The Second Derivative of the Electronic Energy
60
4.7
An Alternative Derivation of the Second Derivative of the Electronic
Energy References
53 54
63 66
CONTENTS
x
66
Suggested Reading 5
General Restricted Open Shell Self Consistent Field Wavefunctions -
-
-
5.1
The SCF Energy
68
5.2
The Generalized Fock Operator and the Variational Conditions
70
5.3
The First Derivative of the Electronic Energy
71
5.4
The Evaluation of the First Derivative
73
5.5
The First Derivative of the Lagrangian Matrix
74
5.6
The Second Derivative of the Electronic Energy
76
5.7
An Alternative Derivation of the Second Derivative of the Electronic
Energy
6
68
79
References
81
Suggested Reading
82
Configuration Interaction Wavefunctions
83
6.1
The CI Wavefunction, Energy and the Variational Conditions
83
6.2
The First Derivative of the Electronic Energy
84
6.3
The Evaluation of the First Derivative
87
6.4
The First Derivative of the Hamiltonian Matrix
89
6.5
The First Derivative of the "Bare" Lagrangian Matrix
90
6.6
The Second Derivative of the Hamiltonian Matrix
92
6.7
The Second Derivative of the Electronic Energy
94
References
96
Suggested Reading
97
CONTENTS 7
Two Configuration Self Consistent Field Wavefunctions -
7.1
8
xi -
-
The Two-Configuration Self-Consistent-Field (TCSCF) Wavefunction and Energy
98
98
7.2
The Generalized Fock Operator and the Variational Conditions
101
7.3
The First Derivative of the Electronic Energy
102
7.4
The First Derivative of the Hamiltonian Matrix
103
7.5
The Second Derivative of the Hamiltonian Matrix
104
7.6
The Second Derivative of the Electronic Energy
106
References
109
Suggested Reading
109
Paired Excitation Multiconfiguration Self-Consistent-Field Wavefunctions
8.1
110
The Paired Excitation Multiconfiguration Self-Consistent-Field (PEMCSCF) Wavefunction and Energy
110
8.2
The Generalized Fock Operator and the Variational Conditions
114
8.3
The First Derivative of the Electronic Energy
115
8.4
The Second Derivative of the Electronic Energy
116
References
118
Suggested Reading
118
9 M ult iconfig urat ion Self Consistent Field Wavefu nct ions -
-
119
9.1
The MCSCF Wavefunction and Energy
119
9.2
The Variational Conditions
120
9.3
The First Derivative of the Electronic Energy
121
9.4
The Evaluation of the First Derivative
123
9.5
The Second Derivative of the Electronic Energy
124
xii
CONTENTS
References Suggested Reading 10 Closed Shell Coupled Perturbed Hartree - Fock Equations -
10.1 The First-Order CPHF Equations 10.2 The Derivatives of the Fa Matrices
126 127
128
128 132
10.3 The First Derivative of the A matrix
133
10.4 The Second-Order CPHF Equations
134
10.5 An Alternative Derivation of the Second Derivative of the Fock Matrix
138
References
142
Suggested Reading 11 General Restricted Open-Shell Coupled Perturbed Hartree-Fock Equations
143
144
11.1 The First-Order CPHF Equations
144
11.2 The Averaged Fock Operator
150
11.3 The Derivative of the ea Matrices 11.4 The First Derivative of the Generalized Lagrangian Matrix 11.5 The First Derivative of the 7 Matrices
153 154 155
11.6 The Second Derivative of the Lagrangian Matrix
156
11.7 The Second-Order CPHF Equations
158
11.8 An Alternative Derivation of the Second Derivative of the Lagrangian
Matrix References Suggested Reading
162 165 165
CONTENTS
xiii
12 Coupled Perturbed Configuration Interaction Equations
166
12.1 The First-Order Coupled Perturbed Configuration Interaction Equations .
166
12.2 The Second-Order Coupled Perturbed Configuration Interaction
Equations
169
References
171
Suggested Reading
171
13 Coupled Perturbed Paired Excitation Multiconfiguration Hartree-Fock Equations
172
13.1 The First-Order CPPEMCHF Equations
172
13.1.1 The Molecular Orbital (MO) Part
172
13.1.2 The Configuration Interaction (CI) Part
177
13.1.3 A Complete Expression for the First-Order CPPEMCHF
Equations
179
13.2 The Averaged Fock Operator
180
13.3 The Second-Order CPPEMCHF Equations
182
13.3.1 The Molecular Orbital (MO) Part
182
13.3.2 The Configuration Interaction (CI) Part
187
13.3.3 A Complete Expression for the Second-Order CPPEMCHF
Equations
189
References
191
Suggested Reading
192
14 Coupled Perturbed Multiconfiguration Hartree-Fock Equations
14.1 The First-Order CPMCHF Equations
193
194
14.1.1 The Molecular Orbital (MO) Part
194
14.1.2 The Configuration Interaction (CI) Part
198
14.1.3 A Complete Expression for the First-Order CPMCHF Equations
201
CONTENTS
xiv 14.2 The Averaged Fock Operator
202
14.3 The First Derivative of the "Bare" y Matrix
206
14.4 The Second-Order CPMCHF Equations
208
14.4.1 The Molecular Orbital (MO) Part
208
14.4.2 The Configuration Interaction (CI) Part
213
14.4.3 A Complete Expression for the Second-Order CPMCHF Equations
215
References Suggested Reading 15 Third and Fourth Energy Derivatives for Configuration Interaction Wavefunctions
15.1 The First and Second Derivatives of the Electronic Energy for a CI Wavefun ct ion 15.2 The Third Derivative of the Electronic Energy for a CI Wavefunction .
217 217
218
219 222
15.3 The Fourth Derivative of the Electronic Energy for the CI Wavefunction .
224
15.4 The Third Derivative of the Hamiltonian Matrix
228
15.5 An Explicit Expression for the Third Derivative of the Electronic Energy
233
15.6 The First Derivative Expression for the "Bare" Z Matrix
234
15.7 The Fourth Derivative of the Hamiltonian Matrix
235
15.8 An Explicit Expression for the Fourth Derivative of the Electronic Energy
246
References
248
Suggested Reading
248
18 Correspondence Between Correlated and Restricted Hartree-Fock Wavefunctions
16.1 Configuration Interaction (CI) Wavefunctions 16.1.1 Electronic Energy and the Variational Condition
249
249 249
CONTENTS
xv
16.1.2 First Derivative
250
16.1.3 Second Derivative
251
16.1.4 Third Derivative
251
16.1.5 Fourth Derivative
253
16.2 Correspondence Equations
256
16.3 Multiconfiguration Self-Consistent-Field (MCSCF) Wavefunctions
258
16.3.1 Electronic Energy and the Variational Condition
258
16.3.2 First Derivative
259
16.3.3 Second Derivative
259
16.3.4 Third Derivative
260
16.3.5 Fourth Derivative
264
16.4 Paired Excitation Multiconfiguration Self-Consistent-Field (PEMCSCF) Wavefun ct ion s
270
16.4.1 Electronic Energy and the Variational Condition
270
16.4.2 First Derivative
272
16.4.3 Second Derivative
272
16.4.4 Third Derivative
274
16.4.5 Fourth Derivative
278
16.5 General Restricted Open-Shell Self-Consistent-Field (GRSCF) Wavefunctions
285
16.5.1 Electronic Energy and the Variational Condition
285
16.5.2 First Derivative
286
16.5.3 Second Derivative
286
16.5.4 Third Derivative
288
16.5.5 Fourth Derivative
290
16.6 Closed-Shell Self-Consistent-Field (CLSCF) Wavefunctions
293
CONTENTS
xvi
16.6.1 Electronic Energy and the Variational Condition
293
16.6.2 First Derivative
295
16.6.3 Second Derivative
296
16.6.4 Third Derivative
298
16.6.5 Fourth Derivative
300
16.7 Energy Derivative Expressions Using Orbital Energies
304
Suggested Reading
307
17 Analytic D erivat ives Involving Electric Field Perturbations
17.1 The Electric Field Perturbation
308
308
17.1.1 The Hamiltonian Operator with Mixed Perturbations
308
17.1.2 The Derivative of the Molecular Orbital Coefficients
309
17.1.3 The Derivative One-Electron AO Integrals
309
17.1.4 The Derivative One-Electron MO Integrals
310
17.1.5 Constraints on the Molecular Orbitals
311
17.2 The Electric Dipole Moment
312
17.2.1 The Closed-Shell SCF Wavefunction
312
17.2.2 The General Restricted Open-Shell SCF Wavefunction
313
17.2.3 The Configuration Interaction (CI) Wavefunction
313
17.2.4 The Multiconfiguration (MC) SCF Wavefunction
314
17.3 The Electric Polarizability
314
17.3.1 The Closed-Shell SCF Wavefunction
315
17.3.2 The General Restricted Open-Shell SCF Wavefunction
317
17.3.3 The Configuration Interaction (CI) Wavefunction
320
17.3.4 The Multiconfiguration (MC) SCF Wavefunction
323
CONTENTS
17.4 The Dipole Moment Derivative 17.4.1 The Closed-Shell SCF Wavefunction 17.4.2 The General Restricted Open-Shell SCF Wavefunction
xvii
326 327 328
17.4.3 The Configuration Interaction (CI) Wavefunction
329
17.4.4 The Multiconfiguration (MC) SCF Wavefunction
331
17.5 The Coupled Perturbed Hartree-Fock (CPHF) Equations Involving Electric Field Perturbations 332 17.5.1 The First-Order CPHF Equations for a Closed-Shell SCF, Wavefunction
332
17.5.2 The Second-Order CPHF Equations for a Closed-Shell SCF Wavefunction
333
17.5.3 The First-Order CPHF Equations for a General Restricted Open-Shell Wavefunction
335
17.5.4 The Second-Order CPHF Equations for a General Restricted Open-Shell Wavefunction
336
17.6 The Coupled Perturbed Configuration Interaction Equations Involving Electric Field Perturbations
340
17.6.1 The First-Order CPCI Equations
340
17.6.2 The Second-Order CPCI Equations
341
17.7 The Coupled Perturbed Multiconfiguration Hartree-Fock Equations Involving Electric Field Perturbations
343
17.7.1 The First-Order CPMCHF Equations
343
17.7.2 The Second-Order CPMCHF Equations
345
17.8 The Coupled Perturbed Equations for Multiple Perturbations
349
References
354
Suggested Reading
354
CONTENTS
xviii 18 The Z Vector Method
355
18.1 The Z Vector Method and First Derivative Properties for the CI Wavefunction
355
18.2 The Z Vector Method and Second Derivative Properties for the CI Wavefunction
358
References
362
19 Applications of Analytic Derivatives
19.1 Unimolecular Isomerization of HCN
365
19.2 Formaldehyde Dissociation 19.3 Dissociation of cis-Glyoxal —t
363
367 H2 +
2 CO
371
19.4 The Hydrogen Fluoride Dimer, (HF)2
373
19.5 Systematics of Molecular Properties
376
19.6 Vibrational Analysis: Anharmonic Effects
379
19.7 Larger Molecules: Structural Certainty
381
References
383
Suggested Reading
384
20 The Future
385
References
386
Appendix A Some Elementary Definitions
387
Appendix B Definition of Integrals
388
Appendix C Electronic Energy Expressions
390
Appendix D Relationships between Reduced Density Matrices and Coupling Constant Matrices
392
CONTENTS
xix
Appendix E Fock and Lagrangian Matrices
394
Appendix F Variational Conditions and Constraints
396
Appendix G Definition of U Matrices
398
Appendix H Relationships among U Matrices
399
Appendix I Definition of Skeleton (Core) Derivative Integrals
400
Appendix J First Derivative Expressions for MO Integrals
402
Appendix K Expressions for Skeleton (Core) Derivatives
403
Appendix L Second Derivative Expressions for MO Integrals
405
Appendix M Definition of Skeleton (Core) Derivative Matrices for Closed-Shell SCF Wavefunctions
407
Appendix N Expressions for Skeleton (Core) Derivative Matrices for Closed-Shell SCF Wavefunctions
409
Appendix 0 Definition of Skeleton (Core) Derivative Matrices for General Restricted Open-Shell SCF Wavefunctions
411
Appendix P Expressions for Skeleton (Core) Derivative Matrices for General Restricted Open-Shell SCF Wavefunctions
414
Appendix Q Definition of Skeleton (Core) Derivative "Bare" Matrices for CI and MCSCF Wavefunctions
417
Appendix R Expressions for Skeleton (Core) Derivative Hamiltonian Matrices for CI and MCSCF Wavefunctions
419
Appendix S Definition of Skeleton (Core) Derivative Lagrangian and Y Matrices for CI and MCSCF Wavefunctions
420
CONTENTS
xx Appendix T Derivative Expressions for Skeleton (Core) "Bare" Matrices for CI and MCSCF Wavefunctions
423
Appendix U Derivative Expressions for Hamiltonian Matrices for CI and MCSCF Wavefunctions
425
Appendix V First and Second Electronic Energy Derivatives
428
Appendix W First and Second Derivatives of Fock and Lagrangian Matrices
433
Appendix X Coupled Perturbed Hartree-Fock (CPHF) Equations for Closed Shell SCF Wavefunctions
437
Appendix Y Coupled Perturbed Hartree-Fock (CPHF) Equations for General Restricted Open-Shell SCF Wavefunctions
439
Appendix Z Coupled Perturbed Configuration Interaction (CPCI) Equations
441
Appendix AA Coupled Perturbed Paired Excitation Multiconfiguration Hartree-Fock (CPPEMCHF) Equations
442
Appendix BB Coupled Perturbed Multiconfiguration Hartree-Fock (CPMCHF) Equations
445
Bibliographical Survey on Analytic Energy Derivatives
448
Index
463
-
List of Abbreviations AO CASSCF CI CISD CLSCF CPCI CPHF CPMCHF CPPEMCHF CPTCHF CSF GRSCF GTO GVB HF LCAO lhs MO MCSCF PEMCSCF PES RHF rhs SCF STO TCSCF ZPVE
atomic orbital complete active space self-consistent-field configuration interaction configuration interaction with single and double excitations
closed-shell self-consistent-field coupled perturbed configuration interaction coupled perturbed Hartree-Fock coupled perturbed multiconfiguration Hartree-Fock coupled perturbed paired excitation multiconfiguration Hartree-Fock coupled perturbed two-configuration Hartree-Fock configuration state function general restricted open-shell self-consistent-field Gaussian-type orbital
generalized valence bond Hartree-Fock
linear combination of atomic orbitals left hand side molecular orbital multiconfiguration self-consistent-field paired excitation multiconfiguration self-consistent-field potential energy (hyper) surface restricted Hartree-Fock right hand side self-consistent-field Slater-type orbital two-configuration self-consistent-field zero-point vibrational energy
A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory
Chapter 1
Introduction In modern theoretical chemistry, the importance of evaluating force fields from reliable wavefunctions has become increasingly apparent. The second derivative matrix (Hessian) of the total energy of a system with respect to geometrical variables is used not only to evaluate vibrational frequencies and related properties, but also to interrogate reactive potential energy hypersurfaces. Following the arguments of Murrell and Laid ler [1] and McIver and Komornicki [2], it is essential to demonstrate that the stationary points for a molecule containing N nuclei have 3N-6 real harmonic vibrational frequencies (3N-5 for linear molecules) for energy minima and 3N-7 (3N-6 for linear molecules) real frequencies plus a single imaginary frequency (corresponding to the reaction coordinate) for transition
states. Explicit implementation of the analytic first derivative of the energy with respect to geometrical variables was first proposed by Pulay [3] in 1969 and the significance of his force method was immediately recognized by the computational quantum chemistry community. In the last two decades, the analytic first and second derivative techniques for ab initio single configuration self-consistent-field (SCF) and correlated wavefunctions have been developed by several research groups and applied to the investigation of a large variety of chemical phenomena. In principle, energy derivatives may be evaluated from total energies by the finite difference method. However, such a scheme is prone to numerical difficulties and is computationally expensive. Thus the advantages in accuracy and inherent efficiency are continually propelling theoretical chemists to further development of analytic derivative techniques. In ab initio derivative studies, Gaussian-type orbitals (GT0s) [4] have been invariably used, predominantly because of the relatively simple recurring character of their derivatives. In an early stage of development, however, only s and p orbitals were included in the derivative formalism, since the derivatives of d or higher orbitals become significantly more complicated. A major breakthrough in this regard was the work of Dupuis and King, using Rys polynomials to evaluate GTO integrals and their derivatives [5]. The important next
3
4
CHAPTER 1. INTRODUCTION
step to the analytic second derivative technique for SCF wavefunctions by Pople et al. [6] was due, in part, to the Rys polynomial technique. It is well established that electron correlation strongly affects the potential energy surfaces of reactive systems. The unitary group approach (UGA) laid out for electronic systems by Paldus [7] and reformulated for practical applications by Shavitt [8], Brooks and Schaefer [9], and Siegbahn [10], provides a remarkably efficient scheme for the evaluation of Hamiltonian matrix elements between configuration state functions (CSFs). With respect to configuration interaction (CI) wavefunctions for closed-shell molecules, analytic gradient methods have been formulated by Pople's group [11] and in our laboratory [12]. Later we also reported the determination of analytic energy first derivatives for open-shell correlated wavefunctions [13]. These studies employed the coupled perturbed Hartree-Fock (CPHF) theory, first outlined in a classic paper by Gerratt and Mills in 1968 [14], to evaluate the changes in molecular orbitals caused by the perturbation of a nuclear position in a molecule. One of the most intensely studied methods in theoretical chemistry today is certainly the multiconfiguration SCF (MCSCF) technique. One motivation for this emphasis is the well-known fact that single con fi guration SCF theory fails to properly describe the separated fragments arising in many reactive processes. Accordingly, it is desirable to use MCSCF wavefunctions. While slow convergence in iterative MCSCF procedures has been a significant barrier to the construction of reliable MCSCF wavefunctions for quite some time, this problem is now practically solved in many cases by employing the Newton-Raphson technique, which is quadratically convergent [15,16].
Within a chosen, finite basis set the exact solution to Schriidinger's equation is the full CI wavefunction. Although recent methodological developments and advances in computer technology have allowed some breakthroughs in computational quantum chemistry, it is highly unlikely that full CI wavefunctions will become available in the near future for systems of greater complexity than 10 to 12 electrons, even if modest basis sets are used. In contrast, using techniques developed to date, MCSCF wave functions including a fairly large number of configurations may be routinely obtained. This procedure is still a difficult one and probably not the best available approach for the quantitative description of molecular systems. Experience does suggest, however, that in most cases chemical systems may be described qualitatively using a small MCSCF reference function. An eminently reasonable approach at the present time, therefore, is to obtain a small MCSCF wavefunction by some quadratically convergent procedure, for example via the complete active space (CAS) SCF technique [17], and then carry out a large multireference CI treatment based on these MCSCF orbitals. The analytic energy second derivative method for MCSCF wavefunctions was first implemented in our laboratory [18] and the first derivative of MCSCF-CI wavefunctions by Page and coworkers [19]. Considering the practical significance of such MCSCF and MCSCF-CI wavefunctions, further efforts toward the improvement of these analytic derivative techniques will be very important. The rest of this book is divided into nineteen chapters. The contents of each chapter are previewed briefly in the remainder of this chapter. The basic concepts involved in the
5 ab initio linear combination of atomic orbitals-molecular orbital (LCAO-MO) method are reviewed briefly and the fundamental equations necessary for future discussion are introduced in Chapter 2. Using these concepts, expressions for the derivatives of integrals and molecular orbital (MO) coefficients are derived in Chapter 3. The results obtained in this chapter are quite general and thus applicable to any type of wavefunction.
Moving from the general to the specific, the first and second energy derivative expressions for the simplest and most frequently used case, the closed-shell (CL) SCF wavefunction, are discussed in Chapter 4, building upon the equations derived in Chapter 3. At a somewhat greater level of complexity, the first and second energy derivative expressions for the general restricted open-shell (GR) SCF wavefunction are presented in Chapter 5. The equations given there are applicable to any type of single configuration SCF wavefunction. Chapters 6 through 9 discuss analytic derivatives for methods that go beyond the single configuration SCF level. Chapter 6 deals with the first and second energy derivatives of CI wavefunctions. The configurations included in a CI wavefunction are constructed using a set of molecular orbitals defined with respect to a reference wavefunction, and the derivatives of the MO coefficients of the reference wavefunction are required to evaluate the derivatives of properties computed from the CI wavefunction. In Chapter 7, the first and second energy derivatives for the simplest case of a multiconfiguration (MC) SCF wavefunction, the two-configuration (TC) SCF wavefunction, are given. For the TCSCF wavefunction, it is important to realize that the MO and CI coefficients must be optimized simultaneously to satisfy the variational conditions. During the evaluation of the first derivative of the energy no information concerning the derivatives of the MO and CI coefficients is required. However, explicit evaluation of such derivatives is necessary to obtain the second derivatives of the TCSCF energy. In Chapter 8 the first and second derivative energy expressions are extended to a more general paired excitation (PE) MCSCF wavefunction. This type of wavefunction is one of the most complicated that does not require a full four-index MO transformation of the electron repulsion integrals. The generalized valence bond (GVB) wavefunction may be viewed as a special case of a PEMCSCF wavefunction. Chapter 9 treats the first and second energy derivatives for even more general MCSCF wavefunctions. The widely used complete active space (CAS) SCF wavefunctions are considered as special cases of these general MCSCF wavefunctions. The determination of the energy gradient (first derivative) at the SCF or MCSCF levels of theory does not require information concerning the first-order changes in the molecular orbitals. However, this is not the case for the Cl wavefunction that uses an SCF or MCSCF wavefunction as a reference. In the CI case, the first-order changes in the molecular orbital coefficients must be evaluated. Usually they are determined by solving the coupled perturbed Hartree-Fock (CPHF) equations for the SCF case and the coupled perturbed multiconfiguration Hartree-Fock (CPMCHF) equations for MCSCF reference wavefunctions. The CPHF and CPMCHF equations are also required to evaluate analytic second derivatives of the SCF and MCSCF wavefunctions. Chapters 10 through 14 deal with the solutions of such coupled perturbed equations for increasingly complex wavefunctions. Chapter 10 begins with the first- and second-order CPHF equations for the closed-shell SCF wave-
6
CHAPTER 1. INTRODUCTION
function. Chapter 11 extends the CPHF equations to the general restricted open-shell (GR) SCF wavefunction. The equations involved with the GRSCF wavefunction are formally equivalent to those for the MO part of the TCSCF and PEMCSCF wavefunctions described in Chapter 13. In order to evaluate the second derivatives for configuration interaction (CI) wavefunctions, it is necessary to evaluate the first-order changes of the CI coe ffi cients. They usually are obtained by solving the coupled perturbed configuration interaction (CPCI) equations. In Chapter 12 the first- and second-order CPCI equations are derived. Chapter 13 discusses the CPHF equations for PEMCSCF wavefunctions. Those CPPEMCHF equations associated with the CI space are formally the sanie as the CPCI equations described in Chapter 12. It should be noted that the CPHF equations for the TCSCF wavefunction are considered as a special case of the CPPEMCHF equations. In Chapter 14, the CPHF equations for more general MCSCF wavefunctions are elaborated. It should be emphasized again that the CPMCHF equations are constructed from elements involving both the MO and CI spaces. While the CI parts of the CPMCHF equations are formally equivalent to the CPCI equations in Chapter 12, a new derivation is required for the MO part of the CPMCHF equations. Up to this point only the first and second energy derivatives have been considered. The next three chapters extend the analytical derivative approach to higher derivatives and to quantities due to other than nuclear perturbations. The third and fourth energy derivatives are important as they are directly related to the anharmonicity of the vibrational motions of molecules. In Chapter 15, analytic third and fourth derivatives for CI wavefunctions are derived using a convenient recurrence technique. The results obtained in Chapter 15 are used to derive analytic third and fourth derivatives for MCSCF, PEMCSCF, GRSCF, and CLSCF wavefunctions in Chapter 16. To interrelate the above-mentioned wavefunctions, a series of correspondence equations is introduced. All derivative equations discussed through Chapter 16 in the text are equally applicable to any "real" perturbations including electric field perturbations. For electric field perturbations derivative expressions are greatly simplified since the derivatives of the twoelectron integrals are not involved. In Chapter 17 analytic derivatives involving electric field perturbations are described in detail. In the first implementation of the analytic first derivative of the CI wavefunction, the CPHF equations were solved for 3N (N being the number of nuclei) sets of nuclear perturbations [12]. However, Handy and Schaefer have presented a technique in which only one set of perturbations, in the present case the CI energy change with respect to the MO rotations, is solved [20]. This new technique is called the Z vector method and proves very powerful whenever applicable. In Chapter 18, the Z vector method is discussed along with some of its applications.
Chapter 19 provides several case studies of the use of analytic derivative methods in applied quantum chemistry. This discussion is in no sense exhaustive but seeks to illustrate the great practical importance of the theoretical techniques discussed in the book to actual studies of chemically interesting problems.
REFERENCES
7
In Chapter 20 we present our perspective on future directions for analytic derivative methods and make some concluding remarks. At the end of the book, a selection of literature references related to analytic derivative methods, mostly directly associated with implementation of these techniques, are collected in a bibliography. The important derivative expressions for each chapter are summarized in the Appendices for the reader's convenience. We encourage readers to carefully follow through the derivations in each chapter with pencil, paper, and patience. References
1. J.N. Murrell and K.J. Laid ler, Trans. Faraday Soc. 64, 371 (1968). 2. J.W. McIver, Jr. and A. Komornicki, J. Am. Chem. Soc. 94, 2625 (1972). 3. P. Pulay, Mol. Phys. 17, 197 (1969). 4. S.F. Boys, Proc. Roy. Soc. London A200, 542 (1950).
5. M. Dupuis, J. Rys, and H.F. King, J. Chem. Phys. 65, 111 (1976). 6. J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Binkley, Int. J. Quantum Chem. S13, 225 (1979). 7. J. Paldus, in Theoretical Chemistry : Advances and Perspectives, H. Eyring and D.J. Henderson editors, Academic Press, New York, Vol.2, p.131 (1976).
8. I. Shavitt, Int. J. Quantum Chem. S11, 131 (1977); S12, 5 (1978).
9. B.R. Brooks and H.F. Schaefer, J. Chem. Phys. 70, 5092 (1979). 10. P.E.M. Siegbahn, J. Chem. Phys. 70, 5391 (1979). 11. R. Krishnan, JIB. Schlegel, and J.A. Pople, J. Chem. Phys. 72, 4654 (1980).
12. B.R. Brooks, W.D. Laidig, P. Saxe, J.D. Goddard, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 72, 4652 (1980). 13. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 75, 2919 (1981). 14. J. Gerratt and I.M. Mills, J. Chem. Phys. 49, 1719 (1968). 15. H.-J. Werner, in Advances in Chemical Physics : Ab Initio Methods in Quantum Chemistry Part IL, K.P. Lawley editor, John Wiley & Sons, New York, Vol. 69, p.1 (1987). 16. R. Shepard, in Advances in Chemical Physics : Ab Initio Methods in Quantum Chemistry Part II., K.P. Lawley editor, John Wiley & Sons, New York, Vol. 69, p.63 (1987).
17. B.O. Roos, in Advances in Chemical Physics : Ab Initio Methods in Quantum Chemistry Part K.P. Lawley editor, John Wiley & Sons, New York, Vol. 69, p.399 (1987). 18. M.R. Hoffmann, D.J. Fox, J.F. Gaw, Y. Osamura, Y. Yamaguchi, R.S. Grey, G. Fitzgerald, H.F. Schaefer, P.J. Knowles, and N.C. Handy, J. Chem. Phys. 80, 2660 (1984). 19. M. Page, P. Saxe, G.F. Adams, and B.H. Lengsfield, J. Chem. Phys. 81, 434 (1984). 20. N.C. Handy and H.F. Schaefer, J. Chem. Phys. 81, 5031 (1984).
CHAPTER 1. INTRODUCTION
8 Suggested Reading
1. H.F. Schaefer editor, Modern Theoretical Chemistry, Vols. 3 and 4, Plenum, New York, 1977. 2. P. Jorgensen and J. Simons, Second Quantization-Based Methods in Quantum Chemistry, Academic Press, New York, 1981. 3. H.F. Schaefer, Quantum Chemistry: The Development of Ab Initio Methods in Molecular Electronic Structure Theory, Clarendon Press, Oxford, 1984. 4. P. Jorgensen and J. Simons editors, Geometrical Derivatives of Energy Surfaces and Molecular Properties, NATO ASI Series C166, D. Reidel, 1985. 5. H.F. Schaefer and Y. Yamaguchi, J. Mol. Struct. 135, 369 (1986). 6. P. Pulay, in Advances in Chemical Physics : Ab Initio Methods in Quantum Chemistry Part K.P. Lawley editor, John Wiley 8./ Sons, New York, Vol. 69, p.241 (1987).
7. W.J. Hehre, L. Radom, P.v.R. Schleyer, and J.A. Pople, Ab Initio Molecular Orbital Theory, John Wiley & Sons, New York, 1986. 8. C.E. Dykstra, Ab Indio Calculation of the Structures and Properties of Molecules, Elsevier, Amsterdam, 1988. 9. T. Helgaker and P. Jorgensen, in Advances in Quantum Chemistry, P.O. L6wdin, J.H. Sabin, and M.C. Zerner editors, Academic Press, New York, Vol. 19, p.183 (1988). 10. J. Gauss and D. Cremer, in Advances in Quantum Chemistry, P.O. L6wdin, J.H. Sabin, and M.C. Zerner editors, Academic Press, New York, Vol. 23, p.205 (1992). 11. T. Helgaker and P. Jorgensen, in Methods in Computational Molecular Physics, NATO ASI Series B293, S. Wilson and G.H.F. Diercksen editors, Plenum, New York, p.353 (1992).
Chapter 2
Basic Concepts and Definitions This chapter briefly reviews the basic concepts and definitions needed in molecular quantum mechanics beginning with the linear combination of atomic orbitals-molecular orbital (LCAO-MO) approach. Readers who require a more elementary introduction to these fundamental topics in quantum chemistry or their applications to specific chemical problems should consult the books listed at the end of this chapter as Suggested Reading.
2.1
The Schrödinger Equation and the Hamiltonian Operator
For a given molecular system the Schrödinger equation [1], the most important equation in modern science, is HT = EW (2.1)
where H is the Hamiltonian operator, AP is the total wavefunction, and E is the energy eigenvalue. The non-relativistic time-independent Hamiltonian operator of concern in most ab initio computations is given (in atomic units) by H= x-, 11
i>i
1 vn.
1
2
2 1
E A>B
2 A
ZAZB
R AB
MA
n
N
EE i A
ZA riA
(2.2)
for n electrons and N nuclei. V is the Laplacian operator for the ith electron, V?,4 is the Laplacian operator for the Ath nucleus, MA is the ratio of the mass of nucleus A to the mass of the electron, ZA is the atomic number of nucleus A, riA is the distance between the ith electron and nucleus A, r ii is the distance between the ith and jth electrons, and RAB is the distance between nuclei A and B. In the Hamiltonian H the first term represents 9
10
CHAPTER
BASIC CONCEPTS AND DEFINITIONS
2.
the kinetic energy of the electrons, the second term the kinetic energy of the nuclei, and the third through fifth ternis are the contributions to the potential energy arising from nuclear-electronic, electronic, and nuclear interactions, respectively. The indices i > j and A > B in the fourth and fifth ternis are necessary to prevent counting the same interaction twice. Using the standard Born-Oppenheimer approximation [2] we consider the electronic Hamiltonian Hetec =
E
n
INL.,
N
1
c.JA
EE — + E — A
71
E IL
h(i) +
z>3
1
(2.3)
2
(2.4)
rui
and solve Helec Telec
= EelecW etec
(2.5)
where the one-electron operator h(i) is N
1
h(i) = — — V? — 2
E— Z/A
(2.6)
A TiA
The total energy at fixed nuclear coordinates is expressed within this approximation as ZAZB Etotal = Eetec
A>B
RAB
(2.7)
It should be noted that the second term in eq. (2.7), the contribution from the nuclear repulsion energy, is a constant at a fixed geometry.
2.2
The Variation Method
satisfying the appropriate boundary conditions the energy For any arbitrary wavefunction expectation value W is described by W
f W*HWd7f 41*Tdr (1111HI(P)
(2.8) (2.9)
(To) where H is the Hamiltonian operator defined in eq. (2.2) and tlf* denotes the complex conjugate of the wavefunction. It may be shown that the expectation value of the energy may approach but never be less (lower) than the true ground state energy, E0 , of the system W > Eo
(2.10)
THE LCAO-MO APPROXIMATION
11
Let us assume that an approximate wavefunction klf can be constructed as an expansion in the "true" orthonormal eigenfunctions of the Hamiltonian operator, i.e.,
= EC1 bj
(2.11)
E c7c, = 1
(2.12)
,
with The expectation value of the energy is given by
EE c7c./(o/ill1./) I
(2.13)
J
= E c;c/E,
(2.14)
using the orthogonality of the "true" wavefunctions. lithe lowest energy level E0 is subtracted from both sides of eq. (2.14), we obtain W — E0 = E cc/(E1 - Et))
(2.15)
E0 is the lowest energy, and therefore El- — E 0 must always be zero or positive. In addition the products of the coefficients C7C1 are all zero or positive. Thus, we can state that
W > E0
.
(2.16)
This theorem was proved first by Eckart [3] and is called the variation principle. It states that the energy expectation value is a rigorous upper bound to the true ground state energy. The essential problem in the variation method is to find a wavefunction which gives the lowest possible value of W. For this purpose the variational condition
W=0
(2.17)
is usually employed. In this equation 6 refers to an infinitesimal change in the given function. If the wavefunctions are chosen with care, it may be possible to approach the true energy very closely. Generally, the more variational parameters the wavefunction (when constructed in a judicious manner) includes, the closer the true energy can be approached as these variables are optimized.
2.3
The Linear Combination of Atomic Orbitals-Molecular Orbital (LCAO-MO) Approximation
In ab initio quantum chemistry the most frequently used molecular model is the HartreeFock approximation [4,5]. The Hartree-Fock wavefunction is usually constructed from an antisymmetrized product of spin-orbitals, 1Pi,
Teiec = A
(2.18)
12
CHAPTER 2. BASIC CONCEPTS AND
DEFINITIONS
where A is the antisymmetrizer and each spin-orbital is a product of a spatial molecular orbital (MO), cki, and a one-electron spin function, a or 0,
0,(1)a(1) Oi(1) = {
or
(2.19)
0, (1 )0(1) Each molecular orbital is conventionally expanded as a linear combination of one-electron basis functions, usually termed atomic orbitals (A0s), which are normally centered on each
nucleus, AO
(2 .20)
=
Co' is the coefficient of the Ath atomic orbital xt, in the ith MO. The simplest antisymmetrized wavefunction for a closed-shell molecule with 2m electrons is described using a Slater determinant [6]
=
1
01(1)0(1) 0Pi (2)a(2)
ckn(1)a(i)
01(1)0(1) 01(2 )/3( 2 )
(kn( 1 )0( 1 )
4),(2)13(2)
0.1(2n)a(2n) 01(2n)0(2n)
(2.21)
0,.,(2n)a(2n) 07,(2n)0(271)
(2.22) 1 1 (kia 60 02a 0213 ... Ona 00 1 V-277. Using a characteristic of determinants, it is easily shown that the Slater determinant in eq. (2.21) satisfies the Pauli exclusion principle [7].
2.4
Basis Sets
The set of atomic orbitals (A0s) appearing in the LCAO-MO formalism given in eq. (2.20) is called a basis set. The most frequently used basis sets are Slater-type orbitals (ST0s) [8] and Gaussian-type orbitals (GT0s) [9]. In these basis sets the angular part is described by powers of cartesian coordinates, while the radial part is expressed by either exp(—(r) for a STO or exp(—ar 2 ) for a GTO. The variables Ç and a are orbital exponents. Thus, the two most frequently used basis sets are STO
XiA
r
=-
x y z expk—(r ivArimn
)
(2.23 )
and xli GTO
(r) =
Nxy mz
n
exp(—ar 2 )
(2.24)
where N is a normalization constant. In these equations the superscripts 1, m, and n denote angular quantum numbers for the associated cartesian coordinates. Very popular
HARTREE-FOCK AND SELF-CONSISTENT-FIELD
WAVEFUNCTIONS
13
GTO basis sets were developed in 1965 by Huzinaga [10]. Later, efficient contractions of these primitive GTO basis sets were carried out by Dunning [11] 7/ G
x CGTO
ii,x,GTo (r)
E c
(2.25)
where nG is the number of primitive (original) Gaussian basis functions included in the contraction. Pople and coworkers [12] have developed a series of minimal basis sets in which STOs were least squares fi t by expansions in Gaussian-type functions, na
STO
(r) =
Ec
GTO •
(2.26)
(r)
The case of nG = 3, i.e., the STO-3G basis, was used with remarkable success in early studies on many organic molecules by Pople and his collaborators as detailed in reference 8 of the Suggested Reading. Today split valence sets, i.e., a double zeta representation of the valence orbitals and a minimal basis for the core orbitals, such as 3-21G and 6-31G* are the most widely employed Gaussian basis sets. Whole books have been devoted to the topic of optimized Gaussian basis sets of various sizes and types for many elements [13,14] and obviously no lengthy discussion is required here. A very helpful review of basis sets has recently appeared [15].
While the STO basis set is better able to describe the motion of electrons, especially near the nucleus, relative to the GTO basis, the evaluation of AO integrals and their derivatives is much more difficult with STO functions. The recurrence relations for GTO make integral and integral derivative computation much easier than for STO [16]. Any deficiency of the GTO basis can ultimately be overcome by using a larger number of basis functions, usually without great increases in computation time.
2.5
The Hartree-Fock and Self-Consistent-Field Wavefunct ions
In a variational sense, the Hartree-Fock (HF) wavefunction is the best wavefunction that can be constructed by assigning each electron to a separate orbital. An orbital is a function depending on the coordinates of only one electron (see eqs. (2.19) to (2.21)). The expectation value (the electronic energy) of the single determinant wavefunction for the closed-shell
system given by eq. (2.21) is (2.27)
Eelec = (41 H FIlleleciT H F)
2
E
Hi +
E
hii
E
-
Kia
(2.28)
-
(2.29)
2
2
E
14
2. BASIC CONCEPTS AND DEFINITIONS
CHAPTER
The one-electron, Coulomb and exchange molecular orbital (MO) integrals are defined by Hi
= hii =
f
07(1)h(1)0i(1)(17-1
(2.30) (2.31)
4);(2)0i(2)dric17-2 f f cYt r ( 1 )04 1 ) -1 r12
=
Jij =
( 1 )0i( 1 ) r12 07 (2)0i(2)drl d-r2
Kii = (iiiii) = f
(2.32) (2.33)
Usually all the orbitals occupied in the Slater determinant in eq. (2.21) are orthonormalized
(odo.i)
= Su =
(2.34)
where Sij stands for an element of the overlap matrix in the MO basis and delta function: =
{ 1
for for
0
i = i 0
is the Kronecker
(
2.35)
The "best" wavefunction is obtained from the variational conditions Eelec =
5
( 41 HFIHeleci* 1-14 =
(
2.36)
under the orthonormafization constraint of eq. (2.34). In this equation (5 refers to an infinitesimal change of the electronic energy. Equation (2.36) leads to the following HartreeFock equations [4,5] (2.37)
FOi = ficki
where ci are the orbital energies and the Fock operator, F, is given by
E (2J,(i) — Ki (1))
F(1) = h(1)
.
(2.38)
The one-electron operator, h(1), has already been defined in eq. (2.6). The Coulomb and exchange operators are J i( 1 ) =
f
r 1 2 d72 07( 2 )0i(2) —
r i 2 d72 K2( 1 ) = /07( 2 )0:( 1 )P12 —
(2.39) (2.40)
In eq. (2.40), P12 is a permutation operator which exchanges electrons 1 and 2. An element of the Fock matrix may be expressed as (2.41)
(0, 1 1103)
(OA Iii3
E (2.ik — ick)102 )
(2.42)
E {2(iiikk) — ( ikiik)}
(2.43)
HARTREE-FOCK AND SELF-CONSISTENT-FIELD WAVEFUNCTIONS
15
The orbital energy appearing in eq. (2.37) is a diagonal element of the Fock matrix fi
(0i1F10i) = Fii •-= hii
(iiiii)}
(2.44)
Since it is never possible in practice to use a mathematically rigorous complete set of basis functions in molecular computations, one obtains approximate solutions to the HartreeFock equations. The best single configuration wavefunction within a finite basis set is the self-consistent-field (SCF) wavefunction. Clearly, as the size of a basis set is increased, the SCF energy and wavefunction will approach the Hartree-Fock limits. For the self-consistentfield (SCF) wavefunction the electronic energy is minimized with respect to changes of the molecular orbital coefficients under the constraint of orthonormality. This problem was solved by Roothaan and Hall in 1951 for the closed-shell case using the LCAO molecular orbital approach [17,18] AO
Ot =
E
(2.45)
The resulting secular equation is
E
=o
[F„„
•
(2.46)
The Fock matrix F and the overlap matrix S are defined in the atomic orbital (AO) basis to have the elements AO
E
FA!, =
Dp,{2(popo-)
—
(APIva)}
(2.47)
Pa
and x;1(1)x,(1)dri
(X A iXy)
(2.48) (2.49)
In eq. (2.47) the one- and two-electron AO integrals are hg, =
x7,(1)h(1)x v (1)dri (Xp.ihiXv)
(2.50) (2.51)
and 7,112 x';(2)x,(2)dri ch-2 (twiPa) =f f 4(1)Xv (1)-
(2.52)
The density matrix D (in the AO basis) is defined by its elements do.
=
E
Cgi*Cvi
(2.53)
CHAPTER 2. BASIC CONCEPTS AND DEFINITIONS
16
where d.o. refers to the doubly occupied molecular orbitals. The normalization condition for each molecular orbital is: AO
E
1
LL
(2.54)
The SCF theory for open-shell cases was rigorously developed by Roothaan in 1960 [19] and later by many others [20,21]. For open-shell systems it is conventional to use the generalized
Fock operator [20,21] occ E (aikJk(1)
Fi(1) = f1h(1)
0ikKk(1))
(2.55)
where f, a, and 0 are one- and two-electron coupling constants (see Chapter 5) and occ stands for both doubly and partially occupied molecular orbitals. It should be noted that there is more than one Fock operator in the open-shell equations. Defining the Lagrangian matrix as (2.56)
(oilFikb.) oc.
+ E (otiok +
(2.57)
oce fihi
E
dik(ikljk)}
{aik(iiikk)
(2.58)
the variational conditions for an open-shell SCF wavefunction are ci) — ic ji = 0
2.6
.
(2.59)
The Configuration Interaction (CI) Wavefunction
In closed-shell SCF theory, a wavefunction is constructed as a single determinant. In the configuration interaction (CI) method [22] the wavefunction is expressed as a linear combination of a finite number of either Slater determinants or configuration state functions (CSFs, which are constructed as linear combinations of determinants to be eigenfunctions of the square of the total electronic spin angular momentum operator, S 2 )
CI
= E c11
(2.60)
The CI are the Cl expansion coefficients and 4)/ the CSFs. Assuming that the real wavefunction is employed, it is common to normalize the CI coefficients such that
E
C = 1
(2.61)
17
THE CONFIGURATION INTERACTION WAVEFUNCTION The electronic energy (expectation value) for such a wavefunction is
(2.62)
(4/c/1Heiec1iFci)
▪
Eelec
ci
▪ E IJ
(2.63)
CJIIIJ
where H ei„ is the usual Hamiltonian given by eq. (2.3). The CI Hamiltonian matrix HIJ =
(4)11Heiec1 4).1) mo Q.!! hij
E
H jj
is
(2.64) MO
E
(2.65)
ijkl
Q" and G" are the one- and two-electron coupling constants between con fi gurations and molecular orbitals [23]. The one- and two-electron MO integrals are q51(1)h(1)¢5(1)dr i
=
(2.66)
and
(ijiki) = f f 0:(1)0i(1)-4(2)01(2)dr i dr2 7'112 The electronic energy is given alternatively in the MO expansion as MO
MO
Eel ec
=E
(2.67)
Qii hi,
+E
ii
(2.68)
ijkl
where the one- and two-electron reduced density matrices [23] appearing in the equation are related to the corresponding coupling constants through CI
=
E
(2.69)
CICAti
IJ
and CI
Gijkl
=
E
(2.70)
CICja lifki
IJ
The following symmetric relationships hold for elements of the one- and two-electron reduced density matrices provided real MOs are used:
(2.71)
=Q and Gijki = Gijlk = G jikl = Gjk = Gklij=
= Glkij = Glkji
.
(2.72)
18
CHAPTER 2. BASIC CONCEPTS AND
DEFINITIONS
Each configuration is constructed using a set of molecular orbitals from a given wavefunction, commonly called the reference wavefunction. In most cases the reference configuration is either an SCF or a multiconfiguration (MC) SCF wavefunction (see Section 2.7). The CI wavefunction is determined by the variational conditions
6 Eetec =
ci) = 0
(2.73)
which lead to the following eigenvalue problem ci
E
— 45/JEet,c)
=o
(2.74)
When all possible configurations (with all possible electronic excitations) are included in the CI expansion (2.60), the procedure is called a full CI. The full CI method is the best possible variational treatment for a given basis set. In practice, however, the full CI wavefunction cannot presently be obtained for systems with more than 10 to 12 electrons using a basis set of reasonable size. Among limited CI methods the most frequently used technique is called configuration interaction with singles and doubles (CISD). The CISD wavefunction includes all possible single and double excitations with respect to the reference configuration. In constructing a CI wavefunc.tion the frozen core and deleted virtual approximations are quite often employed. In the frozen core approximation electrons occupying a limited number of the low-lying occupied molecular orbitals are kept "frozen" and excitations of these electrons are not included in the CI expansion. In the deleted virtual approximation a limited number of the high-lying virtual (unoccupied) orbitals are "deleted" and electrons are not allowed to occupy these orbitals. The frozen core orbitals are usually selected from core (inside valence-shell) atomic-orbital-like occupied orbitals, and the deleted virtual orbitals from the corresponding core-counterpart orbitals. Employing these approximations the number of possible excitations in the CI space may be substantially reduced and great reductions in computation time are expected, usually without significantly lowering the quality of the CI wavefunction.
2.7
The Multiconfiguration (MC) SCF Wavefunction
In constructing the CI wavefunctions described in the preceding section, the set of molecular orbitals is determined in advance. In the multiconfiguration (MC) SCF method [24,25] the electronic energy is minimized with respect to both the molecular orbitals and the CI coefficients. The MCSCF wavefunction is expressed as a linear combination of a finite number of Slater determinants or configuration state functions (CSFs)
CI TmcscF
=E
c 11
(2.75)
19
THE MULTICONFIGURATION SCF WAVEFUNCTION
where the C1 are the CI expansion coefficients and (I)/ the configuration state functions. The CI coefficients are usually normalized as (2.76) The electronic energy for such a wavefunction is Eel ec
(2.77)
( 4' MCSCF1Helecil MCSCF) CI
E
(2.78)
CICJHIJ
IJ
with the CI Hamiltonian matrix
Hij
being
HIJ
(2.79)
J) MO -
MO
yiiIJhii
E
(2.80)
ijkl In the equation above the 7" and FIJ are one- and two-electron coupling constants between configurations and molecular orbitals [23]. The electronic energy is given in the MO basis as
E,,„
mo
=E
mo
.
E
76hij
( 2.81)
The one- and two-electron reduced density matrices [23 1 appearing in the equation are related to the corresponding coupling constants through the expressions ci
=
E
CICJ71/
(2.82)
IJ
and
CI
Fijkl
=
E
(2.83)
lJ
The coupling constants 7 " and r" and reduced density matrices y and for the MCSCF wavefunction have the same meaning as the corresponding quantities for the CI wavefunction. Different notations (Q and G for CI; 7 and for MCSCF) are used in this book in order to avoid any possible confusion.
r
The molecular orbitals are usually orthonormalized giving 823 — 6,3
•
(2.84)
The true MCSCF wavefunction is determined from the variational conditions Eele , =
o (41 MCSCFIHelecIT MCSCF) =
(2.85)
20
CHAPTER 2.
BASIC CONCEPTS AND DEFINITIONS
which lead to equations for the CI space formally the same as eq. (2.74) ci
E c,
(2.86)
— 6/JEeiee
If the Lagrangian matrix for the MCSCF wavefunction is defined as MO
MO
E7j mhim + 2 E Fimid (ind kl)
xi,
(2.87)
mkt
then the variational conditions on the MO space are 27,3 - Xjz = 0
.
(2.88)
These equations are in a form very similar to those for the open-shell SCF wavefunction in eq. (2.59). It should be emphasized that two sets of variational conditions, eqs. (2.86) and (2.88), have to be satisfied simultaneously to obtain an MCSCF wavefunction under the two sets of constraints, eqs. (2.76) and (2.84). Among MCSCF methods, the most frequently used technique is the complete active space (CAS) SCF method [26]. The CASSCF formalism yields MCSCF wavefunctions corresponding to a full CI within a limited configuration space (active space) and a limited number of electrons. This type of MCSCF wavefunction is relatively easily obtained due to the full CI procedure in the active space and proves to be a very powerful starting point in the investigation of a wide variety of chemical systems [26]. The problem with the CASSCF method, however, is that the number of configurations in the full CI increases drastically with the number of electrons. For example, with no spatial symmetry restrictions, CASSCF with 12 electrons and 12 active orbitals generates 226,512 singlet CSFs.
2.8
Energy Expressions in MO and AO Bases
Using the definition of an LCAO-MO in eq. (2.20), the one- and two-electron MO integrals may be related to the AO integrals. These transformations will be extensively used later in the practical implementation of analytic derivative methods.
2.8.1
One-Electron
MO Integral
The one-electron MO integral hii is
=
07( 1 )h( 1 )(ki(1)dri (iii)
(2.89) (2.90)
ENERGY EXPRESSIONS IN MO AND AO
E
21
AO BASES
CC
ov
f x*A (1)h(1)x,(1)dr1
(2.91)
AO
E
(2.92)
The one-electron AO integral h„ is defined by
hp,
=
f Xp* (1)h(1)x,(1)dri
(2.93)
(XAihiXv)
(2.94)
where h is the one-electron operator defined in eq. (2.6). The following symmetries exist for the AO and MO integrals provided real basis functions are used: hp, = hum
(2.95)
= hji
(2.96)
and
2.8.2
Two-Electron MO Integral
The two-electron MO integrals (ijiki) are related to the AO integrals through
(iiiki)
(2.97)
(14 (1)(1)3(1)- 1 0Z(2)(1)1(2)C17-1d72 r1 2
AO
1
E
x(0x,(0-4(2)x,(2)drid72 r12
Ampa
(2.98)
AO
E c;:c;,circ,/,(„0,,)
(2.99)
The two-electron AO integral (pulper) is
(,uviPa) =
1 f 4( 1 )Xv( 1 ) — X*,( 2 )Xe( 2 )dridr2 r12
( 2.1 0 0 )
with 1/ri 2 being the two-electron Coulomb repulsion operator defined in Section 2.1. The following very useful symmetric relationships hold for the AO and MO integrals provided real basis functions are used: (twIP a )
= (p u lap) = ( vpIP0- ) = (i'jultrA) = (Paii-tv) = (Paivia) = ( 6Pittv) = (aPivih) (2.101)
and
(Oki) = (ii ilk) =
= (iil/k) = (Oki) =
=
= (lkiii) • (2.102)
CHAPTER 2.
22
2.8.3
BASIC CONCEPTS AND
DEFINITIONS
The Electronic Energy for a Closed-Shell SCF Wavefunction
The electronic energy for a closed-shell SCF wavefunction is expressed in the MO basis,
E
=2
hi
,
+E
_
(2.1 0 3)
where d.o. denotes doubly occupied molecular orbitals. Using eqs. (2.92) and (2.99), eq. (2.103) can be rewritten as d.o. {AO
2
Eeiec
E E
CC li A ,}
• Av d.o. [ AO r
t2C:C,i C3p * C1, — CI: 0/ Gip * C4 (111)1PH
+ E E ii
(2.104)
AvPa AO { d.o.
E E cixi,} hii,
= 2 kw
i
AO
t
[2 I d
+ tIvPa
Ci:
C7,1 I d t
Op * Or } (kwipo - )
i
i d.o.
d.o.
i
i
_ {E cim*ci } 1E CC .ip*/ (plu pd AO
2
E
AO
DA ,11,,,,, +
AO , D[2h„
[DA,D„a{2(twIper)
+E
D,,,{2(fivipa)
Pa
AO
--- E
E Al/Pa AO
AL,
=E Al,
(2.105)
D„(hp,
-
-
(tipiva)}1
(ppiva)}1
+ F)
(2.106)
(2.107)
(2.108)
The density matrix D and the Fock matrix F in the AO basis are given by
E
CC
(2.1 0 9)
E pp.:4 2 (i1 0Pa)
(2.110)
and AO Ft",
Pa
ENERGY EXPRESSIONS IN MO AND
2.8.4
AO BASES
23
The Electronic Energy for a CI Wavefunction
The electronic energy for a configuration interaction (CI) wavefunction in MO form is MO
MO Eelec
=
+E
E ii
ijkl
where Q, and Gu ki are elements of the one- and two-electron reduced (particle) density matrices [23]. Using eqs. (2.92) and (2.99), the electronic energy may be transformed into the AO basis as AO
MO Eelec
=
E
Qj2
AO
MO
+E
E gv
Gijk1
iikl
E
c7aLcpk-cui(tivIper) (2.112)
Avom ,
AO IMO
EE sj AO IMO
+E E AO
E gv
(2.113)
Gijkl (i41/119(7)
ijkl
wp
AO
Q„,h„,
E
(2.114)
G„,p,(pvi docr)
AvPg
The one- and two-electron reduced (particle) density matrices in the AO basis are: MO
= and
E
(2.115)
MO
G kiv p,o-
=E
(2.1 1 6 )
Coi* CCpk* Cal Gijkl
ijkl
The following symmetric relationships hold for elements of the one- and two-electron reduced density matrices in the AO basis provided real basis functions are used:
=
(2.117)
and G Ai/ prf = G
or p
=Gv
Apo.
=Gvp=
r
= G peru ,4 = G,p„, =
Gappm, .
( 2.118)
CHAPTER 2. BASIC CONCEPTS AND
24
2.9
DEFINITIONS
Potential Energy Surface
The energy expressions described in the preceding sections are defined at certain fixed nuclear coordinates of a given molecule. Using the Born-Oppenheimer approximation [2] the total energy of a system may be described as a function of nuclear coordinates (see Section 2.1). The surface formed by the potential energy vs. nuclear coordinates is usually termed a potential energy hypersurface (PES). One of the most important goals of quantum chemists is to locate equilibrium and transition state structures along a reaction coordinate. In order to locate stationary points, where all the energy gradients (first derivatives of total energy) with respect to nuclear coordinates become zero, the force method by Pulay [27] and/or some standard mathematical optimization techniques [28], e.g., quasi-Newton-Raphson or Newton-Raphson methods, are widely employed. In this light development of the derivative methods in ab initio electronic structure theory has already made significant contributions to the deeper understanding of chemical reactions.
Minimization of the total energy, Etotai, with respect to a set of nuclear coordinates, "a", leads to a stationary value of Et otai, that is, aEtot a l
Oa
=0
.
(2.119)
In order to characterize the nature of the stationary point as a local minimum or local maximum (in one or more dimensions) it is necessary to examine the second-order variation (Hessian) of E tota i with respect to all normal coordinates g,
492 Etotal
a2 q
> 0 for all normal coordinates (local minimum) 2 UAU!i CrA C:C:C!„(prviper
pt Ccr i (AllipC)
TS
MO
MO
vAu ubi c;c,i,cpkcci,(AviPcr
+E
TU
uzupkqcfcptcu'oivipa)
Si
MO
MO
UZUIC Ai C:Ci;C:(AviPcr
+E
utwubi c„ic,,,cptCA /Wipa)
tk
SU
MO
0,:i uzcr,c,5,c,kc!,( k wipu
E
urbi utak cix ii,cptcciolopa)
ri
MO
Urbi U1C;Cii,Cpk Cl:Guvipa
E St
MO
usbj utakci„cfcptcu'oluipo-)
MO
UtjUICith eCpk C2v1Pa SU
E
qUtaaCm C,j,Cpt a:.(iivrper)}
ti
(3.96) The series is terminated to exclude cubic and higher terms in A. The expression (3.96) is meant to implicitly include terms in Ab, Att , and A. Collecting terms by order of perturbation gives:
(ijikl) vert
=
(iilki ) MO
+ )1/4 a { (OW
E
MO
UA(rilkl)
E uz(isiko 8
46
CHAPTER 3. DERIVATIVE EXPRESSIONS
mo
E
mo
E u-;!7 (isiki)
MO
E
Utakb (iilt 1 )
MO
UA(rilkl) b
-F
E MO
E
Utak(iiltl) b
ul(iiiku)b
MO -
F
E
q(isikoa
mo Utbk(iila) a
E
uAu sbi (rsiko
+E
MO
+E
utb,(iiiku)a
MO
TS
UAUtbk(r Ma)
rt
MO
+E
MO
uAu ub,(riiku) +
TU
st
MO
MO
+ E u:i uub,(isiku) + E utwubt(iiltu) su
tu
mo
mo
+ E up.i tvrsiko + E urbi utak (rilto TS
rt MO
NiU:i(rj(klt)
E quiak osito st
MO
MO
+E
usbi u:,(isiku)
+E
utbku:i(iiitu)}
tu
SU
(3.97) The purely AO parts of the MO derivative two-electron integrals (ijjki)a and WIN)" are defined by AO
= AyPg
u) Ci, c,i,ck ci 8(kwaip
" a
(3.98)
and AO
=
E ci,c,i,c ik,c,icr a2(uviPa) 8a0b
twPcr
(3.99)
DERIVATIVE EXPRESSIONS FOR TWO-ELECTRON MO INTEGRALS
47
In deriving eq. (3.97), terms in Ab, A! and At were excluded for the sake of brevity. Since the summations run over all molecular orbitals, the interchange of the indices r, s, t, and u does not alter the equation. Thus, the first- and second-order derivatives of the two-electron MO integrals are
0(ijjk/) 8a
—
mo
+E
(UL(ntjikl)
UTank (ijiml)
U,ani (ijIkm))
(3.100) and
02 (ijild)
acm9b
=
+
(iiiko ab mo E (u,a7ticmilko
+ uta„b_jimiko + uekciiinto + u(7ni bi (ijikin))
mo
+ E (uzi oniikob ▪ qi(railki)
U:a(iiikm) b
+ UZj(inlikl) b
a
mo
+E
nj + Umb
mi
Umb ,Ura,k )(injinl)
+ (Ulanga
mn
Unb i
▪ (uAi unb
Uni b iU:1)(rnjikn)
( 11 jUnb k
•
um b
(U k Unb
• OrtkU77.1)(iilran)]
j
U,71 )(inilkn)
Umb‘i U k )(imlnl)
(3.101) The derivative integrals, (ijild)a and (ijik/)a b , defined in eqs. (3.98) and (3.99), will be called the skeleton (core) derivative two-electron integrals throughout this book. These terms are due purely to AO changes.
3.9.2
An Alternative Derivation of the First Derivatives of Two-Electron MO Integrals
The first derivative expression for two-electron MO integrals may also be derived by directly differentiating eq. (3.93),
a(iilko
a
Oa
--OTt
AO
E 141/02 (7
C CdxfC,17 (pvipo- )
(3.102)
48
CHAPTER 3. DERIVATIVE EXPRESSIONS
AO {aC Ai aa Avpa
•
c!,(thvipa)
7-‘ 4_,
aCi k " Oa CP C(Pv1Pa)
C"
P
k ocl, p-07-1(1w1PasT)
OCk
+ clicil,-wc21-0a (tivipa) + 19(tIVIPO) Cit4C1Cpke
(3.103)
Oa
a
The definitions in eqs. (3.7), (3.93) and (3.98) may be employed to rewrite (3.103) as:
AO I MO
0(ijikl) Oa
MO
E E
AvPa
uTni CIAnCli,CpkCat (AVIPO. )
E
u,ank qc,i,c,:c,i(pvipa)
+E
(IA jCipC": CI;Cly(f411 1Pa)
17Z
MO
MO
+E
wrin ,ci„cl,cpkc:;%Givipa)
772
8( AavalP ' )
+
(3.104)
mo
E
( 3.9.3
(3.105)
The Derivative Expression for the Skeleton (Core) Two-Electron Derivative Integrals
The derivative form of the skeleton (core) derivative two-electron integral (3.98) may be obtained in a manner similar to that in the preceding subsection,
0(ijjkl)a
a
Ob
Ob
AO CiAC pk
49(AVIP(T)
(3. 1 06)
ALL'Pa
AO E
49 (tIvIPa)
ab
pivpa
+
Oa
aowlpa) Oa
1 0 (1-wiPa) C;`,•act ab CpkCc. aa
.
OC,'„ 0 (iivIPa)
p ob
Oa
pk Cul (92(.91-Laval 16)(7) 1
(3.107)
This equation may be modified using eqs. (3.7), (3.98), and (3.99) to yield:
0(ijikl)a Ob
AO IMO
=
EE m
\
u„, b,7,nc,i,cd;cg , c i a CuaviaPcr)
MO
E
ubm3 ci„c-kci „c'cavIPc) Oa
P
DERIVATIVE EXPRESSIONS
FOR TWO-ELECTRON MO INTEGRALS
mo
mo
+E
+E
Oa
1YL
77t
ub
Ci CC k Gn a(luilPc) Oa '
Clpii,cpkc alf92 0-w1Por)} Oaab
(3.108)
mo
▪ E
49
(uLonilkoa + u(imikoa
+
wP„k(zikni)a
+ (3.109)
Derivative expressions for the skeleton (core) two-electron higher derivative integrals may be obtained in a similar manner and are summarized in Appendix K.
3.9.4
An Alternative Derivation of the Second Derivatives of Two-Electron MO Integrals
An alternative derivation of the second derivative expression for two-electron MO integrals may be obtained by directly differentiating eq. (3.105),
02 (ijikl) Oaeb
8 TO(iilk1)1 Ob Oa
A
[
L
mo
u,a„
(miiki)
+ uzi ( imikl)
m
mo
au..
E
0(inijkl) + U,a7z2 ab
06
ar_1;17,3 (inzikl)
▪ ab
+ '914c" (OM!) Ob
8U 1 •
V;ank(iiiird)
11,4,3 0 (irn ablkI) a
UmIc
NO/a) Db
0(ijik771)1
+ Una
Ob
Ob f (3. 1 12)
•
8b
Relationships (3.14), (3.105) and (3.109) may then be employed to give
02 (ijik/) &c ob
mr° [
E
b
mo
—E
umbnU7azi) (milk°
mo U:„-{E [UL(njjkl)
0,i (mnikl)
-
50
CHAPTER 3. DERIVATIVE EXPRESSIONS
+ U(m.7172,1) + 011( 170kn)] + (mjikl) b } + (II `,;*,bi —
(in/1M)
n
mo + U,a,,i{ n, [U i (nmikl) + U„,(inikl) + Ifni' k (iminl) + Unb t (imikn)] + (imikl) b } mo + (11k —
E
0,,,„11k) (iiiml)
71
mo + U7ank tE
+
U(iniml)
+ uPaciiimnd +
(iilmob}
MO
+ (uzbi — E
uu,ad)
MO
+ u77,1 {E
[UP,i ( njikrrt) + U(inikm) UL(Okn)] + (ijikm) b li
MO
+ E futicmilko. + + (7j1k1)ab
Uti(imikl) a + Utk(Ornl)a + Ora(0km) a }
.
(3.113)
Equation (3.113) may then be simplified further to
02 (0k1) NA%
mo
E
mo
E
(utanbi(milko
+
utt(irniki)
+
+ u(iiikm))
(1» i ( rnjikl) b + 11 i (Mlik1) b + U,ank (ijiml) b + Urani (ij(krn) b
+
uti( milk oa +
+ utk ( iiim i)a + On i(iilk771) a)
MO
RULUn6j
Umb
(mni ki)
(UUnbk
Umb i U k )(mjinl)
mn.
• (U,an iUnb i
Um b iU:1)(Mjik71)
• (UZjUnbi
tit JU,i)(imikn)
(11Unb k
UmbJU:,k)(i177171,1)
+ (U,'„kUnb + ULU:1)(0nm)]
DERIVATIVES OF THE NUCLEAR REPULSION ENERGY
51
(ij1k/r b
(3.114)
which equals eq. (3.101). The equality 02 (ijik1)/8aab = •92 (ijjkl)10bOa may be proved simply by taking the derivative of 11(ij1k/)/(9b with respect to the variable "a".
3.10
The Derivatives of the Nuclear Repulsion Energy
As mentioned in Section 2.1, only the electronic part of the Hamiltonian and total wavefunction are considered in detail in this book. The contribution of the nuclear repulsion energy to the total energy of a system is given by ZAZB
En uC
(3.115)
RAB
A>B
Although this term is constant at a fixed geometry, a geometrical perturbation changes the value of Enuc . It is convenient to note the relationships R 21B =
—K
(Y A —
(Z A —
Z4 2
(3.116)
and ORAB OXA
K A — KB
= ORAB =
RAB
axB
(3.117)
for manipulating the derivatives of the nuclear repulsion energy.
3.10.1
The First Derivative of the Nuclear Repulsion Energy
The first derivative of E, is obtained easily by differentiating eq. (3.115) with respect to a nuclear cartesian coordinate KA, 0Enuc
axA
-
0,.
i
N E
zAz8
(3.118)
;
A>B RAB N
= — E (x A _ xB )zAz-B Boil
3.10.2
RAB
(3.119)
The Second Derivative of the Nuclear Repulsion Energy
There are four different types of second derivatives of the nuclear repulsion energy. They may be obtained by directly differentiating eq. (3.119) with respect to a second variable,
XA
Or KB.
52
CHAPTER 3. DERIVATIVE EXPRESSIONS Type I 8 2 Enuc
axi
=
a
(aEn.c)
(3.120)
axA ax A N
-
ZAZB
E ieA B
N
+3
I3,4
E (x,i
xB)
Z AZB 2
BOA
RA 5B
(3.121)
Type II
a2 8 (OEnue) axAax B = axB axA J Enuc
ZAZB
3 (XA
R3AB
(3.122) XB) 2 ZA ZB
R AB 5
(3.123)
Type III 82Enuc
OXAaYA
=
a
(aEnuc)
avA axA ) N = 3 E (xA -
(3.124)
xB)(YA
, —
1B
ZAZB D
(3.125)
'LAB
B#A
Type IV
02 Entic aXADYB =
a
a
(Enuc
(3.126)
OYB aXA
— 3 (XA
— XB) (YA
Y B) ZDAsZB
(3.127)
'AB
The nuclear derivative should be added to the corresponding electronic part to obtain the total energy derivative.
References 1. J. Gerratt and I.M. Mills, J. Chem. Phys. 49, 1719 (1968). 2. J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Binkley, Int. J. Quantum Chem, S13, 225 (1979). 3. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Chem. Phys. 103, 227 (1986). 4. Y. Yamaguchi, Mi. Frisch, J.F. Gaw, H.F. Schaefer, and J.S. Binkley, J. Chem. Phys. 84, 2262 (1986).
Chapter 4
Closed-Shell Self-Consistent-Field Wavefunct ions The quantum mechanical concepts and most elementary derivative quantities have been introduced in Chapters 2 and 3 and allow us to proceed to the simplest practical example of energy derivatives. Closed-shell self-consistent-field (CLSCF) wavefunctions are very frequently used in ab initio quantum chemistry. The electronic energy of the system is determined using only one set of doubly occupied orbitals. Although this single-configuration approximation is oversimplified in a quantum mechanical sense, it is well defined and accepted at least as a conceptual and computational starting point by most chemists. Indeed there are many chemical phenomena, e.g., equilibrium geometries, vibrational frequencies, and infrared intensities of various molecules, that have been explained quite successfully using such methods. In this chapter the energy expression for a closed-shell system is given first and then the Fock operator and the variational conditions are described. Starting from these basic concepts detailed derivations of the first and second derivatives with respect to nuclear perturbations (in terms of cartesian coordinates) are presented.
4.1
The SCF Energy
In restricted Hartree-Fock (RHF) theory, the electronic energy for a single configuration closed-shell SCF (CLSCF) wavefunction is [1]: d.o.
Ee lee = 2
E
d.o.
h1
+E
- (iiiii)}
(4.1)
are the one-electron integrals, and (iiljj) and (u Iii) are the Coulomb and exchange integrals in the MO basis. The summations are over doubly occupied (d.o.) orbitals. Alternatively the electronic energy may be expressed using one-electron integrals and closed-shell
53
CHAPTER 4.
54
CLOSED-SHELL SCF WAVEFUNCTIONS
orbital energies as d.o.
E
Eelec
(hii
+ Ei)
(4.2)
where the orbital energies, q are defined by ,
d.o.
E
Ei =
{2(iiikk) — (iklik)}
(4. 3)
The total energy of the system at the SCF level is a sum of the electronic energy and the nuclear repulsion energy: EscF
=
Eelec
Enuc
(4.4)
It is important to realize that the electronic energy is defined using only the doubly occupied orbitals. Only these doubly occupied orbitals are well defined physically in singleconfiguration Hartree-Fock theory. The virtual or unoccupied orbitals exist as eigenfunctions of the Fock operator (see next section) and are mathematically well defined but of limited value as far as the physical model is concerned.
4.2
The Closed-Shell Fock Operator and the Variational Conditions
In restricted Hartree-Fock (RHF) theory, the Fock operator for a single configuration closedshell SCF wavefunction is [I]: d.o.
F= h
+ E
(24 -
Kk)
(4.5)
h is the one-electron operator and .1 and K are the Coulomb and exchange operators. The Fock matrix elements involve a sum over doubly occupied (d.o.) orbitals d.o.
Fii = (cAilF10j) = hij
E {2(ijikk) — (iklik)}
(4.6)
where hii are the one-electron integrals, and (ijikk) and (ikljk) are the two-electron integrals in the MO basis. Using eq. (4.6), the variational conditions for the system are 15i3 Ei
(4.7)
where bii is the Kronecker delta function and Ei are the orbital energies defined by eq. (4.3). According to the variational conditions (4.7), the CLSCF wavefunction has converged when the Fock matrix in the MO basis becomes diagonal. Each diagonal element is an orbital energy defined by eq. (4.3). It should be noted, however, that the choice of variational condition (4.7) is only possible for the CLSCF wavefunction with one Fock operator for all the closed-shell occupied orbitals. The expression of the variational conditions for an SCF wavefunction of more general form is given by the symmetric character of the Lagrangian matrix, as will be shown later.
THE FIRST DERIVATIVE OF THE ELECTRONIC ENERGY
4.3
55
The First Derivative of the Electronic Energy
Within the Born-Oppenheimer approximation [2] the nuclear repulsion energy and its derivatives are trivial to evaluate at a fixed geometry for the molecule under consideration and were described in Chapter 3. Therefore, only the electronic part of the SCF energy is considered here. An expression for the first derivative of the electronic energy of the system may be obtained by differentiating eq. (4.1) with respect to a cartesian nuclear coordinate "a": aEelec
2
—
d.o. aki E Oa + E `Lc) *
(4.8)
Using the results of Sections 3.8 and 3.9 (or Appendix J) the derivative expressions for the relevant one- and two-electron MO integrals are: all
Ofti• Oa
+ 2
E
(4.9)
all
=
Oa
jr + 2
all
(90.7Ii3)
=
Oa
4- 2
+
E
(4.10)
r
+
E
UAj(inzlii)}
(4.11)
"all" implies a sum over both occupied and virtual molecular orbitals. Using eqs. (4.9) to (4.11), the derivative expression (4.8) may be expanded to yield .
0.Eetec a
..- 2
E h'iLi i
all -
I- 2
E
d.o.
+E
tr„'„ i hmi}
m
all
[2(i ilj j)a + 4 — (ijlij) a — 2
E {u,a.i(niiii) + u,;‘,•,(iiimi)} all
E
UAj (im1ij)}1
This equation may be manipulated in the following manner,
0 Eejec Oa
d.o.
d.o.
= 2
E
+
EE
d.o.
2
E
4
d.o.
all
i m d.o. all
-I-
(iiiii)a}
EE m
+2
all
EE u7-„i hni; j ,n
+
d .o.
4
a ll
EE ij m
. (4.12)
56
CHAPTER 4. CLOSED-SHELL SCF WAVEFUNCTIONS d.o. all
2
d.o. all
EE
— 2
EE vAi ( rniui)
(4.13)
m
d.o.
d.o.
2
E
2
EE
— (ijiij)a}
d.o. all
+
u [h,,,i
+E
2 EE u:2 [hm
E
i rn
d.o. all
-I-
d.o.
—
.
(4.14)
'n
The definition of the Fock matrix, (4.6), can be used to simplify the terms in square brackets: d.o.
OE, lee
2
Oa
d.o.
E
E {2(iiljjr
—
(OP}
ij d.o. all
▪
2
d.o. all
EE
u,ani Fmi + 2
EE m m 3
(4.15)
M
The equivalence of the summations over i or j then yields: aEelec
aa
d.o.
=2
all
+ 4
d.o.
E
E
—
d.o.
EE
UF
(4.16)
j Equation (4.16) is applicable to any CLSCF wavefunction whose Fock matrix is constructed using eq. (4.6). The third term in equation (4.16) may be simplified further due to the diagonal nature of the Fock matrix (4.7): d.o.
Eeiec
Oa
=2
E
-1- 4
E
d.o.
117;
E
—
d.o.
UE
(4.17)
As mentioned in Section 3.7, the molecular orbitals are orthon.ormal
= bij
.
(4.18)
Recalling one of the most important results of Section 3.7, the first derivative of equation (4.18) is U39:i =0 (4.19)
THE EVALUATION OF THE ENERGY FIRST DERIVATIVE
57
The diagonal terms satisfy the even simpler relationship
1 U29,' = — — n 2
(4.20)
Using equation (4.20), the final expression for the first derivative of the electronic energy becomes:
°EaeaI
d.o. "
=
2
d.o.
d.o.
E + E
-
—2
E
(4.21)
One should realize that the first derivative expression (4.21) does not require the explicit evaluation of the changes in the MO coefficients (the Ua defined in Section 3.2). Thus the coupled perturbed Hartree-Fock equations need not be solved if only first derivatives of the CLSCF wavefunction are of interest. Such a simplification is not possible for the second or higher derivatives as will be discussed in Section 4.6.
4.4
The Evaluation of the Energy First Derivative
hi the practical evaluation of the first derivative, expression (4.21) usually is reduced to the AO basis. The first derivative is rewritten accordingly as: 0Eelec
aa
d.o. AO
2
EE ci
d.o. i AOL—I Pu
EE 2
IL
{2c;c1,cip alacilvipa)
aa
P
a( ktopa)1 j aa
d.o. AO EE ci cvi f .as,„ aa
(4.22)
sAo
Defining the density matrix D by its elements, d.o.
D„, =
E
(4.23)
and the "energy weighted" density matrix W by its elements, d.o.
E cc1
W
(4.24)
eq. (4.22) becomes
0Eelec Oa
AO
2
E
D 41
ah ,
AO
+
{2Dpv D„
0 402D ,}
00-/v aa1Pa)
AVPa
AO
—2
E w" aOa sm-
(4.25)
CHAPTER 4. CLOSED-SHELL SCF WAVEFUNCTIONS
58
=5
4 a12fl, are evaluated only for unique In practice the derivative integrals , 2 &a a aa and --4 combinations of the AOs (tt > u and p > a) in order to minimize computation.
Thus, the first derivative of the electronic energy is given in an efficiently programmable form by 0Eete, 8a
AO
D
=-- 4 E (1
2
AO
_
+4
(1 _
)(1
)
>V p > po,
—
x {4
_
6--) w 2
— D ii,D, p } a(PaulPa)
as„„
(4.26)
8a
in which S i, is again the Kronecker delta function. In the second term of this equation the summations run over the indices that satisfy the conditions pv > po- , where /hi/ = p(1.4 — 1) + v and pa = p(p — 1)-f- a-.
4.5
The First Derivative of the Fock Matrix
At this point a first derivative expression of the elements of the Fock matrix will be obtained for future use. Differentiation of eq. (4.6) with respect to a cartesian coordinate "a"
aFt3
ohs.;
Oa — 8a
d.o.
a(ikli k)}
E k
"'
Oa
(4.27)
Oa
is required. The relevant derivative expressions for the one- and two-electron MO integrals are given in Sections 3.8 and 3.9 (or Appendix J) as,
0 hii
a(iiikk) Oa
all
=
E (uihm , + uz,him )
(4.28)
all
—
(ijikk)
E
{IT i (mj(kk)
U(imikk) (4.29)
2Uk(iiimk)} and
0(ikijk) 8a
all
—
E
{U(rnkLik) + Utank
Ijk)
▪ THE FIRST DERIVATIVE
OF THE FOCK MATRIX
r4i(iklmk)
59
147,k (ikijm)}
(4.30)
Inserting eqs. (4.28) to (4.30) into eq. (4.27) one obtains
= Oa
all
itZi -4-
E (u ihn,, + u,-,hi,,) m
d.o.
+E
all
[2(ijikkr + 2
E fui(milko + u(i ikk) +
211k(ij link)}
m
k
Uk(i1nLi
k) (4.31)
+ E {2(milkk) —
(Mkij k)}1
+ E {2(imikk) _ (4.32) Using the definition of the Fock matrix in eq. (4.6), this equation becomes all
all
EU j Fmj +
Oa
Tri
all d.o.
EE
U7an k{4(iiimk) — (imlik) — (iklim)}
(4.33)
m k
all
Ftaj
al l
E uTi Fik
Fkj
all d.o.
EE k
—
(ik1:71)
—
(ilijk)}
(4.34)
1
all
all d.o.
E(UF + ue,Fik) + EE k
karAij,kl
(4.35)
1
The skeleton (core) first derivative Fock matrix appearing in this equation is defined by d.o.
E {2(ijikk)a
— (iklik) a }
(4.36)
CHAPTER 4. CLOSED-SHELL SCF WAVEFUNGTIONS
60 and the A matrix by
= 4(ijiki) — (ikiji) —
4.6
.
(4.37)
The Second Derivative of the Electronic Energy
An expression for the second derivative of the electronic energy is obtained by differentiating eq. (4.16) with respect to a second cartesian coordinate "b",
8 ( 0Eetec)
0 2 Eciec
aa0b
(4.38)
Oa
Ob d.o.
d.o.
— ab ii f au,ai F
2 j
1
+4
2
A
Ob
a aFii i
ab "
ab f
(4.39)
The derivative expressions for the one- and two-electron skeleton (core) derivative integrals were given in Sections 3.8 and 3.9 (or Appendix K) as, all
141' + 2
Ob
E
(4.40)
all
— (iiiii)ab +
ab
2
E {u7bni(miiii)a + u ;( iiim3)a}
(4.41)
+ qi ( imiii).} •
(4.42)
all
=
+ 2
ab
E
Using eqs. (4.40) to (4.42), the first two terms in eq. (4.39) may be written as 2 d.o. 0ahb11.1
d.o.
=2
E
ab f
Ob
ij all
{hclib + 2
E
d.o.
all
+E
+
4
E
+
all
— (ijiij) ab — 2 E d.o.
2
E
+ Utbnj(irniii)a }1
d.o.
h7ib
+E
—
(owl
(4.43)
THE SECOND DERIVATIVE OF THE ELECTRONIC ENERGY
61
d.o. all
EE
•2
E {2onivir —
m
[
d.o. all
d.o.
+ E {2(rniiiir —
EE
+ 2
d.o. E
d.o.
2
E
iØ
(4.44)
_
ij d.o. all
d.o. all
EE
+2
umb i F,ani + 2
EE
2
u6M.7. FaM .7•
(4.45)
m
M
do.
d.o.
E
E
all d.o.
EE
4
(4.46)
uibiF4
In deriving eq. (4.46), we use the definition of the skeleton derivative Fock matrix, eq. (4.36), and the fact that it is immaterial whether the summations run over i or j. The third term in eq. (4.39) may be manipulated using results of Section 3.3 and eq. (4.35) to give all d.o.
4
auft
EE all d.o.
=4
u j ab}
ab'Fij
ta
all
EE
UPI)
—E
all d.o.
+ 4
all
EE i
Fii
k
ii
E
Fibj
(uzi Fki
+
all d.o.
uzi Fik)
•
+ EE k
all d.o.
uZiAii,kii
(4.47)
I
all d.o.
=4
EE
+4
E Fii E 1k
Ultib Fii + 4
all
EE
uzFib,
d.o.
jk
ij all d.o. all d.o.
+ 4
EEEE ij
(4.48)
k
Combining equations (4.46) and (4.48), the expression for the second derivative of the electronic energy becomes d.o.
d.o.
2 E h 16 + E {2(iimab —
.92 Eelec
Octab
ij all d.o.
▪
4
EE
Fij
(0j)ab}
62
CHAPTER 4. CLOSED-SHELL SCF WAVEFUNCTIONS d.o.
all
+ 4
EE (U Fiaj
+ 4
EF E
+ 4
EEEE
U
d.o.
all
d.o all d.o.
all
U
(4.49)
UIA,kI
ijk
Equation (4.49) involves the second derivatives of the MO coefficients, and this equation is applicable to any CLSCF wavefunction whose Fock matrix is constructed from eq. (4.6). In order to eliminate qb in eq. (4.49), the second derivative form of the orthonormality condition of the molecular orbitals described in Section 3.7 (or Appendix L) is used,
Uza3b
1/3"ib
=
o
(4.50)
where u.,P is defined by all tab
'57
U17ri 51
+
771
u u;
s .s.",„
-
(4.51)
It is evident that the diagonal terms of the Ua b matrix satisfy (4.52) Using the relationships (4.7) and (4.52), a final expression for the second derivative of the electronic energy is found: 02 Eelec
=
d.o.
d.o.
2 E
E
0a0b
{ 2(iiiii)ab
iJ d.o.
2
E all
d.o.
q4)
4
EEj
4
EE qutE,
4
EEEE U2
all do.
all do. all d.o.
(4.53)
UIA,kI
ijkl d.o.
==
2
d.o.
Eh + E ij
-
AN ALTERNATIVE
DERIVATION OF THE SECOND DERIVATIVE d.o.
do.
-
2
E szbf, _ all
d.o.
i all
j d.o.
all
d.o.
2
E
ab
63
q
▪ 4 EE @fib; + UZFib• ▪
4
EE J
▪
4
d.o.
all
EEEE qut,{4(iiik1) — ijk
The
7, b
(ikijI) — (ilijk)}
(4.54)
I
matrices in eq. (4.54) were defined in Section 3.7 by their elements: all
ne
+
tuiam ub.
==
)711
1771 j771
—
17%
sb S
Sim
SM.
3971
(4.55)
in
Equation (4.54) is in a form that is symmetric with respect to the differential variables "a" and "b". For the evaluation of the second derivative of the electronic energy, information about the changes in the molecular orbitals due to perturbations Ua is required as seen clearly in eq. (4.54). The elements of the Ua matrices may be readily obtained by solving the coupled perturbed Hartree-Fock (CPHF) equations, which will be discussed in detail in Chapter 10.
4.7
An Alternative Derivation of the Second Derivative of the Electronic Energy
Another expression for the second derivative of the electronic energy of the system can be obtained by directly differentiating eq. (4.1) with respect to two cartesian coordinates "a" and "b", { 2-•-• a 2 Eelec = 2 E(4.56) E 2 a (n133)
0a0b
0a0b
0a0b
0a0b
The relevant second derivative expressions for the one- and two-electron integrals were given in Sections 3.8 and 3.9 (or Appendix L) as, hii
&Lab
all
h719
+
2
E
U;inbihn,i all
all
+ 2
E (tf
i hm 5
Umb
+
2
E mn
Vfl
all
Nab
(iili i) ab 4- 2 E
+ u;;•zbi (iilmi))
(4.57)
64
CHAPTER 4. CLOSED-SHELL SCF WAVEFUNCTIONS all
2
E
+
+ uttamiiiir +
in all
+2
all
E
+ 2 E
mn
mn
all
E (uA. i unb, + umb i u(4,)(imii.)
+4
,
(4.58)
mn
and
Oaab
=(
j jj jj) ab
all
E (u4(miiii) + ut(imiii))
+
2
all
+2
E
17;n1
all
4
-I-
E uz i unbi (Mniij) mn all
+ 2
all
E
mi
+ 2
ni
mn
inn
all
+ 2
E u;,„,unbi (im)in) all
E uziUnbj(mulin)
+2
mn
E
(4.59)
U7aniUnb i(imlni)
mn
Using eqs. (4.57) to (4.59), the second derivative expression (4.56) is then: 192 Eelec
(9a811
d.o.
2
E
all
[117 6 + 2
E
U,thm i
all
-I- 2
all
E (;1•„ihti
ULh !(,'.ni) + 2
E mn
d.o.
+2
E
all
••
at,
ij
/
+ 2E
+ uranb.i(iiimi))
all
+ 2
E
+ 2
E
+ 2 E
inn
mn
+
all
+ qionivir + all
trrzn jUnb
all
+4
E (uAiUnb j + utiu;:j)(imlid mn
d.o.
all
—E
+ 2
ij
E (u7a4(milii) + ra
all
+ 2
/
E (u,anicrnitig +
+
+
qi (iilmj )a)
AN ALTERNATIVE DERIVATION OF THE SECOND DERIVATIVE
65
al/
U,LUybki(mniii)
+4 mn
all
all
ira
+2
U7bti
2
(mu I ni)
mn
all
all
+ 2 E uunbi(iirtini)]
U7a„iU7bL i(rnjlin) mn
(4.60)
mn
do.
E
u,unbi (imiin)
mn
+2
= 2
E
d.o.
h7ib
+
—
E {2(iiIii)" ij
d.o. all
do.
± 2 EE m u,anbi [hrni + E {2(niiiii) _ i
j
d.o. all
+ 2 EE
j
d.o.
u;,,,,,b,{hrni + E {207-tiiii) —
i
m
d.o. all
d.o.
+ 2 E z.g \--- U„,b 114, i + E {2(7-nima _ i
.
.i
d.o. all
± 2 EE
d.o.
u ni
.i m
± 2
[ 4.„,,
+E
d.o. all
d.o.
im
.i
EE u:.,,i [h(;ni + E d.o. all
+ 2 z_a \--‘ z_s V
frni +
{2(miiii)b
—
E
{2(niiiii)b
—
i
d.o. all
EE u;'„i u,b.,,[hmn
d.o.
+E
{2(mnlii)
i
i mn d.o , all
+ 2 EE
_
d.o.
U,ani
.i m
+2
{2(miiii).
i
d.o.
mi Ubn..71 h mn + E {2(mnlii) —
Ua •
i
j mn
do, all
+ 4 EE uz i up,, [4(imun) — (iiimn) — (iniim)] ij mn The definitions of the F, Fa, and A matrices
(eqs. (4.6), (4.36),
and
(4.61) (4.37)) may be employed
to give: 02 Eelec
d.o.
2 E
aa8b
d.o.
E
{ 2(iiiii) ab _
ij d.o , all
+2
EE u;;Pi F,ni s
d.o. all
+ 2 EE u,7,bi Ern, m
66
CHAPTER. 4. CLOSED-SHELL SCF WAVEFUNCTIONS d.o , all
+
2
d.o. all
EE U 6 F° + 2 EE Umb Fa i
j
fil
d.o. all
•
2
EE uZ i Fmb i i
d.o. all
+
2
nt.
EE
d.o. all
+
2
EE
ua.F'6 11J '7713
j
d.o. all
uzi qj F,„ + 2
EE
U7113 a •U713 b F1/7, 11
ran
j
11i 11
d.o, all
4
EE ij
(4.62)
Inn
d.o.
= 2
d.o.
E h7ib + E
—
ij all d.o.
+4
EE
+ 4
EE
utaibF,
all d.o.
•
all
+ 4
E
(utFZ
qn)
d.o.
Fi,
E
u,ak ulik
ij all d.o. all d.o.
▪ 4
EEEE ijk
(4.63)
I
Equation (4.63) is equivalent to eq. (4.49) which was derived in two steps in the preceding section.
References
1. C.C.J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). 2. M. Born and J.R. Oppenheimer, Ann. Physik 84, 457 (1927).
Suggested Reading 1. P. Pulay, Mol. Phys. 17, 197 (1969). 2. H.B. Schlegel, S. Wolfe, and F. Bernardi, J. Chem. Phys. 63,
3632 (1975).
3. P. Pulay, in Modern Theoretical Chemistry, 11F. Schaefer editor, Plenum, New York, Vol. 4, p.153 (1977).
4. A. Kornornicki, K. Ishida, K. Morokuma, R. Ditchfield, and M. Conrad, Chem. Phys. Lett. 45, 595 (1977).
SUGGESTED READING
67
5. M. Dupuis and H.F. King, J. Chem. Phys. 68, 3998 (1978). 6. J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Rinkley, Int. J. Quantum Chem. S13, 225 (1979). 7. T. Takada, M. Dupuis, and H.F. King, J. Chem. Phys. 75, 332 (1981). 8. P. Saxe, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 77, 5647 (1982).
Chapter 5
General Restricted Open-Shell Self-Consistent-Field Wavefunct ions An open-shell wavefunction, which is described by a single determinant (or two determinants for the open-shell singlet case) and is an eigenfunction of the square of the total spin angular momentum operator (S 2 ), is termed a general restricted open-shell SCF (GRSCF) wavefunction. This chapter extends derivative SCF theory to such general restricted openshell systems. The closed-shell system treated thus far may be considered as a special case of general restricted open-shell SCF theory. Once an explicit energy expression and the necessary coupling constants are given, the formalism is applicable to almost any single configuration wavefunction.
5.1
The SCF Energy
Within the framework of RHF theory, the electronic energy for a single-configuration restricted open-shell SCF wavefunction is [1-4 all
Eei„ = 2
E
all
fihii
E
r
+
•
ij
(5.1)
fi is
the occupation number of the ith molecular orbital, and a u and /30 are coupling constants unique to the ith and jth MOs. The term "all" implies a sum over all occupied and virtual (unoccupied) molecular orbitals. The total SCF energy of the system is EscF
Eelec
Enuc
•
(5.2)
The explicit electronic energy expressions and the coupling constants for the three most
68
THE SCF ENERGY
69
frequently encountered cases are: A. Closed-shell: d.o.
Eelec =
d.o.
E
2
+ E {2(kkill)
hkk
— (klikl)}
(
5.3
)
ki
where d.o. denotes doubly occupied orbitals. closed
= virtual
iøj
(5.4)
'
closed 1 2 01 virtual [0
(5.5)
'
and closed 1_1 01 virtual [ 0 0 j
---
(5.6)
•
B. High-spin open-shell: The doublet, triplet, and quartet states possessing unpaired electrons with parallel spins are the most common examples of this case.
E
Ee tec = 2
+E
hkk
k
+E
hmm
in
12(kkiii) — (kcikol
kt
+ EE
{2(kkimm) — (kinIkm)}
k m h.o. +
1,,
f (mnd nn) — ( rani mn) 1
,
(5.7)
where h.o. denotes singly occupied orbitals. closed = open virtual closed = open virtual
1
1/2 0
(5.8)
[2 1 0 1 1/2 0 0 0 0
(5.9)
—1/2 0 —1/2 —1/2 0 0 0 0
(5.10)
and
=
closed open virtual
[ —1
70
CHAPTER 5. GENERAL RESTRICTED OPEN-SHELL SCF WAVEFUNCTIONS C. Open-shell singlet: d.o. Eelec
=2
E
hkk
+
hrnm -1- hnn
d.o.
+ E {2(kkIll) — (klikl)} kl
d.o.
+E
d.o.
E {2(kk Inn)
2(kk Imm) — (krrtikm)}
(mminn)
(mnimn)
— (knikn)}
,
(5.11)
where m and n are singly occupied orbitals possessing electrons with a and )3 spin functions, respectively. closed alpha (5.12) = beta virtual
au
closed alpha beta virtual
[2 1 1 0 1 1/2 1/2 0 1 1/2 1/2 0
0
0
0
(5.13)
0
and
[ —1 —1/2 —1/2 0 —1/2 —1/2 1/2 0 (5.14) Oi; = —1/2 1/2 —1/2 0 0 0 0 0 The above three cases are the most frequently used wavefunctions. However, eq. (5.1) is generally applicable to single-configuration wavefunctions. closed alpha beta virtual
5.2
The Generalized Fock Operator and the Variational Conditions
Within the framework of RHF theory, the generalized Fock operator for a single-configuration SCF wavefunction is [1-4 all
Fi = fih
+ E(a iiji +
(5.15)
13iiKt)
h is the one-electron operator, while 31 and Ki are the Coulomb and exchange operators, respectively. The Lagrangian matrix E was described in Section 2.5: all
=
=
E
{aii(i:71 11 )
021(illi 1 )}
•
(5.16)
THE FIRST DERIVATIVE
71
OF THE ELECTRONIC ENERGY
The RHF variational conditions for the system become
—
=0
.
(5.17)
According to these variational conditions, the general restricted open-shell SCF (GRSCF) procedure has converged when the Lagrangian matrix in the MO basis becomes symmetric. In contrast to the simpler closed-shell case, the "orbital energies" are not uniquely defined for the open-shell system with more than one type of occupied orbital. When an explicit energy expression and the appropriate set of coupling constants are known, all the results of this chapter are applicable. It is important to realize that the Fock operator is defined explicitly for the virtual (unoccupied) orbitals as well, although the coupling constants involving these virtual orbitals are all zero.
5.3
The First Derivative of the Electronic Energy
An expression for the first derivative of the general restricted open-shell Hartree-Fock electronic energy is obtained by differentiating eq. (5.1) with respect to a cartesian coordinate "a" in a similar manner to that for the closed-shell SCF wavefunction in Chapter 4 all
OEelec
ah,
all
= 2 E f, • +
aa
Oa
t au
a(iiIii)1 23 & a I
0a
(5.18)
Using results from Sections 3.8 and 3.9 (see also Section 4.3), this derivative expression may be rewritten as 0Eelec
Oa
all
2
all
E
11-47i + 2
E
all
E
all
i [ai ((ii tj j) a + 2
all
+
+
E (
+
U771,i(imiii)})] (5.19)
This equation may be manipulated in the following manner: all
all
Eciec
Oa
=2
E
E
+
ij
all
all
+2
E E all
2
E ij
all
U7ani kni + 2
E
E uzitinim +
E
all
all
aii
all
f;
2
E
all
(xi.;
E
72
CHAPTER 5. GENERAL RESTRICTED OPEN-SHELL SCF WAVEFUNCTIONS all
+2
an
E E
+ 2
m an
=2
E
m
(5.20)
O(iiiii) a }
tij
E u:,i [fihini
all
+
im
+2
ij
E faiiciiiiir +
all
2
an
all
fiqi +
i
-I-
an
E E
E fai.(milii) +
13ii(miiii)}]
i
an
an
in,.
i
E uzi [ fArn i + E faii(milii)
+ Oi.i(milii)}1
(5.21)
Introducing the definition of the Lagrangian matrix (5.16) and recognizing the interchangeability of the summations over i or j gives: °Ede, Oa
all
=2
E
an
E
AWL
an
+2 E
+
i;
all
u,,Ei,
+ 2 E ua 3 .E.
(5.22)
771
int all
=2
E
all
fh
+E
+
all
+4
E uze,i
(5.23)
The last term of this equation is simplified further by employing the symmetric character of the Lagrangian matrix, eq. (5.17), and the first derivative form of the orthonormality condition given in Section 3.7 (also see Section 4.3): all
4
E
all
U
,,
2
E (u.ia; + u;i) Ei,
(5.24)
all
2
E
(5.25)
Using equation (5.25), the final expression for the first derivative of the electronic energy becomes: pEelec
Oa
all
= 2
E
all
Eij
all
2
E s4fi, ij
(5.26)
•
THE EVALUATION OF THE FIRST DERIVATIVE
73
The first derivative expression (5.26) for the GRSCF wavefunction does not require the explicit evaluation of the changes in the MO coefficients (the Ua in Section 3.2) as was also the case for the CLSCF wavefunction in the preceding chapter.
5.4
The Evaluation of the First Derivative
In the practical evaluation of the first derivatives, expression (5.26) is usually transformed into the AO basis. The first derivative of the electronic energy in this context is all AO
EE
0Eelec = 2 Oa
i
ahg
fict•
aa
pp
all AO
r ic, /pa faijC Civ Cpj C
t4
all AO
EE
— 2
ij
e(Pla/
alPC)
Cja(IIVIP°.)1 aa "
OS 4 3 Oa
(5.27)
It is convenient to define the density matrix DI for the /th shell: all
E c„ici
DiiV =
(5.28)
iv
where the summation runs over all molecular orbitals that belong to the /th shell. The term "shell" is used to describe a set of molecular orbitals that involve the same Fock operator (i.e., share a common set of coupling constants). The total density matrix of the system, DT , is then defined by its elements as: all
E
D =
shell
f1CC
=E
f/DL,
(5.29)
where f1 denotes an occupation number for the /th shell. The elements of the "energy weighted" density matrix for a GRSCF wavefunction are all
147A1, =
E
C4i
(5.30)
Using these density matrices, the first derivative expression becomes AO
2 E
OEdec
Oa
Oa
T
LL. AO shell shell
D tiu,D, twpa
)3IJ D I D,J Y 9(1-41)1Pcr) "
J
fia
AO
-
2
E
w LU „
aa
(5.31)
74
GENERAL RESTRICTED OPEN-SHELL
CHAPTER 5.
SCF WAVEFUNCTIONS
Here all and 8" denote the Coulomb and exchange coupling constants for the I and Jth shells, respectively. In practice, the derivative integrals are evaluated only for unique combinations of AOs v and p > a) in order to minimize computational effort. The first derivative of the electronic energy in a directly applicable form is then OEdec
Oa
AO
= 4
OhPi,
DT
2
"P
aa
AO
Spa )
(
+4
2
p> i-tv > Pa shell shell
EE
x
Aiv,pa.) 2
(1
' Li D'Ai d pa
/3IJ (
)
AP
DJ L'a
+
Di„Dp)}°(Pov iPa)
J AO
—4
45 \ w a S A L,
,j>„ ( 1 —
2
(5.32)
Oa
r
In eq. (5.32)45„„ is the usual Kronecker delta function. In the second term of this equation the summations run over the indices that satisfy the conditions tiv > pa, where Ay = it(11 — 1) + v and pa = p(p — 1)+ a.
The First Derivative of the Lagrangian Matrix
5.5
At this point we obtain the first derivative of the Lagrangian matrix for future use. In particular, differentiation of eq. (5.16) with respect to a cartesian nuclear coordinate "a" ohij
aa =
aa
‘-`-`-; c + 2- VEil
+
8a
I
aa
(5.33)
will be required in subsequent developments. Using the derivative expressions for the oneand two-electron MO integrals from Sections 3.8 and 3.9 (see also Section 4.5), we obtain all
Oa
+
= all
+E
[
E (tr i hm, + uyani him )] all
ctit
E {uAi(milio +
+
2criniciiimo})
all
i3ii
E Wrzni (i/ijm)})]
+ (5.34)
OF THE LAGRANGIAN MATRIX
THE FIRST DERIVATIVE
75
all
E
i3i/(i/jj1)a}
{ctii(61//) a
1 all
all
E
[fi hr
„,
+E 1
all
all
E uf;
{f i hi„,
+E
77L
all
+
E+
+
Ira
(5.35)
mi
The definition of the Lagrangian matrix may be recalled from equation (5.16). The skeleton (core) first derivative Lagrangian ea and the generalized Lagrangian matrices (I may then
be defined by all
=
E
j
(5.36)
and all
c
+
fi
E
{aik(1.71kk)
(31k(ikijk)}
.
(5.37)
Thus, eq. (5.35) may be rewritten as all
all
E
Oa
+
E
UEjm
all
E u,[2aii(orno +
+
(5.38)
ml
Note that the skeleton (core) first derivative Lagrangian ea matrices are not symmetric in general. The T matrix may be defined by its elements:
7.17:1:1 = 2a,„(ijikl)
On,„{(ikljl)
(illjk)}
(5.39)
With this definition equation (5.38) becomes all
Oa
E (uktid, +
=
Ujac jéik)
all
+E
U7,1 [2aii(ijikl)
13i1t(iklj1)
(illjk)}]
(5.40)
kl all ('j
E
all
(wt.; + up„.Eik) + E kl
(5.41)
76
CHAPTER 5. GENERAL RESTRICTED OPEN-SHELL SCF WAVEFUNCTIONS
The following relationship holds between the generalized Lagrangian ( I and Lagrangian matrices when the SCF variational conditions are satisfied:
=
(5.42)
= cti
This equality is easily proven as follows: all
E {aik(ijikk)
= fihij all
E
= fjhij
+
tai k(iiikk)
Oik(ikijk)} = Eii
= fii -
/3,k (ikiik)}
(5.43)
(5.44)
In these proofs the definition of the Lagrangian matrix (5.16) and its symmetric character (5.17) are employed.
5.6
The Second Derivative of the Electronic Energy
The second derivative of the general restricted open-shell Hartree-Fock electronic energy may be obtained by differentiating eq. (5.23) with respect to a second nuclear coordinate ab77 ,
a (a E etec
a2Eelec
Ob
OaOb
(5.45)
aa all
=2
)
E
me
all
fi 791-:- +
i
E { OEij a(iiima Ob ii
a(iiiii)a} +
Ob
all f aUiaj TT a afii } + 4 N47' tj 1 ab E31 + U ii W
(5.46)
Using the derivative expressions for the one- and two-electron skeleton (core) derivative integrals from Sections 3.8 and 3.9 (see also Section 4.6), the first and second terms of eq. (5.46) may be manipulated as follows: all
all
fi --,T I9hli + i
ii
t2a ( ii iiir + 01 .a(iiiii)a } L ai'
all
=2
E
Ob
3
Ob
all
fi th'ilib -I- 2
j
Eumb ik,„i } In
all
+ E [otii
all
(( ii lj i)b + 2
E
onjoilmir l)
all
+2
(5.47)
THE SECOND DERIVATIVE OF THE ELECTRONIC ENERGY all
=2
E fiqb
+2
E
all E i;
I
all
all
E
.fih?ni
im
.1
L
all
+2
(
all
E
77
+E
{fi h?
•
(
5.48)
The definition of the skeleton (core) derivative Lagrangian matrix and the fact that summations over i and j are equivalent leads to an alternative form for these two terms of: all
2
all E
E all
-I- 2
all
E
Ut E + 7 ,,
E umb icjm
2
(5.49)
jm
all
=2
all
E
r
+
+E
all
+ 4
E
(5.50)
ij The third term of eq. (5.46) may be rewritten using results from Section 3.3 and eq. (5.41) as all
4
uai l
f atr 1 ab 3
3 ab j
all
=4
E
all
E upk u
u87ib —
ij all
+ 4
all
E
+ E (04 +
i;
[
+E
Nerli`,k/
(5.51)
kl
all
alt
=4
all
E
+
4
E
ij all
+ 4
all
E quti c,jci + ij k
4
E
U.V4:1741
(5.52)
ijkl
Combining eqs. (5.50) and (5.52), the expression for the second derivative of the electronic energy becomes 02 Eelec
8ctab
all
2
E
fihe
all E ij
• 78
CHAPTER 5. GENERAL RESTRICTED OPEN-SHELL
SCF WAVEFUNCTIONS
all
+ 4
E uzb, ij all
+4
E
(uteji
+ ui;E)ii)
ij all
+4
all
E
+ 4
ijk
E
UtajUbk1 1"41,kl
(5.53)
ijkl
It should be noted that eq. (5.53) involves terms related to the second derivatives of the MO coefficients. For the computational procedure to be practical, these second derivative terms must be removed from the formalism. In order to eliminate the Uza3b in eq. (5.53), the second derivative form of the orthonormality condition of the molecular orbitals described in Section 3.7, may be used. The final expression for the second derivative of the electronic energy is all
all
(9 2 Ede,
0a0b
2
E
2
E
22
Eij
+
+
all
-
ij
all
+4
E (utiqi +
Ua Eb
ij all
+4
E
k
ijk all
+ 4
E
TT& 31
(5.54)
,k1
ijkl all
all
2
E
AKE'
E
2
E szbEi , _
2
all
-
all
r
ab
€ii
ij
all
+ 4
E
+
i3 3 2 )
ij all
4
E
qu ; (.1k
ijk all
+ 4 E.uzuz,[2,i(iiiko + ijkl
+
(5.55)
AN ALTERNATIVE
DERIVATION OF THE SECOND DERIVATIVE
79
eab and
nab were defined in Section 3.7 (see also Section 4.6). Equation (5.55) is given in a form which is symmetric with respect to the variables "a" and "b". The summations run over all (occupied and unoccupied) molecular orbitals. It should be noted that the constants, L, au , and 023 involving unoccupied orbitals, should be set to zero explicitly. It is very difficult to derive a symmetric expression for the derivatives without allowing these
summations to run over all molecular orbitals. For the evaluation of the second derivative of the electronic energy, it is necessary to determine the changes of the molecular orbital coefficients (i.e., the elements of the Ua matrices) as seen clearly in eq. (5.55). These U matrices are evaluated by solving the general restricted open-shell coupled perturbed Hartree-Fock (CPHF) equations, which will be discussed in detail in Chapter 11.
5.7
An Alternative Derivation of the Second Derivative of the Electronic Energy
Another expression for the second derivative of the electronic energy of the system may be obtained by differentiating eq. (5.1) with respect to the two cartesian coordinates "a" and
192 Eei„
all
2
0a0b
I—I
2 i.
,
'&• ` 11 0(01) + L-1 ij
aa ab
0a0b f
(5.56)
Using results developed in Sections 3.8 and 3.9 (see also Section 4.7), the second derivative expression (5.56) may be expanded as: allll
all
82 Eeiec a0b
= —
2
E
fi {
+2
E ul6i hmi m
i
all
E (Umani h.rbni +
+ 2
all
Umbih`:,,,i) + 2
In
UTa„iUnb ihrnd
Mri
all
+E
E
all
aii
[(ii iiir b +
2 E (u(niiii) + u:,,6:7(iiimi))
ij
all
/
+ 2 E (14(rniiii) b + U(iiimg tn + U(miiii) a + Oni(iiimi) a ) all
+2
E
all
UU,6 i (mnljj) + 2
E mn
all
a u6 + umb i U,i)(irriljn)] + 4 E( uminj mn
80
CHAPTER 5. GENERAL RESTRICTED OPEN-SHELL SCF WAVEFUNCTIONS all
all /
I
+
+ E
2
E (u7anbi(niiii) + uranbi(imlii)) flL
all
+2 E
+
+
+
all
+ 4 E uUribi (nitzlij) mn all
all
+2 E
ira
Unb (mj I nj ) + 2
mn all
uTan y(im(in)
mn all
+2 E
U;rzniUnb i(mMin)
+2
mn
E
(5.57)
mn
all
=2
E
all
E
ii
all
+ fiii(iiiii) ab } all
+2
E faii(m/133) all
+ 2
all
U,anbj [f
Oii(m3 ii3)}1
r
+
E
all
+ 2
igni [fi jim
all
+ 2
Onj [f in, all
all
+ 2
r
[fig„i
Oij(mi(ii) b }]
im
all
+ 2
all
ni[fig
(Ji
E
jm
all
all
+ 2 :mn
U6,Ln iU,ti[fi hmn + E
all
+ 2
r
Oij(milni)}}
all
Uibii
hmn
E faii(mnlii)
(imlin)}1
all
+ 4 E u7ani on,[2ai,(imun) + ijm n
(milin)
(5.58)
REFERENCES
81
The definitions of
allow this equation
E,
which are found in eqs. (5.16), (5.36), (5.37) and (5.39), be rewritten as:
Ea , (,
to
and
7,
all
2E
12 9 Eelec
&lob
rjab +
an
▪
2
all
E
+
2
E
um
jnt
all
all
+2
E umb i elm +
2
U1,?„An
,
irn
an
▪
2
E
+
an
2
E
un
jm
an
+2
EUUC +
all
2
imn
E jrnn
all
+
4
E
Jr,
U,ani
(5.59)
ijmn
all
2
E
+4
E
all + E
Ah7ib
all
qb,„
an
▪ 4 E (qqi + u8Pi f) i;
all
▪
4
E ijk
all
uri Nic;fk + 4 E LT Ul rj
(5.60)
ijkl
Equation (5.60) is equivalent to eq. (5.53), which was derived independently in the preceding section. References
1. C.C.J. Roothaan, Rev. Mod. Phys. 32, 179 (1960). 2. F.W. Bobrowicz and W.A. Goddard, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.79 (1977). 3. R. Carbo and J.M. Riera, A General SCF Theory, Springer-Verlag, Berlin, 1978.
82
CHAPTER 5. GENERAL RESTRICTED OPEN-SHELL SCF WAVEFUNCTIONS Suggested Reading 1. J.D. Goddard, N.C. Handy, and H.F. Schaefer, J. Chem. Phys. 71, 1525 (1979).
2. S. Kato and K. Morokuma, Chem. Phys. Lett. 65, 19 (1979). 3. Y. Osamura, Y. Yamaguchi, and II.F. Schaefer, J. Chem. Phys. 75, 2919 (1981). 4. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 77, 383 (1982). 5. P. Saxe, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 77, 5647 (1982). 6. Y. Osamura, Y. Yamaguchi, P. Saxe, M.A. Vincent, J.F. Gaw, and H.F. Schaefer, Chem. Phys. 72, 131 (1982). 7. Y. Osamura, Y. Yamaguchi, P. Saxe, D.J. Fox, M.A. Vincent, and H.F. Schaefer, J. Mol. Struct. 103, 183 (1983). 8. G. Fitzgerald and H.F. Schaefer, J. Chem. Phys. 83, 1162 (1985).
Chapter 6
Configuration Interaction Wavefunct ions The configuration interaction (CI) method has been the most commonly used approach to include electron correlation effects in electronic wavefunctions. Many important chemical problems have been solved by quantum chemists using the CI method. A number of electronic configurations are included in the wavefunction, constructed using a particular set of molecular orbitals, which usually are obtained by an SCF or MCSCF procedure. Of course, the MO coefficients are not variationally optimized in the CI procedure. Thus, the derivatives (to any order) based on the CI wavefunction necessarily require information on the derivatives (to the same order) of the MO coefficients. In this chapter the first and second derivatives of the CI wavefunction are discussed.
6.1
The CI Wavefunction, Energy and the Variational Conditions
The configuration interaction (CI) wavefunction is a linear combination of electronic Configurations 4, ./ that are constructed from an appropriate fixed set of reference molecular orbitals [1] CI AFC/
=E
C14)1
(6.1)
I
The summation runs over electronic configurations. The C/ are the CI coefficients which are to be variationally determined. The wavefunction is normalized, giving the condition
(6.2)
83
84
CONFIGURATION INTERACTION
CHAPTER 6.
WAVEFUNCTIONS
on the CI coefficients. The electronic energy of the CI wavefunction is: CI
E
Eetec =
(6.3)
CICJHIJ
IJ
The CI Hamiltonian matrix appearing in eq. (6.3) is defined by
(6.4)
( 4) ilHeieci 4'.7) mo mo / hi
E 07
+E
,
(6.5)
Qt/ and Gliki are one- and two-electron coupling constants between electronic configurations and molecular orbitals [2]. 1-1,1„ is the usual electronic Hamiltonian. The one- and twoelectron reduced density matrices are defined in terms of their elements by: CI
=
E
cicJe t,
(6.6)
IJ
and
ci
E
CjCjGff1 (6.7) IJ Using these quantities the electronic energy of the CI wavefunction is given in MO form by MO Eeiec
=
MO
+E
E ij
Gijkl(ijikl)
(6.8)
ijkl
The total CI energy of the system is the sum of the electronic energy and the nuclear repulsion energy (6.9) Eci = Eetec Enuc The variational conditions for the determination of the CI wavefunction (6.1) are:
ci
E
c,(H,
-
( 6.10)
67Eelec)
The CI Hamiltonian matrix in eq. (6.10) is defined by eq. (6.5). Sij is again the Kronecker delta function.
6.2
The First Derivative of the Electronic Energy
The differentiation of eq. (6.3) with respect to a nuclear coordinate "a" gives:
xci ---, OCI
OEetec
Oa
= IJ
Oa
I-I LICJ
+
acj
ci
E IJ
CI
Oa
ci
Hij
+
E cic, 0.1-11.7 Oa IJ
(6.11)
THE FIRST DERIVATIVE OF THE ELECTRONIC ENERGY
85
It is immaterial whether the summations initially run over the / or J subscripts and thus this equation becomes: c
Eeiee
aa
=2
ac aai
E
cI
HijCj
+ E
IJ
Oa
IJ
(6.12)
From the variational conditions, eq. (6.10), the following equation is obtained:
EHjjCj =
Edecci
(6.13)
Using this equation eq. (6.12) may be rewritten as: ac
Meiec
= 2Eeiec
Oa
E 1
,
Oa
ci
c, + E
CiCj
IJ
aHIJ Oa
(6.14)
The first derivative of the normalization condition gives:
ci
E
ac, z,
(6.15)
Oa
Using this condition the first term in eq. (6.14) vanishes and a simpler (in fact, deceptively simple) expression for the CI gradient is found: OEetec
_ Oa —
ci
OHL, Oa
IJ
(6.16)
Since the HI., matrix is defined in eq. (6.5) without using the CI coefficients, the first derivative of the electronic energy for the wavefunction, eq. (6.16), which satisfies the CI variational condition (6.10) may be obtained by differentiating the energy expression in only the MO space. An expression for the first derivative of the electronic energy of the CI wavefunction is obtained by differentiating the MO part of eq. (6.8) with respect to a cartesian coordinate "a": MO o mo Ee i ec Ohji 19(ijIk/) (6.17) E=EQi + Gijkl Oa aa Oa ii ijkl Using the results given in Sections 3.8 and 3.9, this derivative expression becomes: aEeleC
aa
mo
=E ij
mo
mo QijIleti +
E
(u:ii hm,
mo
+ E Gij k,[(iiiki)a + E ijkl
+
rthim )}
m
{u,ani (milko + U7 (imlkl)
86
CHAPTER 6.
CONFIGURATION INTERACTION
U,ank(iiiral)
=
mo
MO
ij
ijkl
MO
(6.18)
U7`7,.`
Godoilkoa
E
+ E
WAVEFUNCTIONS
+
Q o uA i h,m )
MO
• E
+
GijkiU(ilnikl)
ijklm GijklU77,k(ijI7n1)
GijklUtanl(iiikM)}
(6.19)
Interchange of i, j,k,l, and m does not change the value of the first derivative as the summations run over all molecular orbitals. Focusing attention on the third and fourth terms in this equation, the first derivative of the electronic energy becomes: MO
(9Eelee
MO
E
aa
Q 3h
+E ijkl
MO
+ E(Qj mUhim + Q mi UZi hmi ) iim MO
+E
{Gi mkiumimiko +
Gm jklU(Milkl)
ijklrn + G kij m U29 (kilint) + G klmjUiaj(klitni)} MO
MO
E
Qijqj ij
ijkl MO
+
MO
2 E (2,,nuzhim + 4 E Gimktu(imiko iim ijklm MO
E
Qijqj
ijkl MO
E
2
/E
MO
MO
+
ij
=
(6.21)
MO
ij
H-
(6.20)
2
E
G,, ki (imiki)}
(6.22)
mid
MO
MO
ij
ijkl
E
MO
2
E ii
uzxi;
(6.23)
87
THE EVALUATION OF THE FIRST DERIVATIVE The particular Lagrangian matrix X in eq. (6.23) is defined by MO
MO
E
Xi•
Qj rn him +
2
E
(6.24)
Gimki(inziko
111 Id
771
This Lagrangian matrix indicates the changes in the CI energy with respect to the changes in the MO coefficients. Note that the symmetry properties of the MO integrals in Section 2.8 and reduced density matrices in Section 2.6 were used in deriving eq. (6.23). The first derivative expression (6.23) requires the explicit evaluation of changes in the MO coefficients or Ua matrices. These Ua matrices are evaluated by solving the coupled perturbed Hartree-Fock (CPHF) or the coupled perturbed multiconfiguration Hartree-Fock (CPMCHF) equations, depending on the reference wavefunction employed to construct the electronic configurations. These methods are nontrivial and will be described in Chapters 10, 11, 13 and 14. The related subject of the evaluation of the third term in eq. (6.23) (using the Z vector method) will be treated in Chapter 18.
6.3
The Evaluation of the First Derivative
In the practical evaluation of the first derivative for CI wavefunctions, the first two terms in eq. (6.23) are usually transformed into the AO basis. The first derivative of the electronic
energy is 1)Eete,
Oa
MO AO
MO AO
= EE Qiicici 8hAv +EE A era I/
ii
.'t
ijkl m vpa
I"
Gi2krCi Cj C k C I a(Pvii9a)
Oa
MO
+2
E uiai xi.,
(6.25)
23
The one- and two-electron reduced density matrices in the AO basis are defined by (see Section 2.8) MO
QEw =
and
E
(6.26)
MO
E
G 1,14, per
(6.27)
Gijkl
ijkl
Using these AO reduced density matrices, eq. (6.25) becomes: aEei-"
Oa
AO
ah
E C 2 kw 19:
AO
rj JLLp
„Lupo-
a(twiPa)
3a
MO
+2
E u4xi, j
(6.28)
88
CHAPTER 6.
CONFIGURATION INTERACTION
CPHF
The third term of this equation requires the solutions to the reference wavefunction. This term may be modified as follows: MO
2
E
MO
MO
UXij
2
ii
E UXU +
2
s>1
E
+
equations for the
MO
2
E uiai Xii
(6.29)
s )
2
k"'
(6.33) MO AO
•
EE (1
2
(6.34)
Defining the "energy weighted" density matrix as MO
E
(6.35)
(1 — '-11 5 )Ca C3 X 2 " "
eq. (6.34) becomes MO
2
E szxi, + i>J
AO
MO
E SX I = 2
E w,.,,
as
(6.36)
1W
j
Combining eqs. (6.28), (6.32), and (6.36), the electronic energy first derivative for a CI wavefunction is given by AO
49 Eel ec
E
aa
AO
ah /Iv
E
aa ll
G
a(PvIP(7) aa
gypa
AO
-
2
E wov
OS v
va
+
mo
2
E u,qx, - xii ) i>j
(6.37)
THE FIRST DERIVATIVE OF THE HAMILTONIAN MATRIX
89
In practice, the derivative integrals are evaluated only for unique combinations of AOs > v and p > a) in order to minimize computational effort. The first derivative of the electronic energy in a directly applicable form is:
°Ed e, Oa
ahm, 2 T4 ' Oa AO
+
4
6„ 2 ) (1
(1
tt› 11 p >
45P.u,P,
2 )
,
> Pu } a(i-wiPa) AO
—4
-
(
y
1—
2
ki>ti
Ai' aa
MO
+ 2
E
x„)
(6.38)
i›i
In the second term of this equation the summations run over the indices that satisfy the conditions yv > pa, where /Iv = p(i.t —1)+ v and pa = p(p — 1) + a. The last term of eq. (6.38) indicates that the CPHF equations need to be solved only for the i-j pairs whose elements of the Lagrangian matrix are not symmetric, i.e., for the pairs with X13 X. When a molecule has point group symmetry higher than C 1 , the number of contributing derivative MO pairs will decrease.
6.4
The First Derivative of the Hamiltonian Matrix
An explicit expression for the first derivative of the Hamiltonian matrix was not required in the preceding section. However, such a result will be necessary for the evaluation of the second derivative of the energy. The explicit first derivative of the CI Hamiltonian matrix is given in this section.
Differentiating eq. (6.5) with respect to a cartesian coordinate "a", gives MO
(911 IJ
EiJ
Oa
jahij
'ceii
Oa
MO
4- 2_, Gtjki Oki
a(ijikl)
(6.39)
Since the coupling constants, Qjf and Gfiki , in eq. (6.5) are independent of the CI coefficients and nuclear coordinates, this derivation is essentially the same as that starting from eq. (6.17) and working to eq. (6.23). Using results from the preceding section: MO
MO
ij
ijkl
E Qt/hz + E
90
CONFIGURATION INTERACTION
CHAPTER 6.
MO {my,
+2
E
E
uia,
(2hin,
+
MO
2
rn
ii
WAVEFUNCTIONS
E
Gil L i (intik1)}
MO
E
2
= frilj +
(6.40)
mkt
(6.41)
uia,xiy
ij
The skeleton (core) first derivative Hamiltonian matrices matrices X" are defined by
mo
ij
ijkl
E Qffh,a, + E
Hy, = and
mo
mo Xit =
Hy, and the "bare" Lagrangian
E chin, +
(6.42)
MO
2
E
(6.43)
•
kl(irniki)
mkt
The relationship between the Lagrangian X and the "bare" Lagrangian
X"
matrices is:
CI
Xij =
E GIGjX iii
(6.44)
IJ
This equation is reminiscent of that for the one- and two-electron reduced density matrices and the coupling constants given in eqs. (6.6) and (6.7). The various "bare" quantities, Q", G", 117j , and X" do not contain the CI coefficients by definition. Using eq. (6.41) it is easily shown that eq. (6.16) is equivalent to eq. (6.23) by: CI
Melee
E
0a
8111j aa
IJ
(6.45)
CI
=
MO
CICJ
+
2
IJ CI
MO
MO
(dith.(1,
ij
+
+E
MO
2
E
UZ Kilt } (6.47)
ijkl MO
E chx, + E ij
(6.46)
ij
E c/c., E MO
E
ijkl
+
MO
2
E
up•.x.• t3
(6.48)
ij
The relationships (6.6), (6.7) and (6.44) were used in this proof.
6.5
The First Derivative of the "Bare" Lagrangian Matrix
It is appropriate to find a first derivative expression for the elements of the "bare" La-
grangian matrix for future use. Differentiation of eq. (6.43) with respect to a cartesian
THE FIRST DERIVATIVE OF THE "BARE" LAGRANGIAN MATRIX
coordinate "a"
a X ill =MO■ aa
7' a 2_,G:ir,',ki
MO
Q I j ahgm
•—•1
+ 2
L-1 m v" Oa
(r irnikl)
mkt
act
91
(6.49)
is required. Using results developed in Sections 3.8 and 3.9 one obtains
aX it aa
mo
mo
..[hm + E { U,ih,,,„ +
m
n
MO
+
MO
E Gq,,J,1(imiki). + E ful(nmiki) +
2
mkt
3
U:m (injkl)
n
+ U k (imjnI) + UZ(imIkn)}1 MO o
. E
(6.50)
mo
+
2
m
E G ki (imikoa
nOci
MO
I MO
MO
+ E uic, i E Q./; h, + n MO
2
221.
E nJi 1
+ E u.:„i {ch,,, +
GI ki (nmjkl)} 3
MO
2
11.7n
E
G.Ii m j ki(inik1)}
kl
MO
+
2
E
a G,,,(iinino IJ unk
+ uvai,G
i (imik.)
(6.51)
milk'
In eq. (6.51), the summation runs over all molecular orbitals. Interchange of the m, n, k, and I indices does not change the value of the first derivative of the "bare" Lagrangian
matrix. Thus,
mo + E u,a„x41 MO
+ E
MO
U 1 Q-jiThik + 2
MO
E G.wnn(ikimn) + inn
kl
4
E mn
(6.52)
• x 23?-4a
mo
mo
E uLx,/,/ + E uzYilâ fd
(6.53)
kl
In deriving eq. (6.53), the symmetry properties of the MO integrals and reduced density matrices and the definition of the "bare" Lagrangian matrix (6.43) were employed. The skeleton (core) first derivative "bare" Lagrangian matrices Xa and the "bare" Y matrix appearing in eq. (6.53) are
mo x!.J. ,— E 7Th
ym
un
+
2
mo E Ggki (imikoa ink!
(6.54)
92
CHAPTER 6. CONFIGURATION INTERACTION
WAVEFUNCTIONS
and
mo = Qfh, + 2
E
mo
E
GY,,„(ikinm) + 4
mn
G.f„ J 1„(irnikn) .
(6.55)
mn
These two matrices are related to their "parent" quantities by CI
E cic.,x,/ta
(6.56)
IJ and CI
Yijkl =
E
(6.57)
IJ
6.6
The Second Derivative of the Hamiltonian Matrix
It is advantageous to have an explicit expression for the second derivative of the Hamiltonian matrix at this point. The second derivative of the Hamiltonian matrix may be obtained by differentiating eq. (6.5) with respect to the two variables "a" and "b":
= MO o ij (92 hii
a2 Hjj aa0b
&lab
‘—d
a aT)
-
MO
ij a2 (ij Ikl) + E G,3,,, aaab
(6.58)
ijkl
(aHIJ)
(6.59)
aa
Recalling eq. (6.41), the equation (6.59) may be rewritten to yield
82 111j aaab
mo
a °Hy,
(6.60)
mo 2 2ij-s
ii Arta
ab
+ 2
ab
MO
axi,1 ij u=a2 a'b3
E
(6.61)
The first term of eq. (6.61) is manipulated as in Sections 6.2 and 6.4. Using results from Sections 3.8 and 3.9,
(6.62) (6.63)
{ Unzb
Umb
}]
THE SECOND DERIVATIVE OF THE HAMILTONIAN MATRIX
mo
+E
mo
+ E fumb i ( miikoa + umb i ( imikoa
G1,1,,,[(iiikirb
m
ijkl " + 11
(i i 'Mir
MO
ij
ijkl
. E vhce + E
Gt.4,(iiikoab
IMO
MO
MO
E up, E
+
Q-'J47,
m
ii
(6.64)
+ U 7bni(Z 3 lkin) a ll
MO
+ 2
93
E
2
G1,7Jkiciinikoal
(6.65)
mkt
MO
E
= H 1c .67 ± 2
qxua
(6.66)
ij
The skeleton (core) second derivative Hamiltonian matrices
mo =E
11 7:11
Hy; are defined by
MO
(2flitc,1
+E
(6.67)
ijkl
The second term of eq. (6.61) may be rewritten using the relationship given in Section 3.3 as MO
2E
aup.
mo
MO
13
Ob
E (uiaib _ E
= 2
U, U 3
(6.68)
ij
Using the results given in eq. (6.53) the third term of eq. (6.61) is
mo
2
axI.7
E u,a, ob i;
"
MO
MO
MO
ij
k
kl
E u,F; xe + E qv + E upayg,
= 2
E
MO
= 2
„,
T Tb
s-'
21 ' ' 2 j
MO
_i_ 2 E / I ,
ij
c'
'To y r./ k‘-' kj -rl-sj
(6.69)
MO
+2
ijk
E qukby
d
ijkl
(6.70) The "bare" Y matrices were defined by eq. (6.55). In deriving eq. (6.70), the interchangeability of the summations over MO indices and the symmetry properties of the integrals and reduced density matrices were used. The explicit expression for the second derivative of the H ij matrix is obtained by substituting eqs. (6.66), (6.68) and (6.70) into eq. (6.61): MO
02 141
+2
0ab
+2
E
mo E E(ue — ij
2, 13 mo
upk uOxit
CHAPTER 6. CONFIGURATION INTERACTION
94
mo
E
+ 2
mo
qicitb
E
+2
ij
upk ul:,X11
ijk
MO
E
2
WAVEFUNCTIONS
qupdyiii 4,
(6.71)
ijkl
=
11 + 2
mo
E ulLibxit ij
6.7
us.aixiIt b)
+ 2
(ubj x ill a
+ 2
Ujai If Pa / ijkl
(6.72)
The Second Derivative of the Electronic Energy
The second derivative of the CI energy expression may be obtained by the differentiation of eq. (6.16) with respect to a second nuclear coordinate "b":
a (a Eelec )
a2Eelec
Oa0b
(6.73)
Oa
ab
a (ci jCj
a 111J)
(6.74)
(-971 IJ CI
= E cic.,
8 2 H ij
+ E `-' E
2 (6.75) &tab Oa IJ 1 Ob J This equation does not appear to be symmetric with respect to the interchange of the variables "a" and "b". However, it may be shown to be equivalent to the symmetric form in the following manner. The differentiation of the variational condition (6.10) gives important relationships between the derivatives of the CI coefficients and those of the Hamiltonian matrix elements CI ci O (0111j a Eelec ) +CJ . (6.76) 61J —(H/J — 61.7Edec) = 0 Oa Oa aa J J
E c.,
E
These conditions are equivalent to the coupled perturbed configuration interaction (CPCI) equations which will be discussed in detail in Chapter 12. Multiplying by (EV/ on both sides of eq. (6.76) gives
ac,/ab)
CI I
ac, CI
fail,
J CI aci. CI acj
+ E 73„ E I
v
J
aEetec ) blJ
--57 (H IJ —
Oa
6 IJEelec)
=
0
(6.77)
▪
THE SECOND DERIVATIVE OF THE ELECTRONIC ENERGY
95
Using eq. (6.15), equation (6.75) may be rewritten as CI
(92 Eelec
CI
a2HIJ
aaab = E IJ
+ 2
aaab
E
ab
, E c, (alb Oa
imeiec)
fia (6.78)
Employing the relationship in eq. (6.77) the second derivative of the CI energy becomes CI
(92 E. lee
E
aaab
2
&t ab
IJ
CI ac, E ac, fia ab
HIJ —
61JEelec)
(6.79)
IJ
An explicit expression for the second derivative of the CI energy results from combining eqs. (6.72) and (6.79): a2Eciec
&Lab
CI
-= E cic, Hp; +
MO
2
E qbxj/
IJ
ij MO /
/b
E utx illa
+2
UlijX31 ) + 2
ij
2 CI E
ijkl
aci ac,
IJ
E qutYg,
fia ab
HI, — 5 IJEelec)
MO CI
(6.80)
MO CI
E E CiCjQfh + E cic,Gfiki(okoab ij IJ
ijkl IJ CI
MO
+
E
2
Ufib
E IJ
MO
+2 +
2
E
CI
CI
IJ
IJ
utj E cic,va + u E cic,vb
MO
CI
ijkl
IJ
E quzi E cic,ya
2 CI IJ
aa ab•
HIJ
— biJEelec)
(6.81)
Equations (6.6) and (6.7) relate the reduced density matrices Qii and Gijki to the coupling constants. The relationships between the "bare" and the corresponding "parent" quantities in eqs. (6.44), (6.56) and (6.57) are used to rewrite (6.81) as 0 2E elec
aaab
MO
MO
= E • + E Jim
MO
+
2
E
CHAPTER 6. CONFIGURATION INTERACTION WAVEFUNCTIONS
96
MO
+ 2
+
E (uibi xz, +
MO
2
E
t,
CI
—2
oc1 oc j
E
(6.82)
— 6 IJEelec)
Oa Ob
IJ
The third term of this equation requires the solutions to the second-order CPHF equations of the reference wavefunction. This term may be modified in a similar manner as was done in Section 6.3: MO
MO
2
E uicy,xi, =
E uribxi, +
2
ij
i>.;
MO
2
E uexi, + i<J
E
usq'xii
(6.83)
i
MO
= 2
MO
2
E (uffx i, + u2c,.,bx, i) + i>i
MO
2
E u,lbxi,
.
(6.84)
i
Using the second derivative of the orthonormality condition of the molecular orbitals from Section 3.7 + = (6.85)
Ur
where
mo
E
S
(u Um
+ up„,trin, — S am. S m — s, s) ,
5
(6.86)
eq. (6.84) is further manipulated: MO
2
E uex i;
MO
2
E t>,
MO
UiPib
i — Xji) —
2
MO
E cpxii — E i>,
. ( 6.87)
The first term of eq. (6.87) indicates that the second-order CPHF equations need to be solved only for the MO pairs whose elements of the Lagrangian matrix are not symmetric, i.e., for pairs with Xi j Xj i. These second-order CPHF equations will be discussed in Chapters 10, 11, 13 and 14. A related subject, the evaluation of the third term in eq. (6.82) by the Z vector method, will be treated in Chapter 18. The derivative of the CI coefficients can be determined by solving the first-order CPCI equations, as will be shown in Chapter 12.
References 1. 1. Shavitt, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.189 (1977). 2. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
SUGGESTED READING
97 Suggested Reading
1. B.R. Brooks, W.D. Laidig, P. Saxe, J.D. Goddard, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 72, 4652 (1980). 2. R. Krishnan, H.B. Schlegel, and J.A. Pople, J. Chem. Phys. 72, 4654 (1980).
3. B.R. Brooks, W.D. Laidig, P. Saxe, J.D. Goddard, and H.F. Schaefer, in Lecture Notes an Chemistry, J. Hinze editor, Springer-Verlag, Berlin, Vol. 22, P. 158 (1981). 4. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 75, 2919 (1981). 5. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 77, 383 (1982).
6. D.J. Fox, Y. Osamura, M.R. Hoffmann, J.F. Gaw, G. Fitzgerald, Y. Yamaguchi, and H.F. Schaefer, Chem. Phys. Lett. 102, 17 (1983). 7. J.E. Rice, R.D. Amos, N.C. Handy, T.J. Lee, and H.F. Schaefer, J. Chem. Phys. 85, 963 (1986). 8. T.J. Lee, N.C. Handy, J.E. Rice, A.C. Scheiner, and H.F. Schaefer, J. Chem. Phys. 85, 3930 (1986).
Chapter 7
Two-Configuration Self-Consistent-Field Wavefunct ions The two-configuration self-consistent-field (TCSCF) wavefunction is a simple but useful correlated wavefunction. Specifically, TCSCF wavefunctions provide reasonable starting points for the description of singlet carbenes and other organic biradicals. They are also useful in the classical problem of the proper description of the dissociation of a single bond in a molecule. Theoretically, TCSCF is a simple case of the generalized valence bond (GVB), paired excitation multiconfiguration SCF (PEMCSCF), complete active space SCF (CASSCF) and general multiconfiguration SCF (MCSCF) wavefunctions. Here, unlike the simpler single configuration SCF cases, one must consider constraints involving both the molecular orbital (MO) and configuration interaction (CI) spaces in deriving the analytic energy derivative expressions. Thus, this simplified multiconfiguration wavefunction provides an excellent introduction to the analytic derivative techniques needed for the more elaborate wavefunctions considered in the immediately following chapters.
7.1
The Two-Configuration Self-Consistent-Field (TCSCF) Wavefunction and Energy
The two-configuration self-consistent-field (TCSCF) wavefunction IPTCSCF = C1 4) 1 + C243 2
kli
[1-5] of interest is: (7.1)
It consists of two configurations (D i . 1 (closed)(open)mrn 1
98
(7.2)
THE TCSCF WAVEFUNCTION AND ENERGY
99
and
4)2 = (closed)(open)nn
(7.3)
where (closed) and (open) designate the fixed closed shells and open shells in molecular orbital spaces, and m and n. are the molecular orbitals with variable electronic occupation. The CI coefficients are normalized according to
+C = 1
C
(7.4)
The electronic energy of the TCSCF wavefunction in CI form is
I
Eelec =
TCSCF Helec
I 4I TCSCF)
(7.5)
2
E
CCjHj
(7.6)
IJ
and the CI Hamiltonian matrix element HIJ
Hij
is
(7.7)
(4)111-Lie c 14)..1 ) MO
2
E
fphi,
+
Al0
41 ( 21 133) +
(7.8)
The summation runs over all orbitals including m and n. fP is the one-electron and all and are the two-electron coupling constants. Explicitly, these coupling constants for a high-spin open-shell TCSCF wavefunction, which would represent the most complicated of the commonly employed TCSCF procedures, are: closed fin
1/2 1 1 0
closed
1
open
r22
aij
open
=
closed open
closed 22
open
1/2 0 1 2 1 1 1/2 2 1 0 0 2 1 1 1/2
(7.9)
(7.10) 2 0 1 0 20
00 0 2 0 1
0000 2 1 02
(7.11)
(7.12)
100
CHAPTER 7. TWO-CONFIGURATION SCF WAVEFUNCTIONS closed -
open
j1
4
closed open
=
?
—1
—1/2 —1 0 —1 —1/2 0 —1
closed open
0 0 0 0
0 0
— 1/2 —1/2 —1/2 0 —1/2 —1/2 0 —1/2 0 0
—1 0 —1/2 0 —1 0 0 0 0 —1 0 —1/2 0 0 0 —1
(7.13)
-
0 0
(7.14)
(7.15)
0 0 1/2 0 1/2 0
The coupling constants involving the virtual orbitals are zero as are all the elements of /12 and a12 . Explicit expressions for the elements of the CI Hamiltonian matrix in eq. (7.8) are:
mo
H11 = 2
mo
E tutu + E
taii(iiiii)
Oii(iiii.i)}
1".7n,vt
MO
E fi t2(iiimm) -
+ 2 Itn,„,, + 2
(imlim)}
4m,rt ▪
(7.16)
(mm)mm)
mo H22
mo
E
=2
E
i*m,n MO
E
+ 2 him + 2
fi{2(iiinn)
(inlin)}
iOrn ,n
•
(nninn)
H12 = H2I =
(7.17)
(mnlirtn)
(7.18)
In the above equations, the summations run over all molecular orbitals. The reduced density (coefficient) matrix elements for the high-spin open-shell TCSCF wavefunction for which the coupling constants were written out above are given by
closed open
virtual
-
1
1/2 C? 0
-
(7.19)
AND VARIATIONAL
GENERALIZED FOCK OPERATOR
=
2
closed open in
2C? 2C1 0
n virtual
A; =
1 2C? 1/2 c? C? C? CI 0 0 0
1
closed open m n virtual
—1 —1/2 —C? —CI
—1/2
—1/2 —1/2C?
-1/2q 0
0
CONDITIONS
2C1 0 CI 0 0 0 CI o 0 0 —C? —CI 0 -1/2c? -1/2c3 o 0 Ci C2 0 ci C2 0 0 0 0 0
101
( 7.20)
(7.21)
The reduced density matrix elements involving the virtual orbitals are all zero. The relationships between the elements of the one- and two-electron reduced density matrices and the coupling constants are 2
E cic.fe
(7.22)
IJ 2
E
c,Cjati
(7.23)
IJ 2
E
/32.7
(7.24)
IJ
Using these equations, the electronic energy of the TCSCF wavefunction in the MO form is
mo Edec = 2
E
mo
fihii
+E
(7.25)
Finally, the total energy is simply the sum of this electronic energy and the nuclear repulsion energy: ETcsvF
7.2
=
Eelec
Entic
(7.26)
The Generalized Fock Operator and the Variational Conditions
The generalized Fock operator for the TCSCF case is formally the same as for the general restricted open-shell SCF wavefunction in Section 5.2 [4], Fi = fih
mo E (ctitit
0i/1C)
(7 .27)
102
CHAPTER 7. TWO-CONFIGURATION SCF WAVEFUNCTIONS
The variational conditions in the MO space are formally identical to those for the general
restricted open-shell SCF wavefunction, Ei3 —
=0
(7.28)
where the Lagrangian matrix is defined by MO
E
(2 =
+
f3i1(ilij1)}
(7.29)
Since the CI coefficients, Ci and C2 in eq. (7.1), are optimized simultaneously with the MO coefficients, the variational conditions in the CI space also are involved: 2
E In the above equation the
c,„(H„, - 6IJEelec) = 0
Hjj
(7.30)
are the elements of the CI Hamiltonian matrix and were
defined by eqs. (7.16) through (7.18).
7.3
The First Derivative of the Electronic Energy
For a TCSCF wavefunction, the variational conditions on both the MO and CI spaces, eqs. (7.28) and (7.30), must be satisfied simultaneously. Thus, the first derivative of the electronic energy can be obtained without explicit information on the derivatives of the MO and CI coefficients. The expression for the first derivative of the electronic energy of the TCSCF wavefunction is equivalent to that of the general restricted open-shell SCF
wavefunction in Section 5.3: mo
— 2
E
mo E
mo
+4
E u4,„
mo
E
-=
2
-
2 E SZEii
(7.31) MO
+E
MO
(7.32)
An alternative expression in terms of the CI coefficients and Hamiltonian matrix elements
is
2 8E,‘„ — E e 0HIJ Oa — 8a and may be compared with the result for the CI wavefunction in Section 6.2. The equivalence between eqs. (7.31) and (7.33) tvill be shown in the next section. The practical evaluation of the first derivative may be carried out in the same manner as was described in Section 5.4.
THE FIRST DERIVATIVE OF THE HAMILTONIAN MATRIX
7.4
103
The First Derivative of the Hamiltonian Matrix
Although the first derivative of the TCSCF Hamiltonian matrix was not required in the manipulations in Section 7.3, it will be needed for the second derivative of the energy. Therefore an explicit expression for the first derivative of the Hamiltonian matrix is derived in this section. Differentiating eq. (7.8) with respect to a cartesian coordinate "a", yields
Oh
MO
OH If
MO
_ 2 E " + Oa Oa
E
f a.r
i) Oa
aa
I
(7.34)
the coupling constants, fP, al/ and 01/, in eq. (7.34) are constant Since (independent of the CI coefficients), the manipulation is essentially the same as that for the GRSCF wavefunction described in Section 5.3. Thus, MO
OH1J
Oa
(
fp hCi
2
+
E
Li
MO
E 2.;
+4
(7.35) MO
+
4
E ij
(7.36)
UP.Eftf 13 3 1
The skeleton (core) first derivative Hamiltonian matrices, HIj , and the "bare" Lagrangian matrices, cif, appearing in eq. (7.36) are defined by A40 2
E
MO
+E
fP/17,
and
MO E ij
+
23
ii
I
E
3
+
(7.37)
.
(7.38)
The relationship between the Lagrangian matrix elements and the "bare" Lagrangian matrix elements is given by
Ez,
2 E cic,cw
(7.39)
IJ
Using equations (7.22) to (7.24) and (7.35) to (7.39), eq. (7.33) can be shown to be equivalent to eq. (7.31): 0Eelec
Oa
_ —
2
E IJ
CICJ
alljj Oa MO
2
E Ii
(7.40)
cicJ [2
E
MO
f2"h":2
+E
104
CHAPTER
7. TWO-CONFIGURATION SCF WAVEFUNCTIONS
MO
+4
E
MO
=2
(7.41)
23 32j
MO
E
E
+
IL
z
MO
+ 4
7.5
E uja,cii
(7.42)
The Second Derivative of the Hamiltonian Matrix
The second derivative of the TCSCF Hamiltonian matrix will be derived here. The second derivative of this Hamiltonian matrix is obtained by differentiating eq. (7.8) with respect to the variables "a" and "b":
aaab
MO
,q2
MO
a2 H
E fi/Jaaab
=2
i."
0
f ct f.iu
" aaab
aaab
r a HLI1
(7.43)
(7.44)
Oa Recalling that the expression in square brackets was given by eq. (7.36), yields
.92H „ aaab ij
MO
[Hy, + 4
= -0-6
E ur.ei-J1 3 32
(7.45)
mo
mo
ii
ii
1J + 4 E 9 Ua biaj E ;7 + 4 E
811 a
(7.46)
The first term of eq. (7.46) is manipulated as was done for the GRSCF wavefunction in Section 5.6, since the f1 .1 , al/ and )(31/ are all constants.
Oxy, ab
a [ mo .
i
=.
MO 2 E ehli
+
4
b
+
mo
E faticiiiiir + oficiiiii)all ii MO + E { 4,1 (iiiii) b + /31/(iiiii)b}
fiuh,iii
(7.47)
MO
E uiw
(7.48)
ij
mo ▪ Hy; + 4
E ij
(7.49)
THE SECOND DERIVATIVE OF THE HAMILTONIAN MATRIX The skeleton (core) second derivative Hamiltonian matrices, derivative "bare" Lagrangian matrices, Elf , are:
Hg,
105
and skeleton (core) first
MO
MO
E fPhrt E
=2
(7.50)
ij
and
mo e
J
a=
E ictf/J(iiiioa + ofricalioal
(7.51)
The skeleton (core) first derivative "bare" Lagrangian matrices are related to their "parent" quantities through the expression 2
E
(7.52)
CjCjcff
IJ
where
mo = fihz, + E
Oil(illii) a}
(7.53)
•
The second term in eq. (7.46) is rewritten using a result from Section 3.3 to become
4
mo
ti f /J ab "
E
MO
mo
E — E uibk (4,
=4
(7.54)
The third term of eq. (7.46) is treated in a manner similar to that in Section 5.5. The derivative expression of the "bare" Lagrangian matrix is manipulated in the same way as its parent. After a similar derivation to that performed in Section 5.5, the third term becomes:
mo
mo E
ly,
cji
tqi
mo
+
+
Trb
J/13
ki
[
(7.55)
mo
mo = 4
E tfia.E/ilb + ii
33
4
E uiai tizi e +
mo 4
ijk
E qutc.f,J, ijk
MO
+ 4
E q UtilRiii:kji
(7.56)
ijkl
The "bare" generalized Lagrangian and
T
matrices are defined by
mo = 11 hii '
E
(ijikk)
(311(ilcijk)}
(7.57)
and mni Tij,k1
= 2ca(ijikl)
13,Injnf(iklil)
(ilijk)}
.
(7.58)
106
CHAPTER 7. TWO-CONFIGURATION SCF WAVEFUNCTIONS
They are related to their "parent" quantities through the following expressions: 2
—E
(7.59)
IJ 2
mn
Tjj,kj
".=
mn IJ
E IJ
(7.60)
Thus, the second derivative of the H if matrix is obtained by substituting eqs. (7.49), (7.54) and (7.56) into eq. (7.46) to yield the fi nal expression, MO
02 "hi
0a0b
Hej + 4
E i;
mo
+ 4
uti c,/da MO
E uzb — E
upk u4
417
ij MO
MO
-I- 4
E
utv,;76
+4E ijk
MO
+ 4
E ijk
MO
▪ 4 >2
UZUktrii l"i
(7.61)
ijkl MO
+
4 E utPify ij
MO
-I- 4
E (uvda +
u,5;
ij MO
+ 4
E
qukjb ciki f)
ijk MO
4
E qukbi-C,
(7.62)
ijkl
7.6
The Second Derivative of the Electronic Energy
An expression for the second derivative of the electronic energy of the TCSCF wavefunction is formally the same as that for the CI wavefunction in Section 6.7: 2 02 Eelec 02H ij 2 ac,- ac, (7.63) nisi — brjE etec) 2 = Ow% Oa ab Om% IJ IJ
E c,c,
E --
( TT
THE SECOND DERIVATIVE OF THE ELECTRONIC ENERGY
107
The last equation of Section 7.5 allows us to rewrite this expression as: 2
(92 Ee i ec act&
MO
CI('j H71 [ .; + 4
E U. E.lif
IJ
ij
MO
+4
E
(UtEr
ij MO
I 4
- -
E
MO /4 1 j Uiajt.Trb ki sik
iik
2
2 E
IJ
A
E
U!113.Ub Till j
ijkl
a cl ac
Oa
6 " E'ec)
Ob
1 ( 7 .64)
•
The relationships between the one- and two-electron reduced density matrices and the coupling constants given by eqs. (7.22) to (7.24) and the relations between the "bare" matrices and their "parent" matrices in eqs. (7.39), (7.52), (7.59) and (7.60) are used to rewrite (7.64) as: MO
02 Edec
2
Oaab
2
EE
MO
CICJA"h g c
i IJ MO
+
4
2
EE c/c4ceficiiiii)ab + ij
2
E
GIGJ uiapf.lif
IJ MO
4
2
EE
c/c.,(Uibi E
ij IJ MO
4
ujai
e)
MO
+4
2
EE
c/c,u,ai ut„
lc IJ
2
EE
c1cJUZULT4!:kji
ijkl IJ
— 2
2_, IJ MO
= 2
Oci
ab
(7.65)
Bid — bIJEelec)
E LN'ib MO
4 Y" MO
4
aci acJ
E
v.-tjbe32
•
MO
+
MO
4
E
qUtirli,k1
2 —
2
E IJ
(7.66)
108
CHAPTER 7. TWO-CONFIGURATION SCF WAVEFUNCTIONS
Ut
In order to eliminate in eq. (7.66), the second derivative form of the orthonormality condition on the molecular orbitals from Section 3.7 may be used:
=
o
(7.67)
where
(7.68) and
=
MO
ua ub
‘—`
M j'M
+
uzb
sq1m
.7/1 IM
—
JM
s
s.7„,)
(7.69)
Thus a final expression for the second derivative of the electronic energy becomes:
mo
Ado
02 Eelec
2
&lab
E
mo — 2
E
+
E
ab
ti
mo + 4
E
+ ulg)
(uibi E7,
MO
+4
E
uzulici (tk + 4
ijk 2
— 2
E ijkl
aCj OCj Oa
— 1.1-Eeiec)
(
7.70)
Iwo
mo = 2 E falib mo — 2E
vio
E
mo
—2
ii
E 12
MO
+
4
E (qqi + ij
MO
+ 4
E ijk
MO
I
- -
4
E
Ui75 1.1{2ai1(ijjkl)
)32 1{(ikij1)
(iljjk)}]
kl
— 2
2
aci ac,
L./
Oa Ob
IJ
His — biJEelec)
(7.71)
It should be noted that the equation for the GRSCF wavefunction in Section 5.6 and the equation for the TCSCF wavefunction, eq. (7.71), differ formally only in the last term of
REFERENCES
109
the latter. The last term in eq. (7.71) involves the derivatives of the CI coefficients. It is necessary to determine the changes of both the MO and CI coefficients with respect to perturbations for the evaluation of the second derivative of the TCSCF electronic energy. These quantities come from solving the coupled perturbed two-configuration Hartree-Fock (CPTCHF) equations, which will be discussed in detail in Chapter 13.
References 1. G. Das and A.C. Wahl, J. Chem. Phys. 44, 87 (1966).
2. A.C. Wahl and G. Das, Adv. Quantum Chem. 5, 261 (1970). 3. A.C. Wahl and G. Das, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.51 (1977). 4. F.W. Bobrowicz and W.A. Goddard, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.79 (1977). 5. R. Carbo and J.M. Riera, A General SCF Theory, Springer-Verlag, Berlin, 1978.
Suggested Reading
1. S. Kato and K. Morokuma, Chem. Phys. Lett. 65, 19 (1979). 2. J.D. Goddard, N.C. Handy, and H.F. Schaefer, J. Chem. Phys. 71, 1525 (1979). 3. Y. Yamaguchi, Y. Osarnura, G. Fitzgerald, and H.F. Schaefer, J. Chem. Phys. 78, 1607
(1983). 4. Y. Yamaguchi, Y. Osamura, and H.F. Schaefer, J. Am. Chem. Soc. 105, 7506 (1983).
5. M. Duran, Y. Yamaguchi, and H.F. Schaefer, J. Phys. Chem. 92, 3070 (1988).
Chapter 8
Paired Excitation Multiconfiguration
Self-Consistent-Field Wavefunct ions A paired excitation multiconfiguration self-consistent-field (PEMCSCF) wavefunction is another special case of more general multiconfiguration SCF (MCSCF) wavefunctions. The PEMCSCF wavefunctions provide useful simple orbital representations for describing molecular bonding and chemical reactions. The PEMCSCF molecular orbitals typically resemble those expected for bond and lone-pair orbitals. For the PEMCSCF procedure, the only configurations which are employed are derived from excitations of pairs of electrons within a limited number of molecular orbitals (MOs). The excitation of two electrons is termed "paired" in that both (one with a and one with 3 spin function) are excited from one MO and remain paired in another MO. The simplest case of a PEMCSCF wavefunction is the two-configuration SCF (TCSCF) wavefunction described in the preceding chapter. A generalized valence-bond (GVB) wavefunction where valence-bond-type paired electronic excitations have been employed is a more complicated example.
8.1
The Paired Excitation Multiconfiguration Self-Consistent-Field (PEMCSCF) Wavefunction and Energy
Consider a high-spin open-shell paired excitation multiconfiguration self-consistent-field (PEMCSCF) wavefunction [1,2]. Obviously the closed-shell PEMCSCF wavefunction is
110
111
THE PEMCSCF WAVEFUNCTION AND ENERGY a special case of this example. We partition the molecular orbitals into four sets:
C 0 P V
(Closed) doubly occupied MOs (Open) singly occupied MOs (Pair) paired excitation MOs (Virtual) unoccupied MOs
The P set involves those orbitals with fractional occupation number and is commonly called the "active space". For each configuration, there is a subset of n molecular orbitals that are doubly occupied and belong to the set of paired excitation orbitals:
(8.1)
={i1,i2, i33 • in}
n is the total number of doubly occupied MOs and equals half the number of electrons involved in the P set given the paired excitation restriction. Using the above terminology, the PEMCSCF wavefunction is: CI
PEMCSCF
= E
(8.2)
Cij
where CI is the coefficient of the /th configuration and each configuration state function DI may be described by (
(8.3)
= A 1 (closed)(open)(Si)
where A is the antisymmetrizer. The CI coefficients are normalized according to the usual convention as
(8.4) The electronic energy of the PEMCSCF wavefunction in CI form is Eelec
=
(41PEMCSCFIlletecIlliPEMCSCF)
(8.5)
E
(8.6)
CI-C.111u
IJ The elements of the CI Hamiltonian matrix remain relatively simple: I-1/j
(8.7)
= (4)/Il1etecl(DJ) mo
=2
mo r
E ettii + E i.i
ali(iiiii) + (311(0./)}
•
( 8.8)
The summation here runs over all orbitals including the paired excitation MOs. The quantity ft" is the one-electron, and all and /3[34 are the two-electron coupling constants. These
112
CHAPTER 8. PAIRED EXCITATION MCSCF WAVEFUNCTIONS
coupling constants are determined by the diagonal elements H11 of the CI Hamiltonian matrix,
closed
f
1/'
—
_I/ 1-/Lii
1
Si
[ 1/2 1 1
closed
2
1
open
open Si
-
closed open
OU
S,
2 1 1/2 1 2 1 2
(8.10)
—1 —1/2 —1 1 —1/2 —1/2 —1/2 —1 —1/2 —1
(8.11)
For the off-diagonal elements II jj, the only exchange type and are
closed /3 3.1
=
open
Si Sj
(8.9)
0
nonvanishing O
0
coupling constants are of the
0
0 0 0 0 0 0 0 1/2 0 0 1/2 0
(8.12)
In eq. (8.12) the configurations I and J are constrained by the PEMCSCF model to have only one paired orbital different. Those coupling constants not defined above are identically zero. The CI matrix elements in eq.
(8.8)
are given explicitly by
:
a) For the diagonal elements,
p c,o
H11 =
H00 +
2 iE es,
2
E E fi f2(iiiii) iEs, j
E
-
(ow}
(ow}
(8.13)
i,3E S1
where
Hoo is H00 =
2E
c,oc,0
EE
fihii +
+ Ai(iilii)}
•
(8. 14)
h) For the off-diagonal elements with one paired orbital different, Hij =
(pqlpq)
,
where p E SI and q E Sj
In the above equation, p and q are the two unique Sj sets.
MOs
(8.15)
that are not common to the Si
and
THE PEMCSCF WAVEFUNCTION AND ENERGY
113
c) For the off-diagonal elements with more than one paired orbital different, (8.16) The one-electron reduced density matrix used in the above energy expression is
closed open pair virtual
1/2 1 I
(8.17)
wir
0
An array cdf is defined by CI
(wf
)i = E
(5(i E S
(8.18)
The function 6 is unity when an orbital i is in the paired excitation set; otherwise it is zero. Using this definition, the Coulombic two-electron reduced density matrix is
2
closed open pair virtual
aij
1 2w1 0 1/2 wf 0
1 2w 1 0
(.4) f
Lace
0
0
(8.19)
0
with the additional matrix w defined by CI
=2
E
j E Si)
CY
(8.20)
where i and j must both be orbitals in the set of paired excitation orbitals. The exchange two-electron reduced density matrix is closed open pair virtual
Oij
—1 —1/2 —
Ca f
—1/2 —1/2 —
1 /2w1
0
0
—cof
—1/2w1 0 —wo 0 0 0
(8.21)
where the matrix wo is defined by CI
(cos)..? =
E
CI t
6(i,j € Si)
—
E
b(i E
E SJ) CICJ
(8.22)
IJ
in the first term i and j must be in the same set of paired excitation orbitals and in the second term one must belong to each of the two sets. In this equation Ev* means that the summation runs over only those pairs of configurations that differ by at most one electron pair.
114
CHAPTER 8. PAIRED EXCITATION MCSCF WAVEFUNCTIONS
The relationships between the elements of the one- and two-electron reduced density matrices and the coupling constants are
CI
fi =
E ci c,fp
(8.23)
IJ CI
E
(8.24)
CI Cjaft
IJ
and
CI
/3jj
(8.25)
C1C.41 IJ
Using these equations the electronic energy of the PEMCSCF wavefunction in MO form is Eelec = 2
MO
MO
E
+E
+ 0
(8.26)
•
The total PEMCSCF energy of the system is: EpEMCSCF
=
Eelec
(8.27)
Enuc
The total energy and its derivatives for the PEMCSCF wavefunction are formally the same as those for the TCSCF wavefunction, the difference being the contents of the f", a", 0" and f, a and 13 matrices.
8.2
The Generalized Fock Operator and the Variational Conditions
The PEMCSCF generalized Fock operator [1] is formally the same as for the general restricted open-shell SCF and TCSCF wavefunctions, the difference being the elements of the one- (f) and two-electron (a and 0) reduced density matrices:
Fi = fih
ma E(a&zi + 0i1K1)
(8.28)
The variational conditions on the MO space are formally the same as for a TCSCF wavefunction: (8.29) — =0 The Lagrangian matrix is
MO
= (0,1Fil4 i) = fihi; + E
t3it(illjt)}
.
(8.30)
THE FIRST DERIVATIVE OF THE ELECTRONIC ENERGY
115
The variational condition on the Cl space is
E c, (Hi./
(8.31)
buEelec)
The elements of the CI Hamiltonian matrix appearing in eq. (8.31) were defined by eqs. (8.13) to (8.16).
8.3
The First Derivative of the Electronic Energy
The expression for the first derivative of the electronic energy of the PEMCSCF wavefunction is formally equivalent to that for the TCSCF wavefunction:
mo
mo
19Eele, Oa
2
E{jjjjy +
E
Oij(iiIii)a}
mo + 4
E
(8.32) mo
mo
2
E + E mo (8.33)
—2 The first derivative of this energy in CI form is metec
Oa
Ha 1,
=E
(8.34)
Oa
IJ
where the first derivative Hamiltonian matrices are OHL] a = H + Oa
MO
a IJ 3 32
The skeleton (core) first derivative Hamiltonian matrices, matrices, ELT , in the preceding equation are
(8.35)
Hy.,, and the "bare" Lagrangian
MO
H j= 2
E fpwl, +
and
(8.36)
MO
" = fjj hij
E tan iiin) + d' oiii)}
•
(8.37)
116
CHAPTER 8. PAIRED EXCITATION MCSCF WAVEFUNCTIONS
The relationship between a Lagrangian matrix and the "bare" Lagrangian matrices is CI
E
(8.38)
IJ
A proof of the equivalence of eq. (8.32) and eq. (8.34) was given in Section 7.4. The practical evaluation of the first derivative may be performed in the same manner as was described in Section 5.4.
8.4
The Second Derivative of the Electronic Energy
The form of the second derivative of the electronic energy for a PEMCSCF wavefunction is formally the same as that for a TCSCF wavefunction, the difference being the elements of the one- (f) and two-electron (a and 0) matrices: 02Eetec
Octab
CI
= E cic, IJ
02 111J
Octeb
CI
2
°CI OCJ ( II LI — 6 1JEe1ec 1J 8a Ob
E
The second derivative of the Hamiltonian matrix is
a2 H
(8.39)
_
frab IJ
8a8b
(8.40) The skeleton (core) second derivative Hamiltonian matrices Hy.b, are defined by
mo HJ
=2
E
ph,q)
mo + E
+
(8.41)
Similarly the skeleton (core) "bare" first derivative Lagrangian matrices ( ma are jja
=
mo E
+
(8.42)
Finally the "bare" generalized Lagrangian and T matrices are
mo (1/1 = fiuhij
E {41(iiikk) + ski(ikiik)}
(8.43)
117
THE SECOND DERIVATIVE OF THE ELECTRONIC ENERGY and ran" T3 k!
= 2a,/;,/n (ijiki)
(iilj IC)}
4:17,1(ikiji)
(8.44)
These "bare" quantities are related to their "parents" through: CI
E cic.,E1.74°
=
(8.45)
IJ CI
E cic,c
=
(8.46)
IJ
and
CI
Tirkl =
E
n1 J
C1C,TriTk i
(8.47)
IJ
Combining equations (8.39) and (8.40) gives the final working expression:
Ç2 j ' aa0b
MO
MO
= 2 >2 Lid +
E
+
MO
— 2
E
tab
Sij Cii
ij
MO
+ 4 >2Ue77,
+ q)
mo
+4
mo
E
+
ijk CI
-
2 7 -1J
MO
4
E
11,5"i 0
ijkl
OCJ ( Ob 111j
Oa
p
2 E fiwi
(8.48)
buEelec)
E
MO
ij MO
MO
2 E
2 E ii
ij
MO
4
E (utE7, +
MO
4
E ijk
MO
4
E
UZUL[2aii(ijikl)
#i1{(ikij1)
(ilijk)}1
ijkl CI
-
2 7 -IJ
OC/ OCJ Oa Ob
( His
bisEciec )
(8.49)
CHAPTER 8. PAIRED EXCITATION MCSCF WAVEFUNCTIONS
118
In these equations the 4'36 and ya b matrices were defined in Section 3.7 as: ah rSij
where
MO
ab Thj
(TTa nb imu jm
oab ij
rïb Tra trn u rn
ab
— .57„,
(8.50)
ST, )
(8.51)
and satisfy the following relationship, (8.52)
It is necessary to determine the changes in both the MO and CI coefficients with respect to perturbations in order to evaluate the second derivative of the electronic energy. These quantities are by no means trivial to determine but may be found by solving the coupled perturbed paired excitation multiconfiguration Hartree-Fock (CPPEMCHF) equations. The latter equations will be discussed in detail in Chapter 13. References 1. F.W. Bobrowicz and W.A. Goddard, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.79 (1977). 2. R. Carbo and J.M. Riera, A Genera/ SCF Theory, Springer-Verlag, Berlin, 1978.
Suggested Reading 1. S. Kato and K. Morokuma, Chem. Phys. Lett. 65, 19 (1979). 2. J.D. Goddard, N.C. Handy, and H.F. Schaefer, J. Chem. Phys. 71, 1525 (1979). 3. Y. Yamaguchi, Y. Osamura, and H.F. Schaefer, J. Am. Chem. Soc. 105, 7506 (1983). 4. M. Duran, Y. Yamaguchi, and H.F. Schaefer, J. Phys. Chem. 92, 3070 (1988). 5. M. Duran, Y. Yamaguchi, R.B. Remington, and H.F. Schaefer, Chem. Phys. 122, 201 (1988).
Chapter 9
Mu lticonfiguration Self-Consistent-Field Wavefunct ions The multiconfiguration self-consistent-field (MCSCF) method is the most powerful method to improve the quality of the molecular orbitals and to take electron correlation effects into account simultaneously. The complete active space SCF (CASSCF) wavefunction is a very important special case of still more general MCSCF wavefunctions. In the CASSCF method a full CI procedure (complete optimization of both the MO and CI spaces) is performed within a limited orbital set (usually called the active space). Quadratically convergent procedures developed in recent years have made it feasible, although still fairly computationally expensive, to apply MCSCF methods to a wide variety of chemical problems. In this chapter, the first and second derivatives of MCSCF wavefunctions are described.
9.1
The MCSCF Wavefunction and Energy
The multiconfiguration self-consistent-field (MCSCF) wavefunction is a linear combination of electronic configurations (Di that are constructed from a set of reference molecular orbitals
[1-3] CI
iimcsaF = E c/4, /
(9.1)
I
The wavefunction is normalized, giving the condition
ci
E0.1 /
on the CI coefficients.
119
(9.2)
CHAPTER 9. MCSCF WAVEFUNCTIONS
120
The electronic energy of the MCSCF wavefunction in CI form is CI
E
Ede,
(9.3)
CICJHIJ
IJ
The CI Hamiltonian matrix element in eq. (9.3) is defined by H/J
=
(9.4)
(4)/Ille1ec0J) MO mo EE
ij
(9.5)
F./4(JAN)
ijkl
1/
in the above equation are the one- and two-electron coupling The quantities 7 and constants between the electronic configurations and the molecular orbitals. The one- and two-electron reduced density matrices [4] are defined in terms of these coupling constants by cr Cxyll (9.6) = E IJ CI
=
E
cicAti„,
(9.7)
IJ
Using these quantities the electronic energy of the MCSCF wavefunction is given in MO form as Eelec =
MO
MO
ij
ijki
E
E
(9.8)
r
for the The coupling constants -y" and rif and the reduced density matrices 7 and MCSCF wavefunction have the same meaning as for the corresponding CI wavefunction. Different notations are used in this book in order to avoid any possible confusion. The total energy of the system is finally the sum of the electronic energy and the nuclear repulsion energy (9.9) Enuc EMCSCF = Eelec
9.2
The Variational Conditions
The variational conditions in the CI space are formally the same as described for the CI wavefunction in Section 6.1: CI
E
—
5/./Eeicc)
=o
(9.10)
The CI Hamiltonian matrix in eq. (9.10) is defined by eq. (9.5) and 6u is the Kronecker delta function.
THE FIRST DERIVATIVE OF THE ELECTRONIC ENERGY
121
Another constraint on the MCSCF wavefunction (9.1) is the variational condition on the MO space. This condition requires a symmetric Lagrangian matrix at convergence: — where
MO
=0
(9.11)
MO
E Yj 7nhim + 2 E ri mki(imikt)
(9.12)
mid
If the Fock operator for the MCSCF wavefunction is defined as MO
= 7h + 2 EFijki J kl
(9.13)
kl
then the relationship between the Fock and Lagrangian matrices is given by
mo Xj
(9.14)
=E rn
MO
MO (eyj m
=
h+
2 E
rimoki) ickm)
(9.15)
kl
In eq. (9.13) h is the one-electron operator and J kl is the two-electron operator which satisfies ( 9.16) (0i1Jk1i0j) = ( iiik/) •
9.3
The First Derivative of the Electronic Energy
The first derivative of the electronic energy of an MCSCF wavefunction can be obtained without explicit information on the derivatives of the MO and CI coefficients. The variational conditions on both spaces must be satisfied simultaneously as in eqs. (9.10) and (9.11). Thus, these coefficients are optimized explicitly in the MCSCF procedure. The first energy derivative expression in CI form is CI
"elec —
—
Oa
C/Cj
H j 8a
(9.17)
The first derivatives of the Hamiltonian matrices were obtained in Section 6.4: OHIJ
MO
= Ha + 2
E mix-Y s3 ti
(9.18)
ij The skeleton (core) first derivative Hamiltonian matrices Hyj and the "bare" Lagrangian matrices x ij are MO
MO
E
E 1131,(iiik1Ya
i)
ijkl
(9.19)
122
CHAPTER 9. MCSCF WAVEFUNCTIONS MO IJ X i3 =
MO
E
+ 2
E
.
(9.20)
The following relationship holds between the Lagrangian and the "bare" Lagrangian matri-
ces: x i,
ci =E
c/c.,xif
(9.21)
IJ
This expression may be compared to that for the coupling constants in eqs. (9.6) and (9.7). It should be noted that the "bare" quantities, -y", 111j, H7j , and x" do not contain the CI coefficients in their definition. ,
Using these equations, an expression for the first derivative of the electronic energy becomes 0 Eeiec
Oa
CI
= E cic, IJ
CI
= E cic, IJ
a 111,1
(9.22)
Oa IMO
MO
-yilif i qj + E rtiki iilr la (
ij
mo
ijki
Iwo
= E 7ii kiii + E ri,ki (iiikoa + ij
MO d- 2E 12 ii
Uiajx
(9.23)
mo 2
ijkl
E
Uiai xii
(9.24)
sj
Alternatively the energy first derivative expression is obtained by differentiating eq. (9.8) with respect to a cartesia-n coordinate "a": 0Eelec
Oa
MO
MO
E
da
ii
E
0(iiiki)
rijkl
ijkl
Oa
(9.25)
A more explicit expression for eq. (9.25) is equivalent to that obtained for the CI wavefunc-
tion in Section 6.2: melec
Oa
mo _ E
-Nifei`j
ii
mo E ijkl
MO
+2
E
(9.26)
ii
which is identical to eq. (9.24). The Lagrangian matrix x in eq. (9.26) was defined by eq. (9.12). Using the variational condition (9.11), eq. (9.26) is simplified further to give Eelec Oa
mo mo E 7 3 h + E roki (iiikoa ii
ijkl
MO
+ E (q +
(9.27)
THE EVALUATION OF THE FIRST DERIVATIVE MO
123
MO
= E -yocti + E ii
ij/cl
MO
E
(9.28)
In the last step, the first derivative form of the orthonormality condition of the MOs described in Section 3.7,
=
(9.29)
was employed. It should be emphasized again that the explicit evaluation of the changes in the MO and CI coefficients is not necessary for the first derivative of the energy of MCSCF
wavefunctions.
9.4
The Evaluation of the First Derivative
In the practical evaluation of the first derivative, eq. (9.28) is usually transformed into the AO basis. The first derivative of the electronic energy is
OEetec
MO AO
•
MO AO
•
. E E Ni ct c3 all"v ij
Ai/
MO AO
- E E ij
+
EE
rij kic,i cf,c,okcE
ijkl /iv Pa
• ci
c.7. 's"- x. •
a( liv 1 Pa)
Oa
(9.30)
Ai/
The one- and two-electron reduced density matrices in the AO basis are defined by (see Section 2.8) MO
=
E iJ
(9.31)
and MO
E
Flu/pa
(9.32)
ijkl
The "energy weighted" density matrix for an MCSCF wavefunction is MO
w1. =E
(9 .33)
Using these density matrices, the first derivative expression becomes OE el--
Oa
" =
AO
E Au
-yAv
aa
0(twiper)
p
+
AWPC7
Al/Pa
aa
. (9.34)
124
CHAPTER 9. MCSCF WAVEFUNCTIONS
In practice, the derivative integrals are evaluated only for unique combinations of AOs > y and p a) in order to minimize computational effort. The first derivative of the electronic energy in a directly applicable form is Melee
2
Oa
AO E
Oh4
(544 ,, \
2
th>t,
)
,
Oa
411
AO
+ 4
(1 -_ 8Pcr )(1
E > V, p > LV > pa
2
2
8tw
' Pa )
2
x {21"„pc a(AaValPa) 6AL ,)w
OSAm
(9.35)
"" Oa
2
In eq. (9.35) 6,„ is the Kronecker function. In the second term of this equation the summations run over the indices that satisfy ,av > pa, where AV = — 1) + y and Pa
= iP(P — 1 ) +
9.5
a.
The Second Derivative of the Electronic Energy
The second derivative of the electronic energy of the MCSCF wavefunction is formally the same as for the CI wavefunction in Section 6.7:
02 Eelec Nab
CI
GI
02 Hjj = E c/c., -
2
IJ
E IJ
aCI
Oa
acj ab
—
5 IJEelec)
(9.36)
The second derivative of the Hamiltonian matrix was derived in Section 6.6: MO
(92111J
HP.; + 2
Oaab
mo 2
E
E uFox-v IJa
MO
2
E UUtayi i
(9.37)
ikl The skeleton (core) second derivative Hamiltonian matrices HVi and the skeleton (core) first derivative "bare" Lagrangian matrices x ua appearing in eq. (9.37) are defined by Hjj
mo
MO
ii
ijkl
E 7.,f/h731? + E rtyki (iiiknab
(9.38)
125
THE SECOND DERIVATIVE OF THE ELECTRONIC ENERGY
and
MO
Iva
Xii
=
E
tm +
7:,jit hci
MO
2
ln
E
.
(9.39)
mid
The "bare" y matrices are ypict=
7Jhk
+
MO
MO
2
+4 mn
E ri/mJin(irnikn)
•
(9.40)
mn
These "bare" xa and y matrix elements are related to their "parents" through CI
E
GIG jXfi a
(9.41)
Ii
(9.42)
IJ
and
CI
E
=
Yijkl
f/I
IJ
Combining eqs. (9.36) and (9.37), gives
02 Eeiec &Lab
MO
MO
=E
+E
ij
ijkl MO
E uiaibxi,
+2
MO
E
+
mo 2>
qUidYijki
+2
ijkl
CI
2 \---%
aci acJ Oa
IJ
( -
06
(5./.7 Eelec)
•
(9.43)
In deriving eq. (9.43), equations (9.6), (9.7), (9.21), (9.41) and (9.42) were employed. Using the variational condition for the symmetric Lagrangian matrix (9.11), the Uab term in eq. (9.43) is conveniently eliminated: 92 Eelec
4
0a0b
mo
mo
ij
ijkl
E + E
MO
E ij
ab
"
tj t3
126
CHAPTER 9. MCSCF WAVEFUNCTIONS MO
+
2
E
UU l ykj
ijkl
CI OC' OCJ - 2E aa ab
is — 6./JEelec
(9.44)
1J
MO
MO
ij
ijkl
E
MO
E 70c + E riikt (iilkoab
E ?ex
MO
ij
MO
+ 2 V (U
+ 1/4
MO
2 V UetYiiki ijkl CI
In these equations the
2 V IJ
Oa
Cab and nab matrices
—
ab
ab
mo ,
=E
- (9.45)
were defined as in Section 3.7 by, zib
where
61J-Eetec)
+UU
ab
(9.46)
-
- sLs;.,.)
(9.47)
and satisfy the following relationship:
rie
Pjlib + eab = 0
.
(9.48)
It should be emphasized that the first-order changes in both the MO and CI coefficients are required to evaluate the energy second derivative of the MCSCF wavefunction. These quantities must be evaluated by solving the coupled perturbed multiconfiguration HartreeFock (CPMCHF) equations, which will be discussed in Chapter 14. References 1. H.-J. Werner, in Advances in Chemical Physics : Ab Initio Methods in Quantum Chemistry Part II., K.P. Lawley editor, John Wiley & Sons, New York, vol. 69, p.1 (1987). 2. R. Shepard, same book as reference 1, p.63 (1987).
3. B.O. Roos, same book as reference 1, p.399 (1987). 4. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
SUGGESTED READING
127
Suggested Reading 1. P. Pulay, J. Chem. Phys. 78, 5043 (1983). 2. R.N. Camp, H.F. King, J.W. McIver, Jr., and D. Mullally, J. Chem. Phys. 79, 1088 (1983).
3. M.R. Hoffmann, D.J. Fox, J.F. Gaw, Y. Osamura, Y. Yamaguchi, R.S. Grey, G. Fitzgerald, H.F. Schaefer, P.J. Knowles, and N.C. Handy, J. Chem. Phys. 80, 2660 (1984). 4. M. Page, P. Saxe, G.F. Adams, and B.H. Lengsfield, J. Chem. Phys. 81, 434 (1984). 5. T.U. Helgaker, J. Alml6f, H. J. Aa. Jensen, and P. Jorgensen, J. Chem.
(1986).
Phys. 84, 6266
Chapter 10
Closed-Shell Coupled Perturbed Hartree-Fock Equations Although the analytic derivative expressions presented in the preceding chapters are complete, many of these expressions require the solution of various coupled perturbed equations that only have been alluded to thus far in the text. The set of simultaneous equations known as the coupled perturbed Hartree-Fock (CPHF) equations provides the derivatives of the MO coefficients of the SCF wavefunction with respect to perturbations. Since the MO coefficients are determined using the variational conditions on the SCF wavefunction, the derivatives of the MO coefficients may be obtained from the derivative expressions of the variational conditions. In this chapter, the first- and second-order CPHF equations are derived for the simplest but most frequently used case, the closed-shell SCF method.
10.1
The First-Order CPHF Equations
For a closed-shell SCF wavefunction, the Fock matrix elements in restricted Hartree-Fock (RHF) theory [1,2] are
d.o. = hii
E {2(iiikk) - (ikljk)}
(10.1)
where the sum is over doubly occupied orbitals. The variational condition is quite simply:
F
=0
for
i=
doubly occupied and
j=
virtual
(10.2)
Since the SCF energy is invariant to a unitary transformation within the occupied space, a diagonal Fock matrix normally is used to define the "canonical" molecular orbitals: = bi3 ci 128
(10.3)
THE FIRST-ORDER CPHF EQUATIONS
129
In eq. (10.3) (5,i is the usual Kronecker delta function. The orbital energies at convergence (i.e., at the conclusion of the SCF iterative procedure) are given by El;
=
(10.4)
Fri
and those for the occupied orbitals may be interpreted through Koopmans' theorem [3]. The first-order coupled perturbed Hartree-Fock (CPHF) equations [4] are the first derivatives of the variational condition in eq. (10.2). They are derived by differentiating eq. (10.1) with respect to an arbitrary cartesian coordinate "a":
a
d...
ah, aa
Oa
f 2 a(ijikk) E k Oa
0(ilcijk)1 Oa f
(10.5)
This derivative was obtained in Section 4.5 (see also Appendix N) as all
all
+E
aa
u:i Fk;
+E
U:jFik
all d.o.
EE k
(10.6)
I
The elements of the skeleton (core) first derivative Fock matrices are d.o.
=
E {2(iiikkr -
h"ti
(ikli O a }
(10.7)
and of the A matrix:
(10.8)
= 4(ijikl) — (ikij1) — (ilijk)
Using the variational condition eq. (10.3), eq. (10.6) is reduced to all d.o.
= F2, Oa
EE
u
3
k
(10.9)
I
The important first derivative form of the orthonormality condition on the molecular orbitals was given in Section 3.7 as
Uiai
+ S.73 = 0
(10.10)
Using this relationship, eq. (10.9) is reformulated as
aa
Fiai
(—
— Siai)E j
UiaiEi
all d.o.
EE k
uzAii ,k/
1
— (Ej — Ei)(Tri — all d.o.
EE
uLAi i,k/
(10.12)
130
CHAPTER 10, CLOSED-SHELL CPHF EQUATIONS
The fourth term in eq. (10.12) may be manipulated in the following way: all d.o.
all d.o.
EE k
EE u1t1 {4(iilki) _ (ikiii) _ (auk)}
(10.13)
k 1 d.o. do.
I
EE u 1 {4(iiiki) k 1 virt.LL
-
-
(auk)}
_
.
(10.14)
/
k
Using the first derivative of the orthonormality condition, eq. (10.10), allows this term to
be rewritten as: d.o. d.o.
all d.o.
EE k
UtaiAij,ki
=
{4(ijik1) — (ikiji) — (iiijk)}
(1.7":1 +
1 vin. d.o.
+EE
ug,{4(ipk1) -
— (iiijk)}
(10.15)
k d.o. do. Sic:1{ 11(iiik1) —
I
= 2 k vint. d.o.
+EE k
=
_ EE vint.
- (auk)}
u{4(iiiki) -
I d .o. d.o.
Oki 1 ) — (illik)}
-
k d.o.
+ E E u44(iiiki) -
(10.16)
014;01
-
(10.17)
k
In these equations virt. denotes the virtual (unoccupied) orbitals. It should be noted that in the first term of eq. (10.16) the exchange of the k and 1 indices does not alter the summation.
Combining eqs. (10.12) and (10.17) yields the following expression:
aa
Fiai — (Ei — Ei)Utai
—
d.o. do.
EE sh{2(iilki) k virt.d.o.
+EE k
ug,{4(iitki)
(ii(ik)}
(10.18)
/
When the Fock matrix is diagonal, then the derivatives of the Fock matrix are also diagonal: Oa
= u" Oa
(10.19)
THE FIRST-ORDER CPHF EQUATIONS or
131
OF,3
Oa
for i j
=
(10.20)
Using these conditions, eq. (10.18) becomes virt.
do.
- EE
ei )
—
(i l ijk)}
k
d.o. d.o.
=
— SZE j
-
EE k
-
(10.21)
1
This result is written commonly as virt. d.o.
(Ej
—
—
EE k I
(10.22)
Aij,k1 u11 = -13(1,ij
The A matrix was defined by eq. (10.8) and the fig matrices are do. d.o.
—
S.:13 E j
—
EE k
-
.
(10.23)
1
The simultaneous equations given in eq. (10.22) are the first-order coupled perturbed Hartree-Fock (CPHF) equations, which involve the unknown variables Ua. The dimension of these CPHF equations is the product of the number of doubly occupied orbitals and the number of virtual orbitals. The CPHF equations usually are solved iteratively [5]. When the dimension is not too large (less than 1000), the CPHF equations may be solved using any of the standard methods for simultaneous equations. The first-order derivative of the orbital energy involves a diagonal term of eq. (10.18), and the results obtained from solving the CPHF equations (10.22) are
afi Oa
OFii Oa
(10.24) d.o. d.o.
= Fia2 —
- EE k
sq2(iiiko -
1
vin. d.o.
+EE k
Ukti t4 (iiikl) — (iklil) — (illik)}
.
(10.25)
1
When the SCF condition is satisfied, the SCF energy is stationary with respect to any (real) perturbations. In other words, an infinitesimal (real) perturbation of any kind at this stationary point cannot change the SCF energy. Consequently, elements of the Ua matrix between two doubly occupied orbitals or between two virtual orbitals are not determined uniquely. Such pairs of molecular orbitals are usually called "non-independent". On the other hand, occupied-virtual pairs are uniquely determined and are termed "independent pairs".
132
CHAPTER 10. CLOSED-SHELL CPHF EQUATIONS
Elements of the Ua matrix for the independent pairs of molecular orbitals are obtained by solving the CPHF equations (10.22). The non-independent pairs may be evaluated using virt.
d.o.
EE
1 (
(10.26)
U1:1
k
The right hand side of eq. (10.26) contains the elements for the independent pairs of the Ua matrix. It also should be realized that eq. (10.26) has been defined explicitly using the diagonal nature of the Fock matrix. Thus, this equation is not valid when a molecular system has degenerate orbital energies either by symmetry or accidentally. However, the explicit evaluation of the Ua matrix for non-independent pairs (including degenerate pairs) by eq. (10.26) can be avoided for many applications, such as second derivatives of the SCF energy and first derivatives of the Cl energy [6-101.
10.2
The Derivatives of the Fa Matrices
Before studying the second-order CPHF equations, it is useful to find derivative expressions for those quantities involved in the second derivative of the Fock matrix. Examining eq. (10.6) it is obvious that these quantities include the Fa and A matrices. The next two sections are devoted to finding the derivative expressions of these two matrices. The derivatives of the Fa matrices are developed in this section. Using results from Sections 3.8 and 3.9, the derivative of eq. (10.7) with respect to a second variable "b" is:
aFia Ob
f 2 0(ijIkkr Ob
Ob
a(ik I jk)". 1 Ob
(10.27)
all
E
tee
(u,b„,h7„, + u,bni htn )
7/2
d.o. [
+E
all
..
2(13 1kk)ib + 2
E fonainiikkr
k
+
+ 20„ k (iiintk)a}
in
all
— (ikli k) °b
E futi(mkuk)" + or,k(inzlik)i+ u(ikimoa
+ UP„k (ikijm)a}1
(10.28)
d.o.
lee
E {2(iiikk)ab
(iklj kr b }
d.o.
all
qi[hnz aj
E {2(miikk)a —
(rnklik)ali
THE FIRST DERIVATIVE OF THE A MATRIX all
133
do.
+ E u,bn,[h7,7,
E {2(imikk)a — okinikr}]
all d.o.
+ EE 0,44(iiimoa _ (imijoa — cikijnoal 771
(1 0.29)
k
Using the definition of the skeleton (core) first derivative Fock matrices from (10.7) and writing the skeleton (core) second derivative Fock matrix as
d.°. + E {2(iiikkrb
=
(ikii krb}
(10.30)
eq. (10.29) may be written as all E oniF,70
Fitz",
all
E
umb
7/1
In
all d.o.
+ EE upe,[4(iiikoa — k
—
k)a]
(10.31)
I
The skeleton (core) first derivative A matrices are A ictj,k1 = 4 (i ilki) a
(ikli 1 )a —
kr
(10.32)
Thus eq. (10.31) can be written as OFiai
all
= Fiajb Ob
(umb Flan.
In
all d.o.
+ EE utActi,ki k
(10.33)
I
It is noteworthy that the derivative expression for the Fa matrices in eq. (10.33) has a similar form to that for the F matrix in eq. (10.6). Derivative expressions for the skeleton (core) higher derivative Fock matrices are found in a similar manner and are summarized in Appendix N.
10.3
The First Derivative of the A matrix
In this section the derivative of the A matrix is obtained. Using the results from Section 3.9, the derivative of the A matrix (10.8) is: 0Aii,ki
Oa
=4
OW I kl) Oa
0(iklil) Oa
a(illik) Oa
(10.34)
ail
4[(i jiklr
E
U;17,j(imikl)
134
CHAPTER 10. CLOSED-SHELL CPHF EQUATIONS
▪U7L(ii1 7711 )
U7anl(ii 110.11)11
all
E
[(ikli
+ +
+ all
-
[(ilijk) a
E
u,(iklin)}1
+
7n
▪ U°11' i(illMk)
utank(imiii)
U7n1(inellj k)
UZ k (illjm)}1
(10.35)
— (ikliir — (illjk)a all
E
(mkij1) — (mlijk)}
all
A-
E
_
mj
(iklml) — (illmk)}
all
E
uLf4(iiimo _
all
+E
ua„,,{4(iiikm) —
(10.36)
— (imlik)}
Using the definitions of the A matrix from eq. (10.8) and the AΠmatrices from eq. (10.32) gives
-= ..tj,kl AI all
E (u,an i Amj,kl
U7an
A irn ,k1
Utan kAij,m1
Aij,km) (10.37)
Derivative expressions for the skeleton (core) higher derivative A matrices are obtained in the same way and the results are summarized in Appendix N.
10.4
The Second-Order CPHF Equations
The second-order CPHF equations are the second derivatives of the variational condition in eq. (10.2). They provide the second derivatives of the MO coefficients with respect to perturbations. These second-order CPHF equations may be derived by differentiating eq.
▪
THE SECOND-ORDER CPHF EQUATIONS
135
(10.6) with respect to a second parameter "b": 02 Fii
0 (0F Oa
0ab
(10.38)
a
=
all
all
/Jul;
+E
all d.o.
+ EE
utFik
k all
0F
a
(--5-r aUkai Fkj + (ouzi
all
all do .
k
)
Fki
k ab
0F.k)
E — ik Ob F
-1 Ob
( OUa A
+ EE
(10.39)
ETA Ai j,k1
1
+ uk, 0Aii,ki) ab
(10.40)
1
The derivatives appearing in eq. (10.40) were given explicitly in Section 3.4 and in eqs. (10.6), (10.33) and (10.37). Substituting these results into eq. (10.40) yields:
0 2 F1
all
E
(umb irrt:ii
all d.o.
umb iFin )
0a0b
EE k
all
N I A?j,ki
1
all
f —
E up
Fkj
k all
all
+ EU
Fpci + E
,
all
um b k Fm,
+E
all d.o.
urn b,F,,n
+ EE m
all
(
all
+E
—E
all
UlbcmUtanj Fik all
+ E Ukait Fi&k + E
all
um b i Fmk
+E
um' k Fini
all d.o.
+ EE rn
all d.o.
+ EE
um bnAki,77n
n
Um b nAik,mn
n
all
u,a,p
—E
Ut m Um a 1 Aij,kl
k all d.o.
all
+ EE k
all
E
um biAmi,k,
Um b j Aim,k1
1 all
all
+ E umbkA,,m, + E all
Fab
+E
all
E UFk3 + E
all d.o.
▪ EE
ubAij ,kt
um b,Aii,km
Fik
(10.41)
CHAPTER 10. CLOSED-SHELL CPHF EQUATIONS
136
(10.42) k
I
Some of the summations run over all MOs while others are over doubly occupied MOs only. Using the variational condition eq. (10.3), eq. (10.42) may be reduced to
o 2 Fi .
all d.o.
3 =
+
&lab
j
EE
Ultjb Ei
k all
+ E (ut +
UeAij,ki
I
+ uti Fiak )
, +
utFg.
all
+ E (u:i N; + u1 U)Ek all d.o.
EE uz ubin Aij,k1 kl m all d.o.
(ututn, + utUiam )A kj ,Im EE kl all d.o.
(ugi utbm + EE kl all d.o.
01 Ulam ,3
)
Aik ,i m
+
EE k I
(10.43)
The second derivative of the orthonorma,lity conditions on the molecular orbitals was presented in Section 3.7 as U ia3-b
= 0
U.72
(10.44)
where all
s^6
E (u21„ (Jib, + uibm Urn,
—
531 m — Si?7, 57m )
(10.45)
THE SECOND-ORDER CPHF EQUATIONS
137
Using eq. (10.44), eq. (10.43) may be manipulated in a similar manner to the first-order CPHF equations to yield:
8 2 Fii Fab
aaab
(Ei
ci )uirtib
Gtjb Ej
_
(aril)}
d.o. d.o.
-
EE k I virt. d.o.
• EE k I all
+ E (ugi Ft; + uzi ni + uvibk +
U Fiak)
all
+ E (uzi uz; + upci Ugi) Ek all d.o.
• EE kl m
U,
UAjj ,kl
all d.o.
• EE kl
(ugi utbn,
quiam )Akj,Im
(Uf:JUP,,,
Ukillianz )Aio nt
all d.o.
• EE kl all d.o.
• EE k I
(UZA6 i
UtiA tc i, k1)
.
(10.46)
The second derivatives of the Fock matrix are diagonal due to the diagonality of the Fock matrix at convergence. Thus, 02Fi3.
0(01)
82 c .
=
(10.47)
Oat%
Or a2Fii
for
0a0b =
i
j.
(10.48)
By exploiting eq. (10.48), eq. (10.46) is rewritten in a familiar form as virt. d.o.
(E, — €.)119..b _ EE 13 t
'Tab Dab A --ij,k1= Lflo,ij
1
k
where d.o. d.o.
—
j—
EEecat2(iiiko _ k
all
1
+ E (ugi Ftc; + ut1 F4 +
uf:i Fibk
+ uki Ftk )
(10.49)
▪ 138
CHAPTER 10. CLOSED-SHELL CPHF EQUATIONS all
• E (uuti + Ni ur,i) Ek all d.o.
EE kl
Ulm Ul Aij,kl
in
all d.o.
• E E(u:ut + uzi ugn) kl rn all d.o.
+ EE (u;upn,
+
kl m all do.
+ EE k
(U:IA Z
+
utAzi,ki) .
(10.50)
I
The second-order derivative of an orbital energy, 02 (i /aaab, is obtained from a diagonal term in eq. (10.46), using the results from the CPHF equations (10.49). The elements of the U ab matrix for non-independent pairs are evaluated using the independent pairs as Uab — —
vir t .
c.1
d.o.
EE
1 C2 )
k
A„, k i
ue
+
(10.51)
I
(10.51) contains elements involving the independent pairs of the (Tab matrix. Note that eq. (10.51) has been derived explicitly using the diagonal property of the Fock matrix in eq. (10.3). This equation is therefore not applicable when a molecular system has degenerate orbital energies either accidentally or by symmetry. However, explicit evaluation of the Uab matrix for non-independent pairs (including degenerate pairs) can be avoided for many applications, such as fourth derivatives of the SCF energy or second derivatives of the CI energy [6-11]. It should be emphasized that the right hand side of eq.
10.5
An Alternative Derivation of the Second Derivative of the Fock Matrix
An alternative expression for the second derivative of the Fock matrix may be obtained arduously but more straightforwardly by differentiating eq. (10.1) with respect to the two cartesian coordinates "a" and "b":
a2
j
aaab
E d.°.
92h j aaab
f 2 a2 (ijikk) &Lab
a2(ikiik) aaab
(10.52)
The relevant second derivative expression for the one-electron MO integrals was given in Section 3.8 as: all
= h`1.7
E (uy bi
an hm ,
in
+
Uianbihim )
ALTERNATIVE DERIVATION OF THE F MATRIX SECOND DERIVATIVE
139
all
E
+
+ U.;`„j hL
mi
Umb j fq,)
min all
+ E
+ firbni //Tau ) hm ,„
(10.53)
inn
For the two electron MO integrals, the second derivative expressions from Section 3.9 are -
02 (ijikk) 0a0b
(ijikkr 6 all
E
[u,aniamiikk)
+
uzbi (imikk)
all
+E
[u,ani(niikk)b +
+
+ 2 U7anbk(iiintk)]
tr ;( imikob
2u(iiimk)b
+ or,; (imikk)a +
0(milkoa
all
+E
[( utani unbi
+
umbiu)(Tnnikk)
+
+
(uoLk
Umb iU:k)(milnk)
mn
(UUL Oni Uta,k )(imikn)
(UUnb
Ura b y77,k )(imink)
(Uk0,k
UtkUr7k)(iilmn)] (10.54)
and 02 (ikijk)
0a8b
( ikijk)a b all
+E
uxicrakDk)
+
uzbk(imuk)
+
u,anb,(ikimk)
+
uzbkokiimd
all [
E + uz.i(nkiik)b +
4- q i (m ic l jk )a all
+E
Ru:Lunb,
Ui(ik(ntk)b
k) b
umb
umb k (im i ioa
+ umb i u:k )(mniik)
+
mn
rn kr
m i ni
(P(ikli7n)b
umb k Umb i ti,ay.i)(Mkink)
(ULUn6 k
//ni b it/77,k) (Mklin)
(1-41kU7b j
Umb kU) (i Mink)
(Uian kOtk
OnkU7a1k)(iMlin)
(Utan jUlbl k
Umb j U k )(iklmn)] (10.55)
▪
▪ 140
CHAPTER 10. CLOSED-SHELL CPHF EQUATIONS
Using eqs. (10.53) to (10.55), the expression in (10.52) may then be expanded as
a
2 Fi
•
aaab
all
= h
+
(U:17,11hm i
Ultim )
all
+ E (uzign, + ail
U b 7n ih7n j
Ub
U?an j
7n j hilm)
+ uti U;',i )h,n„
+E mn
+2
E
all
[(iiIkkrb
+E
(uei(nAkk)
all
+E
+
u,t(imikk) + 2(Jxk (iiirnk))
+ uzi ( imikob + 2u7ank (ijirnk)b
(u76;zicniok)b
+
+
0„,(inzIkk). + 20(iii1ak)a)
all
+ E 1.
ULU:i )(mnikk)
mn
+ 2 (UVI,ba + + (I/A k0k
Umb J U k )(imInk)
iU:k) (nli Ink) + 2 (UUnbk Umb klia)(iiimn)}1
d.o. [. (2k1 kr b
- E
all
+ E (uei(mklik) +
uxk(irnlik)
+
u,a,t7 (ikInto
+
u77,6k(iklim))
all
+ E (u i( inkiik)b + ▪ U(Inkli
w(inilik)b + u(almk)b + u,ankokiirrob
qk(iinli
UP„ j(iklink) a
Utk(ikli Mr)
all
+ E
(u,any,b, + Umb iUa)(rnnlik)
(uZiUnbi
mn
▪ (UlaniUnb k
OnA lik)(Mklin)
▪ (Ukti ttk
Utbn kU7aik)(iralin)
(UAkU 7bli
Umb
U!'nkU7L)(intink)
Umb jUk)(iklmn)}1 (10.56)
d.o.
E
t2(iiikoab
—
Oki krb}
ALTERNATIVE DERIVATION OF THE F MATRIX SECOND DERIVATIVE all
141
d.o.
E
E {2(milkk)
all
E {2(imikk)
— (nikij k)}]
d.o.
E ue, [him
_ (ikimk)}1
all d.o.
EE
rx,44(iiimk) -
- (ikliM)]
ra k
all
[
E
d.o.
+ E {2(milkob
all
(midi k) 6 11
d.o.
E oni h7„, + E {2(milkkr m
— (mkij kr}]
all [iim
+
all
117„, + m all
E
{2(i.Inikk) b — (iklmk) b }1 k d.o.
E {2(imikkr
— (ikImk
do.
u,ani u,b,,[h,nn
+ E {2(mnikk)
—
(mkInk)}]
—
(mkink)}]
mn
all
d.o.
E
+ E {2(mnikk)
mn
all d.o.
EE
UkUP,k [4 (iilinn) — (imiin) — (midi rn,)]
EE
U,an ilf.rth [4 (milnk) — (mnlj k) _ (mkij
k all d.o.
mn
mn
k
all d.o.
EE
Umi U:k[4 (rnilnk)
EE
11:0 U,tk [4(imink) — (inlmk) — (iklinn)1
EE
UTbni U,a,k [4(imink) — (inlink) — (ikintn)]
EE k
UTan k[4(ijimk) b — (imli k) b — (ildj 77)1
(mnl k) _ (mkij n)]
k all d.o.
mn
k all d.o.
mn
k all d.o.
mn
713
all d.o.
EE k
0k[4(ijimk) a — (imli k) a — (ikij
•
(10.57)
CHAPTER 10. CLOSED-SHELL CPHF EQUATIONS
142
Finally, the definitions of the F, Fa, Fab and Aa matrices given previously allow (10.57) to be rewritten as:
a2
all
all
E ur„% i + E U
= Fajb
0a0b
F1k
all d.o.
• EE U f A j,kl k I
all
+ E (uziFp,, + u all
E (but
+ u,
,
i + u Fiak)
+ u,b,, uiaj ) Fkl
kl
all
d.o.
▪ EE
ukLinULAi j
,k1
kl all do.
-E
(uzi EE kl
+ uzi ug„) Akj,i m
all d.o.
+ N.; Urm ) Aio ni
va
• EE kl m all
(uzi utb
d.o.
EE k 1
.
(10.58)
Equation (10.58) is equivalent to eq. (10.42), which was derived independently in the preceding section. References 1. C.C.J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). 2. G.G. Hall, Proc. Roy. Soc. London, A205, 541 (1951). 3. T.A. Koopmans, Physica 1, 104 (1933). 4. J. Gerratt and I.M. Mills, J. Chem. Phys. 49, 1719 (1968). 5. J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Hinkley, Int. J. Quantum Chem. S13, 225 (1979).
6. Y. Osamura, Y. Yamaguchi, P. Saxe, M.A. Vincent, J.F. Gaw, and H.F. Schaefer, Chem. Phys. 72, 131 (1982). 7. Y. Osamura, Y. Yamaguchi, P. Saxe, D.J. Fox, M.A. Vincent, and H.F. Schaefer, J. Mol. Struct. 103, 183 (1983).
8. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Chem. Phys. 103, 227 (1986).
SUGGESTED READING
143
9. J.E. Rice, R.D. Amos, N.C. Handy, T.J. Lee, and H.F. Schaefer, J. Chem. Phys. 85, 963 (1986). 10. T.J. Lee, N.C. Handy, J.E. Rice, A.C. Scheiner, and H.F. Schaefer, J. Chem. Phys. 85, 3930 (1986).
11. J.F. Gaw, Y. Yamaguchi, H.F. Schaefer, and N.C. Handy, J. Chem. Phys. 85, 5132 (1986).
Suggested Reading 1. T. Takada, M. Dupuis, and H.F. King, J. Comp. Chem. 4, 234 (1983).
Chapter 11
General Restricted Open-Shell Coupled Perturbed Hartree-Fock Equations In this chapter, the first- and second-order CPHF equations are described for general restricted open-shell SCF wavefunctions. These equations are obtained by differentiating the variational conditions on the open-shell SCF wavefunction. The method is applicable to almost all single-configuration SCF wavefunctions and real perturbations.
11.1
The First-Order CPHF Equations
Within the framework of RHF theory, the generalized Fock operator for a single-configuration SCF wavefunction [1-3] is ail
Fi
= fih + E(aiiJl + 131 K1)
with the Lagrangian matrix c defined in Section 5.2 by all Cij
= ( Oil F ikki )
= fihij
E faii(ijill)
(1 1.2)
In these equations, "all" implies a sum over all molecular orbitals, occupied and virtual, although the elements of the a and coupling constants involving virtual orbitals are always zero. Using this definition, the variational condition for the general restricted open-shell SCF equation is
Eii — cii = 0 144
,
(11.3)
THE FIRST-ORDER CPHF EQUATIONS
145
a nontrivial result, since the order of indices is crucial. The first-order CPHF equations are derived by differentiating eq. (11.3) with respect to a cartesian coordinate "a":
_ 0
(11.4)
—
Oa
The first derivative of the Lagrangian matrix (11.2) was obtained in Section 5.5 (see also Appendix P) as all
all
E uzid3 + E
19Eij
Oa
Uktj E i k
all
E
(11.5)
kl
The skeleton (core) first derivative Lagrangian matrices Ca, the generalized Lagrangian matrices (I , and the r matrices are defined by all
= fihZi
E
=
E {aik(iiikk)
(11.6)
all
fihii
Oik(ikijk)}
(11.7)
and
(ilijk)}
71.7y,/ = 2an,„(ijikl)
(11.8)
Substituting eq. (11.5) into eq. (11.4), gives Ocii Oa
Ofii Oa
all
all
E
23
+
E
Uc
all
+E
Uglr1
kl
all
E
all
uic:4 —
E uzii f k
all
_E
(11.9)
ufLeirlii,k1
ki
= eli
—
all
r
+E
a Eji ,
Luzi V/tj
f 3'is)
—u
k
+ E ur,,(7:j,k, _ T it ) all
ji,k1
Id
146
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL
=
CPHF EQUATIONS
E ii — all
E
(cikj
—
(ik)
((i. -
( ik)
31 Tji,kl
Tij,kl
I
kl (11.11) all
E
6u((ik:
cik)
6 ki((j
f i t)
-
— 61i
, i • Tj i,k/
Cik
,31
k>1 all
E (flak
jk ik fit) + Tij,lk - T3i,1k
k>1 all
E
Ubc 6ki((k
( jk)
(Ski (Ivi
Using the first derivative of the orthonormality condition from Section 3.7
1/Zi
Cji
=0
(11.13)
and the following equation for the generalized Lagrangian matrices, given in Section 5.5,
= this expression may be rewritten as:
acji
lia
Oa
▪ Etqj
—
all
+ E urc , (5„((ti _ k>1 all
- E
6k,
61i(1 all
—E
scta bki(c1; —
—
43
fi)
Tlik,1k1
k>1 1
all
—•
2
• Sit:/c {bki (Czkj
( jk)
45 kj (ciki
-r
Cik
r
ik
jk
kk
Tji,kk
(11A5) -
f ii
all
+E
3t
— ( ik)
6 12
il
Tij,k1
k>l
— bki (Îj — Cjl + bkj
— Cil
— Tij,lk +
7 ij lk]
,31
THE FIRST-ORDER CPHF EQUATIONS all
___
re
2..., o ld roki (
/..i
147
— 6kj ((L.. — fit)
—
rj k ] ,. ik + ' ij,lk _ jz,lk
k>1 all -
— 2
oa kk
21,
ik °
Rearranging equation (11.16), gives: all ik
E
ij,k1
-
_L
' ji,k1
k>I
— Eii) — 5ki (ch —
+ (Ski
— bIi (cki — Eik)
6/
a a = Eii — Eii
all
szt
- E k>l
1
-T1k +
ki clj
b kj
621
)1
a11
- -2 E
S kcik
k
k
Using the definition of the r matrices in eq. (11.8), equation (11.17) may be formulated more explicitly as al/
EU
[2 (aik — Q jk — (xi/
k>1 (Oik
-F
—
((J
/3jk
(ilijk)}
{
— Li) — 44(4 — fit) — bti (di — Eik)
(da:
Elk)]
a a Eii — Eii
-
all
E
szi [ 2(aik -
aik)(ijiki) + (Pik — Pik){(ikl,g)
(i/Ijk)}
k>! lci (dj
— zr \-all-7
ki ((L.
S ka k[2 (aik —
C il)]
aik)(ijikk) + (Pik — Pik){(ikijk)
In deriving eq. (11.18), the following equalities of the fmn
= T inn
31,k1
rpm
sj,lk
T Fpn
21,/k
r mn kl,ij =
(ikljk)}1
T matrix elements, mn
mn
mn
(11.19)
were used. For the general restricted open-shell SCF wavefunction elements of the Ua matrix between the same shell (which share a common set of coupling constants; see Section 5.4)
148
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL
CPHF EQUATIONS
are not determined uniquely. Such pairs of molecular orbitals are usually called "nonindependent". On the other hand, the MO pairs between different shells are uniquely determined and are termed "independent pairs". These independent pairs for the three most common single-configuration SCF wavefunctions are illustrated by the shaded areas in Figure 11.1. The two-electron coupling constants, a and 0, are the same among the orbitals in the same shell and they are zero if the virtual orbitals are involved. Thus, eq. (11.18) may be rewritten for the independent pairs in the simpler form indep.pair
A.ii,ki UTI =
Bg, i;
(11.20)
k ). 1
where "indep.pair" denotes the independent pairs, and the A and by j,k1
= 2(ctik —
ajk —
+
+
(Oik — Oik — Oil + 034){(ikljd)
+
6ki
— (it) —
Bg matrices are defined
kj
(i/Ijk)} — fik)
Cil)
— (ik)
(11.21) and €7i —
Iii
all occ
-
EE S [2(ct1k — ajk)(ijiki) + (Oik — Aik){(ikisi l) + (illjk)} k>1 Ej1)
Sk[(aik
ajk )(ij I k k) + (Oi k - Op) ( ik I j k)1
occ
- E
Skj ((ti
Ea)]
+ ki j -
(11.22)
In the above summations, "all" implies a sum over both occupied and virtual (unoccupied) molecular orbitals, while "occ" denotes a sum over only occupied orbitals. These simultaneous equations usually are large in dimension and are solved iteratively. When the dimension of the eq. (11.20) is not too large (less than 1000), the equations can be solved using any of the standard methods for simultaneous equations.
I
1
......-1111 ■-
doubly occupied
1
nowam.4111111.
doubly occupied
doubly occupied
k singly I
1
virtual
closed-shell
k
k
singly.{ virtual
virtual
high-spin open-shell
open-shell singlet
Figure 11.1 Independent pairs (indep.pair) of molecular orbitals in the coupled perturbed Hartree-Fock equations for GRSCF wavefunctions. Only the tIc' matrix elements shown as shaded areas may be solved for uniquely.
150
CHAPTER il.
11.2
The Averaged Fock Operator
CPHF EQUATIONS
GENERAL RESTRICTED OPEN-SHELL
In the general restricted open-shell SCF theory, a unitary transformation within unique shells, i.e., (doubly occupied-doubly occupied), (a spin-a spin), (fi spin-0 spin), and (virtualvirtual), does not change the total energy of the system. It is therefore advantageous to redefine these orbitals uniquely for use in a later process in which the entire Ua matrix is necessary. One of the most common procedures for this purpose is to employ the averaged Fock operator all
a
=
h
+
E
f,(2J _ K)
(11.23)
where fk is the occupation number of the kth molecular orbital. The elements of the averaged Fock matrix are given by all
(0iipatrio3)
E f,{2(iiikk)
hij
(ikiik)}
(11.24)
After self-consistency is achieved, the molecular orbital coefficients and their "orbital energies", e, are defined using the averaged Fock operator for each shell (e.g., closed-shell, a-spin shell, 0-spin shell, and virtual shell). For each shell of non-independent pairs the following diagonality is established: 15 3 ei
where the indices i and
( 1 1.25 )
j belong to the same shell.
The corresponding CPHF equations for the averaged Fock operator are derived in a similar manner to that described in Section 10.1. Differentiation of eq. (11.24) with respect to a cartesian coordinate "a" ij
all
ahij
fk 2
aa
Oa
ikk) Oa
0(ikijk)1
aa
f
(11.26)
is required. Using the results presented in Sections 3.8 and 3.9, eq. (11.26) is expanded as all
hz; E
Oa
+
all
E
all
fk[2(iiikk)a + 2
E {u7ani(niikk) +
uz,(imikk) +
214zn k(ijinetk)}
71L
all
— (iklik) a —
+ =
u(ikimk)
E fuz i (nikuk) + uz,(imiik)
+ utl k (ikiim)}1
all
+E
fk{2(iiikk).
(ikuk)a}
(11.27)
THE AVERAGED
151
FOCK OPERATOR
all
all
E u7ani [h,„ + E fk{2(railkk) — (mkiik)}] all
all
E
[hi,
+ E fk t2(imikk)
(ikimk)}1
all
+ E ckfk{4(iiimk)
(11.28)
(imlik) — (iklim)}
ink
Using the definition of the averaged Fock matrix in eq. (11.24), this equation becomes all
0€1ii
/a
+ E
E
ij
Oa
all
all
E
Ug1f1Aij,k1
(11.29)
kl
The skeleton (core) first derivative averaged Fock matrix appearing in this equation is defined by all
E
E
/a
ha ii =
fkl2(iiikk)° — (ikuk)a}
(1 1 .30 )
and the A matrix by A,kt
= 4(ijikl) — (ik1j1)
(ilijk)
(11.31)
.
Note that the A matrix in eq. (11.31) is a different quantity from the A matrix in eq. (11.21). Using the diagonal nature of the averaged Fock matrix within the same shell, eq. (11.25), and the first derivative of the orthonormality condition of the molecular orbitals, eq. (11.13), equation (11.29) may be modified as othershell
other shell
194
/a
f ij
Oa
+ u;ie, + uzet + E
Ugifl kj
E
U/aGjEl ik
all
E
(11 .3 2)
UZI fiAij,k1
kl
=
othershell
st
—
a.) )
othershell u jacie ik
(JAE/0
+ E
all
(11.33)
fiAij,k1 kl othershell E
/a
(Eli
— E i)
— SZEI j
E
other shell (IL( ' kj
E
uîci fiik
all
uajciftAij,k1 kl
(11.34)
152
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL
CPHF EQUATIONS
Recalling again the relationship in eq. (11.13) the last term in eq. (11.34) may be manipulated in the following way: all
E
all
=
UlacifiAij,k1
kl
E UZI fiAii,ki
all
+
k>1
all
UZI ftAij,k1
—
k>1 _
E (utc,
-2 E
(11.35)
all
— 2
SZI) fkAij,lk
szkfkAii,kk all
(fi ▪E k>1 -
+
k>1 all
1
1
E urck fkAij,kk
_ fkA--ij,lk + //I_
k>1
all
. E
all
E
-E
fk)uzki,ki
f k SlAij,k1
k>1
all
E
(11.36)
It should be noted that the exchange of the k and 1 indices does not alter the value of the A matrix. The occupation number fk is identical for all orbitals in the same shell and it is zero for the virtual orbitals. Thus, equation (11.36) is further reduced to indep.pair
all
E
uz,fiAi i,k/
E
=
kl
AnAti ,ki
(fi k>1
k>1 occ
- E
- ( iklik)}
(11.37)
Combining eqs. (11.34) and (11.37) one obtains
_ (ei -
actij
aa
othershell
other shell
-
+E
indep.pair
if ki
E u4ei,
occ
▪ E k>1
E fkSZIAij,k1 k>1
(fi - fk)uzAiJ,ki
fi,442(ijikk) — (ikjjk)}
(11.38)
The final expression to evaluate elements of the II matrix for non-independent pairs is =
1
othershell [fa
—
SILjeri
E
othershell
//Lek
indep.pair
+E k>1
,
+E occ
fi — fk)UglAij,ki
—
E k>1
fknAij,kl
occ
- E
fk 5-4{2(iiikk) - (ikiik)}1
(11.39)
THE DERIVATIVE OF THE a MATRICES
153
When the entire II matrix is required in the later process, the molecular orbitals must be subjected to a unitary transformation within unique shells using the averaged Fock operator in eq. (11.23) prior to solving the CPHF equations. Employing these redefined molecular orbitals, the elements of the (fa matrices for the independent pairs are then obtained by solution of the CPHF equations in eq. (11.20). Finally the elements for the non-independent pairs are determined in terms of the independent pair Ua elements from equation (11.39). An alternate approach to this subject has been discussed in References 4 and 5.
11.3
The Derivative of the
Ea
Matrices
Before proceeding to the second-order CPHF equations it is useful to find derivative expressions for quantities involved in the second derivative of the Lagrangian matrix. Examining eq. (11.5) it is clear that those quantities include the fa, (I, and r matrices. This section and the next two sections are devoted to finding the derivative expressions of these quantities. The derivatives of the ea matrices are discussed here. Using the results from Sections 3.8 and 3.9, the derivative of eq. (11.6) with respect to a second variable "h" is written as:
0E4.
av
all
Obi
E
fi
oil {
a(iailbj
}
(11.40)
fi ab
all
= fi [IzZlb + all
+E
Ub -ha
+ U,bni gn. )]
m'
[ cei/((iimab +
all
+ all
+ oni onitioa +
E
+
Oil
+
ut i( il l int)a} ) ]
((illil) ab
+
all
= filei13 + E failoilio" + o i,(iiiii)b} I all
all
+ E qi [fihr;, + E in
1
all
all
+E .
on [Ann
+E
3
{aii(miiiir + t ii(milil)a }1 {ai/(imiur
+ oi,(iiimo-}]
1
all
+ E u,t 1 [2ai/(iiim1)a + i3i/{(timiii) a + (illim) a }1 ml all
=
ab
Eii +
E Onici,;:i
all
+ E Usbniccitn
(11.42)
154
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL
CPHF EQUATIONS
all
+E
+
(11.43)
+ ut,E7k + E uZ
(11.44)
Ora
mi all
all
E
//Lc:,
kl
The skeleton (core) second derivative Lagrangian matrices, Ea ', the first derivative generalized Lagrangian matrices, Cla and the first derivative T matrices, Ta, are ( 7:
all
E
b
,
fiqj
all
(I;
E
=
{aucciiikk).
+
rilk(ikijk) a }
7
and 2amn (tijiki) a
mna
rij,k1
Omn t(ikiji) a
(illjk) a }
The derivative expression for the ea matrices in eq. (11.44) has a similar form to that of the E matrix in eq. (11.5). Derivative expressions for the skeleton (core) higher derivative Lagraugian matrices are not worked out explicitly here, as they are found in a comparable manner to the derivatives of the Ea matrices just discussed. These expressions are summarized in Appendix P.
11.4
The First Derivative of the Generalized Lagrangian Matrix
In this section, the first derivative of the generalized Lagrangian matrix, ( I , is obtained. Using the results from Sections 3.8 and 3.9, the derivative of eq. (11.7) with respect to the variable "a" is:
act, Oa
=
ah, ft—L + Oa
all
E
0(iik) jk
{
alk aa
E
+
(u i hm,
all
+E
aokiik)1
k
all
= fi[h.lii +
+ Olk
Oa
f
(11.48)
rp,ini him)]
all
iikkr [crik((i
+E
{ui ( miikk) + W(iMikk)
m
k
2 U,ank(iiirnk)D
all
Olk((ikijk) a
+ (I( iki7rt k ) +
E {G(nkiik) + tr(imuk) (11.49)
THE FIRST DERIVATIVE
OF THE T MATRICES
155
all
E {crik(iiikkr + Oik(ikijk)a} all
+ E falk(rnilkk)
+ filic(Inklik)}]
all
+ E lc:m ein-4k) U1,[2alk(iiintk)
131k(ikimk)}1
014(inliik)
(ikiiin)}]
•
( 1 1 .50 )
Using the definitions of the ( and T matrices in eqs. (11.7) and (11.8) yields all
0(ti Oa
all
E
ulannrIT,mn
mn
In eq. (11.51) the skeleton (core) derivative generalized Lagrangian matrices, defined by eq. (11.46).
11.5
The First Derivative of the
T
were
Matrices
In this section the derivative of the r matrix appearing in eq. (11.5) is determined. Using results from Section 3.9, the derivative of the r matrix, (11.8), is:
Ori7,2t Oa
= 9
aa
,
Pmn
00/Ijk)} a(ikiji) Oa + act
(1 1.52)
2a, [(ijikl)a all
+E
{u;i ( pilko + tvipiki) + ITA (ii I P1 ) + U;i(iiikP)}1 all
▪ Ann[Okliir
E
fu;Apklio + U;k(iPlil) + U;j(iklpn + U;1(ikliP)}1
ail
▪
Omn
E {uwpilik)
+ U;i(iPlik) + U;j(i/Ipk) + U4(i/liP)}1 (11.53)
Omn{(ikiji) a all
+E
u42«,,n(pilko +
Ann {
(Pilik)}}
156
CPHF EQUATIONS
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL all
+E
U;3[2 amn(iPikl) + Xrtn{(ikiP 1 ) + (ilipk)}1
m all
+E in
U;4ni 2a n (ii IA + /37714 (iPlil) +
(illi P)}1
ail
+ N.e__., U42 Cirrin(iiikP) + Omnf(ikli P) + (ipijk)}1
.
(1L54)
m
The definition of the
T
matrix, eq. (11.8), may be used to rewrite this equation in the
simplified form 8-1-27,,,
ran'
0a all
,
+ u;k7-47,7,•, +
E (uArmi +
(11.55)
in which the r a matrices were defined by eq. (11.47).
11.6
The Second Derivative of the Lagrangian Matrix
The second derivative of the Lagrangian matrix is required for the derivation of the secondorder CPHF equations and is obtained by differentiating eq. (11.5) with respect to a second variable "b":
a2fi Oaab
[Ofiji Ob Oa 0 = ?i
(11.56)
a aciii
[ ouckL.
Ukz Ob
E raUkij 1:7979-fik
a Ocal UkJ -- 0 Ti
ouL it 19 Tt
all
E ki
(11.57)
kl
all
ab
all
all
all
E uzidi + E v4Ei, + E
rtj ,ki
rk`c
01j1; k1
(11.58)
Results from Section 3.3 and the preceding sections allow eq. (11.58) to be rewritten as 02
a aa b
all
all
E uzi' u —E m kl
+
\--"` r T a ,..ilb u kl 1 ij,kl kl
all
+
umbnr;licm.
mn
ui‘mu,,;,,
il
rtj,kl
all
1 b Til + Umi b .7"fl 1- N--"' (Umi. m3,ki trn,kl m
Um b k Talii, m.1
+ Umb 4,km)
]
(11.59)
•
The expressions for the first derivative of the generalized Lagrangian matrix (11.51) and of the 7 matrix (11.55) finally permit the reformulation of (11.59) as
02ci
all
all ab
ij
Ow%
+E
uedi
+E
Uj:jbEik
all
+E ki
IcCib t ,kt
al
+ E (ugi uibi + iii Uri) di kl
all
+E
ukimuitnriAl
kim
+
all
uti u,7,„
kmn all
+ E (u;:j umb n +
in
kmn
all
E
a jib
b kir21
klt
kl all
+E
(uLd.b,
+
UkajEibk
b ia Uki(ki
Uj: jE7k)
(11.60)
Note that the first four terms of the right hand side of eq. (11.60) have forms very similar to those of the first derivative of the c matrix in eq. (11.5).
158
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL
11.7
The Second-Order CPHF Equations
The second-order CPHF equations are a cartesian coordinate "b":
CPHF EQUATIONS
derived by differentiating eq. (11.4) with respect to
a
=0
Nab aaab Substituting the Lagrangian second derivative eq. (11.60) into eq. (11.61) we have (9 2 q
(9 2
aaab
aaab all
ab
E
all
(uj,ibdi
U/c:.1;fik)
E kl
all
+E
upi +
kl all
+E
uLu ttn r
f
(uf:i umb i,
+
klm all
+E
all
u,bci u;:,,,)7,i,7,,,„
+E
(//j umb n
+
upci uz.n.
p•
TaLmn
kmn
krrtn all
+E
uLT1j,bki
kl
+
Uta riii,akl
all
+ E( ulcikbi + utA, + (I(; + up,i (c1„) all
all
ab
E
+
uec.ik
•
n.„6 kl
kl all
—E
(u,,:i u,b, + up,, q) Cjki
kl all
_E
un Uibm 71::lici
klm all
— E (uu,b„„ +
all
utc,u,L)7-,i:,„
kmn
—E
u:i umb r,
+ uziuma n) rikn„
kmn
all
— E ([4,7-;c , +
Ukb erl rkl)
kl
all —
.6
E uzi ( cti + ugiEbik
+
UZielk)
(11.62)
THE SECOND-ORDER CPHF EQUATIONS
159
Rearrangement of terms gives 0 2 ei
02
&tab
OccOb ai'
—
2 t3
all
+ E [IV (ciki _
— Uite.bi (Ciki _
fik)
Eik )
I
k
all
+ E
i-i ) Uficib (74,k1 — T 3t,kl
kl
all
+ E (uAupi + oci u)(cik , —
j ) kl
kl
all
+ E
unup,,, ( Ts7,1k1
ii. r,k I)
—
r
kiln all
+ E uzionn +
in ) jk,n 71,7,mn — Trn
ULU
kmn
all
+ E (uzymb n + upc,u7a„n)(T iz„
—
kmn
)
all
+ E
Tki,mn in
all
utd(4.:,bki —
„r il e' )
+E
7.1Lbki
kl
UZl rt, akt — ' ji,ki
kl
all
+ E uic:, ((lb.; — c!ik
+ Uli Cik;
— f 9:k
k :b — (4i C — f'lk — UZi 1 +
—
Eji)
—
—
THE SECOND-ORDER CPHF EQUATIONS o cc
E
161
kd
Wk{(ctik
all
+ E (ukti o; + uturj)(cik
,
— :Uab [fi hi,
+E
E all
pii ( iiiioabl
+
all
fitz(ilim1)}1
all
U,anbi [2aii(ijina) + Oil
ml
+
164
CHAPTER 11. GENERAL RESTRICTED OPEN-SHELL
CPHF EQUATIONS
all ran all
mn all
mnl all
' mz
/71 [2aii(mj1nl)
mnl all
Umb iUtaa [2ctii(mijnI) mni al/
UZ,j Unbi mnl all
Umb j Ul [2 aii(imin 1 ) mn1 all
GIn1[2ail(iiimOb
(illim) b}]
ml all
U1,42«it(ijimi) a
Oilf(imifi) a
(iiiim) 3 11
+
Oi/(milil) b}1
ml all
all
E
Uli [fignj all
all
+E
ij
+
all
[fihli all
[fitei'm
(11.73)
The definitions of c, ( a , cab , c , (a, rand ra from eqs. (11.2), (11.6), (11.7), (11.8), (11.45), (11.46) and (11.47) are finally used to rewrite (11.73) as:
02Eii 0a0b
LU
ab
E13•
E
ugpok;
all
E kl
u1274,k1
trizbi ik
REFERENCES
165 all
+ E (uLut; +
Niq)( ki i
kl all
+ E ukin,
ibm ;i7,11
klm all
+ E (uzi u + up,i u,L) 717,„,„ kmn all
+ E
+
krrol
uz.; uz n) TIT,mn
all
+ E kl
+
all
+ E (uzickb, + TA + ut,c,t; + ()
NiE tc k)
(11.74)
Equation (11.74) is seen to be equivalent to eq. (11.60).
References 1. C.C.J. Roothaan, Rev. Mod. Phys. 32, 179 (1960). 2. F.W. Bobrowicz and W.A. Goddard, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.79 (1977). 3. R. Carbo and J.M. Riera, A General SCF Theory, Springer-Verlag, Berlin, 1 978. 4. LE. Rice, R.D. Amos, N.C. Handy, T.J. Lee, and H.F. Schaefer, J. Chem. Phys. 85, 963 (1986)5. T.J. Lee, N.C. Handy, J.E. Rice, A.C. Scheiner, and H.F. Schaefer, J. Chem. Phys. 85, 3930 (1986).
Suggested Reading 1. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 77, 383 (1982). 2. Y. Osamura, Y. Yamaguchi, P. Saxe, M.A. Vincent, J.F. Gaw, and H.F. Schaefer, Chem. Phys. 72, 131 (1982). 3. Y. Osamura, Y. Yamaguchi, P. Saxe, D.J. Fox, M.A. Vincent, and H.F. Schaefer, J. Mol. Struct. 103, 183 (1983). 4. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Chem. Phys. 103, 227 (1986).
Chapter 12
Coupled Perturbed Configuration Interaction Equations The analytic evaluation of the second derivatives of the configuration interaction (CI) energy requires the solution of a very challenging set of equations. These coupled perturbed configuration interaction (CPCI) equations are those simultaneous equations which provide the derivatives of the CI coefficients with respect to perturbations. These CPCI equations are obtained by differentiating the variational condition on the CI space under the constraint of normalization on the CI coefficients. In this Chapter the first- and second-order CPC1 equations are described.
12.1
The First-Order Coupled Perturbed Configuration Interaction Equations
The variational condition for the determination of the CI wavefunction is [1]
E
cJ(H/J
-
051JEelec
-=
(12.1)
where the CI Hamiltonian matrix element was defined in Chapter 6 as MO
MO
Hjj
=
E
Q
hi;
E
ij
.
(12.2)
ijkl
The CI coefficients are normalized according to the usual convention as ci
E cy
(12.3) 166
167
THE FIRST-ORDER CPCI EQUATIONS
The one- and two-electron coupling constants, Q" and G", appearing in eq. (12.2) are related to their "parent" quantities, the one- and two-electron reduced density matrices f2), through CI
=E
CICA 21.31
(12.4)
IJ CI
E
(12.5)
IJ
The coupled perturbed configuration interaction (CPCI) equations may be obtained by directly di fferentiating the variational condition (12.1) under the constraint given by (12.3). Employing Lagrange's method of undetermined multipliers, the two equations are combined to give
c/
o
— .s/JEelec) + g (— 1 + E
E
(12.6)
The factor 19 may be chosen arbitrarily. It is convenient to set this factor to unity and thus
c/
c/
E
c.,{H„ -
—
Sij (Ee 1„ + 1
E GI)}
(12.7)
The differentiation of eq. (12.7) with respect to a variable "a" leads to the CPCI equations,
ct
E J
0(7J II/J — 6" [ aa
+
{
CI
OE
aeaiec
"
a ll aa
>C'
i —
Eelec +
2
E
c .
)1
acK cK-79-7 )}]
=0
(12.8)
This equation is manipulated further to yield
ci ac
CI
E
E
— 6 IJEelec)
Oa ECI
(
Cj
(5IJ (1
aEeiec)
(51.7 Oa
Oa
GI)]
Cl
+
CI
E cps, (2 E
acK) =0 CK aa (12.9)
The normalization condition and the interchangeability of the summations over J and K may be used to rewrite (12.9) as
ci ac, TT _
E
Oa
(
_
2CICJ)
CI
(aHiJ
aa
,/,/
a Eeiec) Oa
6
=0
(12.10)
168
CHAPTER 12. CPCI EQUATIONS
Then equation (12.10) is rewritten as CI
E
CI
49Ci
(Hid — buEetec
=—
Oa
(mu
2_,
(
Oa
bij
OEetec
Oa
)Cj
. (12.11)
The simultaneous equations given in eq. (12.11) are the first-order coupled perturbed configuration interaction (CPCI) equations, which involve the unknown variables OCJ/Oa. The dimension of these CPCI equations is the same as that for the corresponding CI wavefunction. It should be noted that eq. (12.11) does not suffer from a singularity. The first derivative of the normalization condition (12.3)
aci aa
ci
E
(12.12)
0
is explicitly included within it. The first derivative of the CI Hamiltonian matrix appearing in eq. (12.11) was derived in Section 6.4 and is mo
r,
dIIjj
Oa =
ij
J ah .; ,
ti Oa
mo E
---sj kl ijkl
a(ijikl)
(12.13)
aa
MO
=
+2
E
(12.14)
The 1/7,1 and X" matrices are defined by MO
=E
MO
Qtyhz, -F
ij
and
XII
E
(12.15)
ijkl
MO
MO
= E Q.14 hi, + 2
E
.
( 12.16)
mkt
The "bare" Lagrangian matrices, X", are related to their "parents" through CI X23 =
E IJ
MO
MO
E
(12.17)
c1c.,xilij
Qinz hi, + 2
E
G.i mk,(imikr)
(12.18)
mki
The first derivative of the CI electronic energy, 0Eetech9a, and the El' matrices must be obtained prior to solving the first-order CPCI equations. The evaluation of the CI energy first derivative was described in Chapter 6, and of the 1/ 3 matrices in Chapters 10 and 11 with more to come in Chapters 13 and 14.
THE SECOND-ORDER CPCI EQUATIONS
12.2
169
The Second-Order Coupled Perturbed Configuration Interaction Equations
The second-order CPCI equations may be obtained by a further differentiation of eq. (12.11) with respect to a second variable "b": Hjj — IJEeiec + 2C1CJ
ci (OHL., +
j
• aEelec
( 62 Hij
CI
E
61.1
O w%
cr (oHij
-E
) aaab ac,) ac, Oa b
+
Ob
Ob
)a2c,
0 2 Eelec) 0a0b
Cj
• j (9Ee l ec )ac, Ob Oa
Oa
(12.19)
Rearranging both sides of this equation leads to )o2c.,
II ij — 8IJEelec + 2C1Cj
) &Lab
CI ( \--, 82 111j ..' 0a0b J
=
E cr (0 il: a , J \
aci -
--
ab
)Ci
vij Oa0b
51JOEelec Oa )
— —2
(92 Eetec)
ac
2C1
EJ c' aa
CI E (oH/J
acJ b CI CI
acJ acJ
J
ab aa
E
Ob
J
61.1
aEei„)acJ Ob
O aa
(12 .20)
Eq. (12.12), the first derivative of the normalization constraint, is used to rewrite (12.20) as
ci
EJ (Hi, =—
Cl ,
-
6/JEeiec
(02 HI.)
E
E Cj I J
(
Oa
(a.HIJ
a2EZ; :
451.7
Om% J cr /aHij
—
+ 2cic,(92Cj a
OEelec(9clab
Oa
bij
aEezec
ab
ab
+C1
+ ci
OCJ)8CJ
Oa
ab
ac,) ac, Ob
Oa
(12.21)
The simultaneous equations given in eq. (12.21) are the second-order CPCI equations, which The second derivative of the normalization involve the unknown variables,
a2cJiaaab.
170
CHAPTER 12. CPCI EQUATIONS
condition
CI
I o2c1
ac, aa ab
aaab
E
0
(12.22)
is explicitly included in the CPCI equation. The second derivative of the CI Hamiltonian matrix appearing in eq. (12.21) was derived in Section 6.6 and is
mo
02Hij Oab
MO
a2hi.
=
0! 4;1
ii
&Lab3
+
ij 02 (i,j1k1)
Gsiki
ijkl
(12.23)
aaab
MO
Hg +
2E
uzbxit
ij
MO
▪ 2
E
(utxya
+
uf.i xt,t)
ij
MO
•
2
E
(12.24)
ijkl The 11 1 6 j, X L" , and Y u matrices are MO
=E ij
MO
W31 ,IhrP? 23
+E
MO
=E
(12.25)
qjki(ijikl) ab ijkl
MO
+2
E
(12.26)
mkt
and vIJ iijkl
MO
=
+2
MO
E
G(iklinn) + 4
mn
E
G
(12.27)
in (imIkn) .
mn
The "bare" quantities, X I." and Y", are related to their "parents" through CI
X:13
=E
(12.28)
IJ MO
E
MO
+ 2
E
(12.29)
G jrnkl(inliklr
mkt
7ri
and CI
Yijkl =
E
(12.30)
CICJYgi
IJ
mo
MO
Q jlhik + 2
E mn
Gji„,„(iklmn) 4- 4
E Gjm,n(intikn)
.
(12.31)
mn
The first and second derivatives of the CI electronic energy, 02 Eei„I 0a0b, and the Ua and (Jab matrices must be evaluated before solving the second-order CPCI equations.
REFERENCES
171 References
1. I. Shavitt, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3,
p.189 (1977). 2. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
Suggested Reading 1. D.J. Fox, Y. Osamura, M.R. Hoffmann, J.F. Gaw, G. Fitzgerald, Y. Yamaguchi, and H.F. Schaefer, Chem. Phys. Lett. 102, 17 (1983). 2. T.J. Lee, N.C. Handy, J.E. Rice, A.C. Scheiner, and H.F. Schaefer, J. Chem. Phys. 85, 3930 (1986). 3. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Theor. Chim. Acta, 72, 71 (1987).
Chapter 13
Coupled Perturbed Paired Excitation Mu lticonfiguration Hartree-Fock Equations The coupled perturbed paired excitation multiconfiguration Hartree-Fock (CPPEMCHF) equations provide the derivatives of both the MO and CI coefficients with respect to perturbations. These CPPEMCHF equations are obtained by differentiating the variational conditions in both the MO and CI spaces subject to constraints. The equations consist of three parts: MO-MO, CI-CI, and CI-MO (or equivalently MO-CI) terms. Since the TCSCF wavefunction described in Chapter 7 is a special (the simplest) case of the PEMCSCF wavefunction presented in Chapter 8, the coupled perturbed equations will be derived only for the more general PEMCSCF wavefunction in this chapter.
13.1
The First-Order CPPEMCHF Equations
13.1.1
The Molecular Orbital (MO) Part
For a PEMCSCF wavefunction described in Chapter 8 the generalized Fock operator is the same as for the GRSCF wavefunction [1,2]:
mo Fi = fih
E
(ai/J/
0i1K1)
(13.1)
The variational condition on the MO space is formally the same as for the GRSCF wavefunction, namely, -
=0
172
(13.2)
THE FIRST-ORDER CPPEMCHF EQUATIONS
173
where the Lagrangian matrix is defined by
mo
+E
fi hi,
=
faii(0//)
.
(13.3)
The Lagrangian matrix is defined alternatively by an expression including the Cl coefficients and the "bare" Lagrangian matrices: CI
E
=
E3
(13.4)
CICJE!jj IJ
The "bare" Lagrangian matrices are IJ
MO
=
(i.i111) ± /3 11 (illj1)}
(13.5)
The MO part of the first-order CPPEMCHF equations is obtained by differentiating eq. (13.2) with respect to the cartesian coordinate "a":
aa
aa
=
0
(13.6)
The required first derivative of the Lagrangian matrix is found by differentiating eq. (13.4) with respect to a cartesian coordinate "a":
ci a E cicf,efl Oa
aco Oa
(13.7)
IJ
CI
.
n,--,
x-■ (Yu , '
ij
2-, IJ Oa CJE i3 CI
= 2
CI
CI
nz
-
>2I ci EJ
>2 IJ
+ r
CPC, ..,/
ij
-aa E.,
C1
ci
aej 1j
ECJCj
Oa
IJ
Ci
+E IJ
C1CJ
aci!
"
aa
OE $3 Y Oa
The first derivative of the "bare" Lagrangian matrices, ell, appearing in eq. (13.9), is derived in a similar manner to that of the Lagrangian matrix of the GRSCF wavefunction, since the coupling constants f L, ot" and 13" are independent of the CI coefficients and nuclear coordinates. Differentiating eq. (13.5) with respect to the variable "a" gives OEIJ 11
fiij— aa +
Oa
mo
E
= MO
d-
E
U1:1 r 721ii,1 kl
MO
ait
a(ij1/1) Oa
+
Oil
mo
+E
tqf Ell
ôa
J
(13.10)
CHAPTER 13. CPPEMCHF EQUATIONS
174
The results of Section 5.5 were employed in this derivation. In eq. (13.11) elements of the skeleton (core) first derivative "bare" Lagrangian matrices (E I 'fa ), the "bare" generalized Lagrangian matrices (("), and the "bare" 7 matrices (7") are defined by I J' E ij
=
fa til oi m a
+
J i
+
)3Y ( in ioa l
+
ofkj(iklik)}
(13.13)
(illjk)}
(13.14)
7
(13.12)
mo
+E and
,_ mn i
=
lij,k1
+ 0;(4{(ikiii)
mn
These matrix elements are related to their "parent" quantities through CI
(13.15)
C c '•
13
'J MO
r
E
=
(13.16)
+
oi
c`,3
E
=
(13.17)
CIC.1071'j
IJ MO
E
fihij
{alk(iiikk)
+ R t k ( ikli k ) }
(13.18)
and C mn
-E
IJ
C IC j71.7, rki. I
(13.19)
IJ
•
(illjk)}
2a,„(1jikl)
(13.20)
Combining eqs. (13.9) and (13.11), the first derivative of the Lagrangian matrix becomes CI
2
Oa
CI
E ci E MO
▪
€?,
E
oC j j
— Oa Ezi
(cu.; +
Ul€ ik)
( 13.21 )
In deriving eq. (13.21), equations (13.4) and (13.15) to (13.20) were used. Note that the definition of the Lagrangian matrix in eq. (13.4) was employed as a starting equation to obtain eq. (13.21). Since the definition in eq. (13.3) contains the reduced density matrices
THE FIRST-ORDER CPPEMCHF EQUATIONS
175
f, a, and 3 which are functions of the CI coefficients, the derivation starting from eq. (13.3) becomes much more complicated. The MO part of the first-order CPPEMCHF equations is
afii
a(ii —
aa
a -aa == f ii
a
MO
+ E u [bii (d,
- Eik)
o, Wei
- Eik) + 7-4,1j,kl —
1:74.1
}
kl CI
+
2
CI
(1J E c, E OCI Oa J "
I
(13.22)
, 0
.71
It is seen clearly in eq. (13.22) that the derivative of the variational condition in the MO space necessarily has a term due to the perturbation of the CI coefficients. For the PEMCSCF wavefunction elements of the Ua matrix within the same shell (which share a common set of coupling constants; see Section 8.1) are not determined uniquely. Such pairs of molecular orbitals are called "non-independent". On the other hand, MO pairs between different shells are determined uniquely and are termed "independent pairs". These independent pairs for a high spin open-shell PEMCSCF wavefunction are illustrated by the shaded areas in Figure 13.1. In the figure the word "active" denotes paired excitation molecular orbitals, as described in Section 8.1 Equation (13.22) may be manipulated in a similar manner as was done in Section 11.1 and then written as indep.pair A 1j,k1
E
—
(TA =
k>l all occ
- EE 4,[2(0iik -
aik)(ijikl)
(Oik — flik){(ikijI)
(illik)}
k >1
bk i (Clj
Ejl)
cii)]
OCC
-E
sz,[(ct,k
CI
+2
(Oik
CI
WI ( E E — aa
rijk ) (ik Ijk)]
Li
(13.23)
where "indep.pair" denotes the independent pairs, "all" implies a sum over both occupied and virtual (unoccupied) molecular orbitals, and "occ" means a sum over only occupied orbitals. The A matrix in the above equation is defined by Aij,kl =
2 (ail, — Crjk —
(I3ik — 13:7k —
a1
+ Oji){(ikiji)
(iiijk)}
176
CHAPTER 13. CPPEMCHF EQUATIONS
(I k>l occ _ E As;:k {2(iiikk) - (iklik)} —
-
k occ CI
-1- 2>2>2
fkShAij,k1
c1 aC' LI/ {2(ijikk) — (ikljk)}]
k IJ
Oa
(13.58)
When the whole ua matrix is required in the later process, the molecular orbitals must be transformed unitarily within unique shells using the averaged Fock operator in eq. (13.46) prior to solving the CPHF equations. Using these redefined molecular orbitals, the elements of the Ua matrices for the independent pairs are then obtained by solution of the CPHF equations in eq. (13.40). Finally the elements for the non-independent pairs are determined in terms of the U a elements for the independent pairs from equation (13.58). An alternative approach to this subject has been discussed in References 3 and 4.
13.3
The Second-Order CPPEMCHF Equations
13.3.1
The Molecular Orbital (MO) Part
The MO part of the second-order CPPEMCHF equations is obtained by differentiating the variational condition in eq. (13.2) with respect to the cartesian coordinates "a" and "b":
a2 fii
(92 cii
&Lab
8aab =
(13.59)
The second derivative of the Lagrangian matrix requires differentiating eq. (13.9) with respect to a second variable "b":
a2(ii &cab
=
a
(afii )
aa
(13.60)
THE SECOND-ORDER CPPEMCHF EQUATIONS CI
8 — ab ( 2 E c
Oa
Ob
E aa +
CI
aci acf,' - C,E Oa3 1J ab
+E
CICj
IJ
+
E IJ
CI
CI
E E
(9 2c
Oa0b
c"
'3 acca b
(13.62)
ci ac 1 ac,
_ E— — E.Y + Oa Ob 13 IJ
CI
E /
C1
affi cr ac, ab Oa
02E1
=9
2
2
(13.61)
Oa
IJ
aelf E c, E ac, Oa Ob
2
CI
+
Ea IJ
+ E
CI
CI
+
"
oc
CI
▪
CI
ÔC
,j j
CI
E
2
CI
183
cr 432 cj 1 j
CI
2
E c, E I
0a0b Esi
J
c/ (ac, j off" off; E Oa ab + oc, A Oa J
CI
0 2, 47
13 + E c,C .1 Oa0b
(13.63)
'J
The second derivative of the "bare" Lagrangian matrix is derived in a similar way to that of the GRSCF wavefunction, since the coupling constants f", a" and 0/./ are independent of the CI coefficients and nuclear coordinates. After a similar derivation to that in Section 11.6, the second derivative of the f" matrix is
0 2 c .fi aaab
a (8€9 Ob
(13.64)
Oa MO
MO
= E ft" + E ul:ibc.:; + E k
MO
u;:cte
k
MO
+
+ E (uAutb,
.I J
upci u,a3)(2,,
kl
kJ
MO
•
+E
ufici qn, + uti /47., nrfa' MI n +
kmn
MO
E
(u,zi umb n + UZiii::,,,
kmn
MO
b
+E
cif:174 ; 1i
kl
+E
IJ
+ E u,,:m u,bm ilrki klin
MO
MO
il" + E4_,vat)kl rij,k1
/
b
+ utdriii,ikj:
(uixcicik1; +
rya ,./Jb
t., kicik
TTbki..,ki til -7- %., j_
'la
_L -I-
TT 6 .-1 .1° ) 1... kicik
(13.65)
The elements of the skeleton (core) second derivative "bare" Lagrangian matrices the first derivative "bare" generalized Lagrangian matrices (C 1j 0 ), and the first derivative
184
CHAPTER 13. CPPEMCHF EQUATIONS
"bare" 7 matrices (7 /Ja ) in the above equation are defined by
mo E
phz".
Eff ab
f
(iiiioab
0.11J (au oab}
(13.66)
13 iikj (iki k) a }
(13.67)
MO
ijJ
E
=
and I Ja
oa
=
+
{au(iiikkr
orn/Jn f(jklioa
( illi k)a}
(13.68)
These matrix elements are related to their "parent" quantities through CI
E cic,Effab
ab
(13.69)
'J
=-_
MO
+
E
fih#
oabl
(1 3.7 0 )
1 CI
dr
E
l
all occ
—
—EE
cejk)(ijikl)
—
(Oik
Ojk){(ild:71)
k>1
+
(ski
(j CI
+4
MO
E c, E
CI
.ab IJ
+
MO
E E actii
2
+ CI
i>j CI
—4
MO
ECj E (*.ti ij CI
—4
MO
ECjE
492 Edec
Oa0b
r + u?j E.iib)
•I J MIZL j(lk
ijk CI MO
—4
E c„E
UU j rj
ijkl
6/./ 151.7
OEeie, Oa 0Eetec
Ob
OCJ ) 0Cj Oa Ob +
acJ).9c, Ob
(13.87)
Oa
It should be noted that the second term in eq. (13.87) contains the CI-MO interactions.
13.3.3
A Complete Expression for the Second-Order CPPEMCHF Equations
In the preceding two subsections, the second derivatives of the variational conditions for both the MO and CI spaces were obtained. The second-order CPHF equations for the PEMCSCF wavefunctions are A
it A 21t [
A21 A22 [
uab 02C/0a4913
Bel
=
Bec
I (13.88)
▪ CHAPTER 13. CPPEMCHF EQUATIONS
190
The A" in this equation is the MO term. Each element of this matrix was given in eq. (13.24) as A 11 1-1-ij,k1
2 (ocik — ajk — ait + (tit) (ij I kl)
-I-
(Oik — Op, — Oil + Oil){(ikij1) + (ilijk)}
-I-
( 6icicii — fil — 6ki (4 —
fit
di — fik
— bu
+ 61j(qc . —
tik)
(13.89) The
A 22
term represents the CI part and its elements from eq. (13.87) are 1 Al2.7 = — (H/J — 5/./Eetec 2
2C/CJ)
•
( 13 .90 )
Finally, the A 21 (or A 21t ) terms contain the CI-MO (or MO-CI) interactions. Each element is given by CI
A
21
4-2•17,1j
2
E
(fu —
(13.91)
The perturbation term for the MO space cornes from the MO part of the rhs of eq. (13.77): Bab.l. 0,23
— ce.ik)(ijik/) -I- (Oa — Oik){(ikijI)
45 kj
Eli)]
occ
E ezt kaik —
(13ik — 1 3ik)(ikijk)]
aik)(iiikk)
MO
+ E (u,aci u,b, + u/bciu,a;) (di — kl MO
+ E uLryib„,, T1.71, — 4771' klm MO
n rtz,,„ _ rj.k,mn
+ E (uumb n + kmn MO
+ E (u:i u,b + uzi u,ann kmn
in )
—
(i/Ijk)}
REFERENCES
191 MO
a
ukl
MO
)
+E
rij,k1
kl MO
Ukii
((kJ
Ili (CZ —
— E bi k) +
[U ib
zb — ,ik
— Ujici((ki
2
UZ1(7.1j,ak1
— Tjit,akl
kl
elk)]
- upci
cl ac ac E ( f sjJ Oa ab
E ir k)
IJ I
IJ
CI
+
2
CI
E c, E
{a2
ac1,1) 3, ab
ab
Oa
aa ) f (13.92)
The perturbations in the CI space arise from the CI terms on the rhs of eq. (13.87):
Bgbr2
—
1
C ./
CI
MO
>CJ>
CJH+ 2
rab IJ Sij Eu
i>j CI
+ ECj CI
-
2
MO
E
ab
IJ
1 a2 Eei„ Ci 2 aaab
+
MO
>2Cj> ( upjf5f
a
+
ij
CI
MO
• 13 UtEgiNj(lk
ijk Cl
-
2
MO
ECj>2 CI ( O HL),
Y"
aEe,„ IJ
Oa
1 CI (aHIJ ab
61.1
Oa Eelec
ab
(13.93)
Note that the CI parts, the A 22 in eq. (13.90) and the BO in eq. (13.93), were halved in order to adjust the coefficients of the CI-MO interactions, i.e., the third term from the last in eq. (13.77) and the second terra in eq. (13.87). References
1. F.W. Bobrowicz and W.A. Goddard, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.79 (1977).
192
CHAPTER 13. CPPEMCHF EQUATIONS
2. R. Carbo and J.M. Riera, A General SCF Theory, Springer-Verlag, Berlin, 1978. 3. J.E. Rice, R.D. Amos, N.C. Handy, T.J. Lee, and H.F. Schaefer, J. Chem. Phys. 85, 963 (1986). 4. T.J. Lee, N.C. Handy, J.E. Rice, A.C. ScheMer, and H.F. Schaefer, J. Chem. Phys. 85, 3930 (1986).
Suggested Reading 1. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, J. Chem. Phys. 77, 383 (1982). 2. Y. Yamaguchi, Y. Osamura, and H.F. Schaefer, J. Am. Chem. Soc. 105, 7506 (1983). 3. M. Duran, Y. Yamaguchi, and H.F. Schaefer, J. Phys. Chem. 92, 3070 (1988). 4. M. Duran, Y. Yamaguchi, R.B. Remington, and H.F. Schaefer, Chem. Phys. 122, 201 (1988).
Chapter 14
Coupled Perturbed Multiconfiguration Hartree-Fock Equations We turn now to the case of the arbitrarily general MCSCF wavefunction. The analytic evaluation of the first derivatives of the MCSCF energy may be accomplished without consideration of the associated coupled perturbed equations. However, analytic MCSCF second derivatives do require the development and solution of this challenging set of equations. The coupled perturbed multiconfiguration Hartree-Fock (CPMCHF) simultaneous equations provide the derivatives of both the MO and CI coefficients with respect to real perturbations. These CPMCHF equations are obtained by differentiating the variational conditions on both the MO and CI spaces. The equations consist of three parts: MO-MO, CI-CI, and CI-MO (or MO-CI) terms. In this chapter the first- and second-order CPMCHF equations for a general MCSCF wavefunction are derived. Although we have attempted to define all terms as they appear for the first time in this text, this is a good place to review some terminology that is critical to our discussion but not yet part of the lingua franca of molecular quantum mechanics. For example, the word "bare" (as in "bare" Lagrangian matrices) is used for the quantity that does not include the CI coefficients in its definition, while the word "parent" is used for the corresponding quantity that explicitly involves the CI coefficients in its definition (as in Lagrangian matrix). These two words, "bare" and "parent", are exploited in the derivative expressions only for correlated wavefunctions. The word "skeleton (or core)" (as in "skeleton" derivative Lagrangian matrices) indicates the derivative quantity that includes the AO (basis function and Hamiltonian operator) changes but does not involve the changes in the MO coefficients. This word "skeleton (or core)" is employed in the derivative expressions for all wavefunctions throughout this book.
193
194
14.1
CHAPTER 14. CPMCHF EQUATIONS
The First-Order CPMCHF Equations
As mentioned in Chapter 9 an MCSCF wavefunction is constructed by optimizing simultaneously the MO and CI coefficients. Thus the derivatives of the MO and CI coefficients with respect to nuclear coordinates appearing in the energy derivative expressions may be obtained by solving the simultaneous equations which involve the derivatives of the variational conditions on the MO and CI spaces and the interacting term between the two spaces.
14.1.1
The Molecular Orbital (MO) Part
For an MCSCF wavefunction the variational conditions in the MO space require a symmetric Lagrangian matrix at convergence:
—
=0
(14.1)
where the elements of the Lagrangian matrix are MO
E
MO
ryjm him + 2
E
(14.2)
mki
The Lagrangian matrix also is defined by the CI expression
ci E
(14.3)
CICJX11
IJ
where the "bare" Lagrangian matrices are
mo xty = E 7I
i
im + 2
mo E
r.f4k1(imikl)
(14.4)
•
mk1
in
The MO part of the first-order CPMCHF equations is obtained by differentiating eq. (14.1) with respect to the cartesian coordinate "a":
= 0
(14.5)
The required first derivative of the Lagrangian matrix is obtained by differentiating eq. (14.3) with respect to a cartesian coordinate "a": CI a E
Uxi
aa
IJ
ac,
ci
=
CI =
CI
aci X , cix;.' + E ci— E ,aa 23 IJ IJ CI ,,,-,
2 E C1 I
J
uk,jv
Li
Oa
"
CI
+
ci E IJ
aXij
+ E C/CJ " Oa IJ
C IC j
axiJ sj
[la
(14.7) (14.8)
THE FIRST-ORDER CPMCHF EQUATIONS
195
,x",
The first derivatives of the "bare" Lagrangian matrices and are
mo
oxf ,7
MO E
+ 2
,3
Oa x ij
o(imikt) imki
a IJG
were derived in Section 6.5
MO
MO
E
+E
(14.9)
Oa
mid
(14.10)
Uby
kl
The skeleton (core) first derivative "bare" Lagrangian matrices matrices (y") are defined by MO
714 Mtn +
= E
Xfija
,
(x /Ja)
and the "bare" y
MO
E rgki(imikoa
2
(14.11)
mid
and MO
MO
41,1 = .y32f hik +
2
E
+
E
4
mn
rg,„(imikn)
(14.12)
mn
These matrix elements are related to their "parent" quantities through CI
Xj
=E
ci c,xwa
(14.13)
IJ MO
E 7jm h ,am
MO
+
2
E
(14.14)
mid
and
ci
Yijkl
E
(14.15)
IJ MO
7j1hik + 2
MO
+
E
4
mn
E
r imin (irnikn)
.
(14.16)
mn
Combining eqs. (14.8) and (14.10), and noting the relationships in eq. (14.3) and eqs. (14.13) to (14.16), the first derivative of the Lagrangian matrix becomes
ax a i Oa
CI
CI
2
E
C1
E
MO
MO
E
j ij
ULXkj
E
The MO part of the first-order CPMCHF equations is now Osij
aXii
aa
aa
(14.17)
▪
▪
196
CHAPTER 14. CPMCHF EQUATIONS mo
+ E ukLiefix ki - (51,xki +
yi3 kr — yiikr)
kl
ci
CI
+ 2
E c, E I
J
ac, (,.,
Li ) - .X 31 • = 0 •
_ x i37 Oa
(14.18)
It is evident from this equation that the derivative of the variational condition in the MO space necessarily involves a term due to a perturbation of the CI coefficients. In order to eliminate linear dependencies among elements of the Ua matrix
+U +
S = 0
(14.19)
the third term on the rhs of eq. (14.18) may be manipulated as follows: MO
E uz(b u X kj
bzjxki
Yijkl
kl MO
E uz,(,51ix kj
blixki
Yijkl
//A bkisu - bkiXli
+ Yijlk
Yjikl
k>1 MO
+E
k>1
Y j ilk)
(
MO
+E
U (kxk
ökjXki
+ yi,kk
yJikk)
(14.20)
Introducing the relationship eq. (14.19) into eq. (14.20) one obtains mo
E uzeuxk i -
+
Yijkl
Yjikl)
kl MO
E
biixki -
bljxki
Yijkl
Yjikl
k>1 MO
_ E (u +
sz/)(bkixi ;
- 5kix, +
Yijlk
Yjilk)
k>1 „
-
2
MO
E szk
6kixk i
-
_kiski
Yijkk
6
Yjikk
(14.21)
MO
E
6,ixk i
- 6jj Xkj -
bkixti
-
Yijlk
kixti
Yijki —
k>1 MO
_E
Yjikl
Yijlk
Yjilk
Yjilk
k>1 ,
-
2
MO
E
Sa bkixkj
-
S
vkjxki
Yijkk
Yjikk
•
(14.22)
THE FIRST-ORDER CPMCHF EQUATIONS
197
Combining eqs. (14.18) and (14.22), the MO part of the first-order CPMCHF equations becomes
axij
axii
aa
aa mo
—
+ E
(5,k3 - öljXki - kiXE 2 + bkjxii + Yijkt —
Yjikl
Yi
Yjilk
k>1 MO
E
s 1 skixi ; -
2 E
k j Xli
Sic:k (5ki X kj —
Yijlk
bkj X ki
Yjilk
Yjikk)
Yijkk
—
CI
2
CI
E c, E
ac, ( 1,
— x Oa
Xji
=
23
0
.
(14.23)
This equation may be modified in a similar manner as was done in Section 11.1 and then written as MO
E
CI
Ajj,k1 14/
=
BlVij
=
bliSkj
2
k>i
CI
E
E
_
1L -zi Oa
23
(14.24)
where
Yijki
— 15/j X Yjikl
—
bki X ij + 5 kj X li
Yijik
Yjilk
(14.25)
and
Bvij
xzi + mo + E 51(bkixii - 4k, x, +
Yi ilk
Yjilk
k>1
mo
— E 5a(6kixk 3 2
-
bk3 xki + Yi 3 kk — Y 3ikk)
(14.26)
Equation (14.24) simultaneously contains both the unknown variables Ua and the partial derivatives 0C/10a. The second term on the rhs of eq. (14.24) represents the CI-MO interactions. These CPHF equations may be solved uniquely for the "independent pairs"; i.e., (i = active, j = core), (i = active, j = active), (i = virtual, j =core), and (i = virtual, j = active). The independent pairs for a MCSCF wavefunction are illustrated by the shaded areas in Figure 14.1. Determination of the "non-independent pairs" of the Ua matrices will be discussed in the following section.
CHAPTER 14. CPMCHF EQUATIONS
198 1
core
active
virtual
MCSCF Wavefunction Figure 14.1 Independent pairs (indep.pair) of molecular orbitals in the coupled perturbed Hartree-Fock equations for MCSCF wavefunctions. Only the Ua matrix elements shown as shaded areas may be solved for uniquely.
14.1.2
The Configuration Interaction (CI) Part
The variational condition on the CI space is CI
E c,(H"
5,JEei„) = 0
(14.27)
where the CI Hamiltonian matrix element was defined in Chapter 9 by MO Hjj
=
MO
E 7fh + E 1110 (iiiki) . ij
ijkl
(14.28)
THE FIRST-ORDER CPMCHF EQUATIONS
199
A constraint exists on the CI coefficients, since the wavefunction is normalized according to
(14.29) The CI part of the first-order CPMCHF equations is derived by differentiating the variational condition (14.27) with respect to the variable "a" subject to the constraint in eq. (14.29):
a
ci [ c,(Hij - (51,E,1 „) aa E j
+
)1
ci
e (- 1 +
E
0
(14.30)
In this equation, 0 is a Lagrange undetermined multiplier. These differential equations were developed further in Chapter 12 and the result was CI
(H ij
5IJ EeleC + 2CIC j
)
CI
=-E
)aC
(5 1J
( a aa
aa
0Eeiec
aa
The first derivative of the Hamiltonian matrix was shown in Chapter 9 to be 19Hjj
aa
MO
+2
=
E
The skeleton (core) first derivative Hamiltonian matrices matrices (x") in this equation are
mo E
H1,
(Hy.,) and the "bare" Lagrangian
mo E
1,111-qi
ij
(14.33)
iJkl
MO
E
(14.32)
ij
MO
-ylmJ
E
+ 2
(14.34)
mk1
The "bare" and the "parent" Lagrangian matrices are related by CI
E
(14.35)
CICjSti
IJ MO
=
MO
7j. t
.
2
E
(14.36)
mkt
in
The one- and two-electron reduced density matrices are defined by C/
=
E IJ
CliCyyji
(14.37)
200
CHAPTER 14. CPMCHF EQUATIONS
and
CI
E
rijkl =-
(14.38)
C1CJrf.4t
IJ
Inserting eq. (14.32) into eq. (14.31) results in CI
E
(His — biJEciec + 2CiCs °C '7
aa
)
J CI
.—
E
HI,
MO
+
E wi.x./ t, ,..7 -
2
Oa
ij
J CI
=—
aEe,„)
(5/./
E c.,Hy, -
CI
MO
J
ij
Cj
(14.39)
E c„ E uia.xii + ci OEa i
2
:-
J
(14.40)
3
There is a linear dependence among the elements of the Ua matrix appearing in eq. (14.40):
+
+
=0
(14.41)
In order to eliminate this linear dependence the second term in eq. (14.40) is modified as follows. The summation over the MOs is rewritten as MO
MO
E
u5L.xw tj 1j
. E 1>j
ij
MO
MO
i>j
i
qx.ifi + E u,cfri x.fj +
MO
. E
i>i
MO
. E
uilx.ti-i 1
-
usti
E
(14.42) MO
(Ul`j + S*31-1 — --- ,__, N--‘. Saiixii n
(p.(xl ,/ -
MO
.7$
1
(14.43)
MO
- E szi x./d - .i E
s?ixfiJ
414.44)
i>i
Using this equation, the second term on the rhs of eq. (14.40) becomes
CI
_ 2
MO
E c„ E
uiai xff
J CI
=— 2
E
MO
c,
E
u-,5;(xiff
-
J
IJ
)
CI
MO
Cl
MO
J
i>j
J
i
P.xf:1
+
Thus, the CI part of the first-order CPMCHF equations is expressed as
ci
E
(Hi,
- 6 IJEelec
2CiCs (9Cyj aa )
(14.45)
THE FIRST-ORDER CPMCHF EQUATIONS
CI
-E
-
CI
2
201 MO
E E u?,(xily - 2= i>j
CI
+2
MO
E Gj
CI
MO
EE sia x132 i>j
E nxiiI J +
..ri
C1
19Eelec Oa
.
(14.46)
This equation contains simultaneously the unknown variables OCl/Oa and the Ua, as was the case for the MO part discussed in the preceding subsection. It should finally be noted that the second term on the rhs of eq. (14.46) contains the CI-MO interactions.
14.1.3
A Complete Expression for the First-Order CPMCHF Equations
In the preceding two subsections, the first derivatives of the variational conditions in both the MO and CI spaces were obtained. Since these two variational conditions interact, the simultaneous equations have the form
A" An t [ Ua Bta — A n A22 ac/aa _
(14.47)
[
The All term contains the MO part. Referring to eqs. (14.24) and (14.25) elements of this matrix are A 11
xki
=
ki — Yjikl
Yijkl
Yijlk
6 kjXU
ki Xlj Y jilk
•
(14.48)
The A22 term represents the CI part and each element, from the lhs of eq. (14.46), is of the form (14.49) A32, = HIJ — j jEe1 ec -I- 2CIC j Finally, the A21 (A21') term contains the CI-MO (or MO-CI) interaction. Referring to the second terms in eqs. (14.24) and (14.46), each element of this matrix is CI
Afi; =
2
E
(14.50)
C J(44
The perturbation term for the MO space, from the MO part of eq. (14.24), is al 0,23
—
x7j
11,10
+ E sk((skisi; -
bkixii + Yijlk_ Yjilk)
k>1
1 2
+ —
MO
E siick (bkixkj k
— bkixki + Yijkk
Yjikk)
(14.51)
CHAPTER 14. CPMCHF EQUATIONS
202
and that for the CI space from the CI part of eq. (14.46) is CI
B 1
-
Mo
CI
E CjH, + 2 E
Cj
E
Sx
i>j CI
MO
+EE 14.2
0
„
+Ci Eel Oa
(14.52)
The Averaged Fock Operator
The two MCSCF conditions, eqs. (14.1) and (14.27), do not provide the unique variational criteria for the core and virtual orbitals as a unitary transformation among these orbitals does not change the electronic energy. Therefore, it is useful to redefine explicitly the core and virtual orbitals. One of the most common ways to define these orbitals is to use the averaged Fock operator
F" = h +
1 m°
E
(14.53)
7k 1(2-Tki — Kkl)
kl
where the elements of the averaged Fock matrix are given by
E
—
MO
= (oilFav) = li ii +
(14.54)
The two (core and virtual) parts of the Fock matrix are diagonalized and consequently the following diagonality is established for the core and virtual orbitals: (14.55)
bijE'i
where f1 are "orbital energies" and i and j belong to the same shell. The corresponding subsets of the original MCSCF molecular orbitals must be rotated using the resultant eigenvectors. This new set of molecular orbitals should be used for the CI procedure. The corresponding CPHF equations for the averaged Fock operator may be derived in a manner similar to that described in Sections 11.2 and 13.2. Differentiation of eq. (14.54) with respect to a cartesian coordinate "a" E , ij
Oa
L12_ 14Oa + 2
1
MO
2
awl/a) aa
kl
Z--N O
ki
Z-I
aa
ki
{ 'Al
{2(jilkl) — (ikIji)}
.9(ikliol Oa (14.56)
THE AVERAGED FOCK OPERATOR
203
is required. Note that the derivative of the one-electron reduced density matrix y ki does not vanish, since its definition, eq. (14.37), explicitly involves the CI coefficients. The derivative of 7ki is obtained by differentiating eq. (14.37) with respect to the variable "a": CI
0 7k1 Oa
Cl
ij
▪ E
crui Oa
{— a
IJ
CI
2
E IJ
(14.57)
CICTJ11(
1,E IJ
Oa
(9C j if
+
Oa
7ki}
C, g— c- 'y 1J( Oa 7k1
(14.58)
(14.59)
Using the results from Sections 3.8 and 3.9 and eq. (14.59), eq. (14.56) may be expanded as
mo
E
12,Zi
aa
+
(uA i hm ,
mo
u:tni him)
mo
+ E
mo
+ E u(milko
+E
urani (imiko
kl
MO
MO
+ E uciiimo + E MO
1 mo 2
E kl
u,araciiikm1
7k/[0:kiiir
MO
+E
mo
E
mo
+ E u77,3 (ikimo
+E
in MO CI
OC,
aa
kl IJ
UZI(ikliTH
m jj{
7k1
.. ( 1k1) — (ik1j1)}
.
(14.60)
Rearranging terms in eq. (14.60) gives
1 mo
af' aa
E kl
7kif2(ijild)a — (ikijI)a}
MO
+
• +
MO.
E u, [kn.; +2 kl mo mo E u,a„, him + - E in 2 kl 1
MO
E
(4.7.,k 7k4
2(railki) — (inkli1)}]
7 42(imiko _ - (indir)
mid
1 MO E 2 mid
uAryk/{2(iiikm) -
(ik1M1)}1
CHAPTER 14. CPMCHF EQUATIONS
204 MO CI
cl ac
aT 11{2(iiiko -
EE kl IJ
(ikij1)} .
(14.61)
Using the definition of the averaged Fock matrix in eq. (14.54), this equation may be rewritten as MO
MO
= Eitti•
aa
k3 MO
1 M°
+
EUgjeik
U:1
kl MO CI
E ac
11,44(ijikm) — (ikljm) — (imijk)}
,
-
E c, aa
(14.62)
kl IJ
In this equation the skeleton (core) first derivative averaged Fock matrices are defined by 1 MO
la
E
ij =
-yk/{2(iiikir
kl
(14.63)
- okijoal
Using the diagonal nature of the averaged Fock matrix for the core and virtual orbitals, eq. (14.55), and the first derivative of the orthonormality condition of the molecular orbitals, namely eq. (14.41), equation (14.62) may be modified as OE'ij
Oa
othershell
la
= E ij + U;iEl j + UiajEl i +
E
othershell
ugieki +
k 1 M° U:1 + 7 ) " Id
cr., kl IJ
MO
E 7,44(iiikm) - (ikiim) - (imuk)} m
+E 7 k,{2(iiiki) _ (ikiir)}
(14.64)
Oa
othershell
in
= E ij
(f', —
(')U
— s?jel; +
E k
1
E m
—2 z_., UL
ugiekj
+ k
IJ
71mAij,km
,j
oc
EE CI n Id
oth E ershell uzieik
MO MO v.
Id MO CI
+
uzeik
k
acj
MO CI
+
E
"
7d 1 12(iiikl) — (iklii)}
(14.65)
The A matrix is defined by Aij,k1 =
—
— (illjk)
(14.66)
Note that the A matrix in eq. (14.66) is a different quantity from the A matrix in eq.
(14.25).
THE AVERAGED FOCK OPERATOR
205
Recalling again eq. (14.41), the second term from the last in eq. (14.65) may be manipulated in the following manner:
E E
MO MO
MO MO
MO MO
. E ue,
71mAij,km
kl
+
k>1 MO
E
+E
E7imAi3,km m
Ulak
E
-,,cniA„,,,n
in
k>1
MO
uek E
(14.67)
l'kmAij,km
in k MO MO
MO
MO
k>I
in
. E ue, E 71mAij,km _ E (riz + sr,i) E m
k>1 _
MO
1 M° Skik
2
71,,,Ai 2 ,bn
E in
(14.68)
7kmAi3,km
mo mo
, E ulic , E
(7/mAii,km — i'kmAij,ltn)
m k>1 MO MO _
—
E k>1
1 —
2
.51
E m
MO
MO
E
7kmAij,Im
sckik
E
(14.69)
"IkmAij,km
m
k
Elements of the one-electron reduced density matrix involving the virtual orbitals vanish. Thus, equation (14.69) is further reduced to MO
MO
all occ
E u,aci E
71mAij,km
m
kl
EE k> 1
occ
1
(71mAij,km
7kmAij,1m)
OCC
OCC
E k>1
E
u,aki
E
sz1
in
occ
oce
sickt E
(14.70)
7kmAii,kni
in
Combining eqs. (14.65) and (14.70) one obtains
Oa
(ei ei) uiaj —
la = c ii
Sc'
+
E k
+
1
2
1
1
all occ
EE
occ
(.41
E
(7inzAii,km —
m
k >1
occ
ca
E
7kmAij,lm
occ
cya
E
7kmAij,km
othershell
other shell
7kmAij,1m)
uLek, +
E k
UiVik
▪
CHAPTER 14. CPMCHF EQUATIONS
206
occ + EE
ac C1- -- -y1 2(i3jkl) - (ikij1)}
(14.71)
.
"
kl IJ
The final equation for the non-independent pairs of the Ua matrices is Ufti
1
=
othershell
1.3
E ute k +
— Ei
1 2
all occ
2
1 -
4
E
U/(:j
k
occ
E
EE + >
(7/7nAii,km —
likmAiJam)
in
occ
1 ‘--., "c —
other shell
'1 1
5
E
7/cm Aij,/m
k>- 1 occ
occ
E szk E
7km Aii,k„,
in
k
occ CI
E E kl IJ
aCj C I — 7ki Oa
..
2(ulkl) — (iklil)}1
(14.72)
for (i = core, j = core) and (i = virtual, j = virtual). In the above formalism the molecular orbitals have to be first transformed unitarily using the averaged Fock operator in eq. (14.53)prior to solving the CPHF equations. Using these redefined molecular orbitals, the elements of the Ua matrices for the independent pairs are then obtained by solution of the CPHF equations in eq. (14.47). Finally the elements for the non-independent pairs are determined in terms of the Ua elements for the independent pairs from equation (14.72).
14.3
The First Derivative of the "Bare" y Matrix
It is necessary to derive the first derivative of the "bare" y matrix for later use. The first derivative of the "bare" y matrix is obtained by differentiating eq. (14.12) with respect to an arbitrary cartesian coordinate "a": ayIJ,_, unt
aa
Li
=71
mo
ah,k
+
IJ a(ikIntn) + 2Er., 3 77171. aa
/ aa
mn
MO
4
T ., 1 j
a(irrilkn)
E mn - jmin
aa
. (14.73)
Recall that the one- and two-electron coupling constants, -y" and r", do not involve the CI coefficients. Therefore, the first derivative in eq. (14.73) is obtained easily using the results from Sections 3.8 and 3.9: If
.7
7;f1
aa
MO
tk + E
h,,
(upai hk
2
E riA,„ mn
upak hip)}
MO
MO
▪
+
(iklmn)a
E fu;i(pkim.) +
Wipint71)
THE FIRST DERIVATIVE
OF THE "BARE" y MATRIX U;m(iklpn)
LTA,,(ikimP)}1 mo E {g(pinikn)
MO
+4 E mn
r
i
207
[(imIkn)a
U;k(imipn)
U77,n (iplkn) (14.74)
U;,,,(imikP)}1
The following relationships for the two-electron coupling constants
[Ilk/
r.ffki
r/.d,k
rflik
rfli;
rfti = rffij = riikii
(14.75)
(14.74). The resulting expression is
simplify equation IJ
MO
IJ L a
nik + 2
aa
MO
E F3/1mn (iklmn)a + 4 mn
Frim J in (imIkn)' mn
mo
MO
Upai {'yjil hp k + MO a
1 1J 7. f ( hi
MO
2 E Fljf(pkInin) -I- 4
E
mn
mn
MO
MO
in (prnIkn)
+ 2 E r.fiJmn(iplmn) + 4 E rg in (imipn) mn
mn
MO
Up% { F311/7n„(ikipn) + 2T3I-41n (iplkn)}
+2 mnp MO
Upar,{ rYm jikImP)+ 2I"' 1„(imIkP)
+2
(14.76 )
mnp
MO
„,I.1"
E (upaiypiAi + UpakYillp1)
Yijkl
MO
2 E upan,
+ 2r,„(ipikn)
7nnp
+
r.f iJnm (ikInp) + 211„ Jim (inIkp)
(14.77) }
MO
Y4C 1-
12 (g4A1
+ upak ygi )
mo
4 E upan,
-I- F31-4 in (iplkn)
. (14.78)
rnnp
In the above equation the skeleton (core) first derivative "bare" y matrix, y" a , is defined by IJ'
Yijkl
IJ
= 7j/
MO
MO
hclic + 2 E r.iirm „(ikimn)' + 4 E rlj in (iMikn) a mn
mn
im
(1 4. 7 9)
CHAPTER 14. CPMCHF EQUATIONS
208
14.4
The Second-Order CPMCHF Equations
14.4.1
The Molecular Orbital (MO) Part
The MO part of the second-order CPMCHF equations is obtained by differentiating the variational condition, eq. (14.1), with respect to the cartesian coordinates "a" and "b":
a2xii &cab
a2xii
&Lab
=
(14.80)
The required second derivative of the Lagrangian matrix is found by differentiating eq. (14.8) with respect to a second variable "b" in a way similar to that demonstrated in Section 13.3:
a2xii aaab I-,
=
0 (Oxij) Ob Oa ) 8 ( 2
(14.81)
E c, E J
I
=
G.' n,--,I n,--t J
E
2
Ill/
IJ
ci
+
E
2
C1
1
ci [ E J
Oa
J ij X
"
CI
E
+
CIC J
IJ
OX4-4) 23 da
(14.82)
CI J aaab "
1
acJ axff + aa ab
at aacb, a;!
0 2xIJ
Cl
4
ij
(.1l.,
Cl l..,
_ E c,c., " Oaab
(14.83)
The second derivative of the "bare" Lagrangian matrix is derived in a similar manner to that of the PEMCSCF wavefunction, since the coupling constants 7" and r" are independent of the CI coefficients. Using results from Section 14.1, the second derivative of the x" matrix is expanded as
a2x /3/ ' aaab
=
=
=
0 ( 0xf1) Ob \Oa) mo _, mo E Ha ,.,.I J _i_ %., ki.vki -r- E uri,y141) m
(14.85)
M0 mo ---ab E U+ `A si + ----:b r, k kl
(14.86)
a (.4.14 Ob
4. ij
OXIja s;
Ob
(14.84)
k
kl
8
UkaiYiljjkl
The first term on the rhs of eq. (14.86) is derived by the same sort of manipulation that led from eq. (14.9) to eq. (14.10):
Oxfr Ob
=
MG.
sfrib +
mo
E utixlr + E utz; kl
(14.87)
THE SECOND-ORDER CPMCHF EQUATIONS
209
The skeleton (core) second derivative "bare" Lagrangian matrices (x /Jab ) and the first derivative "bare" y matrices (y/Ja ) are: MO
'jab
Xij
7-7
MO
E
a IJ ht,
+2
E r"raid (imiklyb
(14.88)
mkt
and IJG
IJ La = 7 jt nik +
MO
MO
2
E
• IJ Fitnin (tkimn)a +4
E
ri.s " ,n (imiknr j771
.
(14.89)
mn
mn
These matrix elements are related to their "parent" quantities through the relationships CI
E c/ c,silab
(14.90)
IJ MO
E,
MO
Yjni,ht + 2
E
(14.91)
mkt
and CI
E
Yijkl
(14.92)
C JYtikjat
IJ
Mc)
=
7i1hcik + 2
MO
E
+4
E
an (i7721k7Z) a
1jn
.
(14.93)
mn
mn
Employing the result in Section 3.3 and eq. (14.10), the second term of eq. (14.86) may be rewritten as
a mo Ob
E ujicisL-; _ k
mo v ., OUZi i j
mo
°xi.' + E ufici 063
01) x k3
iLmd
k MO
(14.94)
k MO
= E ue — E ukm u i 41 m
k MO
MO
MO
+E
IJb
E urrb, nyilçlm„
+ E umb
(14.95)
mn
MO
MO
= E uexU
+
MO
E
+ E uzi/4).YLfmn.
. (14.96)
kmn
Using again the result in Section 3.3 and the results of Section 14.3, the third term of eq. (14.86) is manipulated to yield
a mo ukaryiiiki = MO
—
Ob
kt
kl
04,7 mo E u a •Ys3k1 Ob
kt
" Ob
(14.97)
CHAPTER 14. CPMCHF EQUATIONS
210 MO
MO
= E - km E
m1Yk1
MO
MO
I Jb
yiJ ki + E
+E
nb
" pis etipl
.Yp kl
kl
MO
E
+ 4
ri;rfam (inikp))}
(14.98)
mnp MO
MO
MO
E u;:i 4yir;31ki
TTÇL
▪ rrab I J
U ° Yijkl
ki Yi
kip
kl
kl
mo
+4
+
Up6 m (rijfmn(ikiPn)
+
tpopbm Irvnin (ikipn)
E
FI34 in (iplkn)
yn im
(inikp)}
klmnp
( 14.99)
of the "bare" Combining eqs. (14.86), (14.87), (14.96), and (14.99), the second derivative Lagrangian matrix is MO
. x,
1.1'lb
+E
MO
Nie + E uLywka, kl
k
MO
MO
MO
+ E uce./k i + E
+ E urciumbriYgnn kmn
uxif
k
k
MO
MO
MO
E Ne y .13.1k, + V"u a I Jb + 4..., Ll klYijkl id kl
+
MO
+
E
4
E Uitcl Uit jYrnijiki klm
j in (iplkn) + r il i.n im (inikp)} (14.100) Uriti pb m Ir jl (iklpn) + f: lim
klmnp I J'b + = x ii
MO
MO
k
kl
E uexik -1 + E u.i tyfiii MO
MO
+ E nxiiir + E utsi-la k
k
MO
MO
+ E urciyilich, + kl
kl
MO
MO
+ E
UtUmb nYllimn
MO
4
UPciU:ennejmn
kmn
kmn
-I-
+ E
E klmnp
Wall « bmn
1" Jip (imikp) 4- F ilpJEn (ipikm)} /ridr,p (ikimp) + r in
( 1 4.101)
-
THE SECOND-ORDER CPMCHF EQUATIONS
211
Inserting this equation into eq. (14.83), and noting the relationships in equations (14.3), (14.13), (14.15), (14.38), (14.90) and (14.92), the second derivative of the Lagrangian matrix is
n,-, n,,,-, E „,_,,u,
02 xii
CI
= 2
Oab
+
Ob
aa
IJ CI
CI
I
±
+ E
'3
2
E ci E I
Oa
J
mo
ab X ii
+
[ac.,19Xfj3
E c, E
2
x
CI
CI
j i j
ulic.ibX /c.i
+
J
+
02
,_,
L Xi j 8a0b 13 ' j
aCj OX1
Ob MO
Ob
Oa
treyi i ki
k
kl
MO
MO
+ E u-gi xt, + E uzi 4i k MO
k MO
E ut YZikt
+ E
Liz lit kl
+ E
UAU m b nYk jrnn
+
kt MO
kl MO
+E U - Pc iUma n Y k jmn
krnn MO
+ 4 E
kmn
uum bn{riinp(ikimp) + rinip(imikp)
+ rjpi n (ipikM)}
klmnp
(14. 1 02) The MO part of the second-order CPMCHF equations, eq. (14.80), is now
a2xii
Ow% mo
Oabb
ab ab = X — X i
+ E ukiib
siisk, — ôl j Xki + Yijkl
Yjikl
kl MO
—
ukt,xbki )
MO
MO
+ E (upci xz,
— Ut4i)
MO
E
ulq4ki
—
+E
kt
utd(yiaiki — 4k1)
kl
MO
MO
U
E
E
— UZiyki nin )
kmn
U n (ULYkjmn
kmn
MO
-I-
4
E
riinp(ikimp) +
rintp(imikp) + PiPln(iPikM)
k/mnp —
Fi inp (jklmp)
—
fi nip (jmikp)
—
ri pin(iPlkm)}
Ykimn)
-
-
212
CHAPTER 14. CPMCHF EQUATIONS ci acioc,( ij X - — + 2 g -23 Oa Ob
X li) 32
IJ
+
CI
CI
J
1 cr
CI
E
2
+
C
.,
1
2 ci ( I _ x•J — "
2_
X ill)
(041 2x— _oc, Oa Ob
Ox 3l-t7) 2
ci v-. aCj axli
Ox .ld )
Ob
J
CI ±
49
E c, E &LA —
2
E
CI L..., —
I
Oa
Ob
J
= 0
Oa)
(14.1 0 3 )
This equation shows clearly that the second derivative of the variational conditions in the MO space contains second-order terms due to perturbations by the CI coefficients. Equation (14.103) may be manipulated as was done in Section 14.1 and eventually written in a familiar form as MO
E
ij k1
Urib = —
k>l
MO
+E
tab
k>1 MO
1 +2 E
g (vkiXij —
al) g
Yijlk
bkjXki
kk okiXkj
MO
—E
Yijkk
Yjikk)
MO
(urci xt,
uej xL)
MO
E (0xL — uz,xL)
MO
—E uzi 4 ki kl
E
ejikl
kl
MO
—
MO
Um b ULYkjmn
UkijYkimn
kmn MO
– 4
Yjilk)
E
Un
uZiAjmn
E urci umb n rpnp(ikimp) + r.inip(imikp) +
rjpi n (ipikM)
kirnnp
– ritnp(ikimP) – rinip(imikP) – ripin(PIkrn.)} ci
– 2 IJ
– UitYkimn)
kmn
aci ac.1 Oa Ob"
CI CI _ 2 E c,JE A20 — /(xy xi") " — .71 0a0b
213
THE SECOND-ORDER CPMCHF EQUATIONS Cl
—2
{
CI
E cl E
(ax y Oa
ac
Ox/I) 2.
Ob
Ob
1/
(Ox Ob ea
+
ax.f;11
Oa ) (1 4.104)
The A matrix was defined by eq. (14.25) and the 6 ab matrices were developed in Section 3.7 and satisfy uzb + u2a1b + =o (14.105) Note that the second term from the last in eq. (14.104) contains the CI-MO interactions.
14.4.2
The Con fi guration Interaction (CI) Part
The CI part of the second-order CPMCHF equations is derived by differentiating the firstorder equation (14.31) with respect to a second variable "b":
a
CI
Cl
aCj
(0111J
2CICJ)--g-; —
— 6IJEe1ec
Meiec
aa
0IJ-)
.1 U
•
(14.106) These differential equations were developed in Section 12.2 and the result was
a2 67 j Hjj —
Cl
—E _
(a2Hij br
8a0b
(OHij z-• CI
—
2C1C.I)
5 IJEe1ec
Oa
(5 1J
a2 Eelec) O w%
8Eelec
Ob
C
wcj CI -W
Oa
) aoCbj
ac,\ OC
aEde, + Ob
(aHij
L—d
&Lab
Ob
ea
(14.107)
The second derivative of the Hamiltonian matrix element from Chapter 9 is a2H IJ
aaab
MO
= 117j + 2
E
ab IJ Ux
ij
MO
+2
E
(uei?i xtfa
+
X ifjj
ii MO
+ 2
E ijkl
(14.108)
CHAPTER 14. CPMCHF EQUATIONS
214
The skeleton (core) second derivative Hamiltonian matrices, lirj, in this equation are of the form MO
H rj
I J I. ab
/Li;
MO
+E
.
(14.109)
Using eq. (14.108), the first term on the rhs of eq. (14.107) is rewritten as
(a2 Hij
CI
.51 j a2Eelec) Oaab 0a0b
E CI
=-E
+
MO
MO
2
E qbx!" +
E
2
X 1.16 )
MO
+2
E qutyliki
82 Eciec actob
I
CJ
ijkl Cl
CI
-
—
2
(14. 110 )
MO
E E uosi,1 tj ij
—2
CI
MO
CI
MO
E Cj E (up,xfia + ulti xtt) ij
—2
E Cj E
UP, / y471,
Cj
a2 Eelec
ijkl CI
=
MO
Cl
EC . Hj -
2
(14.111)
Octi9b
E Cj E u,7,b(xw - x,i1.1) i>j
CI
+2
MO
E Cj E
MO
CI
ex/ Ji'
+ E
c.,
E ci6xf/
i>j CI
—2
MO
E c E (u-,Pi xr +
U441 6.)
ij CI
—2
MO
Eeiec E Cj E qutaym,, + c, a2(9a0b
(14.112)
ijki
In manipulating the second term of eq. (14.111), the technique used in Section 14.1 (see eqs. (14.42) through eq. (14.45)) was employed. Combining eqs. (14.107) and (14.112) the CI part of the second-order CPMCHF equations is now
— 6j J Ee rec + 2C1CJ
ci
- E cwir; —
2
oi E
mo c.,
E
a2c, Oaab
u?"
-
32
THE SECOND-ORDER CPMCHF EQUATIONS
cr MO CI + 2 >CJE gPxjf -F 12
215 MO
C1.7 12
cab
IJ xii
i>j
CI
—2
E
>JCj
ij MO
E c, E
CI
aaab
MO
CI
—2
d-
UZJ ULyjiki j
ijki o Hij
-E
451.1
Oa
CI
-E
61J
ab
Eeiec
OCj OCJ Oa ) ab
+
Oa OE,ie, Ob
cl ac,)ac., ab ) ûa
(14.113)
It should be noted that the second term of eq. (14.113) represents the CI-MO interactions.
14.4.3
A Complete Expression for the Second-Order CPMCHF Equations
In the preceding two subsections, the second derivatives of the variational conditions on both the MO and CI spaces were derived. The second-order CPHF equations for the MCSCF wavefunctions are
A n A nt [ A21 A22
uab 02 ciaaab
=
(14. 114 )
rtab2
The A" term contains the MO part. Elements of this matrix were given in eq.
6 1ixki — skixki — •
Yijkl
Yjikl
bkixi;
Yijlk
(14.25)
as
+ bkjxti Yjilk
-
(14.1 15 )
The A22 term represents the CI part and each element was defined in eq. (14.113):
= HIJ — 6 IJEelec
2C1CJ
(14. 116 )
The A21 (Ant ) term contains the CI-MO (or MO-CI) interaction. Referring to the second term from the last in eq. (14.104) and the second term in (14.113), each element of this matrix is CI
ALli3 = 2
E Cj (xf;
-
The perturbation term for the MO space, from the MO part of eq. (14.104), is Dabl
= — x`iiî
x 1( .ib
(14.117)
▪ 216
CHAPTER 14. CPMCHF EQUATIONS mo
+ E vcr
-
Yijlk
Yjilk
k>1 MO
1 — 2
+
2
Gii(skiski
Yijkk — Yjikk)
MO
-
MO
E (usti - u4sti ) - E MO
UPciXiaci)
MO
E
uz(ytki
- E tt(eiki -
Yjikl)
kl MO
E
MO
-
u,b„„(ugakinzn
kmn
E
uki,onnfri mp(ikinzp)
klmnp —
C/
2
1J CI
-
9
uzn(UtiYkjmn
kmn
MO
4
-E
u4ykim,i)
Fihip (jklmp —
CI
E
j 1( x I
j
Ob
Oa
rinip(imikp)
r•n ip imikp) —
)
n/-1 ar
+
(
+
— UZjYkimn )
rh,incipikno
ripin (jp ikrn)}
Ij) — X
{ ac (asti
aa
axt,i )
ac, Oxf/
ab
Ob
Oa
Ox \) (9a ) f (14.118)
Referring to the rhs of eq. (14.113), the perturbation term for the CI space is given by CI
CI
Baby
=_ E
trab Jnjj
+ 2 E c
, MG, E
X
MO
E abxtI
IJ
i>j
CI
-
2
-
2
(aHl, k. Oa
CI
Oa0b
(f/ 2
23
-F
Uia2 ;131b )
MO
>.Cj>ijkl
CI
a2Eetec UI
MO
>2Cj E ij CI
+
(allu k. Ob
b 1J r ia, 23
klYikj
451J öjj
aEetec Oa aEeiec
ab
ac.oac., +
+
Oa
ab OCj
oa
( 1 4.11 9 )
The solutions to the first-order CPMCHF equations are required in order to construct these second-order CPMCHF equations.
217
REFERENCES References
1. H.-J. Werner, in Advances in Chemical Physics : Ab Initio Methods in Quantum Chemistry Part II., KT. Lawley editor, John Wiley & Sons, New York, vol. 69, p.1 (1987). 2. R. Shepard, ibid., p.63 (1987). 3. B.O. Roos, ibid., p.399 (1987). 4. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
Suggested Reading 1. Y. Osamura, Y. Yamaguchi, and R.F. Schaefer, J. Chem. Phys. 77, 383 (1982). 2. P. Pulay, J. Chem. Phys. 78, 5043 (1983). 3. R.N. Camp, H.F. King, J.W. McIver, Jr., and D. MuBally, J. Chem. Phys. 79, 1088 (1983). 4. M.R. Hoffmann, D.J. Fox, J.F. Gaw, Y. Osamura, Y. Yamaguchi, R.S. Grey, G. Fitzgerald, H.F. Schaefer, P.J. Knowles, and N.C. Handy, J. Chem. Phys. 80, 2660 (1984). 5. M. Page, P. Saxe, G.F. Adams, and B.H. Lengsfield, J. Chem. Phys. 81, 434 (1984). 6. T.U. Helgaker, J. Alml6f, H. J. Aa. Jensen, and P. Jorgensen, J. Chem. (1986). 7. R. Shepard, Int. J. Quantum Chem. 31, 33 (1987).
Phys. 84, 6266
Chapter 15
Third and Fourth Energy Derivatives for Configuration Interaction Wavefunctions An energy expression that explicitly includes the orthonormality condition for the configuration interaction (CI) wavefunction is presented in this chapter. First this equation is used to derive the first and second derivatives of the electronic energy. Then the technique is extended to third and fourth derivatives. Such third and fourth derivatives of reliable correlated wavefunctions could add much detail to our knowledge of anharmonic effects in the rotational and vibrational spectra of polyatomic molecules. Anharmonicity is clearly one of the important frontiers of modern physical chemistry and chemical physics. It is demonstrated that these third and fourth derivative expressions are symmetric with respect to the interchange of differential variables (nuclear coordinates). Formulae for the energy derivatives of the CI wavefunction up to fourth-order are presented in full detail to encourage the computational implementation of these methods. Once the derivative expressions for CI wavefunctions are obtained, it is relatively easy to find the corresponding derivative expressions for other variational wavefunctions, such as multiconfiguration self-consistent-field (MCSCF) and restricted self-consistent-field (SCF), by using the relationships for the density matrices among various types of wavefunctions and imposing constraints on the molecular orbital space. This subject will be discussed in detail in Chapter 16.
218
FIRST AND SECOND DERIVATIVES
15.1
OF THE ELECTRONIC ENERGY
219
The First and Second Derivatives of the Electronic Energy for a CI Wavefunction
The first and second derivatives of the electronic energy for a CI wavefunction have been described in Chapter 6. In this section a new derivation of these derivatives is presented in order to demonstrate the power of the new method, which will be extended to the higherorder derivatives in the following sections. The electronic energy of the CI wavefunction is usually expressed by [1] CI (15.1) Eetec = EC 1CjH1j IJ where the CI Hamiltonian matrix is defined by (15.2)
(4)/lHeieci'DJ)
HIJ
mo
E
MO
E
Q11 hii
ii
.
(15.3)
ijkl
Q lj and G" are the one- and two-electron coupling constant matrices and they are related to the one- and two-electron reduced density matrices Q and G through [2]
cr
= E
(15.4)
IJ
and
CI
=
ECjCjG j
(15.5)
IJ
Using these quantities the electronic energy of the CI wavefunction is given in MO form by MO
mo
Eciec =
EQih + E ij
(15.6)
ijkl
The variational conditions for the determination of the CI wavefunction are ci
E
—
6/JEdec)
=o
(15.7)
and the wavefunction is normalized, giving the condition CI
E
C,
=
1
(15.8)
While the electronic energy in CI form is usually given by eq. (15.1), the following alternative form will be used in this chapter:
ci E cic,(Hif — IJ
61jEelec) = 0
(15.9)
220
CHAPTER 1,5. THIRD & FOURTH DERIVATIVES FOR Cl WAVEFUNCTIONS
so that the equation explicitly includes the normalization condition (15.8). The first derivative of the CI electronic energy may be obtained by differentiation of eq. (15.9) with respect to a nuclear coordinate "a": Cl
E
vIJ
(—
IJ
Cl a ci CI
a Eeiec)
c
+2
aa
Oa
E aa E —
(H/J
Ede)
=
.
(15.10)
The second term of eq. (15.10) vanishes due to the variational conditions (15.7), and thus,
a [ci .-E c/c.„ Oa
Cl
E c„,„
Ee,e,)
-
IJ
(aH,
IJ
Melee
( 51j
aa
Oa
) = 0
. (15.11)
Using the normalization condition, eq. (15.8), an expression for the first derivative of the CI electronic energy is Cl a Eetec alb, (15.12) Oa Oa IJ
= E c,c,
which is identical to the result presented in Section 6.2. The first derivative of the Hamiltonian matrix was found in Section 6.4 to be
a ll " Oa
mo =
+ 2
(1 5. 13)
The second derivative of the CI electronic energy may be obtained by further differentiation of eq. (15.11) with respect to a second nuclear coordinate "b":
[el Y .' CICJ(Hij — bijE ei„)] &Lab 7.1 (92
a [ci
aH" m eiec )] E cic., 15/., ( aa Oa
Ob IJ CI
E cic., IJ CI
+
aci
(02H ij
&Lab
(51.1
ci „., (alb,
E1 ab — Ej
, J2
Oa
(15.14)
(92 Eelec) = Oa0b
6u
aEelec )
Oa
. 0
(15.15)
Note that eq. (15.15) is not symmetric with respect to the differential variables a and b. The differentiation of the variational condition (15.7) gives relationships between the derivatives of the CI coefficients and those of the Hamiltonian matrix elements
ci
E J
CI
ôa\
ljj —
Eeiec)
E
(alb, Oa
melec) aa
=
(15.16)
FIRST AND SECOND DERIVATIVES OF THE ELECTRONIC ENERGY
221
These conditions are equivalent to the first-order coupled perturbed configuration interaction (CPCI) equations which were discussed in Chapter 12. Substituting eq. (15.16) into the second term of eq. (15.15), one obtains (92 [ CI
E cic,(Hij
0a0b
— 81JEelec)]
IJ
CI
E cic,
(02 H ij
Oa0b
IJ
ci
ac,
_ 2 v, IJ
(51 .1
_
Oa ab
a2 Eeiee )
&Lab
61JEciec
(15.17)
= 0
Using the normalization condition in eq. (15.8), the second derivative becomes
a2 Eelec Oa.% =
E
a 2 H1 CICJ
2
j OaOb
E
aci ac, Oa Ob
IJ
HIJ
(15.18)
6IJEetec)
which is identical to the result given in Section 6.7. The second derivative of the Hamiltonian matrix was found in Section 6.6 as
mo
82 HIJ
+2
8a8b
E ux.ty
mo + 2
+
E i3 (wx s,jfia +
mo 2
E uiaj uty41,
.
(15.19)
ijkl
The "bare" quantities such as XLI used in the above equations were also explicitly defined in Chapter 6. At this point the second derivatives of the variational conditions, eq. (15.7), are introduced for future use. They are easily obtained by differentiating eq. (15.16) with respect to a second variable "b":
a2c,
E &Lab (His
15 1JEelec)
CI
(aHIJ + E ac, Oa Ob
61.1
+
aEeiec Oa
2_,
ac, (aHIJ aa
ab
jj
8Edec)
ab
CI
E
C1,1( .92111 '1 Ow%
61.7
82 Edec)
&cab )
=0
(15.20) These conditions are equivalent to the second-order CPCI equations described in Chapter 12. The first derivative of the normalization condition, eq. (15.8), is also presented here:
ci „ oci = Oa
0
(15.21)
CHAPTER 15.
222
15.2
THIRD & FOURTH DERIVATIVES
FOR CI WAVEFUNCTIONS
The Third Derivative of the Electronic Energy for a CI Wavefunction
The third derivative of the electronic energy is obtained by further differentiating eq. (15.17) with respect to a third variable "c":
a3
[ di
OCLOb0C
E
cic.,(H/J
Cl
(03 H IJ
▪ E c„c.,
SI
&Lab&
oc, E ci ac
+2
-
(51J Eelec)]
IJ
83 Eelec) .1
&tab& )
(02 11u &tab
—2
aci ac, (aHIJ acI
0 2 C3
—2
aa
abac
—2
act
Oa ab
61 .1
ac
02 C
ab
6 IJ
Hjj
—
(92 Eelec)
Occab
aEelec
Oc 61JEelec)
Hij — 61 3 Ee te c) = (15.22)
Using eq. (15.20), the last two terms in eq. (15.22) may be rewritten as
ci 02cji„
-
E— aa E oboe (H " - IJ Eelec) CI oc CI 0 2 c j - 613 Eelec) — 2 E ab E acaa
2
▪ 2
E
aci[ci ac, aa ab
ac
CI
+ E c., (02 I.1 LI abaC J
Cl acI
I
CI
u"
[x_, C/
ac,,
J
=
Oc
0 2 Eetec 6Ij
CI
+
OCJ
(51.1
OC
Ob
)1
e" °Ed"
aa
ac,
(.9HIJ
)
ac
aa
a2 Eeiec )] 61,78E0elec)
oIj
&Ede , Oc
)
(15.23)
6 IJ acoa
el- [49CIOCJ (49111.1 2 j [ aa ab ac
aEeiec)
ObOc Li
(49HIJ
J
2 HU + E c., (0acaa
451.1 0 Eelec)
IJ
OC, aCJ aa ac
Ob
aE 15.[Jdec)] ab
THE THIRD DERIVATIVE
c' +2E
[aclac, (aHi, Ob ac Oa
IJ
ai 8c,
+2
E
Oa
1
CI
+2
OF THE ELECTRONIC ENERGY
oci
ci
E , E c, acaa
aEe,„)] (51.1 ac )
02 Eel ec )
6/.1 Mac
abOc (02 Hij
J
ac, ac, (aHii ab aa ac
6 m aEciee) aa +
(8 2 Hij
GI
E c,
223
61.7
02 Ee j ec )
(15.24)
acaa i ' J Inserting eq. (15.24) into eq. (15.22), one obtains ai
°3 [E 0a0b0C
- 6 LiEelec)]
Ci- CJ (His
IJ
CI
,E
cic,
(03 Hij
&Lowe
I
aCI CI
CI
-I- 2
Oa
z-a I
( 02 Hij
aci CI L-1 Ob E I J
I 2 \--' CI
WI
ac
c'
L'acaa
E c,
(02 HIJ
CI
J ci OCI aCi 1J aa ab
mac
( A.-., Hij
(0111J
----
6IJ
ac
CI
02Eelec ) OCOCt
192 Eelec + 2 \-
)
(51j &Lab
&L ab
I
+2
02 Eelec )
EJ cJ awe
CI
- -
03
Edec) 61j OCI,ObaC)
aEelec )
151.1 Oc
OCI 0C,I ( 0-11 1J Oa IJ Ob ac
0.Ee tec ) ku Oa
CI
aEei„ 451.1 Ob ) = 0
+2
-----
OCI aCj ( OHIJ
+ 2 N-z_., IJ ac aa
ab
(15.25)
The terms including the second derivatives of the CI energy, such as 5/ja2Eetec/aaab, may be eliminated using a condition in eq. (15.21). Thus the final expression for the energy third derivative becomes ci 03 Eese. 03 Hjj &tab& =
E cicJ aaabac IJ
CI
+2
CI
E c1 E
ac, .02 Hij Oa abac
J CI [aci ac, (aHiJ + 2 N--.' ab Oc IJ I
ac, (aili, + aci _____ ac aa ab
ac, a2H, ac, a2HIJ + acaa ac aaab
+ab
61 J OEetec) + aciac.i (oHif ac ) Oa 01) ac
,. aEeiel 01.., ab )
biJ
0.Eelec
)
fia )
(15.26)
224
CHAPTER 15. THIRD & FOURTH DERIVATIVES
FOR Cl WAVEFUNCTIONS
Note that this derivative expression is symmetric with respect to the interchange of differential variables (nuclear coordinates).
15.3
The Fourth Derivative of the Electronic Energy for the CI Wavefunction
The fourth derivative expression arises from differentiating eq. (15.25) with respect to an-
other nuclear coordinate
"d": 84
[ci
E
(His —
abcd IJ Cl
cr E
oc, ECI c,,
r ad J cr oc' cr E +2 Cj r aa J
E
CI
61
( 80:8 11b 18JC
1 aa J ad cr 8 2 CI 82c1
abac
E
&La
E
Cj
E
E
1 8 2 Hij
Obac ( 03 H, j
cl
c/
(0 2 /hi
J
acaa
a2c/ E cJ Er Obad
cr + 2 \--/
+ 2
CI
9_ ci aCad
r a
+2
cl E IJ
Ocaaad 82 E i ) ‘51J
61 J
J
E/ ac E E
Eetec)
51J02 Obec
(
__ Cl
E J
ac, aCj aa Ob
02H,.,
ad Cj
aaab
61.1
Oaab
aced
45IJ
O3E e l ec ) aaabad
h2F, , ) 51j •-• 4-,eec i aab a
(02 H,.,
02 11-1.1
ecc an%
on%
( oc, a3Hjj Oc E c., aaabad
J
+ 2
451 j e92 Eciec) abac
brJ82"Eelec)
CI
Cl aci CI ac j
abaced)
bij 03 ECteC )
Cj
acaaad r ab J cr E o Ci cr 8c,j 82 H, j + 2 ad acaa r ab E J + 2
a3 Eetec ) j0 6 3aE abeiaecC )
451..1
Obacad
f o2H,,
1 J cr oc , Cl
+ 2
(03.111J
Cl oc, E oc,
+2 E + 2
brj 0a0b0cOd
aaabaced
IJ
+ 2
mEejec
84 Hjj
E
t5/JEelec)]
02 E i ) e" eaab
02 Eelec)
°ceci )
THE FOURTH DERIVATIVE OF THE ELECTRONIC ENERGY
+ 2
OC, 82 C., OHL' fia abaci ac
+2
02 C, OCJ (OHIJ &cad ab ac
+2
(a2 HIJ ab ac &tad
aEelec ) 0C
ab Ocad
5IJ
0Eejec 51J
ac 02 Edec)
5 1.1 aaad
acia2c, OHL,
+2
Melec)
(51.1
lia
Oa
+2
02 C, OCJ (OH, lia abaci ac
51J Oa
+2
OCIOCJ (02 I1L, abad Oc fia
5,1J abad
+2
ac,a2c, Oc aaad ab
51.1
+2
02 C, OC., (OHIJ ab acad fia
51J Ob
0Eelec)
02 EeIec)
aE,jec )
ab aEetee )
Collecting terms in this equation, the expression may be rewritten as [CI
94
4
aaabacad
(HIJ
G./
( 04 HIJ aaabacad
IJ CI
+ 2
+ 2
E
+2
oc
,
ci
aa CI aci
c.,
oc' CI
OC 1 CI oc , GI
ad'
0311
(
E E CI
+ 2
— 8/JEdec)]
IJ
E
E
actabacad
(03 HIJ
Ocadaa C
(a3HI J adaaab (03 H1J
Oaabac
J
WI OCJ ( 02 HIJ
Obad
bij a3Ee ec
abacad
Obacad
c / [OCI OCJ (0 2 H/J + 2 z.... Oa ab acad IJ
lia ac
a4Eetec
1.1
45rJ
03 Eetec) acadaa
a3E /c, )
51J °dam e%
03 E,1„)
5IJ &Lab& s 02 Eeiec) vij acad
s a2 Eelec) abaci
225
(15.27)
226
CHAPTER 15.
THIRD & FOURTH DERIVATIVES
WI 0C3
FOR CI WAVEFUNCTIONS
a2E,,,)
+ aa ad ( 02 Hi., abac aciac, (821113 + ab ac aaad aci ac, (82H1J ± ab ad &Lac
(51j abac a2E) dec
(51.1 &Lad
a2 Ee j ec )
6 " aaac
a2Eelec )1
ac, ac., (492 H1.,
ad aaab (aoHilij aaab CI (9 2 c1 x--. / C { ac, aEd„) + 2 61.1 E1 acad J Oa ab Ob ) + OC
ac, + ab
(alb.,
aa
ci a2ci ci
aa
aaad
1
(011-13
ab
J
6.0
ac
151J
151J) 8Eelec+
ab
ab
C
02 Ee tec )]
aaab
aEeie c) Oc
)
.51.)-
02 Eeiec)]
adac
aE,(91:9;
ac ) (02HI3
aci (O H13 + ac
ac
aaab
aa
v , [(xi.,
E
bt,f
( a2 Hij
61 j aEelec) + C j (9111j 2
OCJ ( (9H1J + Oc Oa CI a 2 c , CI
+ 2
aa
) -1- Cj
[ac,
abadE
4- 2
( Melee
45./../
CJ
aba
bIJ
a2Edec
)]
ab0c (15.28)
Using eq. (15.20), the last three terms are simplified to yield
04
[_ c/
achabacad
IT
c/ . E cic, IJ
CI
+2 E I CI + 2 7
( ,,q4 H I
C.,
ac, cI E
C3
J
ci CI 7 ac, E I
c/ -1- 2 E
/
Oc
K./
j
u aaabacad
ac, CI 57 aa -7
I
±2
61.7Ee1ec)]
cic,(1113
Cj
J
cl
E c, acl J
a'4 Eeiec 61.1 Oaabacad)
( a3111J c
abaad a3HIJ aCadaa 03111 3
adaaab (83 H13 &lab&
(5,,
6IJ
(51.1 Sjj
a3 Eelee
)
Obacad 03 Edec) acadaa) 03 Ee i„)
&Waal) a3 Eelec aaabaC
)
THE FOURTH DERIVATIVE OF THE ELECTRONIC ENERGY CI
-I- 2 N--' L...,
ac, (02 1-hi
[aci
02 Edec)
151.1 Ocad
aa ab acad
IJ
aci ac,
(a2HIJ
+ aa Oc
abad
02
Eelec) 51.7 abad
aci ac, (.92 HIJ
6 1.1
+ aa Od Obac
aci ac, (a2 HiJ &tad
bIJ
aci ac, a2 HIJ + ab ad adac
(51J
+ Ob ac
227
(
aci acJ (a2 H1.7
a2 Edec) abac 02 Eetec) aaad
a2Ee tee) aaac
02 Ee i ec )]
6'J ad &Lab &tab [ ,2 ,,,, a2c, j a2c, a2c 1 a2cl -aaab - acad + a2c1 + &cad abac Oa& abad
+ OC CI
—2
E IJ
(H LI — (SLIEetec) = o
.
(15.29) In the sequence of differentiation, each term contains the Hamiltonian matrix and energy in the form of (H/J — 8/JEciec) and their derivatives. This structure has the advantage of being able to obtain the higher energy derivatives by differentiating eq. (15.9) instead of eq. (15.1). In eq. (15.29), the terms involving the third derivatives of the CI energy, such as bi.P93 Eeiec/actabac, are eliminated using the derivative of the normalization condition, eq. (15.21). Thus a complete expression for the fourth derivative is
a4 Edee aaabacad
CI
=
E
CICJ
IJ
04 H LI Oaabacad
CI
CI
I
J
+
+ CI
+ 2
E IJ + + +
OCJ
[
ac, 83 HIJ Ob acadad
O CJ 03 HIJ act abOcad
03 111
,
+
ac, a3 H1 ad &c ab&
adadab r ac1 ac ac, (a2 111J Oc
[
a2 Eei„)
Ob acad Oa
OC, OC (02 HIJ Oa ac
OW
aci ac, (0 2 1-11.1 aa ad mac aci ac, (a2 HIJ ab ac &Lad
&ad
a2 Eei„)
61J abad 6 1.1 5 .1.1
a2 E,1„) Obac
a2 Ee i ec aaad
)
228
FOR CI WAVEFUNCTIONS
CHAPTER 15. THIRD & FOURTH DERIVATIVES
aci flCj (fl2H jj Oa& Ob Od OCI OCJ Oc Od CI
2
02 Ee tec 0a0c (92 Eelec )]
a2 1-11.1
IJ
OccOb
&tab
02 CI 02 CJ O2 C1 02CJ Oa& abed 0a0b OcOd
E IJ
6.1.7
02 C/ 82 CJ
6IJEeiec)
OccOd
[
(15.30) Note that this derivative expression is symmetric with respect to the interchange of differential variables (nuclear coordinates). Several of the terms appearing in eqs. (15.26) and (15.30) have not yet been elaborated and this is the task for the remainder of Chapter 15.
15.4
The Third Derivative of the Hamiltonian Matrix
The third derivative expression for the Hamiltonian matrix appears in eqs. (15.26) and (15.30) in the previous two sections. This third derivative expression comes from differentiating eq. (15.19) with respect to the variable "c":
mo
ô3 H jj
&dab&
ijkl
mo E
(2 1j
(93 (ijikl)
23 ki (9a0b0c "
UObYIJ
ij
mo
mo +2
E (upi xix. + uzix,r) + ij
-I- 2 E Oc 2,
( auk OC
11:" b
Oc
3
3
Oc
axWa
Otriaj xij b
11 ac13 + + Ui3
OU a* b IJ
,a
—i1UklYijkl
-I- u i i
( Oc
(15.33)
OX,P)
3 x,LT +
1 ,a
E qu 1 y4,-:, ijkl
(auto
MO
OHrj
2
Oc
+ uta,
auk'b yr-f.. oc
at:
oxwb) Oc
. (15.34) )
The four terms in this equation are now considered separately. The first term in the above equation is manipulated as was done in Section 6.6, to yield
a Oc
mo
E ij
tj
z3
mo +E ijkl
G flki(ijikl) ab
(15.35)
THE THIRD DERIVATIVE OF THE HAMILTONIAN MATRIX MO
Ole
E
Oc
ij
MO
+E
J 19
Gti3k1
229
(iillarb ac
(15.36)
ijkl
MO
E uvcir
117.bic + 2
(15.37)
tl
The skeleton (core) third derivative Hamiltonian matrices Hytic and the skeleton (core) second derivative "bare" Lagrangian matrices X /Jab are defined by MO
Hy? = E
MO
Qt/htzir
E affici (iiikoabc
ij
and
xiij ab =
MO
E
(15.38)
ijkl
MO
+ 2
Win,I'
E
koato
kl(iMI
(15.39)
mkt
Tm
The skeleton (core) second derivative Lagrangian and the derivative "bare" Lagrangian matrices are related by CI
— E c/ c,xpab 3
vat)
(15.40)
IJ
Using the results in Sections 3.3 and 6.5, the second term in eq. (15.34) may be rewritten as MO 2 v• (auo X11 L7.‘ i3
=2
ufah
'JaC
Oc (
fr oc
MO
-.3 _ E
Iffk
U i:.' X jjj )
+ 2
=2
2b
MO
k
kt
E uLx,f;' + Eu,,yiiiiici
(X j j4
uaibc4I
MO
(15.41)
MO 2
E ufk ucejiit ijk
+ 2
UbXJc
MO
+2
E uzbuLxli +
MO
2
ijk
=2
uzibc x4./ + 2
il
MO
2
E uzbu iyii.„-:,
ijkl
The third term in eq. (15.34) is manipulated to give
mo 2
E 13
(15.42)
ijki
MO
E ubxiit. +
E utibu i yiii .kr,
( auibi LP 79—, xii + Ut''';' aXalci + acr 3 au' ai x"!jb
eV' + ulal ac
.
(15.43)
▪ 230
CHAPTER 15. THIRD & FOURTH DERIVATIVES MOrt ,
E
2
+
FOR CI WAVEFUNCTIONS
m0
up; _ E
Uickuti
k MO
ut (xrc +
E
x ilij° t
/me
MO
+E
k MO
E
+ ( u2
U raYiliC
kl
uick u) X I J' MO
E
(ebc
(15.44)
(J/ifii'kJib)] kl
MO
=2
E 2,
—
MO
2
E urijni xiita ijk
MO
▪ 2
MO
E
+
2
MO
E utuxua +
E utu iyiN,a
2
ijk MO
2
▪ 2
E rzcxfib 2,
2
ijk
MO
MO
2
MO
E uPPCiljja + MO
+2
MO
E uri uLxict +
2
ijk
ij
=2
E uick wi xe
ij
E uzxybe +
ijkl
MO
E Uc X I
E Ut X!tac +
p v./Jb
(15.45)
MO
E
2
ij
ijkl
MO
MO
+2 E
ij
U_
ijkl
MO
2
E u,E;
ulve`
+
E
2
ij
UUYC
U kt
f././ ijklb
.
(15.46)
ijkl
Using the results in Sections 3.3 and 14.3, the last term in eq. (15.34) is rearranged to yield
2
mo (au. E
+
ac
ijkl MO
=2
„
Ir
AT Tb
kl
+ uza4,
ijkl
123ki
Oc
MO
E
utfif c - E
UzÇn Uiani UPaYilj `kli
m
ijki
MO
+ lili( tlf -
E m mo
MO
+ quibc, (ya,` +
E u;in 1 + E u,c,,,y.,1;/,
mo +4
E Upcm fen,n (ikipn) +
77171p
G
in (ipikn)
GI,L m (in Ikp )} )]
(15.47)
▪
THE THIRD DERIVATIVE MO
=2 + 2
E
OF THE HAMILTONIAN MATRIX MO
uty uty,,‘ / , —
cni
ijklm
MO
MO
E
uzutfyiii L - 2
ijkl
ijklm
MO
+ 2
E ui uzj utyilA
2
ijkl
E
231
E
mo + 2 E qutupck yi/j pji
MO
uzupaygic
E
+2
ijkl
ijklp
ijklp
MO
+
I UZI Upc {G ijm n (iklpn)
8
in (iplkn)
Gim(inikP)/
ijklmnp
(15.48) MO
=2
E
uftuta yii„Ja
MO
+
2
ijkl
E
quzf1'Ç 1
+
MO
2
E
qU 1 1T4 ic
ijkl
ijkl
MO
E
+ 8
Uif'i ULU,n e
G ip (iMikp)
G.IipJin (ipikm)}
ijklmnp
(15.49) MO
=2
E uzic
MO
+
2
ijkl
+2
ijkl
MO
E
+2
MO
E
E uz1/1),IYaic ijkl
uzqu,enn Zg imn
(15.50)
ijk Irnn
The "bare" Z matrices Z u introduced in this equation are defined by
mo
=4
Z"
E {Gtinp(ikimp) +
Gg,p(imikp) + qr1,17,(iplkm)
(15.51)
and are related to their "parents" by CI
E c/c,,Z
Zijklmn
linn
(15.52)
IJ MO
4
E
{Giinp(ikImp)
+
Ginip(imikp)
+
Gjpin( iPi
kin) }
(15.53)
Thus, the last term of eq. (15.50) may be rewritten as MO
2
E
uzutE urenn zif ik ,,,„
ijklmn
mo 8
E
(fiai
u1 u n
//7ann
+
mo
uivizumbn)E GYnp(ikimp)
ijklmn
(15.54)
232
CHAPTER 15. THIRD & FOURTH DERIVATIVES
FOR CI WAVEFUNCTIONS
Combining eqs. (15.37), (15.43), (15.46) and (15.50), the following third derivative expression is obtained: MO
03 H j
Hy.bf + 2
aciObac
E
MO + +
2 2
E
4
ii ii
+
ij
MO
MO
2
ii
E ut9:txitb +
E uticir +
2
ijkl
E utu,:,y4c
MO
2
E ijkl
MO
2
J
MO
2
ij
E utvitya +
UU I Y
ijkl
E u4x,re +
MO
E ijkl
MO
2
ij
+
2
ij
MO
+2
E uzpx!". + ij
E uibpciir +
MO
MO
2
E uiwklya, + ijkl
MO
2
E qup„ya,c ijkl
MO
+
E uliuta timc n Zilikimn
2
(15.55)
ijklmn
This expression is rewritten in a simpler final form as MO
03 HIJ
abc l IJ
acu9b
2
E uiapx,"/ ii
MO
+
2
E (u.ribxiitc
upfxt..y.
ui7xivb)
MO
+2
E (uiai xir +
Xilij "
UiCi Xfij a
MO
+2
E (uzibu i + u; u1 + uiciPuz1)yiv,, ijki
MO
+
2
E (uri utyiii -kf,c +
+ quzyg,b)
ijkl MO
+8
MO
E (uzuLuTenn + UJU 1 U + quumb n ) E
ijklmn
(15.56) Note that interchange of the i,j,k,l,m,n, and p indices does not alter the equation as the summations run over all molecular orbitals.
EXPLICIT EXPRESSION FOR THIRD
15.5
DERIVATIVE OF Eei„
233
An Explicit Expression for the Third Derivative of the Electronic Energy
An explicit expression for the third derivative of the electronic energy is provided by combining eq. (15.26) for the energy third derivative and eq. (15.56) for the Hamiltonian third derivative:
mo
MO
MO
+ 2E
E Q 3 h jc + E 2, ijkl
03 E
0abc
MO
•
2 V'
+
2
+
2 E (Ur t/. /
/7tj51.b A tj
ii
u
ij
+ UPPCZ-
Uf; ,C0
MO
E
+ tifi
(U449 +
Ilf,(4 1)yijki
U!';UL
ijkl MO
+
2
E
(UtTL
Ut-bj Ub lif.jik1
kl
UicjUk41 11;k1)
ijkl MO
-I-
8
mo
E
(qq,u,cnn +
UPJUIIUZ„
+ qULU,bnn
)
E
qiinp(ik imp)
ijklmn
+
2
+
2
CI CI E E
CI IJ
ac, 02H IJ
02HIJ
( Oa Oa Ob0c
[ aci ac , (alb
Ob
,
61.7
Oc
aa Ob
aCIOCI (OHIJ
Ob 8c
ac, oc,
51.1
Oa (
&Oa
OCJ a 2 HIJ\ OC Octal) J
8Eetec)
Oc
0.Eciec )
6. 1J OE aealec)1
OH
ac &a k, 81)
(15.57)
Ob )
The first and second derivatives of the Hamiltonian matrices in this equation were defined explicitly by eqs. (15.13) and (15.19). The skeleton (core) derivative Lagrangian and Y matrices are
mo E Qâm q,„„, +
xitst = MO E
,ab
2
Afo E mki
Gimk,(imikoa
,
(15.58)
mo +2
E
mkl
Gimkj(iMlki) °b
(15.59)
234
CHAPTER 15. THIRD Sc FOURTH DERIVATIVES
FOR CI WAVEFUNCTIONS
and MO
= Q1hk + 2
+
E inn
15.6
MO
4
E
G im in (imilcn)"
.
(15.60)
mn
The First Derivative Expression for the "Bare" Z Matrix
It is convenient to find the first derivative expression for the "bare" Z matrix before proceeding to the section on fourth derivatives. This derivative is obtained by differentiating eq. (15.51) with respect to a variable "a":
az1,tjklmn 1
MO
=4
IJ (Giinp(ikimp) +
E P MO
=4
E
(Gy,„a( ikaimP)
+
G _17 ;,fdp °(imikP)
Guir, 6(iPikm) ) (15.62) lia
P MO
=4
MO
E Gri "[( ikimp)a + E Ulain (ikirP)
MO
+4
(15.61)
+
Ut4
fu,(rkimp) + UMirimP)
U7c.zp ( ikiM7)11 MO
E
+E
• Ulr-lk(iMITP)
{uA(rmikp) + U;tm( ir i kP)
Uir-tp(iMikr)}]
MO
MO
+ 4 E Gii;',/n[ciplkiroa + E fumrp i km) + U(irlkM) (41:k(iPirrn)
UAn (ipikr)}1
(15.63)
MO
= 4 E (Ggp(ikimp). +
+
+
Gij;73(n(iPlknt) a)
MO
E
pr mo
+ G jp(rmikP) + q7,(rPikm)}
4 E U:k{G Ynp (irimP) + Gjeap(imirP) + Gn(iPirm)} pr mo G.Win(iplizr)} +4 U-A4G iliinp (ikirP) + G.Ii?jzip(irikp)
+
E
THE FOURTH DERIVATIVE OF THE HAMILTONIAN MATRIX
235
MO
+ 4
E
Urap {Ggp(iklmr)
+ G31,L p (irnikr) +
3pi
n (• ITik TT/ )1
Pr
(15.64) The skeleton (core) first derivative "bare" Z matrices are defined by MO
Zijklmn 1 ja
=
E
4
{G 31np( I, J ik I MP) a
3np l (imIkPr
lpn I (iPikra) a }
(15.65)
Using this definition, the following considerably more compact expression is possible: azi/khnn
Z 141n1 42 n
Oa
MO
E
Errak zirjjr1rran
(UraiZrlikhrin
trAn 2Vurn)
MO
+
4
E
u:par ,,J,p(ikimr)
+ G .14 2 (imIkr)
G137,1in (irikm)}
pr
(15.66)
15.7
The Fourth Derivative of the Hamiltonian Matrix
The fourth derivative expression for the Hamiltonian matrix is critical for obtaining Cl energy fourth derivatives and follows from differentiating either eq. (15.55) or eq. (15.56) with respect to the variable "d": MO
HI
OctObOcOd =
E cdzigj aaabacad ij
a -
MO
a4
ijkl
194(iilki)
u kt act0b0cOd
(a3H1J)
(15.68)
Od 0a0bOc
MO
+2
E (uzibx,[/` + up;xiir + u27x2tb) ii
MO
+ 2
E
2j 23
uibi x ivca
ii MO
+ 2
(15.67)
E (u19ibu 1 + u2);(41 + (1,7 ut)yg
236
CHAPTER 15. THIRD & FOURTH DERIVATIVES
FOR CI WAVEFUNCTIONS
mo
E (uutotaic + u
+ 2
uyi li‘t4
UfjUllY)
ijkl
MO
E
+ 2
(15.69)
viajutu;nnzir,iimr,
ijklmn
mo 2
E
+2
ad +
a II:c j ad X13
MO
01-1fr
( au,ab ITLIC aci
uabc ox
:3 ad
Out/
uabaxille
ad
+
-F tf,b3c
.z.7
OXP G
lid
au; xi J b + Ili. axi,Jb) , 3 ad ad 1 MO
aua
bc
ad
MO
2 ijkl
jab (If
ad A ij (a ulkb3 rT e ad ki
3
ad
OUP
kl
MO
2 ijkl
ad
ki
kl)
ad
+
3
ad
11 '
ay m
(If; UZI)
tjki ad
n b ( 0114 Trb --IJ (i vTTki,v1J` --6c-i— v klYijI k + Ufi - Ti 1— I ilk(
▪ ad
+ u,bcaura
_Lua
ijkl
+
3 ad
17-b
+ up,F (4, +
+2
13
axwca 13
ad3 ad
ki
MO
ad
utb
axijab
t Tab
°UT., u b
•
auP. s3 xi,"
:3 ad
3
auis
+
ax Ube 3
--M- Xi" + U a
+2
IJc
ki
+
8d
± 0. (41
UklYijkl
3
aUic.
irc
Uasjkl Yi ad kl
1-)2
3
aYil.ja Iki
ad
au.g, „
Li b 3 ad 1.0ki + 149j Urd -ta: )
MO
+
( aug..
Ub ad13 ki
2
TTC
ran
+
ijklmn MO
+
uia;
2 ijklmn
u,enn
az ,ffam „ slid
aUti TT ,
7 :9;7 1 . I} mn
+ ua.0klb ad
Z" ijklmn (15.70)
Each of the nine terms in eq. (15.70) will be rewritten separately and then reassembled into the final full equation for the fourth derivative. The first term in this equation is treated
THE FOURTH DERIVATIVE OF THE HAMILTONIAN MATRIX
237
in a manner similar to that used in Section 6.6:
a [m- 0
a H ylie
MO
E
lid
+ E af31,(iiikirbc
(15.71)
ijkl MO
E
Q2 3
ij
H cilicd
aftcic
MO f-jj a(iiikl)abc E
ad mo
ijkl
3
2
E
ad
(15.72)
oxi 3 $ jJab`
(15.73)
The skeleton (core) fourth derivative Hamiltonian matrices li tc f d are MO
MO
E Qiihcli?cd
H tcd =
E
iiikoabcd
(15.74)
and the skeleton (core) third derivative "bare" Lagrangian matrices are
= MO E
X '° '2
MO
h sm °
-I-
2
E
(15.75)
mid
The skeleton (core) third derivative Lagrangian and derivative "bare" Lagrangian matrices are connected by -
yak (1 ij =
CI
E IJ
CICjX[jabc 3
(15.76)
The second term of eq. (15.70) is rearranged to MO
2
( a pi
ôxJ 1, - 3 )(" 4- b 7Lbc ed 13 3 ed 13
mo
=2
-E
+2
U2aibc ( X lid +
tduz,c) X IJ MO
E uliciv + k
MO
1 ij lcl
MO
=2
E ti
+2
(15.77)
kl MO
ulti-xli d
Hi- 2 :E: U sjOc Ukl d Y.I.J sjkl ijkl
The third term of eq. (15.70) may be treated as MO
2
2.s
(liu ad
au ad
li XYC "i3
23
axit) 3
xijb 3
ad
-1- v 23
ad
+ au, ad X 83
vbc +
IL 3 '
anja lid
.
(15.78)
238
CHAPTER 15. THIRD Se FOURTH DERIVATIVES mo
MO
E
=2
FOR CI WAVEFUNCTIONS
— E uidk uivi
J
83
mo E
mo urzxtla i
yI J "
kl tjkl
kl
mo (u cad
quiz) xyb
—E
mo
+ uicia ( x.lbd + MO 2
E
uiaibd xuc
rrd y 1..1b k
(15 .79)
mo UiaibXrd + 2 E
+ 2
oki mo
23
MO
2
E ui6,Pdx,fr +
2
mo 2
E U8 3X2'3' +
.
2
ij
The fourth term of eq. MO
2
E
(15.70) may be manipulated as
( a (ilk
r ybc
i3
3 ad
3
ad a
aX 1 '1
bc
tc
3
ad
3
ad
MO
f
=2 E [ uiajd
au P z i x!'"` + ulb
OXI'Pb
r 74b uf•
Xti ad 3
MO
+
—
E uidk u x,vbc
ij
k MO
+ Uiai (X lib" +
E ukiivbc
MO
d „urijbc) L—d Uk li ijkl kt
Ic
MO
x8V"
+ ( up,fi — E uidk u k
)
mo E qixt31" + E U611:Wa
MO + 0.3 ( X !fad +
kl
ax ljea
2j 3 ad
( 15 .80 )
THE FOURTH DERIVATIVE OF THE HAMILTONIAN MATRIX MO
E
+ (
239
xilij" mo Trd
+ liz i ( X Y abd +
y IJab
mo
ud vIJab)]
(15.81)
kl MO
= 2
E
+
u25;dxybe
MO
MO
2
E
+2
2
mo
E u26,5ixiit" +
MO
E
+ 2
2
ij 2
u6.udY.1-jca ij kl zjkl
ijkl
MO
+
d
Lib. kiYijki
ijkl
ii MO
+
ri qt./
MO
MO
E v qxb iif + ii
E ijkl
+ 2
2
"
c Trd -171,1" kl 1
.
(15.82)
The fifth term of eq. (15.70) is treated as
(aui a.b ad3 uica
MO
2
E ijkl
ab
+ U2
b
▪ ad =
MO
2
>2
[
au
3d
+
au!)
aua
kl ria _t_ u 12)c ad kl 3 ad
TT a aUPcl)v IJ
°
'T
2
3
ad
22 11/10
MO
( uzi bd
"V" uidm uranbj ) u
+ ulzib(uoi — E mo
MO
—
E
• (uir
U117,11,6,0(1;1,1
UP3F (UV
tridm u7c.nai Uzi
u icjp (u
MO
- E
ukt„,,u n,)
mo
- E rn
(15.83) MO
MO
= 2
E uf.ibdub
MO
—
+ (1(1—
ijk( MO
+
Ujz„ —
▪uzsaduZi
E
ij km ml
MO
E
uid,,U,b;j Uiad
MO
ut.dm u rza.i upci
UP2PUizji rn
—E
uic; ubi
UUim Uii
MO
E
uicruiLu77,6, (15.84)
11/1 0 = 2
E (uiaibduL + upjour,, + u,7oupc1)ya, ijki MO
+2
E ijkl
(ultibud
+ u,; u1
+
ut')
240
CHAPTER 15. THIRD & FOURTH DERIVATIVES FOR CI WAVEFUNCTIONS MO
— 2 E
UeUk m 7er 1 + idm jU1,14
(Uidm I nb j fc I
Uibi ULU:a
ijklm 11,4m UZUL ± U
Using the result in Section
14.3
7U m
E
(15.70) may be handled
as
nv-IJ
(us9;buL + utfu,a„ +
rrspa Trb ) 1' lijkl 3
V
kl
ijki
mo =2
(15.85)
.
the sixth term of eq.
mo 2
q 1 )yg,
E
(uiaibuz
+
ad MO
+ (Tut)
+ E (u7dorpge1 +
ijkl
E
+4
upd,{G„(ikipn)
mnp MO
=2
E
+ ari;,f,in (ipikn) +
(15.86)
+ u,; u 1 + uffiut)Yiljijcid
ijkl MO
+ 2
(U,Pib UZUpdi
f&ITa Uptri-
ij ki -p MO
+ 2
E
(u4buLu,
+
f/FP ukl, 1 3
pt
pjkl
07uzi updk + ufrutUpdk )Yiljp`ii
ijklp MO
▪ 8
(U,91 Ur,/ Ugm U
tr U1:1 4
Uri
ut1 4m)
ijklm - np
G31-7,, J in (iplkn)
X{
MO
=2
E
(ufiuL
+
ijkl
+ 2
MO z V _s
u6ii9uz1
G hn (inlIcp)}
(15.87)
+ uî; Ujci)lfgad
(Ullib UL u + fit; /PLUpdi
Uff ULU pdi)Yplijki
j kip MO
+ 2 E (u13oucklUpdk +
UZUpdk
W3Pui:1 Updk)Yi 3I-pji
ijklp
MO
2
E
(uiaibu i qn
+
up,Fuzu,nd„
iiklmn The seventh term of eq.
mo 2
E ijkl
[ A utai b
+
(15.70) may be processed jc
ad Ukl ijkl + U13
aub ki Y ad '3k1
uicrutqn)431,1,nn as
IP-Ub
aYiT j
kl ad
(15.88)
THE FOURTH DERIVATIVE OF THE HAMILTONIAN MATRIX OUP
iJa • --g(T (lckiYtjkl ,P
E
=2
037.1.Ja .92
(121 11,1A G
a 11 + ad kl Uazikl mo mo
, Tc
aug, „b
t"
ad
aYilisic;) Ute3 1412 ad
Ytjkl
(15.89)
mo
— E uidny)upd y4,J; +
[(uft
241
—E
ijkl
Utn,q1 Yi ;1kfcl
mo + ut3,, ubki (1 + mo E u d./-4 . + E u y i cd ( y; i.J.d pjkl 7k tjp1
mo + 4
E updm {Glj f,„.(ikipn)c
in (iplkn)c
mnp +
Mo
E
(u
+
uidmumb i )uY.44,a
b
(
G im (inlicp)c}) MO
ukicd —
1m
mo
mo
P
7)
C—•
2_,
d
c
Ukm Umi
IJ a
m
+ upy,c,, ( vad + E u:iyc + E uldA Yiliii: MO
+4
E
mnp
• ( uisd
Updm
jijmn( ik IPTI ) a
_ mo E uidm umc3 ) uicia yi3/C
, 7 ,.F rTa (
jbd
1- Us3 ( kl
I
-1
mo
E ut„u 1) 1/-Aij/C
ulc3
MO
j
q,jin,(inikp)a})
G.ljni j in (iPI k72) a
MO
_ r". ird . v-IJb _L V• ud v.I...0
1-
Zr
L'pt 4 pjkl
m
LspLi pk À ijpl
MO
+4
E ugm{GYmn(ikiPn) b
mnP
G gln(iPlkn) b
MO
=2
E (uilduzi +
U U,P,I)Y4C + 2
ijkl
E ijkl
MO
+2
E ulij uzi rjin zgc,,,„
ijklmn MO
+2
E (u u 1 + u2 U )
MO
+
2
ijkl
E ijkl
MO
+2
E
ijkltnn
3
MO
+2
E @lug, + qUe)Yili; ijkl
MO
+2
E ijkl
Uiai UP,/
Glrn(inikP)b })]
Ygri
(15.90)
242
CHAPTER 15. THIRD & FOURTH DERIVATIVES FOR CI WAVEFUNCTIONS MO
E uzi uzU mn d
4- 2
likhan
(15.91)
ijklmn
The eighth term of eq. (15.70) may be treated as (oultj
MO
E
2
ad
ijklmn
I
MO
E
=2
b
a OUL
ad
UklUmc n 4" Ui3
(qd
—E
,
b
Umn + Ulaj Ilk I
Uidr UT.i)
(It (IL +
°U.M,n)
7IJ lijklmn
ad
uz (NI
MO —
E
r
ijklmn
U4U76.1)14c,in
r
MO
E
+ UiajUjd(UZia —
u:i,u,;,)] zgimn
(15.92)
r MO
E
2
mn
klmn
+ uzuziturc
+ qu11 u4) vki „„
MO
E (ug,. uziuTc„„ +
—2
u:,,u,(!r urbi u,%,n + u27,u,bd uind
UCFIJ
ijklmnr
(15.93) Employing the result in Section 15.6 the ninth and final term of eq. (15.70) becomes
mo 2
E
u?3_ubkluc mn
azIJ jklmn Ocl
ijkirnn MO
=2
E
upc, urcnn Zgdi
n
+
MO
,J u .zriki „,, +
— ;Tr lmn
UrdmZtjjklrn)
ijklmn MO
+4
E urdp
G./4 (ik I mr)
4Jap (inzikr)
G5„(irlkm)}]
(15.94)
pr
MO
=2
E
Ijd Ulzill,baU7c7,„Zijkl mn
ijklmn MO
+2
E (uiai uzi
Urdi Z rij klmn
+
Ut
Urdk Zlijr /7,
ijklmnr (JiajUP,IU7Cnn Uid Zilk irn) . ra
MO
8E
Q.,
u,;,„ Urcip {GY(iklmr)
G.fLp (imikr)
G.Win (irlkm)}
ijklmnpr (15.95)
Combining eqs. (15.73), (15.78), (15.80), (15.82), (15.85), (15.88), (15.91), (15.93) and
▪
243
THE FOURTH DERIVATIVE OF THE HAMILTONIAN MATRIX
(15.95), yields the following equation involving 44 terms:
H IT . actebacOd 84
+
mo Hef d + 2
E ii uid.xpab. 3 3
MO
2 E ut A d-‘2-i,.[J + 2
ii +
MO
MO
E uzbujid u + E
MO
ccuLy.ii;k7,
2
ij
ijkl
MO
MO
2 E ue dx ij IV' + 2 E uexued
+ 2 V-.. ( Tib Ukdi Yi rJ
c
iiid
Mo
+2
E
Uibice 1 X
fia -h
ii
mo +
2
Mo
Mo
2
E vibicx,vad + ii
2
mo
MO
E uicrde +
2
E u,7xii-ed +
2
tij
2
2
E uibtx,v c. +
+
2
ai
ii
E utlxitab + ii
ijkl
MO
MO
2
MO
+
E ui xiii" + 2 E quey4Lb. ij
MO
2
uffugi vie
MO
MO
E ucxe . + ii
+
E
ijkl
MO
+
E eu/da yiii ,Jaa iiki
2
E utxrd +
2
E uuLya,
ij
ijkl
MO
MO
E uivciabd + ii
2
E ufi uki,y,./4," ijkl
(ubd U); ---1 + Uibj9d Ulac1 + UtP3OUL)Ifiii la 2 MO \--z_. ijkl
MO
+2
E (ueuj + ut.,PuLd + uf; [IV) Yili .li ii ki
mo _ 2
E (uidm Uei tTica + UZib (TL U,en, I + Uli,„ Utcj Uel
ijklm
U mo
+
2
E
u2 1
Ulim UZUL + 11,7 Ujimi oni)
+ uibiu +
ijkl
MO
+ 2
E (uzpUlc i U/i
071.11:4
fr
[tupdi ) Ypi
i
ijklp
MO
•
2
E
ijklp
+ q9uf ( u4 + uicrUtUpdk)If 1
244
CHAPTER 15. THIRD Se- FOURTH DERIVATIVES
mo +
E (ueibubucj,n
2
jktmn MO
+2
E
(u î iduz i
+
up.,,ufurnd
FOR CI WAVEFUNCTIONS
Uic; Ut/ Um d eki mn
U NI) Y
ijkl MO
+
2
EUU 1 YÇ +
MO
ij kl
utgypc,udm ,ZiWim ,
ijklmn
MO
+2
E
2
E
(up,dub +
qua) Yaia
ij ki MO
+2
MO
E
E
2
bYi ilfdad +
ijkl MO
+
2
ub udmn zuka,mn
ijklmn
E
(uicilukti
+
ijkl MO
+2
E tif; ukt, yji 1:4 + ijkl
+
MO
E
2
quumd n Ziiikbimn
ijklmn
MO
2 E (ur u tumc n
+ U U,f U, En
Uiai Ut,/ UZi.„) ekimn
ijklmn
MO
(ut u,a.,
—2
/ci g,n
NI u:1„,u7;,) zgimn
ijklmnr MO
+
2
E uei uzi n zeran
ijklmn
MO
+
E
2
(u,F; u7;i.n u,d.,
+ urj
u n u7d.,c zt.V, (mn
ijklmnr
▪ Uifij Ut,/ U,cnn U7d.m 44kIrn MO
E UUb ii ki
+8
umc urdp (GYnr(ikin-tr)
ip (irnikr)
ijkimnpr
+ G
n (irikm))
(15.96)
Collecting terms within (15.96) reduces the expression to 15 terms: MO a 4 Hij H 71/cd + 2 E u p.bcd t.7 13 0a0bacad
ij
mo 2
E ij
(ut9ibcxe
„exiir
uicticd x
Uibld Va )
▪ THE FOURTH DERIVATIVE OF THE HAMILTONIAN MATRIX
mo +
ub xi
2
ux
/cc(
vibxxillad + mo +
2
+
2
-ird
utPfxr
u d xirc
+
uffx//db)
uibi x2/3.../c.24
(utal xird
mo
245
us.c3 )(tiff('
uicsiduji
uicsicdupci
+ u27141 +
u,î) Uij 141
+
Uibid UL)Ya l
ijkl MO
+
ijkl MO
+
no.bucd kl
2 2
u tj
(ubuit
▪
MO
2
IJd
u: U
UP;U:1
ijkl
+
IJ
Td ra
(
(uiPib uL + utduel +
Ujaid Uti)Yai c
(u u,
Uiaid UjOYi jl e
ijkl MO
+2
▪ UU
ijkl MO
+
(u u 1
2
▪ WFUd i.ja sj kl )YIjkl
ijkl MO
+2
1Jed (qUkiYijkl
v.IJI'd kl'ijkl
ira TTC
+
ijkl
US d Yi• jab sj kl sjkl
Uibi
MO
-I-
2
E
(u46141 U4n
,
YiVaCa
fit 14/Y
d)
+ u4cuti u,dnn + uiaiduti uL,
ijklmn +
(Jib; 14/ Winn
+
UP/
MO
+ 2
tiff
ti7cm .,
E
+
Um b n)'ijk1mn
uiluta u,dn„Zir mn
ijklmn UUbUm d n Zlijkbi mn
UtiUliU4n Z1 ahnn )
MO
+ 8 E
(uilutu,,n ugp +
(14
ijklmnop UiajULU m b n Uic)p )G.ffnp (ikimo)
(15 .9 7)
246
FOR CI WAVEFUNCTIONS
CHAPTER 15. THIRD lz FOURTH DERIVATIVES
Substitution of this equation into eq. (15.30) will give the desired result in the next section.
15.8
An Explicit Expression for the Fourth Derivative of the Electronic Energy
By combining eq. (15.30) for the fourth energy derivative and eq. (15.97), which contains the fourth derivative of the Hamiltonian matrix element, an explicit expression for the fourth derivative of the electronic energy is straightforwardly but laboriously obtained:
mo Qiihz bj cd
mo
E
04 Ec
8aabacad
ij MO
2 E
Ad
o
E
+ 2 E uiajbcd
ijki
ij
(uia,bc xidi
u,b3cd x i.„i )
uia,cd xibi
ii MO
2 E
uxibjd
x
ii
ij
+ UXJ + MO
+
2
E
2
MO
+
(qxto
+
u.ib,xiF3ad
(u,5;bco,1
+
ut5.k,bdq
t)
t) )
u2clixiajbc)
X icif
ij
E
+
uvut,
+
utouL)Yijkl
ijkl MO
▪ 2 E (u1;bud + u4cuti +
U4d Ula)Yijkl
ijkl MO
+ 2 V (qb 14/ + UibpJA +
U2701)Yi 34 ki
ijk -i MO
+ 2
E
+ uu i + uiaiduti)ys.Jkl
ijkl MO
•
2
E
+
uzyq,
+
qduOyijb k ,
ijki MO
+ 2
E uffuz, +
+
ijkl MO
+2
E (quya, +
umyibi t
+
quyib,ek ,
ijkL
• 11,9JULIZI
Uibit4iYi7k i 1)
EXPLICIT EXPRESSION FOR FOURTH DERIVATIVE OF Eelec mo
E
2
(qbu,,Umd, + uu1u7 L
247
+ uzdu lu,c„„
ijklrnn -Lu b;i
Ifib34 U1I7cni,
11,5;1 11Jc:1UL) Zijklmn
MO
+
E
2
(UULUL4i klmn
IlUtiUm d „Zi klmn
ijklmn UUjd PiOnn Zic'ljklmn)
mo
E
+ 8
(uzuz,utcnn ucj,
+ quomb n Uodp
ijklmnop t lid On, U:p ) G jinp(ikiM0) CI
▪
2
CI [
E E J
I
+
aCJ a3 Hij aa abacad
aCJ 83 111,7
ab acadaa
ac, a3H,,, ac, a3 H IJ + ac adaaab + ad Oaabac ( 02 Hij 6,1 j 82Edec \ CI [WI aCs 2 7 L.—• aa ab acad aCad ) IJ aci ac./ HIJ Eeec l ) + aa ac 02abad 61,7(92abad acl ac, (a2HIJ 61.1 abac et" + Oa ad abac , aci ac, (02HIJ 6,j a2 Ee. + A ac aad a aaad ) aci ac, (02 1-11.1 a2 Eeiec) 61,1 aaac ) + ab ad accac i ac, (.9 2 HIJ a2Eeiec) + ac biJ] ae ad &Lab &cab ) -
CI — 2 IJ
1- 02C1 &Cs Laaab acad
02 C a2 Cj &Lac abad
02C/ 02 Cji
aaad abac
(His — 61JEe(ec)
(15.98) The second and third derivatives of the Hamiltonian matrices appearing in eq. (15.98) were defined explicitly by eqs. (15.19) and (15.56). The skeleton (core) derivative Lagrangian, Y, and Z matrices are
xi7ibc
A,40
.E ni.
Q j m it'Imbe + 2
mo E mid
G3•
ki (i rn ikoabc
(15.99)
248
CHAPTER 15. THIRD Si FOURTH DERIVATIVES MO
Yijq i = Q1h
FOR CI WAVEFUNCTIONS
MO
-I- 2 E Gi imnokimnr + 4 E mn
,
(15.100)
mn
and MO
27jk1mn — 4 E {Gj ,„,(ikimpr + qinipcimikpr + G.ipin(ipikm)a}
.
(15.101)
It is seen clearly that the solutions of the first- (//a), second- (U"), third- (U abc ) and fourthorder (Uabcd) CPHF equations and the first- (aCi/Oa) and second-order (82C110a0b)CPCI equations are required to evaluate the fourth derivative of the electronic energy for the CI wavefunction. Needless to say, this will be a formidable computational task.
References 1. 1. Shavitt, in Modern Theoretical Chemistry, H.F. Schaefer editor, Plenum, New York, Vol. 3, p.189 (1977). 2. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
Suggested Reading 1. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Theor. Chim. Acta 72, 71 (1987).
Chapter 16
Correspondence Between Correlated and Restricted Hartree-Fock Wavefunctions In terms of the complexity of the analytic derivative expressions, the configuration interaction (CI) case, for which the molecular orbitals are not determined variationally, is by far the most difficult of those considered explicitly in this text. Analytic higher derivative expressions for CI wavefunctions can be used to obtain the corresponding energy derivative formula for multiconfiguration self-consistent-field (MCSCF), paired excitation (PE) MCSCF, general restricted open-shell (GR) SCF, and closed-shell (CL) SCF wavefunctions by explicitly imposing variational and orthonormality conditions on the molecular orbital (MO) space. The reduction procedure used follows the steps CI —> MCSCF —> PEMCSCF —> GRSCF CLSCF The equations expressing the correspondence between correlated and RHF wavefunctions are presented in this chapter to interrelate explicitly the conditions and definitions involved in various energy derivative expressions.
16.1
Configuration Interaction (CI) Wavefunctions
16.1.1
Electronic Energy and the Variational Condition
The electronic energy of a CI wavefunction is given by CI
Eelec =
E
CiCjHi
IJ
249
(16.1)
250
CHAPTER 16. CORRESPONDENCE EQUATIONS MO
MO
=E
.
+E
Qh 3
(16.2)
ijkl
ij
The Hamiltonian matrix element 11 ij between configurations I and J is (16.3)
H/J = ( 4) /iHeiec0J)
mo
E
mo
Qwhi ,
+E
ii
(16.4)
ijkl
The one- and two-electron reduced density matrices, Q and G, are related to their "bare" quantities, the one- and two-electron coupling constant matrices, Q" and G", through
E
=
ci c.,Q1)7
(16.5)
IJ
and
CI
E
Gijkl
CjCjGf j
(16.6)
IJ
The variational condition in the CI space is
E c, (His
— 6 IJEelec) = 0
(16.7)
and the CI coefficients are normalized in the conventional manner: CI
E 16.1.2
1
(16.8)
First Derivative
The first derivative of the electronic energy expression was derived in Chapter 6 and is
°Ed e, Oa
CI
aHisi Oa
E IJ
(16.9)
MO
MO
+E
=E ij
ijkl
MO
+ 2
E
U
:3
23
(16.10)
The Lagrangian matrix X in this equation is defined by
mo =
E
Qj mhim + 2
mo
E
(16.11)
mid
Note that the first derivative expression (16.10) requires the solution of the first-order CPHF equations for Ua.
CI WAVEFUNCTIONS
251
Second Derivative
16.1.3
The second derivative of the electronic energy expression was derived in Chapter 6 and is 49 2 Eelec
0a0b
ci
(92 H I j
=E
CI
2
0a0b
IJ
aCj E — — Oa Ob IJ
nIJ –
IJ Eelec)
(16.12)
where the first term in eq. (16.12) may be expressed as CI
MO
IJ
ij
HIJ E c1c, a20a0b
MO
E
+E
Gijkl(iilkir b
ijkl MO
+
2
E ux MO
MO
2
L ( ub, X a
qxpj
)
+2
E
UUPaYijkl
ijkl
ii
(16.13) The skeleton (core) first derivative Lagrangian matrices in this equation are
mo 1; = E
X
gim.h`tyn. + 2
mo E
,
(16.14)
mkt
and the Y matrix is given by MO
Yijkl =
Qjihik
+2
E
+ 2Gimin (imikn)}
(16.15)
ran
It should be noted that the second derivative equation (16.12) requires the solution of both the first- and second-order CPHF equations (for //a and Uab ) and the first-order CPCI equations (for OCII0a).
16.1.4
Third Derivative
The third derivative of electronic energy expression derived in Chapter 15 is Hjj E ciCj 03 0a0b0c
03 Ede,
Oa0b0c
IJ
CI
+
2
E E CI
+
Cl (
2 IJ
act, a2 HIJ Oa
Ob0c
[ ac, ac, aHLT
acf a2HIJ Ob
Ocaa
6, JaEd") a ab
a2 HIJ) 0e 0a0b
252
CHAPTER 16. CORRESPONDENCE EQUATIONS &CI OCJ (OHL/ ab Oc Oa
45 IJ
aci oc, (aHIJ Oc
451.i
Ob
Oa
aEelec ) Oa OEciec )]
(1 6 .16)
Ob
The first derivative of the CI Hamiltonian matrix was developed in Section 6.4 as MO
oHif =
E
+2
Oa
(16.17)
23
where the Ha and XL/ matrices are
mo Alo Hy,= E Qh a ±E Gffk,(iiikoa 2J U ij
MO
=
,
(16.18)
ijkl
MO
E
Qhi, +
E
2
Glimj kt(imikl)
(16.19)
mkt
The second derivative expression for the Hamiltonian matrix was derived in Section 6.6 as mo
li2Hj j
E uPox!,/
Hy; + 2
Oa0b
Ji
tj
ij
MO
+ 2
E
UPi X
jj
a
+
Xjijb
ii MO
+ 2
E
(16.20)
U
ijkl
where MO
E
I-11j =
ij
MO
QW11..(i'î
E awk,(iiikoab
MO Xlija
=
E
(16.21)
ijkl
MO
E
Q h'Ini + 2
G!jm j
(16.22)
mid
xrIJ
ijkl
MO
=
hik
+ 2
E
2Gg in (imilcn)}
(16.23)
mn
It is useful to define the modified first derivative Lagrangian matrices as MO
E kl
(16.24)
CI WAVEFUNCTIONS
253
Using these quantities and the result from Section 15.4 the first term in eq. (16.16) is explicitly given as CI
03 111,1 u fuj 0a0bac IJ
MO
MO
▪E
E
Qiih731?'
MO
2
E
G jkl(ijikl) abc + 2
uiaibcx ij
ijkl
ij
+
MO
E
(utPib4]
+
ut-b;xlcii
+
ui74])
ij
MO
+
2
E
(ux,b; + ut.i xicr +
uivc,1))
ii MO
2
E
ILV Or ke
ijkl MO
+ 8
E
(u,f1;
uzi u +
upj (41u,ann
ijklmn
+
uici uqn)
MO
x E Gii(ikimp) .
( 16 .2 5 )
The skeleton (core) second derivative Lagrangian matrices in this equation are defined by MO
E
X
MO
Qfra ld + 2
E
Gimki(imikoab
(16 .2 6)
mkt
and the skeleton (core) first derivative Y matrices are given by
mo = QiNk -I- 2
E
{Gj imn(ikimnr
2Gjmin (irnikn)a}
(16.27)
mn
Note that the third derivative equation as formulated above requires the solutions of the first- (Ua), second- (Uab) and third-order ((M c ) CPHF equations and the first-order (OCl/Oa) CPCI equations.
16.1.5
Fourth Derivative
The fourth derivative of the electronic energy found in Chapter 15 is 194 Eel ec
CI
a4HIJ
E
CICJ 0a0b0cOd IJ Oa0bcd CI
2
E
CI
+ E [ac-' Oa ObOcOd
a311" Ob 0c8dOa
254
CHAPTER
ac 9 3H,,
a3HIJ
Oc adaaab I [ a c, ac,
2
ad aaabaci s 02 Eetec)
HIJ
act ab U9côd ( (92H ij
IJ
16. CORRESPONDENCE EQUATIONS
acad )
-1j
2 Eeiec) 15 .ti 0abad
°bad (82111j
6.1.1 4n 2 LJelec .
-/
abac
abac
( 82 II ij
02E 1
6.1.7 aaad )
&Lad
(a2H,,
a2Eeiec)
aaac
Oa&
ac, (a2HIJ
s (92 Eeleci
'ij aaab )
Oc ad &Lab
a 2c 82c, 8 2c, 021 E j + + IJ 190.ab aCad act& abaci aaad abac 2c
— 2
a
(5/JEciec)
(16.28) The third derivative expression of the Hamiltonian matrix was developed in Section 15.4 as
Hrt +
aaabac
mo 2
E
(P ubcXW
ij MO
+ 2
E
(ula3b x iI2Jc
MO
+
2
E
utbicx iIta
+
mo ▪
2
E
+
ii ii
ii
+
ufrx,r)
+
+
ijkl MO
+ 2
E
+
utt4i yi/x
+
IficiULYab )
ijki MO
▪
8
E
MO
(uutu7,„„
ijklmn
+ u u 1 u + u.,,uzu,bnn) E
GY.,(ikImp) (16.29)
where the
HIT, X IJa6 , and Y "a matrices are defined by MO
H7.1.)7c
MO
E (2 1''ek u +E ij
ijkl
(1 6. 30)
▪ ▪
Cl WAVEFUNCTIONS
255 MO
MO
Q IJ
3m tm6 +2
E
(16 .31)
ki(iTnikl) ab
mkt
and V IJa
QjliJhik a
MO
E
+2
{G3111„,.(iklmn) a + 2G
in (imiknr}
.
(16.32)
mn
It is then useful to introduce the modified second derivative Lagrangian matrices as ,[ab] A
ij
MO
_L_ E X/ 2.1
b Va. Uk1' ijkl
MO
ab nkl ijkl
E u i unb,„Zijklmn
kl
(16.33)
klmn
where the Z matrix is defined by
mo Zijkimn = 4
E {G3,„p(ikimp) + G in,p(imikp) + Gion(ipikn)}
(16.34)
Using eqs. (16.24) and (16.33) and the result from Section 15.7, the first term of eq. (16.28) is explicitly written as CI
E IJ
CICJ
actObacOd
mo
MO
t34H jj =
E
hy.76cd
ij
MO
(ii ikoabcd +
2
ua ijbcd xii
ijkl MO
2
Gi
uibri xi[V
E
trip 4] +
b xs[jc])
MO
2
E
usajc 4bd]
+ uibpdr] MO
2
E
u sad x [be]
3
3
:3
+ upfx ,[71 + u2c3 ix13 h1 )
(u- x1630 23
uspi xicila
u-fi xdab
uxaibc)
MO
2
E (uiaibud + uia.tupi +
Urjd UZ)Yijkl
ijkl MO
2
E (uFLyivc, + uzu,oriv,, + qUgiYini ijkl
+ u3 UficciltiVa +
vac
+
u2ci ug1y4 ti )
MO
▪
UU5Umd n Zriki mn
2 ijklmn
256
CHAPTER 16. CORRESPONDENCE EQUATIONS tifiU LULZ.tiki m ,
U4UVim& nZZi klmn)
MO
8
E
u_ jaiuP _-
Uodp
u tI
nnU :122
ijklmnop
fia m b n ii4)G ii np(ik17110)
.
(16.35)
The skeleton (core) derivative Lagrangian, Y, and Z matrices in this equation are defined by mo mo xiaibc E Q,mh,:mbc + 2 qi„, k1(imikoabe (16.36) 7
E
mki
ni
Yi
7tr =
mo + 2 E {Gi,mn (ikimnrb
2Gimin (intikn) c1 }
(16.37)
mn
and Zi kl mn
mo = 4 E {Giinp(ik IMPr
G iniP(iMikP) a
G jpin(iPlkM) a }
.
(16.38)
Note that the solutions of the first- (Ua), second- (Uab ), third- ((Me) and fourth-order (uabcd.) CPHF equations and the first- (aCdaa) and second-order (02 C110a0b) CPCI equations are required in the above formulation to evaluate the fourth derivative of the electronic energy of the CI wavefunction.
16.2
Correspondence Equations
Higher energy derivative expressions for configuration interaction (CI) wavefunctions described in the preceding section can be used to obtain the corresponding energy derivative formula for multiconfiguration (MC) SCF, paired excitation (PE) MCSCF, general restricted open-shell (GR) SCF, and closed-shell (CL) SCF restricted Hartree-Fock (RHF) wavefunctions by explicitly imposing variational and orthonormality conditions on the MO space. The general structure of the reduction procedure used in the following sections is CI
MCSCF
PEMCSCF
GRSCF
CLSCF
The equations expressing the correspondence between correlated and RHF wavefunctions are presented in detail in each section to interrelate the various conditions and definitions involved in energy derivative expressions. Table 16.1 on the next page summarizes the correspondence equations for some basic quantities.
Table 16.1
Wavefunction Matrices
CI
one-electron density two-electron density
God
Correspondence Equations for Basic Quantities
MCSCF
PEMCSCF
ri
26iili
rijkl
Sij 8klaik
X i;
x ij
Derivative Lagrangian
Y
Yijkl
Derivative Y
Yi;k1
Yijkl
CLSCF
28ii6ici
+-1- ( Sik 6J1
-11(Sik 61i/ Lagrangian
GRSCF
— gSikSji
2eii
2Fij
2f 1 . Ji
iai2F
Sitbik)
28j1Cik
+ 2 71],ki
25ikFik
2Aii,ki
Cia
2r1 r1
28jk Fit
2.24.7j,ki
258
CHAPTER 16. CORRESPONDENCE EQUATIONS
16.3 Multiconfiguration Self-Consistent-Field (MCSCF) Wavefunctions 16.3.1
Electronic Energy and the Variational Condition
The electronic energy for an MCSCF wavefunction is given by CI
Eel ec
=-
ECiCj Hjj
(16.39)
IJ MO
MO
ij
ijkl
E
E
(16.40)
where the Hamiltonian matrix element H11 between configurations I and J is defined by (16.41)
( 4) 11lielecI tkl)
mo
MO
E ij
+E
(16.42)
ijkl
The one- and two-electron reduced density matrices, 7 and r, are related to their "bare" quantities, the one- and two-electron coupling constants, 7" and through CI
=
E
CIC.rdij
(16.43)
IJ
and CI
'ijkl
= E ci c,rff,„
(16.44)
IJ
All the equations in Section 16.1 are applicable formally to an MCSCF wavefunction. However, two additional conditions in the MO space must be satisfied for this type of wavefunction. First, the variational condition —
0
(16.45)
with the Lagrangia-n matrix x defined by MO
E
MO
7jrn him
+2
E
(16.46)
mid
must be met. Secondly, the orthonormality condition on the molecular orbitals
sii =ôi
,
.
must be imposed. Here .5 is the usual Kronecker delta function.
(16.47)
MCSCF WAVEF UNCTIONS 16.3.2
259
First Derivative
The first derivative form of the orthonormality condition for the molecular orbitals was obtained in Chapter 3 as Sf'.7 = 0 . (16.48) Using eqs. (16.45) and (16.48), the first derivative expression for a CI wavefunction, eq. (16.9), may be specialized to that for a MCSCF wavefunction as follows:
Equation (16.52) shows that the first derivative methods for an MCSCF wavefunction do not require solution of the coupled perturbed MCHF (CPMCHF) equations. The variational parameters (Cp and CI) are optimized simultaneously for the MCSCF wavefunctions as stated clearly in eqs. (16.7) and (16.45). Thus, the MCSCF analytic derivative evaluation is much simpler than the CI case with the same set of configurations.
16.3.3
Second Derivative
The second derivative form of the normalization condition of the molecular orbitals was found in Chapter 3 as U46 + Urib =0 (16.53) The augmented S matrix, cab
se3
mo
in the above equation is defined by
+ E (uia,„uab,„, +
- slim s3b
—S
tm 57m)
(16.54)
CHAPTER 16. CORRESPONDENCE EQUATIONS
260
The second derivative of the MCSCF energy is obtained from eqs. (16.12) and (16.13) as Eelec
actOb
CI
ac, ac.,
8202 Hij
CI
aabb =Cj IJ
2 E 1.1
. '"Eet-)
(16.55)
The term involving Uab in eq. (16.13) may be rewritten in terms of elb using eqs. (16.45) and (16.53) as mo
2
E uexii
mo
.
+ U`li)xii
(16.56)
mo
= — E eilibxii ij
(16.57)
Combining eqs. (16.13), (16.55), and (16.57), the second derivative of the MCSCF energy is given by a2Edec
0a0b
mo
E
-yijh.71
Alo E
MO
E
xi,
mo + 2
E (uibi xy, + q 4) +
—2
E 1.1
CI
aCI 0Cj ffIJ 8a, Ob k
mo
2
E
UU j yIkI (16.58)
61JEelec)
The skeleton (core) first derivative Lagrangian matrix, xa, appearing in eq. (16.58) is defined by
A40
E
mo
7jm
m+2
E rimk,(imikoa ,
(16.59)
and the y matrix by MO
j ki
+ 2
E Yi
+ 2rin„,,,(irnikn)}
.
(16.60)
mn
Note that the second derivative of the MCSCF energy requires the solution of the first-order CPMCHF equations. These equations provide the derivative information for both the MO coefficients (Un) and the CI coefficients (ÔC l/ 29a).
16.3.4
Third Derivative
The third derivative of the electronic energy may be obtained from the CI third derivative expression presented in Section 16.1. For this purpose it is necessary to introduce the third
MCSCF WAVEFUNCTIONS
261
derivative form of the orthonormality condition on the molecular orbitals as ua.bc sj
uabc Js
I
'
tatic s43
=0
(16. 61)
where the augmented S matrix, eibc is defined by •
qabc MO
+ E (uiamb u 77, + + U, u! + u.gyia + UsÇ U p + 7. mo — E (stsï + .571 sf + stcn s.7,„ + sl;„s?„, +57,a„stf +
u)
mo
+E mn (5s1;n s,cnn + ssbin s,c7. + stm s;„s„ +
S m S nS n
+ Sf„,,S1„Sm b „ + S.7„,574,,g,„)
(16.62)
It is further required to introduce the first derivative form of the variational condition on the MO space: OS •
OX
Oa —
Oa
(16.63)
0
It is useful to introduce the modified first derivative Lagrangian matrices as ,„[a]
MO
E
=
Ugiyijkl
(16.04)
kt
The first derivative of the Lagrangian matrix in eq. (16.63) may be expressed using the result in Section 14.1 and eq. (16.64) as 0Xu
Oa
=
+ E ugixk; +
[a]MO
X tj •
CI
2
E IJ
CI
OC Oa
x '3
(16.65)
with the "bare" Lagrangian matrices x" MO
X
=
MO
E 71„,'I
+ 2
E r5mj
(16.66)
mkt
In eqs. (16.16) and (16.25), those terms potentially including the second and third derivatives of the MO coefficients are
mo
MO
2
E
qbcxii + 2
CI
+2
CI
Ec,E 1
E (uztxti + up;xt9 + 1/274)
0c, j 0 2
Hij
Oa Mac
A.ry j 0 2 Hij
j a2Hij
+ —0; OcOa + ac 0a0b
. (16.67)
CHAPTER 16. CORRESPONDENCE EQUATIONS
262
The third derivative of the MO coefficients in the first term of eq. (16.67) may be eliminated using eqs. (16.45), (16.61) and (16.62) as MO
MO
2
E
E
ut5Libcxi,
(uiarpc
ur c) x .,
_ MO E cçijabc
(16.68)
j
(16.69)
ij
_ MO E se cxii mo
-
E
2
+
ufkbu;k
uibink
ijk AIO
+
uict:14k x ii
E (s s;1, + SSJk +
2
ijk Af0
-
E( ss1s +
2
+
cj
sansti )sii
(16.70)
ijkl
Referring to eq. (16.20), the last term of eq. (16.67) involving Uab elements is explicitly written as CI
2
CI
aCja2 Hij
E c, E
ac [
CI
CI
=2
ac Om%
E E
MO
+ 2
E u2Fi4 bzw +
MO
2
E
b)
ij MO
-I- 2
E
U j Utyijki
where the H ab, x Ija , and
y"
(16.71)
matrices are MO
MO
Hy; = E 7,1/14 ij
E
(16.72)
ijkl
MO
MO
E 7j h m ± 2 E G,J k,(imikoa
=
(16.73)
mki
MO
+ 2 E
=
+ 21-141n (imIkn)}
(16.74)
ran
Combining eqs. (16.70) and (16.71) with eq. (16.67), the terms involving the second-order U matrices are manipulated as follows:
mo 2
E
mo
qbpic)
–E
oi
UiccjXki +
2
E
ci —acJx1
ac
(16.75)
MCSCF WAVEFUNCTIONS
rii
mo
E ro OC 23
263
E
= 2 u,cci X kj + UfcjS ki
MO
—
MO
\---, eb O; ii 13 ac
E
[
MO
=—
u, i x, .;
+ uki Ski
MO
E ic p[)_ E L.
k
ij
(16.76)
CI
(16.77)
ac., i
l
Uk i X k • + 2 \--.` z_..u1, 0c x " IJ
(16.78)
The de fi nition (16.65) was used for the first equality from eq. (16.75) to eq. (16.76) and the relationships (16.53) and (16.63) were introduced to obtain eq. (16.78). The skeleton (core) second derivative Lagrangian matrices are defined by MO ab
=
E
Lab + 2 -yim uirn
m
MO
E
r.i mki(imikirb
(16.79)
2r,m.in(imikn).}
(16 .8 0)
inkl
and the skeleton (core) first derivative y matrices by MO
+ 2
=
+
E mn
Using the result in eq. (16.78) and definitions in eqs. (16.79) and (16.80), the third derivative expression for the MCSCF energy is found from eqs. (16.16) and (16.25) to be MO
a3Eeiec
E
8a0bac
— E
MO
+
j
i.j
MO
E
ij
ijkl MO
E
+ 2
ses;,,
+ SS,ak
+
SgSlik xii
ijk MO
-
E
2
+
Sliks.71,51
Sfk571,511) X ij
j kt MO
•
E
2
(uzst;; + uti xf1
Uiej x
ii MO
▪
E (_u _ ,7.1ici ki
2
UibjnyZiki
+
+ uu 1u
.
u_ elYtkl
ijkl MO
+
8
E
(qut u„
ijklinn MO [
MO
E (xc) — E
MO
E
E r;,„„(iklmp)
MO
UkiX ki)
z3 ef; (x
MO
+
jr ki)
bc eii X —
E
UjcljS ki
264
CHAPTER 16. CORRESPONDENCE EQUATIONS CI
2
CI
E ciE
(ac,02H,i, Oa
ac,(9214.1
Ob0c
021-/-;,) Oc Oaeb
Ob OcOa
CI
2
OC1 OCJ (19HIJ [ Oa Ob Oc IJ OCIOCJ (OHL! Ob Oc kOa
E
9 C1OCJ (011.1.1 Oc Oa Ob
451,1 °Eelec Oc ,61.7 61.1
0Eelec )
Oa E aebi „ )]
( 1 6 .8 1)
where
— MO rbxlJ + 2 E E
mo
MO
02 .11;j 8a0b
ij
22
12
Uibi X 131a + UlLi
Tb
E ut'Lj Ut Yitki
) +2
ij
. (16.82)
ijki
Note that the third derivative of an MCSCF wavefunction requires only the solution of the first-order CPMCHF equations.
16.3.5
Fourth Derivative
The fourth derivative of the electronic energy may be obtained by manipulating the CI fourth derivative expression presented in Section 16.1. For this purpose it is necessary to introduce the fourth derivative form of the orthonormality condition of the molecular orbitals as Eriaibcd u;ibcd clad = 0 (16.83) ebcddefined by where the augmented S matrixis ',ii
sec.,/ MO
E (uitc u.cjim
u;mbc uidm
uitcnd u-rm jm
+ UtÇ U
+ u;:inauibm + u U + upnbut,F,n )
+ E(uitug, + —E
uia,,,
ur,bn
uicmd + uiancy.ibnzd +
+ uldi ut;„ + ul,d„ut)
+ srifscini + st;,dslr„ + scimda s.ibm
slçmda
sdab s;n1
Sjtb Sfni
mo
E (siamb s;md + slmb sf + sg. s.b4 + sg,std„ + ss + sg st) jiaibcd
(16.84 )
MCSCF WAVEFUNCTIONS
265
The w matrices in eq. (16.84) are defined by abcd
- m0 E
jntb r ciind
+
sand
s.im c c il)rn d
d Sir jt ,n
Sr
Lf cn
In
s6cnr .arnd
sticnr arnd
mo [ E + (s-r.nbn,
sbnzd r .77cn _4_
(Sfm Sin
mn
(.5:'„c„
irtarcol.)
(457ndn
71- 171.11
ad
S Siln
cç
cb
)
5.40 r qb
sbd r ac jrn
tyn
scd r ab
3 771
JM SM
Sidn)
S,bitn4n) I s ni s fn )
"1171. " 31/
+
71-'gn)(sli s.ln + slm slin )
+
- irm"n)(514m5;n + 57m5fn)
+
- rzin )(nn s.9,., + sïm s)1 (16.85)
where
mo
7r ab
E
=
MY&
+ s;.„ky,lk)
.
(16.86)
In eqs. (16.28) and (16.35), those terms that potentially include the third and fourth derivatives of the MO coefficients are MO
2 E wiba xi
MO ,+ 2 E (u,a.p cxv.] +
ij
ij CI
-I- 2
E I
CI
C1
E J
ur x(c.iI ] + uicid a x [bi] + uidia b x lci])
3
.Q,,,^r _Q,
Ell-, j
u3„,,
Oa
abacad
Er
a3H" acJ 03 H I J + + ac, ab Ocadaa + ac &law%
OCJ ad
&Ill.')
aaabac (16.87)
The fourth derivatives of the MO coefficients in the first term of eq. (16.87) may be removed using eqs. (16.45), (16.83) and (16.84) as
mo
2
E
mo
E
( ultibcd
_F u;bed) x ij
E abcd _ MO ii MO
Esiapd xi, ij
(16.88) (16.89)
266
CHAPTER 16. CORRESPONDENCE EQUATIONS MO
E (uiakbcri, +
— 2
ijk MO
-
E
2
Uibic,d 116A + [la
+
ufkbuo,
MO
+
E (stcsik
2
+
uzyuî,(!
uadt47,
b Ulk ) X ij
xi,
scga stik
,511 r1 Sjk
stdkab s.c7k ) x
ijk
MO
E
+2
+
(szcbsv,
stsvsv
+
SAd SIA)Xij
ijk MO
- E
abcd ij
(16 .9 0)
X
Referring to eq. (16.29), the last term of eq. (16.87) involving U abc elements is explicitly written as ci
2
CI
ô3 H E c, E ac, ad Oa0b0c CI C ac, [Hgc ± 2 C1
=2
jj
E
Od
MO
>7ij uP!'ex'T4 ii
z3
MO
+ 2
E
(ulxtr
Ui17414
(1,7416 )
(utPi slybe
UfiXfica
(Tic jXli ab )
mo + 2
E
mo +2
E
(qb(1 1
UP;UL
ijkl MO
+ 2
E tilt(
tP-U"jc tj klYijkl
+ u-tut y,W; + uici uty fikbi )
(
mo
MO
+ 8
uic;(11) yii,,fk ,
E
uziuLt
+ ut ubuAn + uici utumb n)E r(ikimp)]
ijklmn
(16.91) where the Hyijc, x mab , and y lja matrices are MO
117.bic
IJ Labc
=
7ij ij
1.1-6
Xij
mo
E
r1311(iilki)abc
(16 .92)
ijkl MO
+2
E .f mj ki(i7111k1) 2b
mkt
(16.93)
MCSCF WAVEFUNCTIONS
267
and MO
711J hrlk +
y1.44/ =
E {rymn (ikimnr +
2
2T i14 /„(irnikn)a}
.
(16.94)
mn
Combining eqs. (16.90) and (16.91) with eq. (16.87), the terms involving the third derivative U matrices are treated in a similar manner as for the second derivative U matrices in the preceding subsection: MO
ci
—E
+ 2E IJ
MO
2
E ij
3
ij
E
ad
MO
+
qi xk i
(16.96)
MO abc[aX ij
=—
(16.95)
ad Xt3
mo
mo E uia bc [axij
=2
ac,
(U4Xk 3
8d
ij MO
E
[d] _
MO
CI
+
E
2
E
CI
IJ
ij
(16.97)
UkliXki)]
OCJ
(16.98)
ad
The definition (16.65) was used for the first equality from eq. (16.95) to eq. (16.96) and the relationships (16.63) and (16.83) were employed to obtain eq. (16.98). It is useful to define the modified second derivative Lagrangian matrices as {ab]
X•
MO
MO a
= .L, *3 • • -r-
E
TT& „,a (Ld kiyiiki
rra „,b, kly zj kl
kl
E
(16.99)
u;Lc,Um bnZij klmn
klmn
where the z matrix is given by MO Zijklmn = 4
+ ri nip(imikp) + ri p,„,(ipikm)}
E
.
(16.100)
The skeleton (core) third derivative Lagrangian matrices are defined by MO abc
E
73n., 71
7.mbc + 2
MO
E
jrnkl(iMIko
abc
(16.101)
mkt
the skeleton (core) second derivative y matrices by MO ab
7j1hikb
+ 2
2r,min(imikn)°b}
E
(16.102)
mn
and the skeleton (core) first derivative z matrices by Iwo 41;kimn = 4
E
{riinp(ikimp).
+ rinip(imikp)a + r,pincipikm).}
.
(16.103)
268
CHAPTER 16. CORRESPONDENCE EQUATIONS
Using the result in eq. (16.98) and definitions from eqs. (16.99) to (16.103), the fourth derivative expression for the MCSCF energy is found from eqs. (16.28) and (16.35) to be MO
MO E Ni ha:cd + E riiki(iiikir bed _ MO E szpcd x
a4 Eelec
8a0bcd
i.;
ijkl MO
-
E ugui, +
2
Uiakc ql,
Uiakd ul.g
sbiccd s,iak
sala
Xij
ijk MO
,
+
2 E sg c 51
+
2
MO
E
s.ibk
+ sTkcs_bikci + stFes.bg
stdicab E;k
Xij
ijk MO
-
E
▪
2
Wia,bcd xi
ij
MO
E
ui
(ufibx!cil
afx[i
+
U,Pid
lbj cl
ij
+
U1
xt
; d1
+
+ uic,f' x!";b3 )
MO
2
E (qxtir + upi xf.la + utP,.(if + ulli eec) ii
MO
+ 2
E
+ uzicuu +
(uzbud
qdUtf)Yijkl
ijkl
▪ 2 E (qUtyai MO
Q.;
YVkl
ijkl
+ rrbijvTrd
Ici Yia3 MO
▪
2
ac
ti
E (quz,
+
fra.rull bc zjL kYijkl
„c Trgiyaibici ukuc t, kl ud mn tjklmn
ijklrnn
UZ.,,
Um b n Zijklmn
kimn
MO
8
E
(Ut5LJUP,IU,,Uodp
UtPitILIU,bytn Uodp
Uiajqum b
ijklmnop
X fiinp (ikimo) MO [
- E cp(xi ij
MO -
E
qxki)
+
etild (x ta]
MO
2_, Ic
UIC:i X ki
MCSCF WAVEFUNCTIONS
269
(b
ccda
-r 4,ii
MO
E uz,ski)
i—
[c]
d ab
x„
mo
— E
(4i xki)]
CI CI E E acJ ab acadaa aa abacad a3H1,1 i, 49C j a3pr Oc adaaab ad aaabaci [
▪
2
+
2
CI
[•-•— .1.
ou Oa
E IJ
02E le )
(02 1ab aced
IJ
(a2 HIJ
1.1
abad
Oa ac
acl ac, (a2HI , • Oa ad abac (a2 HIJ • Ob Oc &tad (a2 HIJ
45 /J
6. /.7 .61.7
Ob ad &Lac
aci ac, 02 11.0 Oc
CI -
2
ad
bIJ
aaab
r a2cia2c,
1J Laaab acad
ec acad
a2 Eelec)
abad
0 2 Eelec)
abaC ) (92Eelec )
&Lad (92Eelec )
aaac 02 Eelec)1
&Lab
0 26,1 ,92c
02c 1 a2c j ]
aaac abad
aaad abac
6 IJEelec)
(16.104)
where
mo
E
te abc
aaabac
rabc IJ X ij
ij
uox14` 2, 2,
+ 2
+
up;xfia
+
ui7xff)
(
+ 2
(a ii ii
+ qzfr a +uici xWa b ) + U1b; u1 +
+ 2
(uzuZ,y(„C,
+ 2
'J U2c2aUI1)
+ up„ uLyffica, +
mo 8
E
(uzut,1u7c
+
IlUZ1U;Inv,
uic„uAyfit) +
mo uf„uzumb n )E
r(ikimp)
ijklmn
(16.105)
270
CHAPTER 16.
CORRESPONDENCE EQUATIONS
Note that the fourth derivative of an MCSCF wavefunction requires the solutions of both the first- and second-order CPMCHF equations.
16.4
Paired Excitation Multiconfiguration Self-Consistent-Field (PEMCSCF) Wavefunctions
16.4.1
Electronic Energy and the Variational Condition
The electronic energy of a PEMCSCF wavefunction is given by the much less intricate (compared to general MCSCF) expression CI
=
Eelec
E
( 1 6.10 6 )
IJ MO
MO
=
2
E
E
fihii
Oij(iiiii)}
•
(16.107)
Two-configuration self-consistent-field (TCSCF) and generalized valence bond (GVB) wavefunctions are in turn special cases of PEMCSCF wavefunctions. While the equations from the previous section for an MCSCF wavefunction formally are applicable to a PEMCSCF wavefunction, the following correspondence relationships are introduced to further simplify the energy and derivative expressions. For the one-electron reduced density and coupling constant matrices the relations
(16.108)
71j = 26i3 ft and _IJ
hold. For the two-electron reduced density and coupling constant matrices,
rij kl
1 2
= bijbklaik
and
= 45i j (5k1
c ( 0 ik 0 jl
5i14ik)Oij
1
(6 -05 3 + 6il (53k ) 13i1/ 2 These equations clearly show the one-dimensionality of the one-electron reduced density matrix and the two-dimensionality of the two-electron reduced density matrix for a PEMCSCF wavefunction as was described in Chapter 8. tijkt
Using eqs. (16.108) and (16.110), the expression for the electronic energy, eq. (16.107), is derived from eq. (16.40) as follows : mo
Edec
=E ij
mo
+E ijkl
(16. 112 )
PAIRED EXCITATION MCSCF WAVEFUNCTIONS
271
mo
E (2,5„fi )hi, mo +
+
(16.113)
MO
MO
= 2
+
2
ijkl
E
fihii
MO
E aik(iiikk) + E
+
ik
(16.114) MO
•
2
MO
E
fihii
E
(16.11 5)
ij After similar manipulations, the CI Hamiltonian matrix for the PEMCSCF wavefunction becomes =
MO
MO
E
+E
ii
2
(16.116)
ijkl
MO
MO
E
E
(16.117)
+
The variational condition in the MO space for the PEMCSCF wavefunction is given by —
E3 , = 0
(16.118)
where the Lagrangian matrix E is Ado
= fihij
E
.
+
(16.11 9)
Using the correspondence equations (16.108) and (16.110), eqs. (16.46) and (16.119) are related as follows: x i,
mo = E
ntim hin, + 2
rn.
mo E
(16.120)
Ink(
MO
E (2,53„,,f,) him 711
MO
+
2
E
{bino5kieck
1 — (. 15k6 rn 2
64 3 6 k)0 . ,,}(intliol)
(16.121)
mkt
= .2fihii
mo +
2
MO
mo
E a,k(iiikk) + E
+E
fijk(iklki)
(16.122)
▪ 272
CHAPTER 16. CORRESPONDENCE EQUATIONS
mo
16.4.2
=
2fjkij + 2
=
2Ej i
+
E
(1 6 .123) (16.124)
First Derivative
The first derivative of the electronic energy is obtained from eq. (16.52) using the correspondence equations, eqs. (16.108), (16.110) and (16.124):
mo
aEciec
mo
E
Oa
mo
E
ii
-
E
ijkl
(16.125)
ij
MO
=
E
,261,f9 h7;
mo
+ ijkl
s s t oiiskiaik + -21 (uikui/ +
ss
MO
E
Srj (2) mo
•
2
-
2
E
(16.126)
mo
+E
fh
faij(iii.i.i)a
fiii(iiiii) a}
mo
E
szci;
(16. 1 27)
Equation (16.127) shows that, as with any other MCSCF wavefunction, the first derivative of a PEMCSCF wavefunction does not require solution of the coupled perturbed PEMCHF (CPPEMCHF) equations.
16.4.3
Second Derivative
Before continuing on to the second derivative expression, it is necessary to define one more correspondence equation for the y matrix: 1 215j1(*; + 274 tj,kt
Yijkl
The generalized Lagrangian ( I and
T
(16.12 8)
matrices in this equation are
mo =
E
faik(ijikk)
Ak(ikijk)}
(16.12 9)
▪
PAIRED EXCITATION MCSCF WAVEFUNCTIONS
273
and
(idijk)}
= 2a,„(ijIld)
.
(16.130)
Given these correspondence equations, the second derivative of the electronic energy for the PEMCSCF wavefunction is derived from eq. (16.58) as MO
02 Eeiec
MO
E 7i;h71 +
Oaab
ij
rijkl (ijIkl) ab ijkl
MO
E MO
+
+
E
2
MO
q.14,) + 2 E
ij CI
ijkl
E
— 2
IJ
aCr aCj Oa Ob
HIJ
(16.131)
bIJEelec)
An obvious relationship for the skeleton (core) first derivative Lagrangian matrices is si
= 20Jt
(16.132)
which with MO
E
eti
+
(16.133)
can be used to rewrite eq. (16.131) above as MO (92
Eelec
&tab
mo (oik .52/
(50.7k )pi3
}(
iiikoa,
ijkl MO
MO
▪
2
E fug2qi) + u(2,)} ij
MO
+ 2
(2,54 +
E
2r13,1a
ijki CI
-
2
E IJ
aCI aCj
ad
ab
IJ — bljEelec)
(16.134)
274
CHAPTER 16. CORRESPONDENCE EQUATIONS MO
=
2
-
2
▪
4
MO r
E
+E ij
+
MO
E
cbEii
MO
E
ii
ii
+
ij
MO
+
E qup,,c;?,
4
ijk CI
2 v,
_
Z... 'J
MO
+
4
E
rîa
ijkl
(1Pci r: 11 kl
OC/ OCJ ( Hid — b LT Eelec) Oa Ob
(16.135)
The cl.b matrices were defined by eq. (16.54) and II jj by eq. (16.117). Note that the second derivative of the PEMCSCF energy requires the solution of the first-order CPPEMCHF equations.
16.4.4
Third Derivative
The third derivative of the electronic energy may be obtained from the MCSCF third derivative expression presented in Section 16.3. For this purpose the following correspondence relationships are employed: ab = LE ab (16.136) with
ctib
=
.F
'
ij
7
with
=
mo
fihzi
(16.138)
2rfluk1
ejki =
+
+E
(16.137)
duc (tik)jk) a }
(16.139)
and mna Tij,k1 = 2anin(ijikl)a
kr} xii
with 'J
Cu
hii ,IJ
Yijkl
=
mo
E = 26i1(zic
(16.140) (16.141)
ii
+ ofiJoiLio} 2Ti j,kt
(16.142) (16.143)
PAIRED EXCITATION MCSCF WAVEFUNCTIONS
275
with MO
A.T.1
phi,
+ E fou(iiikk) +
1311k 1
(16.144)
k)}
and mni
= 2am un (iilk0 + JJa
+
(16.145)
24ja
(16.146)
with
(16.147) In eq. (16.81) one of the terms involving a product of Ua, Ub , and Uc may be simplified in the following manner: MO
MO
E uzuv,c„,, E r,,„,„(ikimp)
8
j khnn
MO
E qutu„ ait6npain +
=8
6jOin)Oill(ikimP)
Wchnnp
(16.148) MO
E
=8
lki)
Uiam U
ijklm
MO
MO
E u;muziu m onii(iiiik) +
+4
E urauPmu;„zomioklii)
4
j khn
. (16.149)
ijkhn
Using this result, the three terms involving products of Ua, Ub , and Ifc in eq. (16.81) are manipulated as MO
8
MO
E (uiai u,b,,u,c„„ +
Ut UcTra klmn
+ uzuk.i u,b „) E
r j Inp (ikiMp)
ijkhnn
MO
=4
E ÇUiarn U bm U11 + upm u„çm uzi + ufu;0,1) ,
warn
x [2a,a(ijikl)
(3„,a(ikij1)
(16.150)
MO
=4
E
+
ULUJam UL)riria
UtP„,U1,n 141
(16.151)
ijklm
The first derivative form of the variational condition on the MO space is related to
ôxii act
= 2 act
—
aa
=0
(16.152)
276
CHAPTER 16. CORRESPONDENCE EQUATIONS
The modified first derivative Lagrangian matrices in eq. (16.64) are related to the corresponding quantities as MO
[a]
E uyijkl
2E[51.] is = x7j +
=
(16.153)
kl MO
E uz,(26,4 + 2741: kt)
= 2E7i +
(16.154)
ki
Using eqs. (16.124) and (16.154), the term involving x[c] in eq. (16.81) may be rewritten as
mo
mo
Eij [xlc) — E
mo
mo
. E
ee 2€7i +
uki (viicii, + 2riii,ki )
E
mo —
kl MO
=2
U4 (2Eik) (16.155)
k
MO
MO
E ezib E.;,, + E u- ,(
J
I
OCj 03 illv
Oc OdOaab CI
+
2 IJ
ac, Oa ab
act, a3H,„ Ob acad.% +Cj a 031 11 1,1 ad adabacj H IJ
°cad
aci OC 092HIJ 8a Oc ObOd &CI 0Cd (0 2 1-1u Oa ad abac
ac, OCj (02HIJ
6. 1J 81.1 61.1
a2 Eetec)
OcOd )
a2 Eelec) Obtid ) tvEeiec)
Obac ) 0 Eejec ) 61.1 2Oa0c1 02 Eelec) b/.7 aaac
Ob Oc aaad aCd (.92HIJ Ob ad Oa& ôCj (a2H., s 02 Eelec)] Oc ad &Lab 'ij aaab ) [n2rvi (9 2c j a2 c." a2 c (9 2 ci ,92 c j ]
act
C/
— 2
E IJ
a0b OcOd
Oa& abad
+
0a0d Ob0c O
(Hid — (51.1 Ede) (16.196)
CHAPTER 16. CORRESPONDENCE EQUATIONS
284 where MO
03 H}j
HIT
aaabaC
-
E
rabc 1J X ij
ii MO
+2
E
(Ullib Xr
UtX!.744 + 11,7 Xr)
ij MO
xf3 ,bc + upxf i ra + ux wab 2E mo ii ▪ 2 >2 ( uu 1 Ut9 Ujici Ufiz U )Yijkl iikl
-I-
MO
+
2
E (u(4,41,e, + uturayii,c, + ij kJ MO
+8
E
MO
+
(qupci u,c„„
uurdy fikb,)
uibi ubu,ann
+ ufi ubumb,i )E
1-'(ikimp)
ijklmn
(16.197) MO
I/1Y
—2
E
mo
4 E (rrab 1,1`
jj
LI 23;
ji
U.
1.3
E 1- '14'
32
E (q5ifbc
4
b)
+ j.c i.,jab)
MO
+
Vf4 1.3 E31. 24
23
31
23 3 2
ij
MO
+4
E
(uiaibub
1J
+ u; u 1 + uicitut,)(kiic +• rij,k1
ijkl MO
+
4
L-d
j c)
[UUP,/ (6ji(L,
Uib:1
(5j4klja
.1/3a ) sj,k1
ijkl
mo 4
E
(ugn U m U1
+
ifibm
u,
+ U m uï.„, ut1)-riTC
.
(16.198)
ijklm
For the last term of eq. (16.198) a correspondence equation similar to eq. (16.151) was used. Equations (16.196) and (16.198) indicate that evaluation of a PEMCSCF wavefunction requires the solution of both the first- and second-order CPPEMCHF equations.
SCF WAVEFUNCTIONS
GENERAL RESTRICTED OPEN-SHELL
285
16.5
General Restricted Open-Shell Self-Consistent-Field (GRSCF) Wavefunctions
16.5.1
Electronic Energy and the Variational Condition
The GRSCF wavefunction described in this section represents a further simplification of the PEMCSCF approach. The electronic energy for the GRSCF wavefunction is given by mo
mo
E fh ii + E
2
Eeiec =
(16.199)
+
where the summations run over all orbitals. In principle, all the equations from the preceding two sections are applicable to a GRSCF wavefunction when appropriate restrictions specific to the single-configuration open-shell wavefunction are applied. The correspondence equations for the one- and two-electron density matrices formally are the same as for the PEMCSCF wavefunction of Section 16.4:
= 245iifi
(16.200)
and
= 61ibk1Oik
babik )/3ii (16.201) 2 In fact, the f's, a's, and /Ps are much simpler for the single-configuration open-shell case than for the more complicated PEMCSCF case. In a very similar manner as was presented in Section 16.4, the energy expression (16.199) is derived from that for a MCSCF wavefunction, eq. (16.40): MO
Eelec
= E yi
— (OikOjj
MO
hi
+E
(16.202)
ijkl
ii MO
E
mo
+E
si.okictik +
(16.203)
(Sik 45j/
ijkl MO
= 2
MO
E
E
{aii(iiljj)
Oij(ijlij)}
.
(16.204)
The variational condition for this GRSCF wavefunction is
—
=0
(16.205)
where the Lagrangian matrix E is defined by
Ei,
=
mo
+E
+
(16.206)
CHAPTER 16. CORRESPONDENCE EQUATIONS
286
Using the correspondence equations (16.200) and (16.201), the relationship between eqs. (16.46) and (16.206) is obtained in the same way as was presented in Section 16.4:
mo xii =
mo Tinthim + 2 E
(16.207)
mkt
mo
E
(2/5imh)him
tn
MO
{
E
+
+2
5;k577zi
(.
+
45itbm.k)dim}(imik1)
(16.208)
mid
MO = 2fh +
2
E faji(ijIII)
f3ii(i1ljI)}
= 2fii
16.5.2
(16.209) (16.210)
First Derivative
The first derivative of the electronic energy follows from eq. (16.52) and the correspondence equations, (16.200), (16.201) and (16.210): MO
E +
Eelec
Oa
MO
MO
_E
(16.211)
ijkl
MO
E
(26„ii)
mo
E
1 c — (0i/cup + 2
6 klaik
ijkl MO
E
SZi (2)
mo
=
2
E
(16.212)
mo
fihz
+E 13
MO
2
16.5.3
E
(16.213)
Second Derivative
In order to derive the second derivative expression for a GRSCF property from that for a MCSCF wavefunction, the following additional correspondence equation is necessary: Yijkl
=
_L, -e ii,ki
(16.214)
▪ GENERAL RESTRICTED OPEN-SHELL SCF WAVEFUNCTIONS
287
The generalized Lagrangian ( I and 7 matrices in this equation are defined by
mo
E
=
faik(ijikk)
(31k(ikijk)}
(16.215)
(illjk)}
(16.216)
and
7-271kli =
13,,,f(ikij1)
The second derivative expression for the MCSCF, eq. (16.58), is written without the CI contribution as MO
02 E,1„
, MO E
E
aaa b
(ii kl) ab
ij MO
E ii
MO
▪
2
E
(ux?,
+
+
ij
MO
2
E
UUit l yijk i
.
(16_217)
ijkl
The following relationship may be derived in a manner similar to that of the Lagrangian matrix, eq. (16.207), x7i = 2c7i (16.218) with
mo
E
=
(16.219)
is also required to complete the derivation. Using the correspondence equations described above, the energy second derivative expression for a GRSCF wavefunction is manipulated as
mo
E (2bi; ii)1171'
.9 2 Eete.
aaab
mo
E
ijkl
siiesktoeik
+ —2
(6tOit
bilbjk)04(iilkir b
MO
▪2
mo E fr(2e + )
mo ▪
2
E uzup245i,(i?;, + 2,-j!
ijkl
(16.220)
CHAPTER 16. CORRESPONDENCE EQUATIONS
288
mo
MO
2
E
+ E
h
ij
MO
-
2
E
ii MO
+
4
E (qqi +
ii ii)
ii
mo
mo +
4
E uzuwk +
4
ijk
E ijkl
(16.221)
Uij UI:1 741,k1
Equation (16.221) is equivalent to the MO part of eq. (16.135) for a PEMCSCF wavefunction.
16.5.4
Third Derivative
The third derivative of the electronic energy may be obtained from the MCSCF third derivative expression presented in Section 16.3 without the CI contributions. The following correspondence relationships are employed: 2c0 with
Af 0
0.„(i1Isioati
E
c
YZikl
with
(16.222)
Jo,
(16.223)
2rila ij,k1
(16.224)
fark(ijikkr + Otk(ikljk) a }
(16.225)
2 °j/Cik
mo
+E and
717:6 =
(illjk)a}
+
mo
MO
8
(16.226)
E (quz,u,;,„ + upi ue,r7. + quumb E riinp(ikimp)
ijklmn MO
=4
E (ulim u,Lub + onz ul„,ug, + uicni Ujim Uti) 7131
(16.227)
ijklm
The first derivative form of the variational condition on the MO space is related to Oxij
Oa
31 = 2 O"i —
aa
aa
aa
=0
(16.228)
SCF WAVEFUNCTIONS
GENERAL RESTRICTED OPEN-SHELL
289
The modified first derivative Lagrangian matrices in eq. (16.64) are related to the corresponding quantities as MO
X [c;] = 2E[a.1 — x tiii it —
E
+
Ugi yijki
(16.229)
kl
MO
= 2(11 +
E ug,(2b4k +
2r;731,ki)
(16.230)
kl
Using eqs. (16.210) and (16.230), the terni involving xid in eq. (16.81) may be rewritten as
mo
mo
E w [x(c) —E
U4 X ki
k MO
=2
mo
MO
E e.p f.;i + E
u4(tk
+ E ufc1713.'1,k1
Elk)
—
(16.231)
kl
Using the corresponding equations given above, the third derivative in eq. (16.81) is reexpressed for the GRSCF wavefunction as
•93 Eetec
OctabOc
mo -
2
E
Ant-
+
mo r
MO
) a bc + Oij(ii Iii) abc}
E otiA ii I
— 2
ij
ij
MO
-I-
4
E
(ssgs;k + sigslk + sxs.,k)fi,
ijk MO
-
4
E
(seAs,,syd
+
s.bik s;i sz,
+
SaS.aiiSti)Eij
ijkl MO
+ 4
E
+
mo E(uuc; +
4
UteW
Uicj
eg) + u2ci u74)
ijk MO
+
4
E
(uzutick,
Uicjukal r ,bkl)
ijkl MO
+
4
E (uianylim u, + u,Pm ulm Ugi + Ufm tlim Uti)rirki ijklm MO
— 2
E ij MO
— 2
MO
Ca3/E;i +
Ek
MO
u
(
2
-
Eh)
+E
u-,1ri,4,k,
kl
MO
E d.;[€(.4 + E
E stjc.OcE 23
MO
(14( — f ik
)+
E kl
u:iTgki
290
CHAPTER 16. CORRESPONDENCE EQUATIONS MO
E
—2
MO
MO
+ E UP, (k
7
i3
E
Eik)
(16.232)
kl
Equation (16.232) is formally equivalent to the MO part of eq. (16.163) for a PEMCSCF wavefunction. This equation shows that a third derivative property for a GRSCF wavefunction requires only the solution of the first-order CPHF equations.
16.5.5
Fourth Derivative
The fourth derivative of the electronic energy may be obtained from the fourth derivative expression for the MCSCF wavefunction discussed in Section 16.3. For this purpose the following correspondence relationships are used: abc — c•E abc ii
(16.233)
with abc
MO
= fi haubc
E
„,
ab
gijkl =
(16.234) s
lab
-ab
il`tik
(16.235)
2Tij,k1
with MO
E{ ctik(iiikke
(i1;1'
kr b }
(16.236)
and mn
= 2amn(ijila) b
(16.237)
(illi k) b }
The modified second derivative Lagrangian matrices in eq. (16.99) are related to [ab] X 23
MO
=
°91 +
MO
E (utyziki + uladytki) + E kl
b nZijklmn UlccilUra
kb]
= 2€ .ii
=
(16.239)
mo 2e0 Jt + 2
(16.238)
kirnn
•b
m0
E (04: + uTxk ) + 2 E (ut T k
kl
i + 141 1-41bkl)
MO
+ 2
E utc,,Uibra '7:1Z lini + ULU7-471:m
+ ULUI5m -r-47ki
(16.240)
kirn In eq. (16.104) one of the terms involving the za matrices is reformulated for the GRSCF wavefunction as MO
E
ijklmn
Trb Tc tj ki mnZijklmn
GENERAL RESTRICTED OPEN-SHELL
SCF WAVEFUNCTIONS
MO
E
=4
+
quturc„,frii,,,(ikimp)d
rinip(imikp)d
jklmnp
+
291
r.ipincipikm)d}
(16.241) mo = 2 E (u?q7.,/41 +
Uibm
Ulm
Uicm Um
ijklm
(16.242) The terms involving a product of Ua, Ub , if' and Ud in eq. (16.104) may be rewritten in a simple manner: MO
E (u
8
uiaj Uk5 um'
Uodp
quiliUm b ,U4)F .ii„rno) (iki
ijklmnop MO
=4
E
(ut%ont uL,ug, + uiamulmutiutd
+
uianzu,4muLuicr,)7173/
ijklmn
(16.243) Using the correspondence equations given above, the fourth derivative in eq. (16.104) is finally reformulated for the GRSCF wavefunction as MO
04Eelec
2 E fih c:ibed
MO
E
Oaabacad =
ij MO
-
2 E
abcd
MO
-
4
E
(uiakbuo + uxu,b4 +
Ey
ijk MO
+
4 E sabcsA
+
S' 5k
+
sjic as.b,k
+
sttibs;,,
ijk MO
+
4
E
s29,bsv. + sAcsi4 + siakd.
ijk MO
-
2V
abcd
CO • •
E 23 -•
i
MO
+4
E
+ uiajcc[ibidi +
u4dE[ibic3
ij
Utile(5'. d
Uffic .7 :61 )
MO
+ 4 E (7. r a ij
bcd ii ai
rrb cda
4")
eji
uiciEcl!feb
uidicr)
▪ 292
CHAPTER 16. CORRESPONDENCE EQUATIONS
mo + 4
E
+ tizicuti + uri u0 c'
ijk MO
▪
4
E
+
(u,Pibuiv
r Tad ubc) ril
U UP,1
u zj u
kl
ij,k1
ijkl
MO
4
E (u,Pi uzi c:kcd +
•bd
U29:i UQi (L, +
ijk
•ad + ukuc.v k3 mo +
ud
uk.(jac kJ ik
yrb
4 E
d
jibd
LI 23U klrij,k1
13U kirij,k1
U .12
3'1 .ilbc
d
kl rij,k1
ijkl
Utj
4
ja
Ti3
kdi
ut. ud r ipe
i3
4_ f Tc nd
kl ij,k1
ii,k1
mo
E
(ufr.,,u21?7„Uli
▪ UL„ U;n1 Uti ) 7131
(0)„,u;m Ufici
Tsi T kla
ijklm
▪4
MO
E
ijklm
MO
▪
4
E
(u,cuim uz
+ U
(Uidm U;,,,Ujc i
+ t#,„ vim U 7,1)7Zz!kci
ijklm
MO
4
E
U,cm U!l)
ijklm
MO
+ 4
E
(uiP,„u.bi„,u&uid.
UtPmU7mULUI dn
Ufm Uicn r1;71
— 2
— 2
ebcd[a
— 2
eccla [ b
— 2
edab c
uti ( k ii
fik)
(( k — fik)
.
(16.244)
Equation (16.244) is equivalent to the MO part of eq. (16.196) for a PEMCSCF wavefunction. This equation indicates that evaluation of a fourth-order property of a GRSCF wavefunction requires the solutions of both the first- and second-order CPHF equations.
CLOSED-SHELL SCF WAVEFUNCTIONS
293
16.6
Closed-Shell Self-Consistent-Field (CLSCF) Wavefunct ions
16.6.1
Electronic Energy and the Variational Condition
Finally, we turn to the simplest type of wavefunction considered here. The electronic energy for the CLSCF wavefunction is d.o.
Ee i„
2
E
d.o.
h1
+ E
-
(1 6.24 5)
where the summations run over doubly occupied (d.o.) orbitals. Since the closed-shell SCF wavefunction is a special case of GRSCF and/or MCSCF wavefunctions, the energy and derivative expressions may be derived from either starting point. The correspondence equations between GRSCF and CLSCF wavefunctions for the one- and two-electron coupling constants are fi
1 for i = doubly occupied 0 for i -= virtual
(16.246)
2 for i, j = doubly occupied 0 otherwise
(16.247)
{{
Oij
{ —1 =
0
for i, j doubly occupied otherwise
(16.248)
The electronic energy expression in eq. (16.245) is derived directly from eq. (16.199) using the equations (16.246) through (16.248). The variational condition for a CLSCF wavefunction is given by
= 0 for i = doubly occupied and for j = virtual
(16.249)
where the Fock matrix is defined by d.o.
= hii
E 12(okk) - (ikiik)} = Fi/
•
(16.250)
The Lagrangian matrix for the GRSCF wavefunction in eq. (16.206) is related to the Fock matrix for the CLSCF wavefu.nction:
mo
Eii = f/h/i + E hij
+E
+
( 16.251)
{2(iiikk) - (ikiik)}
(16.252)
for i = doubly occupied
(16.253)
294
CHAPTER 16. CORRESPONDENCE EQUATIONS
Since the SCF energy is invariant under a unitary transformation within the occupied space, the diagonality of the Fock matrix is ordinarily used to define the canonical SCF molecular orbitals. in this section, however, in order to maintain generality, the derivative expressions for the CLSCF energy are derived without introducing the diagonality condition of the Fock matrix. The correspondence equations between MCSCF and CLSCF wavefunctions for the oneand two-electron density matrices are
•ij =
0
for i,j = doubly occupied otherwise
(16.254 )
and
=
2 (S.t L.15.1
26ijbki —
bii6jk)
0
for otherwise
= doubly occupied
( 16.255)
The energy expression (16.245) may be derived from eq. (16.40) using eqs. (16.254) and (16.255):
Eelec
mo mo = E 7ii hi; + E r ii,„(iiiko i; ijki
(16.256)
d.o.
= E (2,5i2 )hi,
z ij
d.o.
+
{ 2biibkt — 1 (bikbii + 5 il 6 d.o.
= 2
E
jk)}(i.ilkl)
(16.257)
d.o.
kii + 2 E (iiikk) — —1 Z {(i.iiii) + (OA}
(16.258)
ik d.o.
= 2 Z hii -I- E {2(iiijj) — (ijiij)}
.
(16.259)
Using the correspondence equations (16.254) and (16.255), the relationship between the Lagrangian matrix in eq. (16.46) and the Fock matrix in (16.250) is demonstrated as follows:
mo =
MO
73mhim + 2
E ri mki(iniva)
(16.260)
m kl d.o.
E
(2
m ) him
d.o.
+ 2
{2bjnibkl mkt
bk)}(i 1741) 7. (6jm1 + jIm k 45
z
(16.261)
CLOSED-SHELL SCF WAVEFUNCTIONS
295
d.o.
•
2hij -I-
•
2hi
•
2Fii
E
2
d.o.
2(ijilck) —
d.o.
E (am - E
(16.262)
d.o.
+2
E {2(okk) - (ikuk)}
(16.263)
for i = all and j = doubly occupied
(16.264)
It should be noted that the CLSCF condition (16.249) is included in the variational condition for the MCSCF wavefunction:
=0
xii —
16.6.2
.
(16.265)
First Derivative
Using the correspondence equations and the variational condition described in the preceding subsection, the first derivative expression for the CLSCF energy may be derived straightforwardly from eq. (16.213) for the GRSCF wavefunction as
mo
mo •
2
E
fihti
mo 2
E
(16.266)
ij d.o.
•
2
d.o.
E h7, + E
{ 2 (iiin) a
d.o.
—2
E
(16.267)
t3
Note that the summations run over only doubly occupied (d.o.) molecular orbitals. Alternatively the first derivative of the electronic energy is obtained from the expression for the MCSCF wavefunction, eq. (16.52), using the relationships eqs. (16.254), (16.255) and (16.264) as
0Eetec Oa
mo
E
mo -E
mo +E
ij
SZxij
ij
ijkl
d.o.
=E
(26i.j)leti
ii
d.o. bki ijkl
-
;2
(bik(5j1
+E
6 i1 6jk)}(ijIk 1 ) a
(16.268)
CHAPTER 16. CORRESPONDENCE EQUATIONS
296 d.o.
- E szi (2Fii)
(16.269)
ij
d.o.
= 2
E
-
E i;
d.o.
11.21
E
-
d.o.
2
(16.270)
Equations (16.267) and (16.270) are seen to be identical.
16.6.3
Second Derivative
An expression for the second derivative for the CLSCF energy may be derived from the equation for the GRSCF wavefunction. For this purpose three more correspondence equations must be introduced:
= F4 with
for i = doubly occupied d.o.
= hci`i
= Fii
E {2(okk)a -
(i1c1 :7 Oa}
(16.271)
5
for 1 = doubly occupied
(16.272) (16.273)
and
Tin
for m,n = doubly occupied
=
(16.274)
where the A matrix is defined by
= 4(i.j1k1) — (ikij1) — (illj k)
(16.275)
Using the relationships given above, the second derivative expression (16.221) for the GRSCF may be reformulated for the CLSCF wavefunction as
mo
82 Eelec 0a0b -
mo
2
E fihe
2
mo E
4
mo E (u,i fc31 +
+
E
ii
+
ii
mo +
4
E quzi (iik ijk
Aio + 4
E qutriif,k, ijkl
(16.276)
CLOSED-SHELL SCF WAVEFUNCTIONS d.o.
2
E
297
d.o.
E
It7t
-
ij d.o.
2
E
efFi,
ij
all d.o.
4
EE
4
EEE
Fa
all all d.o.
u.bik Fij
j k
all d.o. all d.o. 4
EEEE ijk I
(16.277)
The ,ab matrices were defined in eq. (16.54). It should be noted that the Fa matrices are symmetric, i.e., FL(16.278) 81 since there is only one Fock operator for a CLSCF wavefunction. In order to derive the second derivative expression for the CLSCF wavefunction from that for the MCSCF wavefunction two additional correspondence equations are necessary: Yijkl = 28 jt Fik
2 Aij,k1
for i,k = all and j,1 = doubly occupied
,
(16.279)
and
x`ti = 2114
for i = all and j = doubly occupied
.
(16.280)
The second derivative expression for the MCSCF energy without the CI contributions is MO
492 Eelec
Oat%
▪ U2a3 43 )
+2
mo E uiaj uziNikl
.
(16.281)
ijkl
Equation (16.281) may be reformulated for the CLSCF wavefunction as follows:
02 Eejec OW%
d.o.
==
d.o.
E (25)Ni + E ijkl d.o.
-
1
(babil
bilbik)}(ijikl)a b
298
CHAPTER 16. CORRESPONDENCE EQUATIONS all d.o.
2
EE
2 (Uti Fiai Uri F)
j all d.o. all d.o.
+ 2
EEEE 2uiayti(6 A Fik + d.o.
d.o. r
2
E + E
2
E upFi,
(16.282)
—
d.o.
all d.o.
4
EE(ui + uF) j all all do.
+
4
EEE
U2 Uk
j k all d.o. all d.o.
+
4
EEEEI
(It Aii,k1
(16.283)
ijk
Equations (16.277) and (16.283) are naturally identical.
16.6.4
Third Derivative
The third derivative of the electronic energy may be obtained from the GRSCF third derivative expression presented in Section 16.5. The following correspondence relationships are employed: EZi b = Fiajb for i = doubly occupied (16.284) with Fiajb
d.o.
E
=
{2(iiikkrb
—
okiikrb}
(16.285)
for I = doubly occupied
= Fiaj
(16.286)
and = A j, kj
for m,n = doubly occupied
(16.287)
with
A'k — 4(ijila r — (iklfir — (i/Ijk)a
.
(16.288)
In Section 4.5 the first derivative vative of the Fock matrix was derived as all
Oa
Fa
— "3
+E k
all
uFk3
+E k
all d.o.
u/aci Fik
+ EE k
l
U2 A
(16.289)
CLOSED-SHELL SCF WAVEFUNCTIONS
299
Utilizing this expression one of the last three terms in eq. (16.232) may be manipulated as follows
E cib[E; , + E u4cik —
q;)
+E
(16.290)
kl all d.o.
EE
all
CP[ Fici +
Ek
all d.o.
all d.o.
all
tfliFik —
Ek
IiiiFik +
EE k l
U111:3 ,kt] (16.291)
all
2 E uAFk ji
e7P[° 3 (9C
(16.292)
k
In the above manipulation note the difference in the summation limit for each index. Using the corresponding equations given above, the third derivative for the GRSCF energy in eq. (16.232) is reexpressed for the CLSCF wavefunction as
a3 Eacc aaabOc
2
d.o.
he c
d.o.
d.o.
+ E { 2(iiiii) abc _ (i i ii irb 1, _ 2 E s ,.bc,,,, ii j3 13
d.o. all
+
4
EE (s1L,,s 5, ij k d.o. all
— 4
EE
+ stsA + sxstik )F,
(sas,„sfa +
s,,s;,szi + sas;,sti) Fii
ij kl all d.o.
+4
EE (uu,b; i
4
Fiajb)
j
all d.ø.
+
[It Fic3n
EE
(u1a,,u21?k Fic,
+
uibk u;k Fri
+
uick
13 )
ij k all d.o. all d.o.
+
4 EEEE (quz,Af,, k , + ui _ bi u _ ijk all d.o.
±
4
EE ijk im all d.o.
— 2
EE
all d.o. — 2 E E
(u,pm u.,,, u,,
+
pabr Fti
2
+ U;Pit] Ti Mj ,ki)
Uibm U:m Uk c + U ic,„U;m Uti) Ai j Ai all
E
I.% Fkj]
k
,bc[a Fij
rij Oa
all d.o.
— 2
i.A.ki
1
EEj `'f db 3
all
2
E
uv-,,,,]
k all
2
E k
uti Fki
(16.293)
CHAPTER 16. CORRESPONDENCE EQUATIONS
300
This equation shows that the third derivative of a CLSCF wavefunction requires only the solution of the first-order CPHF equations. Alternatively the third derivative of the CLSCF energy may be obtained from the MCSCF third derivative expression presented in Section 16.3 without the CI contributions. For this purpose the following correspondence relationships are utilized:
eikt = 26i1F3 ab
for i,k = all and j,I = doubly occupied
2./1.7j,k/
,
for i = all and j = doubly occupied
= 21-'f'.b 23
(16.294) (16.295)
and
8
mo E
mo
+ uf,(1 11) E
(Uti Uti II,enn
ijklmn
r j1
nP( ik
I MP)
ail d.o.
=4
EE
UP,,U;m tlei
Ufm t1;,,„,q 1)
(16.296)
ijk hn
The modified first derivative Lagrangian matrices in eq. (16.64) are related to the corresponding quantities in the following manner: MO
+E
= 2 Fjic!1 = x
(16.297)
kl
all d.o.
EE
= 2 Fiai
u,(25,,Fik +
(16.298)
2 Aii,ki)
1
k
Utilizing this relationship and eq. (16.289), the term involving x[c] in eq. (16.81) may be rewritten as MO
E all d.o.
=2 =2
all
EE i
'
all
d.o.
F,9;
+E
L
Zti[' EE i j
(16.299)
UkiX ki]
+
all d.o. h
k
aFii ac
all
—E
u-LFk, (16.300)
i
all
2
E
uFki ]
( 16.301)
k
Applying the corresponding equations given above in the third derivative expression for the MCSCF energy in eq. (16.81) produces the identical formula for the CLSCF energy third derivative presented in eq. (16.293).
16.6.5
Fourth Derivative
The fourth derivative of the electronic energy may be obtained from the fourth derivative expression for the GRSCF wavefunction discussed in Section 16.5. For this purpose the
CLOSED-SHELL SCF WAVEFUNCTIONS
301
following correspondence relationships are used: abc
for
i = doubly occupied
(16.302)
with d.o.
Fiaibc = hair
E
diab = Fill and
mn ab Tij,k1
with
{2(iiikkr bc _
for
(16.303)
(ikl k) abc }
1 = doubly occupied
,
(16.304)
ab Ai j ,k1
for m,n = doubly occupied
(16.305)
= 4 (ii l koab
( ik ijo ab _ (iiijk)a b
(16.306)
=
.
The modified second derivative Lagrangian matrices are then given by [ab] Eii
MO
=
E uw:
+
MO
u:icr,
+ E Ni r i +
k
UTk1
kl
MO
+ E ugi uitirizin, + uLuibi r.lzikm + =
klm •--,[ab] ij
(16.307) (16.308)
all
.=
U:,,, Ulb„,,r0, /
(
E
F b
all
d.o.
kl
rn
+ EE
+
(uPciFrk
(wiciutnA i k,im
all
d.o.
k
1
+ EE +
uLuibiki,km
+
(utciAzi,ki
+ uzi4,k1)
uLutm Ai,k,)
.
(1 6.309)
Using the correspondence equations given above, the GRSCF energy fourth derivative in eq. (16.244) may be reformulated for the CLSCF wavefunction as 494 Eelec
8a8b0cOd
d.o.
2 — 2
d.o. E hclibcci + E {2(iiiii)a1cd
E
(iiiiirbcd}
d.o. cabcds
ij d.o.
— 4
all
EE (uabu.4 + uacqz + uadu)Fii ij k
do, all
+4
EE
+ sivsTk + sf,ps,k +
ij k d.o.
+4
all
EE ij k
+ szsv + siakds10F,
ssokabs;,,)F1,
302
CHAPTER 16. CORRESPONDENCE EQUATIONS
— 2 all d.o.
uiat
+4 E E
uiaid
+ UP: FiE77 4 + UibfF 1
U 235111.7 b1 )
all d.o.
+ 4 E E (uiai Fibrl
Fic3FI a
+ urici Fidia + usdi
F'
c)
all d.o.
+ 4
E E (uiakbc/ + uacu.b4 +
+ 4
EEEE (uzbud +
uadu.4)Fij
all d.o. all d.o.
Q." upc 1 + u?id
JAI
ijk all d.o.
+4
E E (ulik U.bi k Ficji
U
U25 IV3 +
utak u.cil Fit7
ij k + U kUfk
Fft + Ifibk qk Fiaje
all d.o. all d.o.
+ 4
EEEE
upc , Av,k , +
+ Uti
+
all d.o.
+ 4
+ ickU3dkFiajb )
EE
qUkitit,ki
UPj Uida Acg ki + UtPj U41:ki )
+
+ uicm ua
NI) A
, ki
u1)
,k1
ijk lm all d.o.
+ 4
E E (uL U 7
+ uicm ut„ + uidm Ub
ijk lm all d.o.
•
4
E E (uicm u34„, U 1 +
Uidm U;77,
+ 11,7„U.;,,,Ufid) (
ijk lm all d.o.
+
4 E E (uL
urn.,U 1 + ugn,q,2 14 + u
U;!,m UZI) AfJAI
ijk Im all d.o.
•
4
EE ijkl
(ugn u
.7
77,
all d.o
-
2
Uiam U jçniULUldn
ULUIdn
mn
all
EE
(911.1 eç,c[
ad
2
E
UticiFkj
Utffm Ulm Uibm U icn ) Aij,k1
303
CLOSED-SHELL SCF WAVEFUNCTIONS all d.o.
2
EE all d.o.
_
2 v., z
bcd[ 49 Fij
8a
"
çcda [ oFii
all
2
all
2
>
2
E
3
Oc
•
Ut. Fkj
all
all d.o. 2 E dab L i
E
U,
(16.310)
Fkj
Equation (16.310) indicates that evaluation of fourth derivatives for a CLSCF wavefunction requires the solution of both the first- and second-order CPHF equations. Alternatively the fourth derivative of the CLSCF electronic energy may be obtained from the expression for the NICSCF wavefunction discussed in Section 16.3 without the Cl contributions. For this purpose the following correspondence relationships are used: ab Y iik t = ,„
jiFiakb + 2 A abc
kl
26j1AiOnn + 26 jn Aim,kl
for j, k, in
(16.311) (16.312)
for i = all and j = doubly occupied
= 2Pkbc 13
Zijkirrtn =
,
for i,k = all and j,1 = doubly occupied
251nAij,km
all and j,l,n = doubly occupied
,
(16.313)
.
(16.314)
and z ic !jklmn = 25 .y lAk ,mn + 2jnA,kl + 2'5 ln A cij,km
for i,k,m = all and j,l,n = doubly occupied
The modified second derivative Lagrangian matrices in eq. (16.99) are then of the form [abj X ij
MO
MO
E
ab
X ij
WictYtiki)
(16.315)
UZUm b n Ztij klmn klmn
kl
(16.316)
= 2 F' all
= 217/ + 2
E
(uZi Fiak
all d.o.
Uili
+
2
EE k
all
+ 2
d.o.
EE
(UTJULAik,lin
+ utn up, Ail , krn
I
.
(16.317)
kl m
In eq. (16.104) one of the terms involving the za matrices is reformulated for a CLSCF wavefunction as MO
E
ij larva
nztjklmn
CHAPTER 16. CORRESPONDENCE EQUATIONS
304
mo
E qutuic„,„{rii„,(ikimp)1 + rin,p(imikp)d + ripincipiknod}
=4
ijklmnp
(16.318) all d.o.
=2
EE (u u u + upin ufm UL
+ U„U;m 0:1)4j,ki
(16.319)
ijk lm
The terms involving a product of Ua, the following simple manner:
Ub , Uc
and Ud in eq. (16.104) may be rewritten in
MO
E
8
UtWOLI Utolp
(qutiulcnnUodp
b n Ire U25ij Um
rii np (ik I MO)
ijklmnop all d.o.
= 4
7i Lid„ uu + EE (u,17,q
Uiarn Ujçm lIZn U it
WI 114 /if)
A in )--ij,k1
ijkl mn
(16.320) Using relationships given above, the MCSCF energy fourth derivative in eq. (16.104) is reformulated for the CLSCF wavefunction to produce the identical formula presented in eq.
(16.310).
16.7
Energy Derivative Expressions Using Orbital Energies
When the SCF molecular orbitals are determined in the conventional manner so that the Fock matrix is diagonal, the SCF condition is given by
=
(16.321)
Ei
where the quantity e having a single suffix is specifically called an orbital energy. Then eq. (16.321) is one of the expressions for the variational condition for the CLSCF wavefunction. It should be noted that eq. (16.321) defines orbital energies for virtual orbitals as well as for doubly occupied orbitals. If the condition (16.321) is used, the formulae for the energy derivatives become somewhat simpler. Since the double sum over the terms involving the overlap derivatives may be replaced by a single sum, the first derivative expression for the CLSCF energy, eq. (16.267), becomes 0Eetec
Oa
d.o.
to.
E
E = 2{ 2 (iili
(iiiii) a}
ij d.o.
2
E
(16.322)
Similarly, the second derivative in eq. (16.277) may be written as 02 Edec
Oaeb
= 2
d.o.
d.o.
i
ij
E h`lib + E 1 2 ( ii)ab _ (iiiiirb}
ENERGY DERIVATIVE EXPRESSIONS USING
ORBITAL ENERGIES
305
d.o.
-
2
E
ab
all d.o.
+ 4
EE
(UP
+
Uiaj Fibi)
ii all d.o.
4
EE /71; UP.ifi j all d.o. all d.o.
-F
4
EEEE i jk I
(16.323)
U_IA0,1s1
If the diagonality condition in eq. (16.321) is used to determine a CLSCF wavefunction, then the diagonality in the derivatives of the Fock matrix also has to be satisfied: 19E-
= 6. -i•
Oa
Oa
(16.324)
-
The first derivative of the orbital energy appearing in this equation is given by
61Ei OFii Oa = Oa
(16.325) all d.o.
=
—
+
EE
(16.326)
Aii,k1
k
Enforcing the diagonal nature of the Fock matrix, the third derivative expression for the CLSCF wavefunction in eq. (16.293) is reformulated as
03 Eci„ = 2 d.o. E h t( pc 0a0b0c
_ (iiiii)abc} ij
d.o.
-
2
E
+
4
EE
secc,
d.o. all
j ii
+
+
ii ii
j d.o. all
-
4
EE i
(sf; SPkS;k
.51.iSfk.57k
STiSAS
Ei
jk
all d.o.
+ 4
EE j
(U,Vib3F + F + F jai)
all d.o.
+
4
EE -0
(uAU.bik Ffj + Ubk Uk ,
all d.o. all d.o.
+
4
+ Ufk U;k Fibj )
k
EEEE ijk
I
(uzutAzi,k,
+
tubAzi,k,
+
UrY LAt , ki)
306
CHAPTER 16. CORRESPONDENCE EQUATIONS all d.o.
+4
EE
(u Um
+
Ugt +
(fib?,
Aij,kl
ijk lm d.o.
2
E keei ac do, all
+4
EE
+ et9u4
(16.327)
i
The fourth derivative of the electronic energy in eq. the diagonality of the Fock matrix as a4 Eetec
(9a0b0cOd
d.o.
--
2
E
hem
+
(16.310) may be reformulated using
d.o.
{ 2( ii iji) abcd
d.o.
2 E secci ci
do, all
4
EE (uu,9351
7
+ uiaic ui
uiajduib;) ci
+ sigd s'i; +
sfl ast
j d.o. all
4
EE (stcsi;
4
EE
d.o. all
(sesv +
,s71cs.:!.34
+
sid;b sfj ) ci
+ s.734 s:7) (i
d.o.
2 E cje cd,i
all d.o.
4
EE (u11.74 +
usatFi[:di
i
+ Uibaic all d.o.
4
EE
Fir + UtiFi[r] +
Uf35 Fr )
(u29,F . ir
+
Ficita
Fidia b
+
II,d3 F,a3be)
ii all do.
4
EE
(qbue
+ uiajc utfl +
uiaidu*i
ii all d.o. all do.
4
EEEE ijk
(ub
+ ujaicupcil +
I
all cl .o.
+4
EE ij
lc
(110,U.b ik Fic34
UlgkU;k FP"
ULUAFibic
▪ SUGGESTED READING
307
UPkU;k Fiat + U2bkUlk.F,7
UfkUlkFial)
all d.o. all d.o.
+ 4
EEEE (usviLAi + quïciAtdoci + ijk I qUfa.,4724,ki
UUkilAU,k1
UZU
Agkl)
all d.o.
+4
EE
U6 U, 1 + U1bm Uj9m UA
ijk lm
Ufm.U.jimUta) Ak1
all do.
+
4
E E (uibm U;ni Ugl ijk
UgnU;ImUtgi
all d.o. + 4E E (u,;„ ij k lm all d.o. + 4 din ijk lm all d.o. 4 ijkl mn d.o. [ 2 E ef
dm
E E (ui u;.,
+
Uidm m
+ uiPm
Uiam Uî.m U idd)4,ki
ugi +
E E (uLujim t4„u,d, + pc
+
UidmUjmUO i
lm
+
a a:
UibmU fm. 141) AZj,k1
U a afbi
+
mm
ut„, u)
•g dab
sst ac
i d.o. all [ 4 E E cibc utd i + .ebjcd ri2 i + e.zida uibi 4_ eid3ab uici Ei
i
(16 .328)
•
The equations given in this section are presented in symmetric form to aid the evaluation of the higher energy derivatives of the CLSCF wavefunction.
Suggested Reading 1. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Theor. Chim. Acta 72, 71 (1987). 2. Y. Osamura, Y. Yamaguchi, and H.F. Schaefer, Theor. Chim. Acta 72, 93 (1987).
Chapter 17
Analytic Derivatives Involving Electric Field Perturbations All of the derivative expressions described thus far are applicable to any real perturbations, including the nuclear perturbations that were emphasized. Obviously, most of the motivation for the rapid development of analytic derivative methods has come from the use of energy gradients and energy second derivatives for the optimization of stationary point geometries and the evaluation of vibrational frequencies. In this chapter, however, analytic derivative expressions specifically involving electric field perturbations are discussed. Relevant topics include the electric dipole moment, electric polarizability, and the dipole moment derivatives with respect to nuclear coordinates. This mixed (cross) perturbation method easily can be extended to other types of real variables and higher-order properties. Analytic energy derivative expressions involving electric field perturbations are greatly simplified, since standard atomic orbital basis sets depend only on the nuclear coordinates and not on the electric field.
17.1
The Electric Field Perturbation
17.1.1
The Hamiltonian Operator with Mixed Perturbations
Let us consider mixed (cross) perturbation terms in the following Hamiltonian operator H, appearing in standard perturbation theory: H = Ho + AO( + AfIlif + A.A/H: f + ...
(17.1)
In eq. (17.1) Ho is the Hamiltonian operator without perturbations, H'a is the first-order change in the Hamiltonian operator due to a nuclear perturbation, and 1-1'1 is the first-order change in the Hamiltonian operator due to an electronic field. X. is a nuclear coordinate (a) perturbation and Af is an electric field (F) perturbation. It should be noted that standard
308
THE ELECTRIC FIELD PERTURBATION
309
atomic orbital (AO) basis sets depend only on the nuclear coordinates and not on the electric field. For such standard basis sets, Hai affects the one-electron, two-electron, and overlap integrals as was shown in Chapter 3. However, H affects only the one-electron integrals.
17.1.2
The Derivative of the Molecular Orbital Coefficients
Using the notation of Chapter 3, the molecular orbital (MO) coefficient changes with respect to electric fields and nuclear coordinates are
aci
MO
E
4
OF
a2 Ci 4
="--
OFOG and
cm uf m uS
(17.2)
771
E
MO
UIQCm mip
(17.3)
in
6 2 0 mo
E mi
OaOF
s
(17.4)
171
where "a" is a nuclear coordinate and F and G are electric field perturbations. The relationship between the first- and second-order U matrices including nuclear perturbations was presented in Section 3.3:
OUr:i
(17.5)
Ob
171
The corresponding relationships involving electric field perturbations are
av,f3 OG
uifig
mo
-E
ufm u4,
(17.6)
1777 rraj
(17.7)
and
a
/f.
Oa —
17.1.3
mo ual —
E
The Derivative One-Electron AO Integrals
As mentioned above, the electric field does not affect the overlap and two-electron integrals for standard AO basis sets. The relevant non-vanishing one-electron derivative AO integrals that involve electric field perturbations are dipole moment integrals and their nuclear derivatives. A first-order change in the one-electron integral with respect to an electric field
310
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
has been related to a position operator (see, for example, [1,2]) and the negative of this integral is termed a dipole moment integral:
ahAi, aF
(XkLiViXu) =
(17.8)
— e (XAirflXv) = —
(17.9)
where e stands for the electronic charge. The derivative of this integral with respect to a nuclear coordinate "a" is hm, aaaF
a (Oh„) Oa 8F 29 =—e (XAirfiXv)
(17.10)
)
(17.11)
= — e ( —)Lh nr.fiX ) — e ( X A. I rf 1 oa ) =
hg
.
(17.12)
(17.13)
Derivative one-electron integrals involving second- or higher-order electric perturbations are unimportant and thus are usually neglected (see, for example, [1-7]).
17.1.4
The Derivative One-Electron MO Integrals
The non-vanishing derivative AO integrals described above may be transformed into the MO basis as
AO
E ct c3OF ahAv
irtf
A
(17.14)
11
AO
E-
—
difj
(17.15)
and
AO
ht! =
AO
02 h
E ci" C3vaaaF i"/
=
Av
E cicihaf A
V
AV
(17.16)
Av
The general form for the derivatives of the skeleton (core) derivative integrals is given by (see Chapter 3 or Appendix K) 23
mo
= E (umb i hn," +
+
(17.17)
Thus, the derivatives of the one-electron integral, hifj in eq. (17.14), with respect to a second electric field G and a nuclear coordinate "a", are:
Oh( 3
OG
=
m° (
E
+ ufni hin
.,)
(17.18)
311
THE ELECTRIC FIELD PERTURBATION and AI°
Oa
(
+ u:.„,h(m )
—
hZ1
(17.19)
771
Note that derivative one-electron integrals involving second-order electric field perturbations were neglected.
17.1.5
Constraints on the Molecular Orbitals
The set of molecular orbitals is orthonormal, and in the MO basis this condition is
su i =
(17.20)
where f5 is the Kronecker delta function. The general form of the first derivative of these conditions including nuclear perturbations is (see Chapter 3 or Appendix J)
=0
.
(17.21)
The first derivative form of the orthonormality condition with respect to an electric field perturbation is given by
+ o ,
(17.22)
since the overlap integral in the AO basis is not affected by an electric field perturbation. The general form of the second derivative of the orthonormality condition including nuclear perturbations is (see Chapter 3 or Appendix L)
UfLib
(17.23)
eci`21? = 0
with
mo
s.rib
+ E
(uiam om
+
— Sb
uibm,u,9m —
sa
(17.24)
rn
There are two forms of second derivatives of the normalization conditions when electric field perturbations are involved. For pure electric field perturbations,
Ufg tj
C,f9 = 0 t3
.7 ,
(17.25)
with
e4-,g
m° ( uifm uf, +
ufm uifm )
(17.26)
For mixed perturbations, incorporating one electric field and one nuclear perturbation,
uiaif +
Y
u3
+
=o
(17.27)
with
(17.28)
312
CHAPTER 17. DERIVATIVES WITH ELECTRIC FIELD PERTURBATIONS
17.2
The Electric Dipole Moment
The electric dipole moment is defined by (see, for example, [1-7])
0 Etotal OF
=
a Enuc
o
(17.29)
a Eelec aF
OF
(17.30)
peflec
uc
(17.31)
where F stands for the electric field along the f axis. The nuclear part of the dipole moment for a molecule with N nuclei is given by
=e
E zARfA
(17.32)
A
In this equation Z A is the atomic number of nucleus A and RfA is the distance along the f direction between the Ath nucleus and the reference position (usually the center of mass of the molecule). The electronic part of the dipole moment is expressed as fi efiec =
0Eetec
(17.33)
OF
For the exact wavefunction this equation is equivalent to the expectation value of the dipole moment operator [1-7]: elec f
(17.34)
= — (TigIT)
(17.35)
= — (WIH'f1 41 ) =—
e
(xlilrfilli)
.
(17.36)
Equation (17.34) represents one statement of the well-known Hellmann-Feynman theorem [8,9]. However, the results from eqs. (17.33) and (17.34) need not agree for approximate wavefunctions.
17.2.1
The Closed-Shell SCF Wavefunction
The general equation for the energy first derivative of a closed-shell SCF wavefunction including a nuclear perturbation was given in Chapter 4 as
°Ede,
aa
2
d.o.
d.o.
E
d.o.
—2
E
Sq
(17.37)
ij
where d.o. denotes the doubly occupied orbitals and ci orbital energies. As mentioned in the preceding section, the second and third terms of eq. (17.37) vanish for an electric field
THE ELECTRIC DIPOLE MOMENT
313
perturbation. Therefore
tifelec
d.o. = —2h
Eelec
OF
d.o. 2E diri
(17.38)
1
From the definition of the dipole moment integral in eqs. (17.9) and (17.14), it is evident that equation (17.38) is equal to an expectation value of the dipole moment operator. In the closed-shell SCF case, the two expressions, the energy derivative from eq. (17.33) and the expectation value from eq. (17.34), give identical results.
17.2.2
The General Restricted Open-Shell SCF Wavefunction
The general equation for the energy first derivative of a general restricted open-shell SCF wavefunction including a nuclear perturbation was given in Chapter 5 as 'Melee
mo
mo
E
= 2 E
aa
MO
__ 2 E
+
SZcii
(17.39)
where E is the Lagrangian matrix. The second and third terms in eq. (17.39) vanish for an electric field perturbation. Thus,
elec
jUf
=
a E eiec
F
MO o
=—2
E
mo hit!,
= 2
E
fi dL
(17.40)
The results from eq. (17.33) for the derivative and eq. (17.34) for the expectation value formulae again are identical for the GRSCF wavefunction.
17.2.3
The Configuration Interaction (CI) Wavefunction
The energy first derivative including a nuclear perturbation for a CI wavefunction was given in Chapter 6 as
mo +2
E
qx, j
(17.41)
Q and G are one- and two-electron reduced density matrices [10] and X is the Lagrangian matrix. Since the two-electron integral is not affected by an electric field perturbation, the expression for the dipole moment becomes elec f
OE eie, OF =
MO
2E
Uifi Xi j
(17.42)
ij
mo
E
Qiidtj
(17.43) ij
314
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
The first term in eqs. (17.42) and (17.43) corresponds to the expectation value given by eq. (17.34). For a CI wavefunction, explicit evaluation of the Uf matrices is necessary in order to properly determine the dipole moment. The related subject of the evaluation of the second term in eq. (17.42) (using the Z vector method) will be treated in Chapter 18.
17.2.4
The Multiconfiguration (MC) SCF Wavefunction
The general expression for the energy first derivative of an MCSCF wavefunction including a nuclear perturbation was given in Chapter 9 as Me/ec
MO
MO
aa
MO
E ripd(ijikna — E szx ii
7i3 hiai ij
ijkl
(17.44)
ij
Here 7 and F are one- and two-electron reduced density matrices [10] and x is the Lagrangian matrix. With an electric field perturbation, the second and third terms vanish and the dipole moment becomes
mo
a Eelec
I_Lielec
mo
E = E
OF
dÇ
(17.45)
ij
For an MCSCF wavefunction the first derivative and the expectation value equations produce the same results as was the case for CLSCF and GRSCF wavefunctions. It should be noted that the evaluation of the changes in the MO coefficients is not necessary to determine the dipole moment for these three types of wavefunctions. The CI wavefunction presents a more computationally difficult situation.
17.3
The Electric Polarizability
The electric polarizability is a second-order property with the components of the polarizability tensor given by [1-7]
02 Eeiec OFOG
afg
(17.46)
where F and G stand for the electric fields along the f and g axes. This equation may be rewritten using the definition of the electric dipole moment, eq. (17.33), as
a of
(0Eei
„
)
8G 8F Ô (aEdec)
aF
OG
aPel ec
ac
(17.47)
ggelec
OF
(17.48)
THE ELECTRIC POLARIZABILITY 17.3.1
315
The Closed Shell SCF Wavefunction -
An equation to evaluate the electric polarizability is obtained easily by combining eqs. (17.18), (17.38), and (17.47):
am ef lec fg
d.o.
E
—2
= &G
a h ift (17.49)
OG
do, all
—2
EE (umg ihnifi + ufni kr„,)
(17.50)
rn all
—4
d.o.
EE
(17.51)
j virt.
= —4
,
virt. d.o.
=—4
d.o. d.o.
do.
EEUf, hÇ —
4
EE ufx,
(17.52)
d.o.
EE uriqi — 2 >2 (uri +
g)ht,
(17.53)
virt. d.o.
=
—4
UhÇ >2>2 i
(17.54)
Here virt. and d.o. denote virtual (unoccupied) and doubly occupied orbitals. In the last step of the derivation eq. (17.22) was used. The independent pairs in the Uf or Ug matrices are obtained by solving the coupled perturbed Hartree-Fock (CPHF) equations with electric field perturbations (see Section 17.5). An alternative expression may be obtained from the general equation for the second derivative of the electronic energy of the closed-shell (CL) SCF wavefunction given in Chapter 4: d.o.
82E
&t a b
d.o.
2
Eh + E
2
E
d.o.
—
)
( ii I
2(ii d.o.
2
all d.o. EE (Ut Ficij
11 E abEi all
Uiaj F
all
where the
d.o.
all
d.o.
4 E E i
+ 4
ab}
qu,bj f i
•
d.o.
EEEE
(ikiii) —
(17.55)
were defined in Section 3.7
ï all
ab
E
(cirn 3m +
— Siam S m —
(17.56)
316
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
For double electric field perturbations (i.e., the electric polarizability), the first three terms vanish and the resulting equation is afg
.92 Eelec OF8G
=
(17.57)
d.o.
2
all d.o.
E 7,ifigEi _
4
all d.o.
EE (uri Fifi +
all d.o. all d.o.
4
—
4
EE u4uf.,E,
i •
_ ( ikui) —
EEEE I k
-
(17.58)
The lifg and skeleton (core) first derivative Fock matrices are defined by all nifig =
E uifn uig,„
2
(17.59)
(17.60) and Figi
(17.61)
Although eqs. (17.54) and (17.58) appear to be different, it may be proven that they are in fact mathematically equivalent. Using eqs. (17.22) and (17.59), the first and third terms of eq. (17.58) may be rewritten as ci.o.
2
all d.o.
E
_
4
EE
(17.62)
do, ail
E
=4
all d.o.
uftrn U? tnnEi
EE
4
uifi urj Ei
(17.63)
ii
all d.o.
=4
EE
u4uE,
(17.64) Et
)
Using this result, eq. (17.58) may be reformulated as
r
all d.o.
=—4
EE u4[Figi
—
— (i)
all d.o.
+ EE k all d.o.
4
E
ii
-
Uri
(ik1j1 )
-
(illjk)
}1
I
UF.
ti
(17.65)
THE ELECTRIC POLARIZABILITY
317
In Section 10.1 the first derivative of the Fock matrix with respect to a nuclear coordinate "a" was found as
Oa
=
j all d.o.
+ EE uîc1 {4(iiiki) — k
(ik1j1)
(ilijk)}
(17.66)
1
Referring to eq. (17.66), the terms in the square brackets in eq. (17.65) represent the first derivative of the Fock matrix with respect to the electric field G. Thus eq. (17.65) becomes all d.o ,
fg = —
4
all d.o.
EE
uif,(— aaG3)
i
4
EE
Uf3 Fif3
(17.67)
For a CLSCF wavefunction the off-diagonal terms of the first derivative of the Fock matrix vanish, i.e.,
a
Ft;
ac
for
°
i j
(17.68)
and the diagonal elements of the 11 1 matrix are zero from eq. (17.22)
=0
(17.69)
Therefore, the expression for the polarizability is given by all do.
=
afg
EE Urj hti
—4
(17.70)
ii vir t d.o. —
EE
4
hfi
(17.71)
j
Equation (17.71) is identical to eq. (17.54).
17.3.2
The General Restricted Open-Shell SCF Wavefunction
An equation to determine the electric polarizability for a general restricted open-shell wavefunction is derived in a manner similar to that for the closed-shell case. Using eqs. (17.18) and (17.40),
afs
=
AL ) E Jf( ,2 — OG a
ape) ec
G
=— 2
MO
=
(17.72)
mo mo
E i
fi
E (uh;f7i, + U 9 .h
(17.73)
m
MO
=—
4
E j
fikri g
(17.74)
318
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
This equation may be further simplified to
mo
=
a f-9
4
E
fihirn
-
mo 4
i>i mo 4
E
(17.75)
mo 4
E
-
f,)hLuf,
(17.76)
i>j indep.pair
4
E
(fi
—
fi)hUri
(17.77)
j>i
Eqs. (17.22) and (17.69) and the symmetric property of the dipole moment integrals were employed in deriving eq. (17.77). In eq. (17.77) the summation runs over only the independent pairs, since the factor (fi — fi ) for any non-independent pairs is equal to zero. The Uf matrices are obtained by solving the CPHF equations with electric field perturbations (see Section 17.5). An alternative equation may be derived from the general expression for the second derivative of the electronic energy for the GRSCF wavefunction given in Chapter 5: a2 Eelec act.%
MO
MO
= 2
E
mo —2
fin?)
E Sc vcii
+
+ MO
-
2
mo + 4
E (up;€1, +
E %abG.; +
mo 4
E ijk
ij
MO
-I-
4
E
quk1 [2,,(iiikr) +
+
(17.78)
ijkl
For double electric fi eld perturbations, the first three terms disappear and the remainder is Ot
fg=
02 Eei„ 8F8G
Ado = 2
—4
(17.79)
mo
mo
E 74g€ii - 4 E ( 2 uif,Eli ) - 4 E u4u.,94k ii ii ijk MO E Uli tlfa [2aii(ijikl) + 13ii{(ikij1) + (ilijk)}1
(17.80)
ijkl
In this equation the n fg and skeleton (core) first derivative Lagrangian matrices are defined
THE ELECTRIC POLARIZABILITY
319
by MO
E
f
71;f
(uif„yin, + Urny31,2 )
7
Efj = fiqj
(17.81) (17.82)
and
=
fh
(17.83)
Again, although eqs. (17.77) and (17.80) look superficially different, it may be proven that they are mathematically equivalent. Using eq. (17.81) and the variational condition on the MO space (
ii
—
=0
Eii
(17.84)
the first term of eq. (17.80) may be rewritten as MO
MO
= 2
2
E
(utf,„,u3'%
ufm ulm )cii
(17.85)
ijm
MO
4
E
ufskuq
ez3
(17.86)
uhuzi fik
(17.87)
ijk MO
4
E ijk
Exploiting this result, eq. (17.80) may be reformulated as MO
afg
= 4
MO
E uh(E
MO
ukri qk —
ij
mo
E ki
uz,[2a.ii(ijiko
+
+ (iiiik)}1)
MO
—4
E
(1 7.88)
qe,f.
In Section 5.5 the first derivative of the Lagrangian matrix with respect to a nuclear coordinate "a" was developed as Oq Oa
MO a
C t"
E
MO
urci di
+E
MO
Ugi kl
+ di/
(17.89)
CHAPTER 17. DERIVATIVES WITH ELECTRIC
320
FIELD PERTURBATIONS
Using eq. (17.89), equation (17.88) may be rewritten as
E
4
af g
mo
aEi i)
mo mo
Ugk3•e-k•
E
OG )
u-zi Ejk
mo -4E UP.E-(• *,
(17.90)
ij
The first derivative form of the variational condition on the MO space is
aci;
_a
( ji
OG ÛG Using relationships in eqs. (17.22) and (17.91), equation (17.90) may be further simplified to MO
afg
= 2
MO
E (u
ufik
–
MO
OG
+E
Ili fik]
k
MO
— 4
E
uigi cçi
(17.92)
23
mo
•
—4
•
– 4
E
ufi E;fi
(17.93)
mo
E fi qi ut;
(17.94)
ij indep.pair
4
E
(f,
fi )hiri ur;
(17.95)
Eq. (17.95) is seen to be identical to eq. (17.77).
17.3.3
The Configuration Interaction (CI) Wavefunction
An equation to evaluate the electric polarizability for a CI wavefunction may be obtained by differentiating eq. (17.42) with respect to a second electric field perturbation G, i.e., 807fte.
afg
(17.96)
OG mo
–E
( 8Q ii OG "
+
Oht
067 )
- 2
mo ( au!, f ax • E —xi, + u.•— ij
OG
13 OG
(17.97)
THE ELECTRIC POLARIZABILITY
321
In this equation an element of the one-electron reduced density matrix, Q, is defined by [10] CI
Q, =
E cicAff
(17.98)
1J
In general the derivatives of the CI coefficients do not vanish, although the derivatives of the coupling constants are zero. Thus, the first derivative of the one-electron reduced density matrix with respect to the variable G is given by
OG
ij
= 2 E
(17.99)
IJ
In Section 14.1 the first derivative of the Lagrangian matrix with respect to a nuclear coordinate "a" was derived as CI
2
ci
E ci E aa
mo
mo
E ux o + E
+
kl
(17.100) Exploiting eqs. (17.6), (17.18), (17.99) and (17.100), equation (17.97) may be rewritten as MO [ CI E 2 E ci aG,j Q23 f ij If
°fg
MO
MO
E Qii E
Uti him)]
Ufni
MO
—2 E (f uiig - E — 2
MO MO
—
CI
xii n e-v
MO
j E u42 E ci OG + ij IJ
13
MO
+ E uvck i + E
UfdYij
kl
]
kl
(17.101) In this equation the skeleton (core) first derivative Lagrangian matrices, "bare" Lagrangian matrices and Y matrix are mo = (17.102)
E
MO
E Qfhm +
MO
2 EG
k/ (imiki)
(17.103)
mkl MO
Yik1
= Q jihik + 2
E
+ 2Gi min(imikn)}
mn
Rearranging the terms in eq. (17.101) one obtains
mo =—2
E
12
-
mo 2
E
(u9.x:f. 11 22
+
Usti Xfi
.
(17.104)
322
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
mo -
E (715 /41 Yi ikt
2
ijkl MO
CI r
ac,-2
>J ij 13
f Q i hi 2UX,1
OG
(17.105)
An alternative equation may be derived from the general expression for the second derivative of the electronic energy for the CI wavefunction given in Chapter 6: a 2 Eelec
MO
MO
iLl
ijkl
+
EQh + E MO
E
2
2
MO
+ uzxt) +
E U4- UL Yijkl
2
ij
ijkl
CI
ac, _ E aci ab IJ Oa
—
2
(17.106)
IJ — 6 .1J Eelec
For double electric field perturbations, the first two terms disappear and the resulting equation is
02 Eelec &FOG
Vs
(
(17.107)
MO
— 2
MO
E ufigx,„ —
2
ij
2
EUUj Yji +
ijkl,
(Uf.Xt3 + U4Xfi)
ij
MO
-
E
CI
2
aci ac,
(His — GJEeiec
.
(17.108)
IJ
3
Equivalence of the last terms in eqs. (17.105) and (17.108) may be proven in the following manner. In Section 6.7 the following relationships from the coupled perturbed configuration interaction (CPCI) equations were presented:
CI aci CI
(n xrij
E a. E =
Ob
a Eelec ) IJ
ab
CI eci cl
OHIJ (Mete) E cs,(— (5/.1 Oa Oa
ab ac, tiC j
=_E IJ
Oa Ob
II
6 1J-Edec)
•
Using these relationships the last term of eq. (17.108) may be manipulated as G./
2 E
aci ac.r
IJ
OF 8G
.[113 —
lEelec)
THE ELECTRIC POLARIZABILITY
CI
ac, (aHIJ aF ac, aHIJ
E
— 2
IJ CI
=
E
2
61 j meiec)
aF
au -- O F
IJ
In the last step of the above manipulation, the first derivative form of the normalization condition for the CI coefficients
ci E
aci aG =
0
(17.113)
was employed. In Section 6.4 the general form of the first derivative for the CI Hamiltonian matrix was derived as MO
Oa
(17.114)
E
+2
mo mo E Qwhz, + E Gfikt (iikoa +
mo 2
oci
ii
EUX1J
.
(17.115)
ij
Using eq. (17.115) for electric field perturbations, equation (17.112) becomes CI
OCI OCJ 2 2_, aF aG (HIJ
6 IJEelec)
IJ
CI
—2
ac,
E CI aG
MO
MO 3 ti
If
-1- 2 E U4X11)
(17.116)
t3
This result indicates that eqs. (17.105) and (17.108) are mathematically equivalent. In equations (17.105) and (17.108), it is clearly seen that solutions to both the first- and second-order CPHF equations (see Section 17.5) and to the first-order CPCI equations (see Section 17.6) with respect to electric fi eld perturbations are necessary to evaluate the electric polarizability at the CI level. This is a simple reflection of the fact that the polarizability is a second-order property. The related subject of the evaluation of the first terms in eq. (17.105) and (17.108) (using the Z vector method) will be discussed in Chapter 18.
17.3.4
The Multiconfiguration (MC) SCF Wavefunction
A simpler approach to the evaluation of the electric polarizability for an MCSCF wavefunction is derived using eqs. (17.18), (17.45) and (17.99):
mo
elec f
afg
=
aG
=
E
ulNi f (ac hij
ahf.
(17.117)
324
CHAPTER 17. DERIVATIVES WITH ELECTRIC
mo [
- E
•
CI
2
E
ij IJ MO rMO
- E
ac
FIELD PERTURBATIONS
, hf
OG
E (uz,,h ;f,„ +
Uli ttif,n )1
(17.118)
Lm
ij
MO CI
ac
- 2 EEoy"hi _ , i; i;
•
ij IJ MO
E
— 2
(17.119)
iim
aG ij Itifi
MO CI
EE
—2
f
ij IJ MO
-
E ur,x(i
2
(17.120)
In this equation the skeleton (core) first derivative Lagrangian matrices xf matrix are
mo xfj =
E
(17.121)
112
and -y" are the one-electron coupling constants [10]. An alternative equation may be obtained from the general expression for the second derivative of the MCSCF electronic energy. This second derivative expression was given in Chapter 9: MO
8 2Eeiec actab
MO
MO
MO
= E 71â hz, + E riiki(iiikoab — E stxii — E nztx i; iJ ijki ij ij MO
E (utx7; + uzi x0 + ii
+ 2
—2
CI
aci ac, f
E --, IJ
VI11
MO
2
E
Uiaj 141 Yii kl (17. 12 2)
— bIJ Eelec
For double electric field perturbations, the first three terms vanish and the resulting equation is O
a2Eetec
fg
( 17.12 3)
OFOG mo
E
—
ij
E (ui
+ uif,xf,) —
ij CI
+
mo 2
OG1 OGj
2 E 8F 8G IJ
(n 1 -
mo 2
E
u,,f; uf, Yijkl
ijkl j Ed e.)
(17.124)
THE ELECTRIC POLARIZABILITY
325
where the y matrix is MO
Yijkl = ljthik + 2
E fri imn(ikim.) + 2rin,„(inzikn)}
(17.125)
mn
Although eqs. (17.120) and (17.124) appear different, it may be proved that they are mathematically equivalent. Using eq. (17.81) and the variational condition on the MO space
x
=0
—x
( 1 7. 126 )
the first term of eq. (17.124) may be rewritten as MO
MO
E (utira u!„,,. +
% ii
uf,,, Ulm ) xij
(17.127)
Jim MO
2
==
1, ijk
(17.128)
MO
2
E
(17.129)
UfUxk
ijk
Employing this result, equation (17.124) may be manipulated as
mo 2
cEfg
MO
u { Eufi xik
E ki
Yijkl
MO
— 2 ij CI
2
aCj
E IJ
VI ij
(17.130)
(5/JEciec)
Using eqs. (17.100) and (17.116), equation (17.130) may be reformulated as MO c9's
= 2 ii
+ 4
Uifi [1‘O UZi xik L k
Ago
CI
ij
IJ
CZ ) +
MO U,.i ig. xicj k
acsi E uiti E ci---xt i;
MO
— 2
— 2
E ufi xif,
ii ci ac,(M° E c,— E OG .. IJ
tj
ij f 7- • h• • 4- 2 23 23
MO
E ut_xf4 t3 s3 ii
(17.131)
326
CHAPTER 17. DERIVATIVES WITH ELECTRIC
where the "bare" Lagrangian matrices MO
IJ = Xii
X"
FIELD PERTURBATIONS
are
IJ ttirn + 2 7im
mo
E
rki(im pa)
.
(17.132)
mkt
The first derivative form of the variational condition on the MO space is expressed as
ax ii (17.133) OG OG Using the relationships in eqs. (17.22) and (17.133), equation (17.131), may be further modified to mo uzixi, _ (a ) + u:ti)[E E ( + aG ij k
Mo afg =
)
-I-
Mo k
E uifi E cac,
MO
CI
ij
1J
4
MO
_ 2
E uri x,f,
— 2
M° E aCj CI-.ö-( > E7 I,Thf. + 2 E
ii
CI
MO
„
IJ
13
ij
u,f.x4 23 Sj
(17.134)
ii
mo
—2 E CI
-
2
2, t,
MO
E c, aCj OG E IJ
y ii
(17.135)
ij
This equation is seen to be identical to eq. (17.120). The elements of the Uf and OCi/OF matrices are obtained by solving the first-order CPMCHF equations with respect to electric field perturbations (see Section 17.7). This is clearly a greatly simplified procedure compared to the more general CI case.
17.4
The Dipole Moment Derivative
Derivatives of the dipole moment with respect to nuclear coordinates are necessary to evaluate infrared (IR) intensities, an important goal of many quantum chemical studies. These derivatives are a second-order property and are expressed as apf
_
ô2 Eotai 0a0F
Enuc
a2 Eelec
(17.136) — aaOF aaaF The nuclear part of the second derivative is non-vanishing only when the direction of the electric field and the nuclear coordinate coincides: aterc N a2Enuc = e E ZA (17.137) aa = aaaF A
Oa
THE DIPOLE MOMENT DERIVATIVE
327
The electronic part of the dipole moment derivative is given by afieflec
Oa Here "a" and tively.
17.4.1
8 (0Eelec _ a (aEetec .92Eetec (17.138) = aF Oa ) 8a0F aa OF ) F stand for the nuclear coordinate and electric field perturbations, respec-
The Closed - Shell SCF Wavefunction
An expression for the dipole moment derivative using a closed-shell SCF wavefunction is obtained from eqs. (17.19) and (17.38): ateflec d.o. 02 Eelec —2 (17.139) Oa &OF &a
=—2
U,ani k-fm ) all d.o.
=—4
d.o.
EE u,7i h;ri — i
2
E
Uh — 4
E
EE
d.o.
d.o.
E hve
//N.; — 2
d.o.
d.o.
virt.
—4
(17.141)
tic:"
j
virt.d.o.
=-4
(17.140)
E E utlq
— 2
E
(17.142) d.o.
(uft, +
Eq. (17.21) allows a simplification of the last equation to give
aplec
virt. d.o.
4
aa
E E uiaj hfj +
d.o.
2
E
d.o.
2j
23
—
2
E
i
Note that eq. (17.144) is not symmetric in terms of the variables "a" and F. An alternative equation is derived using the general second derivative expression in eq. (17.55). For mixed perturbations the second and third terms in eq. (17.55) disappear, resulting in the following:
apefiec aa
do.
=
—
2
E +
d.o.
2
E
all d.o.
-
4
EE (uti Fia, +
-
4
EEEE
—
4
qu{4(iiik1)
—
uzF)
all d.o. all d.o. ijk
I
all d.o.
EE UUq
—
(auk)}
(17.145)
328
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
where all
af
= 2
E
Uzln Ujm
(17.146)
Eq. (17.145) is symmetric with respect to the variables "a" and F. It requires the evaluation of both the Ua and Uf matrices, while eq. (17.144) uses only the Ua matrices. Mathematical equivalence of eqs. (17.144) and (17.145) may be proved in a similar manner to that described in the preceding section.
17.4.2
The General Restricted Open Shell SCF Wavefunction -
In an analogous manner, an equation to determine the derivative of the dipole moment with respect to a nuclear coordinate may be obtained using eqs. (17.19) and (17.40):
MO /f1L) 2 79T
02 Eetec
0a0F = MO
MO
—2
(17.147)
E
h`tifj
_
(17.148)
MO
E
2
(17.149) MO
MO
E
fiUli htj — 4
fjqh-jfi — 4
E
fiUtlh;ri ( 17.15 0 )
MO
4
E [-
fi(U,7;
SOh ifi
i>i
MO
2
E
MO
fiS1111;fi — 2
MO
4
E
-
E fiqif mo
fi)Uf`i hifi + 4
i>i
E
E i>,
mo
MO
2
fiSah;fi
2
E
(17.152)
indep.pair
4
E
MO
— fi)UZi hiri + 4
MO
2
E
(17.151)
i›;
fi szhirj
MO
ii ii
- 2
E
J: "ii
(17.153)
329
THE DIPOLE MOMENT DERIVATIVE
The summation in the first term of this equation runs over only independent molecular orbital pairs, since terms involving any non-independent pairs vanish. Note that eq. (17.153) is not symmetric with respect to the interchange of the variables "a" and F. An alternative expression is derived using the general second derivative equation in eq. (17.78). For the cross perturbations, the second and third terms in eq. (17.78) vanish and one finds MO
2
MO
E
+
2
E
af Eij
ij
MO
MO
E (uhfcji +u29,E3f.i )
4
—
4
ij
E ijk
mo
E
—4
UzajUkfi[201j1(iiikl)
13j1{(iklil)
(1 7. 154)
ijkl
where
mo
E
af
(uf,nnuf ,m
+
uif„,u29,)
(17.155)
Eq. (17.154) is symmetric in terms of the variables "a" and F. It requires evaluation of both the Ua and ul matrices, while eq. (17.153) uses only the II matrices. Mathematical equivalence of these two equations may be proved in a similar manner to that shown in the preceding section.
17.4.3
The Configuration Interaction (CI) Wavefunction
An expression for the dipole moment derivative for a CI wavefunction may be derived using eqs. (17.7), (17.19), (17.42), (17.99), and (17.100):
01.2 fe lec
02 Eelec
,
(17.156)
&OF
Oa =
j''
MO
E
=—
[ 2
ij
MO
CI
oc j
IJ
Oa
E ci
QPhi. 3 3
I
,_., — 2 N--' ij
MO
— rd Q ij
ii
MO
_ 2
MO (au!
a " hf3 +Q,3 0ahafi
O
L
E (uLhf„i + U!,`„i hir„,) + 4731 rn
E (Iv ij
--IX.- + Usij -( ) Oa Oa 23
MO
_ E uiln uini)xii In
(17.157)
330
CHAPTER 17. DERIVATIVES WITH ELECTRIC [
MO
-
2
E uh
2
ij
CI
MO
IJ
k
FIELD PERTURBATIONS MO
ij E c, ac, xi . + 10. a3 + E Oa
+E
ULYijkl
kt
(17.158) The two types of skeleton (core) first derivative Lagrangian matrices for the CI wavefunction are MO
MO
E Qi
nign
+ 2
E
mkt
and
mo
E
Q3nikf,
Rearranging the terms in eq. (17.158) gives 0 tl eflec
Oa
mo
• _E
mo
¼t 3
j3
- 2
E
mo -
2
MO
E (uhicr, +
—2
i3 MO CI
— 2
UiajULYijkl ijkl
[
E E c, OC aaJ (ea.12 tj IJ
+ 2uhx4.))]
( 17. 161)
An alternative equation may be derived from the general expression for the second derivative of the electronic energy in eq. (17.106). For the cross perturbations, the second term in eq. (17.106) vanishes and the resulting equation is
ap eflec
mo
mo
-E
aa
Qijh 7if — 2
E
via/
ii
ii
mo -
2
E(u4Xiai UX4) — ij
mo 2
E qukfi yiikt ijkl
ci aci ac
E
2 IJ
-07(H" - 6"Eelec )
(17.162)
Eq. (17.162) is symmetric in terms of the variables "a" and F. Mathematical equivalence of eqs. (17.161) and (17.162) may be proven in a similar manner to that of the preceding section. From the above equations it is evident that, for the formalism presented, the solution of both the first- and second-order CPHF equations (see Section 17.5) and the first-order CPCI equations (see Section 17.6) is necessary to evaluate the dipole moment derivative at the CI level. The related subject of the evaluation of the second terms in eq. (17.161) and (17.162) by the Z vector method will be described in Chapter 18.
THE DIPOLE MOMENT DERIVATIVE 17.4.4
331
The Multiconfiguration (MC) SCF Wavefunction
An equation to calculate the MCSCF derivative of the dipole moment with respect to a nuclear coordinate may be derived using eqs. (17.19), (17.45), and (17.99):
°m elee Oa
MO (
02Eetec
=
E
E
—
Octal'
i
+ 7,3 at
(17.163)
tf-„,) + h.7311
(17.164)
13
MO (
CI
aCj
2 E
ii
7, h Oa 3
IJ
()
[MO
MO
E
E
MO CI
•
—
2
ij f EE c, aCj h• • Oa 12
ii
13
IJ
MO
MO
E
— 2
mi
i2m
MO CI
▪
—2
EE 12
E
2
E
-riih731
(17.165)
ii
ôCj iji
lia
IJ
MO
-
—
a.-
"
12
MO
UxÇ
-E
ii
ii
(17.166)
Y j ii
Note that eq. (17.166) is not symmetric with respect to the interchange of the variables "a" and F. An alternative equation is obtained from the general expression for the second derivative of the MCSCF electronic energy, which was given in eq. (17.122):
aiz eflec
Oa
mo
mo
- E 7iih71 + E 27zif mo — 2 E (uifi zt, + uzxL) -
mo
2
E
uiai ukri Yij kl
ijkl
+
2 v, 19C/ IJ
aCj
lia OF VII
IJ
6IJEelec)
(17.167)
In these equations the two types of skeleton (core) first derivative Lagrangian matrices are MO SZi
E
MO
-6,1477, + 2
E r,mkt(imiki).
mkt
(17.168)
332
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
and
(17.169) It is clearly seen that eq. (17.167) is symmetric with respect to the interchange of the variables "a" and F. The elements of the Uf, Ua, OCII0a, and OCi/OF matrices appearing in eq. (17.167) are obtained by solving the first-order CPMCHF equations with mixed perturbations (see Section 17.7). Eqs. (17.166) and (17.167) appear to be different, but their mathematical equivalence may be proven in a similar manner to that presented in the preceding section.
17.5
The Coupled Perturbed Hartree-Fock (CPHF) Equations Involving Electric Field Perturbations
The coupled perturbed Hartree-Fock (CPHF) simultaneous equations provide the derivatives of the MO coefficients with respect to perturbations. In this section the CPHF equations involving electric field perturbations are discussed.
17.5.1
The First-Order CPHF Equations for a Closed-Shell SCF Wavefunction
The general form of the first-order coupled perturbed Hartree-Fock (CPHF) equations was given in Section 10.1 as
(C
jEi)q
-EE
( 17.17 0 )
Aij,k1Uf ci z =
k
The A and B0 matrices in eq. (17.170) are Aii,kt
= 4(iilki) -
- (ill ik)
(17.171)
-
(17.172)
and d.o.
=
E
—
kl
The skeleton (core) first derivative Fock matrices are d.o
=
h
+ E {2(ijikkr
— (ikijk)a}
.
(17.173)
In these equations virt. and d.o. denote the virtual (unoccupied) and doubly occupied orbitals, respectively.
THE CPHF EQUATIONS WITH ELECTRIC FIELD PERTURBATIONS
333
For a first-order electric field perturbation the CPHF equations are formally the same as eq. (17.170): (E3 — E t•)
trirt.
d.o.
k
1
EE
Ul13
(17.174)
B o,ii f
Aij,k1 Uk fi
The difference lies in the elements of the Bàf matrices, which are = 1-4 = It13
17.5.2
(17.175)
The Second-Order CPHF Equations for a Closed-Shell SCF Wavefunction
The general form of the second-order CPHF equations was developed in Section 10.4 as virt. d.o. (Ej —
Ei) U b
—
EE k
a = Bb 0, ii
where the A matrix was defined in eq. (17.171) and the
Bist6
&Lib Ej —
all
(uFzi
+
(uLut
+
all
• E
Be
matrices are
d.o.
Fait
• E
(17.176)
1
E kl
41{2(iiik1) —
+
u;Fib,
+
(ik(ji)}
Vi ak
all d.o.
EE
iii6m --ij,k1 A
kl m all d.o.
• EE kl
(u,a,i vibm
+
NjUil7r4 ) Akjarn
(14Juibm
+
utj uiam)Aik,Im
tn
all d.o.
EE kl all d.o.
• EE (urc,}16, utAzi,k,) k
(17.177)
1
In eq. (17.177) the ea' matrices, skeleton (core) second derivative Fock, and first derivative A matrices are defined by cab
Sij
(17.178)
334
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
d.o.
E
h ci?
{2(ii)kkrb — (ikuk)b}
and =
—
kr
For double electric field perturbations, the CPHF equations are formally the saine as eq. (17.176): vir t.
(E — i)
ig —
k
The elements of the
d.o.
EE
ufs
Aii,k1 ki =
p h'
.
(17.181)
I
B4 9 matrices are given in a slightly simpler form as d.o.
13 n 1g- = —
E
7
—
kl all
+ E (ukfi
+ ufi Fkfi +
ukf.Fg 3 k + ukg„Fik )
all f f k E (uLuf ., ugkiUkj) all d.o.
+EE kl
Ukfm ugn
m
all do.
EE
(uurrn
+
Ulm ) Aki ,im
(quigm
+
urfi Ulfm)Aik,Im
kl m all do.
EE kl m The
(17.182)
F1 and f9 matrices appearing in eq. (17.182) are (17.183)
and all
utin,
+ ufm uf,„
(17.184)
m
For the cross perturbations, the CPHF equations again are formally the same as eq. (17.176): virt.
(C j — E')U 13 ,I—
d.o.
EE Aii,k, uTraf k, k
1
1J
oi
(17.185)
THE CPHF EQUATIONS WITH ELECTRIC FIELD
PERTURBATIONS
335
The elements of the Boal matrices are do.
Ba
r tj f —
—
tqfj
E ezif{2( i k ii o _ kl
all
+ E
(uf:i Fkf,
+
+ u;;; Fik +
Ifci F
all
( u, + ukfi t4i )Ek + E (u all d.o.
▪ EE
ULUIm Aij,k1
kl
m all d.o.
+ EE
(uAufn,
+
ULUiam ) Akj,I m
771
kl
all d.o.
+ EE (uu,f + kl
quiam)Aik,Ini
111.
all d.o.
• EE k I
(17 .186)
UilAkj
In eq. (17.186) the skeleton (core) second derivative Fock and
ea '
matrices are defined by (17.187)
= 14:71
and all
E (ugn uln,
eig =
17.5.3
+
/Jim
u;,,)
(17.188)
The First-Order CPHF Equations for a General Restricted Open-Shell Wavefunction
The general form of the first-order CPHF equations for a general restricted open-shell wavefunction was given in Section 11.1 as indep.pair
JAI Ujaci = Ba,ij
(17.189)
k>1
where "indep.pair" denotes the independent pairs of molecular orbitals. The A and matrices are defined by =
2 (Crik
ajk
aii
(Oik — Oik — Oit
ail)(iiiki) Oji)
(ikiii)
(iliik)}
Bg
336
CHAPTER 17. DERIVATIVES WITH ELECTRIC +
Ski
— E j l)
bkj
(Cfi
FIELD PERTURBATIONS
— 0514 — Ei k)
(5 1j
@ci
(ik)
(17.190) and a
a
23
31
all occ
- EE sz,[2(aik -
+ (illjk)}
(Oik —
k>1 bki
6kj
(di
ell)]
occ
- E s;:k kaik -
aik)(ijlick)
(f3ik — fijk)(ikijk)]
(17.191)
In the above summations, "all" implies a sum over both occupied and virtual (unoccupied) orbitals, while "occ" denotes a sum over only occupied orbitals. In the last equation the skeleton (core) first derivative Lagrangian matrices are defined as
mo
fiqj E
.
+
(17.192)
For the case of an electric field perturbation required here, the CPHF equations are formally the same as eq. (17.189): indep.pair
Aij,k1 U11 =
Bf
(17.193)
k>1
The difference occurs in the elements of the BI, matrices, which are Dl
17.5.4
= €13 —
(17_194)
=
(17.195)
The Second-Order CPHF Equations for a General Restricted Open-Shell Wavefunction
The general form of the second-order CPHF equations was developed in Section 11.7 as indep.pair
E k>1
Aij,k1U,1
=
B.(a4i
(17.196)
THE
CPHF EQUATIONS WITH ELECTRIC
FIELD PERTURBATIONS
where the A matrix was defined in eq. (17.190) and the pab,
337
Bjb matrices are
abab
E 23
- —
Eii
all occ -
eg[2 (aik —
Ld r k>
+
5kj
— fit) —
(Pik
+
—
(iiii k)}
il)
occ
- E kaik aik)(ijikk) + (fiik — 13ik)(ikljk)J • E (uLut, + ulai ) (di • E un uen, (aim — aim)(i.ilkl) + (ai — aim){(ikli/) + (ilijk)}1 [2 E (pLUmb n + u,, ILL) [2 (crin ain )(kj I mn) all
kl all
klm all
!ram
+
,din ){(kmljn) +
(knijm)}]
all
• E
ri'
+ Uty,ann ) [2 ((tin — ain ) (kilmn)
krnn
-
(knlim)}]
ajn){(krulin)
all
E Urc1 [2 (au E utc, 2(crii
aji)(ijikn b +
—
{(ikijO b
+
—
13; t ) {(i k iiI )a
(i/Ijk) b }1
kl all
+
(illjk)all
kl all ub (pia "ki Skj
j k)
-
f.
k)
—
uibo
elk)
—
Ea)]
(17.197)
In eq. (17.197) the skeleton (core) second derivative Lagrangian and first derivative generalized Lagrangian matrices are defined by
E all
ab
irb}
(17.198)
▪ 338
▪
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
and all
E
=
faik(iiikk)a
+
oik(ikuk)-}
.
(17.199)
For the double electric field perturbations the second-order CPHF equations formally are the same as eq. (17.196) indep.pair
BPii
Aii,k1 Utig =
(17.200)
k>l
and the BP matrices are given in a somewhat simpler form as all
Bfg. 0, tj
occ
—EE
etfkaik
— ajk)(ijik/) -I- (Oik — Oik){(ikij/)
-I- (idijk)}
k> 1 + 15ki (
—Cli CjI )
— (5kj ciii — Cil
eildkaik
— aik)(ijikk) + (j
occ
- E all
+ E (uti uri +
Pik)(ikijk)]
—
kl all
+E
—
aim )(ijIkl)
(aim
Oin-,,){(ikij1)
(iiljk)}]
kim all
Eran k
kt
tirci Ui71.1
mn
[2(cri„ — ai„)(kjimn)
{( kmljn)
+ (pin —
(knijm)}1
all
+ E (uk u;g„n + uef uLn){2(ain — fi
ain )(kijirtn)
kmn (I3in
i3in ){(kmlin)
(knlim)}1
all
uifei Citgj
clkji — E.fpc
- Egik
— [ikj f (Cig— ki
ik
Etk
•
(17.201)
In this equation the skeleton (core) first derivative generalized Lagrangian matrices are defined by
=
fihf;
(17.202)
THE CPHF EQUATIONS WITH ELECTRIC FIELD PERTURBATIONS
339
For the mixed perturbations the second-order CPHF equations again are formally the same as eq. (17.196) indep.patir
E
Aii,k1
Ukajf
(17.203)
=
k>1
and the Boa f matrices are
Bal
— all occ = elacif [2 (Ctik Eaf ij
E af
agk)(ijikl)
3ik —
(1
Oik) { ( iklil ) + ( i lljk)}
k>1
+
bki
— cii)]
(di — E, 1) —
occ
—E
(kilf{(ctik
— aik)(ijikk) + Oa — Ojk)(ikijk)]
k
all
+ E (i4j uif; + ukf,uri ) (di
(L)
—
kl
all
+ E nn uifrn [2(ai7fl —
ajm )(ijik/) + (Am
—
iklil)
+
(illjk)}]
klm
all
+ E(u:u +
quAn)[2(ain. — ain)(kilmn)
kmn
+ (Pin — Oin){(kmlin) + (kniint)}] all
+ E (uïci uhn +
— ain )(kilran)
kmn
13in) {(kinlin) +
Oin —
(knlim)}1
all
+ E uld [2(aii
—
— Oit){(iivijir +
+
kr}]
kl
all
+ E [u,a,i (df,
-
f
— [4:41cfi —
jk) Elk
Uifci (daj
—
Ecik)
—
fa)
.
(17.204)
In this equation the skeleton (core) second derivative Lagrangian matrices are defined by f — fh af 13
Ea t3
(17.205)
340
CHAPTER 17. DERIVATIVES WITH ELECTRIC
17.6
The Coupled Perturbed Configuration Interaction Equations Involving Electric Field Perturbations
FIELD PERTURBATIONS
The coupled perturbed configuration interaction (CPCI) simultaneous equations provide the derivatives of the CI coefficients with respect to perturbations. In this section, the CPCI equations involving electric field perturbations are described.
17.6.1
The First Order CPCI Equations -
The general form for the first-order CPCI equations was developed in Section 12.1 and is
I (
CI
E
(Hy
—
45/./Edec +
19Hij aa
Oa
)
(5 IJ
a Eeiec Oa
)Cj
(17.206)
where the first derivative of the CI Hamiltonian matrices is OHIJ
MO
Oa
MO
MO
1 .7 Qij
+
E
ij
2
E (p_xw 12 13
(17.207)
ii
ijkl
and the "bare" Lagrangian matrices are MO
—E
+
—
MO
2
E
.
(17.208)
mid
For an electric field perturbation the CPCI equations formally are the same as shown in eq. (17.206), with the displacement "a" replaced by the electric field F: CI (
CI
E
(His
— 151- JEdec
2C1Cj s\ ôCj = —
aF
E
OHL,
6. 1.1
aF
Meie
OF
c )CJ . (17.209)
The elements of the derivative matrices on the rhs are of the forms (see Section 17.2) aHis
MO
MO
OF
Qij
+
2
E uf.xJ,„,/ ,„
(17.210)
and 8E,1„
OF
MO
— p fel" =
E
+
MO
2
E
(17.211)
THE CPCI EQUATIONS WITH ELECTRIC 17.6.2
FIELD PERTURBATIONS
341
The Second Order CPCI Equations -
The general form for the second-order CPCI equations was given in Section 12.2 as
E (His —
buEelec + 2c/c,) '02.caJb
CI (02H, j
E
L1
Ow%
ci ( a HI.,
E
(5 IJ
Oa
cl (OHIJ Ob
02Eelec
&Lab
)
CT
8 Eelec
8C1OCJ aa ab
8a
8Eeie, 61.1 Ob +
—E
8C1OCJ
Ob
(17.212)
Oa
where the second derivative of the CI Hamiltonian matrices is mo mo 02 H mo
E
Om%
wih,(1;
+E
ii MO
+2
Gti,a(iiikoab + 2
ijkl
E(
xJa
ii
E ujlbxj! ij
MO
+ uz )(jib)
+ 2
E uiajutifg
.
(17.213)
ijkl
The skeleton (core) first derivative "bare" Lagrangian and "bare" Y matrices appearing in eq. (17.213) are defined by MO
E
Xlija
MO
+ 2
E
mid
and mo
ijkl = Vizik
+ 2 E {Gy,,„(ikimn) + 2G.f.m J in(imikn)} mn
For double electric field perturbations, the CPCI equations formally are [with suitable substitute variables] the same as given in eq. (17.212): CI
— (5 1,7Eelec + 2cic,) aaF2c aG J CI
=—
E
(02 H I j
OFOG
aHIJ CI
_
(
CI (
aF
aH OG
02 Eejec „ 451J
OFOG
)Uj
a Eeiec
'" aF meiec
61J OG
010CJ
`-' 1 8F OG f
., OG
aF
(17.216)
342
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
The elements of the second derivative Cl Hamiltonian matrices in eq. (17.216) are 02
Hjj
OFOG
mo =2
E
+
U/ X'
mo
E (gx,-1/' +uf.xi ,g) +
2
xj z3
ij
mo 2
ii
E
Uzi13
ijkl
(17.217) in which the XIJ i matrix is
mo — E ti —
(17.218)
3mIrr im
The energy second derivative with respect to electric field perturbations was discussed in Section 17.3. For cross perturbations, the CPCI equations again are formally the same as presented in eq. (17.212): CI
E
(His — 451JEetec ci f 32 Hjj
_
192Cj ) aaall
192 Eetec) ri
dadF
(aHij aa
dadF
6 1,1
1.1 (OHL/
_
2CIC's
aEc j ec ale
OCt J)aC J Cjw-
OE ec OF
aF
OF)
(17.219)
Oa
The elements of the second derivative CI Hamiltonian matrices in this equation are given by
mo ▪ E
Qwhzi
+
2
mo E 13
MO
+
2
E ii
(ufixiiya
+
+
MO
2
E
uukf,yijil,
.
(17.220)
ijkl
The energy second derivative with respect to cross perturbations was described in Section 17.4. The first and second derivatives of the CI energy (electronic part), first- and secondorder CPHF equations for U, U 1 , and U hf, and the first-order CPCI equations for Wilda and acilaf must be evaluated before constructing the second-order CPCI equations.
▪ THE CPMCHF EQUATIONS WITH ELECTRIC
17.7
FIELD PERTURBATIONS
343
The Coupled Perturbed Multiconfiguration Hartree-Fock Equations Involving Electric Field Perturbations
The coupled perturbed multiconfiguration Hartree-Fock (CPMCHF) simultaneous equations provide the derivatives of both the MO coefficients and CI coefficients with respect to perturbations. These two derivatives are required to evaluate higher-order properties for MCSCF wavefunctions. In this section the CPMCHF equations involving electric field perturbations are described.
17.7.1
The First-Order CPMCHF Equations
The general form of the first-order CPMCHF equations was given in Section 14.1 as A11 A21 1
[
[
A21
ua
Bal 1332
OC/0a
(17.221)
[
In this equation A n is the MO part and each element is given by —
ii ,kt
5jjXk3 — bijxki Yijkl
Yjikl
bkiXii
ki X1j
—
Yijlk
Yjilk
(17.222)
•
The A22 term represents the CI part and is Ik 22 I,J
6IJEelec
— H IJ —
2C1CJ
(17.223)
The A21 (Ant) term shows the CI-MO (or MO-Cl) interactions with matrix elements CI
A 21 — —
2
E
xff cf,(
_
x/i f)
(17.224)
where the "bare" Lagrangian matrices in this equation are defined by MO
E 7jhim +
Xf!
MO
2
E
(17.225)
mkt
The perturbative term for the MO space may be written as Dal
+ mo
+ E
bkix„ — (5kix,i +
Yjilk
Yijik
k>I MO
•
2
E
Sj:k(bkiXkj —
Yijkk
Yjikk)
(17.226)
344 where the
CHAPTER la
17.
DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
matrices are defined by MO
E
MO
-yindetm, + 2
E rimkt (imiki)°
.
(17.227)
mk1
For the CI space, the perturbation term is CI
CI
B oa,21 =
- ECjH, + CI
2
ECj E
MO
+E
MO
E .5Ti xy +
Sx
aEi> elejc
(17.228)
aa
In this equation the skeleton (core) first derivative Hamiltonian matrices are given by
mo
mo
=E
+E
ij
.
(17.229)
ijkl
For a first-order electric field perturbation, the CPMCHF equations formally are the same as in eq. (17.221), namely,
[
1310.1Bfa2
An A21t 1 [ Uf 21 A22 j OC/OF A
(17.230)
The elements of the 13g1 matrices for the MO space are
Bli = o • •
+
(17.231)
hilm
(17.232)
with the sf matrix being MO
E
For the CI space, the perturbation term is f2 B0,1
CI
=—
E
+
OEeiec
OF
(17.233)
where the 'ILI matrix is given by MO
E
(17.234)
THE CPMCHF EQUATIONS WITH ELECTRIC 17.7.2
FIELD PERTURBATIONS
345
The Second Order CPMCHF Equations -
The general form for the second-order CPMCHF equations was given in Section 14.4 as [
{ An A21t An A22
uab
[
—
02C/Oa0b
Bald B ab2
(17.235)
0
Al2
The elements of the A ll , A22 , and matrices were defined in the preceding section. The perturbation term for the MO space, B'1, are given as O,i j
= — Xiajb +
X
ab
MO
+ E ece
—
15kixii
bkixii +
Yijik
—
xi iik )
k>I
MO
▪
E
2
6kixki
MO
- E
(ugixt,
—
— bki Ski
ugixti)
MO
- E
urci(ytki —
(upcix7c,
—
UtiXiaci)
—E
uti(eiki
_
jikl)
kl MO
E
onn
kmn MO
4
MO
—E
Yjikk
MO
b jtikl
kl MO
-
Pijkk
E
E
UgiAjmn. UjcjYkimn
uman
UteiYkjmn
—
Ufrci ykinvn
kmn
ugi umb n .tr,,„„(ikimp)
+ r,ni
(imikp)
+ r; pin(ipikm)
klmnp
- r ilnp(i kIMP)
-
ci 2 V‘ °CI 8Cj ( L-4 Oa Ob
IJ
— 2
Akin)}
r inip(jMIkP) — r
j
XY )
C/ C/ Ozti E ci E aCj Oa ab
1941 ) Ob
acJ (axf; 06
Oa
ox.f()1 Oa ) (17.236)
where the xab and ya matrices are defined by MO
=E
MO
73,1t7m b + 2
E
r,mk,(imikoab
(17.237)
mid
mo = eyjin, + 2
E fr; imn(ikimur mn
2rimin (imikn)a}
(17.238)
346
FIELD PERTURBATIONS
CHAPTER 17. DERIVATIVES WITH ELECTRIC
The first derivative of the "bare" Lagra-ngian matrix was presented in Section 6.5 as
OxY 13 Oa in which the
MO
jja
MO
+ E u1:i x1 + E
ria IJ
(17.239)
kl
matrix is
X ija
IJa
=
Xij
mro
MO V".• IJ
+2
7jm"irri
E GnJ ki (imikoa .
(17.240)
mkt
The perturbation terms for the CI space, Bgb2 , are
Bel =
CI
CI
—E
Cjiiej
CI
MO
ECj E ablJ + E
+2
MO
CJ
E
cab IJ
02 Eelec
i>j CI
-
2
MO
Cj
E (ut x fj'a + uiai x fib
)
ij MO
CI
-
2
ECj E ijkl
CI if ail if
- E C/
aa
(5/J
aHL,
- E
OE,1„ + Oa
OCJ)ac, Oa ) ab
61JOEeiec
(17.241)
Ohs Oa
Ob
Ob
In this equation the skeleton (core) second derivative Hamiltonian matrices are given by MO "
IJ
___
MO
E
.7
ij
23
E
(17.242)
ijkl
whereas the first derivative of the Hamiltonian matrix was developed in Section 6.4 as ffiVij Lia =
Ado
MO
ij
ijkl
MO
rfiki(i jikir + 2
1: ET.ixf."
(17.243)
For double electric field perturbations the second-order CPMCHF equations are formally the same as eq. (17.235):
[ An Ant A21 A22
[
[ 8 fg1 ufg 02 c/oFoG= Blig2 0
(17.244)
THE CPMCHF EQUATIONS WITH ELECTRIC
FIELD PERTURBATIONS
347
The Bfogl matrices for the MO space are given as
Blo 0,ij
mo = E eff (45kixti
—
bkjSli +
— Yjilk)
Yijlk
k>1
MO ,
+
(0 kiX kj — 6 k j X ki +
—
YijIck
Yjikk
k
MO
—E
(ukfi xt,
MO
—
u13 4
—E
)
k
MO
_
E
qi(Yfiki
kt MO —
—uigc, , )
urci sik ,
k
E un (U
MO
—
—E
v:ki)
_
ufi(Yfikt
f ) Yjikl
kt
MO
— Ukf j Ykimn)
njmn
E
—
kmn
Un ULYkjmn
Ukgj Ykimn)
kmn
MO
_ 4
E uk un {ri,„,(ikimp) + rin,p(imikp) + fi
ripin(iPikm)
klmnp
(i kiMp) — r in ip (intikp) — ripin(iPikm)}
—1 1
— 2
E IJ
(x tj OF O G
ci
— 2
E
ci
oi {
E
Jz
,c,j ( Ox iii
-
+ ac, (axf;
8x7) 3
1
aG
OF \ OG
J
1
-
ti
OG
OS I,f'
a F3
OF
\1
)1 (17.245)
In this equation the yf matrices are defined by ,f
=
(17.246)
and the first derivatives of the "bare" Lagrangian matrices are Oxif
OF
MO
+. E
=
MO
ufki Ski ij
4-
E Ufkl ijkl
(17.247)
kl
with
I Ji
X ij
MO
=
E
Ij
hfim
(17.248)
ni
fg2
For the CI space the B o matrices are Cl
Bàfg12 = 2 E
MO
E i-jr gx/ + i>j
Cl
E
MO
E ;figx.ff +
CI
à2 Eeiec aFOG
-
348
CHAPTER 17. DERIVATIVES WITH ELECTRIC FIELD
oi
— 2
mo
E E (ui,
PERTURBATIONS
+ ufii X23
ij CI
— 2
MO
E c, E
UifjUrciyijjki
ijkl
ci (OH , j
0 Eejec 0C3)0CJ ,51.7 + CI -0F 8F F 8G
- E • C
aHij
- E •
(
01.1
8G
19C aC OG aF
( Eei„
1—
OG
(17.249)
In this equation the first derivative of the Hamiltonian matrix is expressed as
mo
a HIJ
mo +2
OF
E
Ut, i xt;
(17.250)
For the mixed perturbations the second-order CPMCHF equations again are formally the same as eq. (17.235): A
n A21t [ uaf
A 21 A22
(17.251)
02C/OztOF
{
The Ban matrices for the MO space are given by p all
of
of
0,:j = — X ij
mo
+E
4aile
bkj X
—
+
Yj1k
—
+
yii kk
Wilk)
k>1
MO
2
E
—
ez,f, bkiski
f5kj Xk
MO
—
Yiikk
MO
— E(uk0x — uzi xici) — E (ukfi xz, — MO
MO
- E ugi(yi —
yki) —
kl MO
E
-
kl
—E
—
MO
UljYkimn)
kmn
4
EU
n (L)irikiYkjmn
/l c jYkimn)
kmn
MO
-
ufkj X aki )
E
+ rin,p(imikp) +
r jpIn (iplkM)
klmnp —
ritrzp(iklinp)
rinip(jrnikP)
ripin(iPlkM)}
COUPLED PERTURBED EQUATIONS FOR MULTIPLE PERTURBATIONS CI 2 E —
aci oc, (
IJ
1j . )
x3.1
Oa aF xi.i
CI
_ 2
_
1j
CI
E ci E
{oc, (ax.ff
ax .1t ) 8F
aa k, 8F
J
I
349
+
ac,
(axj;
OF Oa
ax.fil
1
&I ) f (17.252)
In this equation the x 41 matrices are defined by their elements as MO
X lc f
-— E
7. 01 3 Tn•M
(17.253)
m
For the CI space the Be2 matrices are Cl
Bô;f12
—
-
2
E cJ I/7.; + E
CI
2
MO
MO
Ci
CI
ECj E
MO
x " + E Cj E
xIJ ii
+ CI
82 Edec
8a8 F
Cl
E (u".xwa + uei xif f) ti
ij Cl
-
2
MO
E Cj E ijkl
.51.1 .51.1
aE,„ Oa a Eelec
ac,)ac,
+ cl Oa + aF
OF
aCj)ac,
aF
Oa
(17.254)
In the equation the 1/75 matrices are given by their elements as MO
1J
17.8
JJ
= E fil 1171
(17.255)
The Coupled Perturbed Equations for Multiple Perturbations
In the preceding three sections, three different types of coupled perturbed equations, CPHF, CPCI, and CPMCHF equations, have been discussed. For any real perturbations, each type of coupled perturbed equation has the same A matrix. It is, therefore, possible and economical to simultaneously solve the coupled perturbed equations for multiple first-order (p, q, r,...) and second-order (pq, pr, qr, ...) perturbations;
FIELD PERTURBATIONS
CHAPTER 17. DERIVATIVES WITH ELECTRIC
350
for the first-order perturbations,
[ A
[
UP
Ur
Uq
= [
BBB
,
(17.256)
and for the second-order perturbations,
A]
Br
UPq UPr Uqr = [
Bir B
.
(17.257)
In these two equations the U matrices denote unknown variables. The corresponding A matrices and the general form of Bio) and Bri g matrices are summarized as follows:
For the closed-shell CPHF equations Aij,ki = 6ik5ji(Ej
-
Ei)
-
14(iilkl)
(iklil)
(il li k )}
(17.258)
d.o.
.13,139
= Fii— Sf i ci —
E sfi t2(iiiko —
(17.259)
kl d.o.
— ajfj —
E 47{2(iitki) — kl
all
+ E (u:
+ uzi Fr, + uri Fiqk + uzip-rk )
all
+ E (uri uzi + uzi qi)Ek all d.o.
+ EE urm Ugn Aij,ki kl m all d.o.
+ E E qt. u,q,„ + uckli Urm Akj,I m kl m all d.o.
+ EE
+ U1. Urm ,
Aiom
kl m all d.o.
(17.260) Note that the A matrix in eq. (17.258) is different from the A matrix in eq. (17.171). For the general restricted open-shell CPHF equations Atj,k/ =
2(aik — ajk -
aii)(ijikl)
COUPLED PERTURBED EQUATIONS FOR MULTIPLE PERTURBATIONS ▪
(Oik — j k —
ki
▪
(illjk)}
(ik1j1)
Oil + 13j1)
— fil) —
Eji) — oki
351
1 k)
Eik)
oli
(17.261)
Bp
— all occ = 412(aik k>
= -
bki (Cij
^
occ
E srk kaik —
+
C . 1) —
Oik ) {(ik ij1)
—
(illjk)}
6 kj (cli
aik) (ij I kk)
(17.262)
(aik —
P9 O,ijEiiP9— Iii all occ
EE
kaik —(jk)(i j1k1)
— jk) {(ikli 1) + (iili k)}
k> I ▪ bki (Clij
Ejl)
6 kj
erkj(ctik —
ajk) (ii kk)
Eil)]
occ
E
I
all
+ E (uq + qui)
(aik
—
aik)(ikijk)]
— qd)
kl all
+ E 0,:nygn [2(aim,
—
aj m )
(ii I ki) +
(aim
—
iklii)
klrn
all
+ E (ufi un + uzi uk„) [2 (cEin —
I
aj n ) (k j mn)
kmn + (pi
13jn) {(kiniin)
(knli M)}1
all
+ E (ufi uln, + uzi u,%) [2 (ai n, — aiii) (ki I mn) kmn + (13in — fijn) {(k7012 , ) all
+ E kl
Uff [2 (fait — «
JO (ij I
(kid im)}}
(13ii
—
i) {( iklAq
(inik)}1
352
CHAPTER 17. DERIVATIVES WITH ELECTRIC
FIELD PERTURBATIONS
all
+ E
[2 (aii
+
—
kl
all
+ E ur,i (cikqi —
+
E k)
— U P .((i ! -
uzi ( cr,
—
Ukj g ( (jP ki
ik)
+
ÇP
EP)] tk
—
(17.263)
For the CPCI equations HIJ
ifir j aEelec op
(OHIJ
Bop ,'
—
ap
CI (82 HI.,
B07,
-E
ap (
_ V .`
OHL' Oq
(17.264)
7
(17.265)
)(J.,'
(92E,, O pOq
OPaq
(a II LI CI
2C1Cj
6IJEelec
CJ
151J
aEeiec op + CI
aC lac j
(51-i
0Eetec + aq
OCJ)OCJ
Op
aq
Oq
Op
(17.266)
For the CPMCHF equations = 5ItZkj —
Yjikl
Yijkl
APJ
6IjXki
= HIJ
—
4-)
Yijlk
E
04 .-, — x xiji mo + E s k',ic (sk ixi;
— A' .kj
6kjXli
_
(17.267)
Yjilk
2C1Cj
6IJEe1ec CI
A21
6 kixlj
(17.268)
7
(17.269)
'4'31
Xli + Yi.j1k —
Yjilk
k>1
▪
1MOE
2
(6kiski — 6 kixki
Yijkk
Yjikk
(17.270)
•E COUPLED PERTURBED EQUATIONS FOR MULTIPLE PERTURBATIONS CI
CI
MO
- E cfmr, + 2 E
B 0,1 P2 =
E
i> j CI
MO
+E — x" Iwo
0,21
E
-
c(6kixi.;
kj Xli
Yjilk)
Yijik
mo
+ -2
E 6
Ski% — bkixki + Yijkk
MO
Yjikk
MO
- E
(ufi xlc,
E
uP kjxq ki
k
(uq.xP kt kj
-
ugkjxP ki
k
MO
EId
_
(17.271)
xPg 32
k>I
1
SPii XL!
+ ci a Edec Op
E
c,
353
-
uTi(Yl iki
MO
- E -O' kl(Yijkl — Y jiki kl
,,q
T
yiiki)
„P
MO
,P )
MO
E ug,„(Ufak inin —
—
UkPiYkimn)
kmn
E
0,:,„(uykitn.
Ykimn)
kmn
MO
E
— 4
(1E,
mn { rj in?, ( ik I rnp) + r j„ip (irnikp) + r jpin (ipikM)
kirnnp
— ranp (jkImp) — ri nip (jmilcp) — r ip rn (i P I k m)} OC/
2
aCj (xlj
La Op 'J
2j
CI l oc j ( 841
CI
9
Oq
-EE
—
Op
I
ax1,1
pc, Oq
Oq
Oq
(041 Op
axl} op ) J (17.272)
B 0,1 Pq2
CI
=
CI
MO
CI
pq _IJ
i> j CI
-
MO
2 E Cj E (Uiqj x sf3jP
+
ij Cl
-
2
ECj
MO
E
UULy
1
i j kJ
at ((mi.,
E
ap
5 IJ
Eae i ec + Op
U1
aCj
ac,
Op
Oq
MO
E
efiqxtf + CI
(9 2 E le ec Op8q
354
CHAPTER 17. DERIVATIVES WITH ELECTRIC ci
-E(
aH
c a Eelec OIJ
Oq
9C
1- ul -
FIELD PERTURBATIONS
Cj
Oq
Op
(17.273)
It should be noted, however, that solutions to the first-order equations are usually required in order to construct the second-order coupled perturbed equations.
References 1. H. Eyring, J. Walter, and G.E. Kimball, Quantum Chemistry, Chapters 8 and 17, John Wiley St Sons, New York, 1944. 2. K.S. Pitzer, Quantum Chemistry, Chapter 12 and Appendix 20, Prentice-Hall, Englewood Cliffs, N.J. 1953. 3. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Chapters 12 and 13, John Wiley St Sons, New York, 1954.
4. A.D. Buckingham and J.A. Pople, Proc. Roy. Soc. A68, 905 (1955). 5. A.D. Buckingham, Quart. Rev. 13, 183 (1959). 6. K. Thomsen and P. Swanstrom, Mol. Phys. 26, 735 (1973). 7. P. Lazzeretti, B. Cadiole, and U. Pincelli, Int. J. Quantum Chem. 10, 771 (1976). 8. H. Hellmann, Einfiihrung in die Quantenchemie, Franz Deuticke, Leipzig, Germany, 1937. 9. R.P. Feynman, Phys. Rev. 56, 340 (1939). 10. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
Suggested Reading
1. R.D. Amos, Chem. Phys. Lett. 108, 185 (1984). 2. Y. Yamaguchi, M.J. Frisch, J.F. Gaw, H.F. Schaefer, and J.S. Binkley, J. Chem. Phys. 84, 2262 (1986). 3. R. Amos, in Advances in Chemical Physics: Ab Initio Methods in Quantum Chemistry Part K.P. Lawley editor, John Wiley Si Sons, New York, Vol. 67, p.99 (1987). 4. C.E. Dykstra, S.-Y. Liu, and D.J. Malik, in Advances in Chemical Physics, I. Prigogine and S.A. Rice editors, John Wiley St Sons, New York, Vol. 75, p.37 (1989).
Chapter 18
The Z Vector Method In order to obtain energy derivatives for the configuration interaction (CI) wavefunction, information concerning the derivatives of the MO coefficients of the reference wavefunction is required. These quantities are usually obtained by solving the coupled perturbed Hartree-Fock (CPHF) equations for an SCF reference function and the coupled perturbed multiconfiguration Hartree-Fock (CPMCHF) equations for an MCSCF reference function. Such procedures require 3N (N being the number of atoms) sets of unknowns for first-order and 3N(3N+1)/2 sets of unknowns for second-order derivatives. The Z vector method considers this problem from a different point of view and allows much smaller sets of unknowns. The method has proven very powerful whenever applicable. In this chapter, the principle of the Z vector method is explained using analytic derivatives for the CI wavefunction as the most common example.
18.1
The Z Vector Method and First Derivative Properties for the CI Wavefunction
The first derivative of the electronic energy for the CI wavefunction was derived in Chapter 6 and is Eelec
aa
mo =E
mo Qiihz9
;
ii
E
+
mo 2
E
utyc ii
(18.1)
ij
ijkl
In this equation Q and G are the one- and two-electron reduced density matrices [1] and hZi and (ijild)a are the skeleton (core) parts of the one- and two-electron derivative integrals. The Lagrangian matrix for the CI wavefunction X is defined by Xij =
mo E
hanhint +
2
mo E Tnkl
355
G jrnkl(irrilki)
(18.2)
CHAPTER 18. THE Z VECTOR METHOD
366
The Ua matrices, which are related to the first derivatives of the MO coefficients with respect to nuclear coordinates (see Chapter 3), are obtained by solving the coupled perturbed Hartree-Fock (CPHF) equations in the following matrix form:
= B8
(18.3)
For a closed-shell SCF wavefunction the A matrix elements were found in Chapter 10 to be 5ik 6 j1(f j
Aij,k1 =
and the
—
{4(ijikl)
-
-
(ikiji)
-
(illjk)}
(18.4)
B8 matrix elements are d.o.
E szif2(iilko — (iklio}
=
•
(18.5)
kl
The skeleton (core) first derivative Fock matrix, Fa, is defined by d.o. =
E {2(iiIkkr —
it clj
k)a}
(18.6)
In eq. (18.4) q stand for orbital energies. The third term in eq. (18.1) may be written in matrix form [2 ] as mo 2 E usyci, = 2XT Ua (18.7) ij where XT stands for the transpose of the
X matrix. Equation (18.3) is rewritten as
Ua = A-1 B8
(18.8)
Combining eqs. (18.7) and (18.8), the third term of eq. (18.1) becomes
mo 2
E
=
2 XT LT'
ij = 2 XT T =2Z The
Z vector in this equation is defined by ZT = X T
which in turn may be written AT Z = X
THE Z VECTOR METHOD AND CI FIRST DERIVATIVES
357
The elements of the Z vector are obtained by solution of the simultaneous equations, eq. (18.13), and the third term of eq. (18.1) may be evaluated through mo
2
E ij
U J XI = 2
mo
EBO ZU ,
,
(18.14)
Ba
where the elements of were defined in eq. (18.5). In eq. (18.3) the simultaneous equations were solved for 3N (N being the number of atoms in the molecule) degrees of freedom, while in eq. (18.11) the simultaneous equations are solved for only one degree of freedom. This is a significant saving in the work involved in solving these equations. The Bt matrices in eq. (18.3) contain the perturbed Fock matrices with respect to nuclear perturbations. The Lagrangian matrix in eqs. (18.2) and (18.13) involves the first derivative of the CI energy with respect to changes in the MO coefficients.
The first derivative of the CI energy with respect to an electric field perturbation, i.e., the electric dipole moment (electronic part), was obtained in Chapter 17 as
kt
EeleC
elec
(18. 15)
(18. 1 6) (18.17) Here the matrix cif stands for the dipole moment matrix described in Section 17.1. The corresponding CPHF equations to evaluate the elements of Uf are
A
Uf Bt
(18.18)
For a closed-shell SCF wavefunction the Bt* matrices were given in Section 17.5 by Bof
•
—
f F — ii hf
ij
(18. 1 9)
In a similar manner as for the nuclear perturbations, these equations for the dipole moment may be manipulated to yield mo
2
E
usv i,=
2 X T Uf
(18.20)
= 2 X T A -1 K
(18.21)
= 2 Z T Bfo
(18.22)
ij
The Z vector here is the same quantity defined in eq. (18.12). In eq. (18.18) the simultaneous equations were solved for three degrees (x, y, and z) of freedom, while in eq. (18.22) they are solved for only one degree of freedom.
CHAPTER 18.
358
THE Z VECTOR METHOD
In the Z vector method, corrections due to the first-order changes of the MO coefficients are reformulated in general as MO
MO
E
E
(18.23)
A superscript "p" stands for any real first-order perturbation. The Z vector is obtained by solving the simultaneous equations AT Z = X .
(18.24)
The corresponding A matrix for the reference wavefunction (other than the closed-shell SCF wavefunction; see Chapters 11, 13 and 14) is used in solving eq. (18.24). With the Z vector evaluated, any first-order correction terms, expressed in the general form of eq. (18.23), with respect to different types of "p" perturbations may be evaluated simultaneously. However, the corresponding Bg matrices must be available. The above argument is not limited necessarily to the CI wavefunction. In principle, the Z vector method is applicable whenever equations of the form of eqs. (18.23) and (18.24) are valid. In practice, the details are somewhat more complex because of the relation
o
(18.25)
due to the first derivative of the orthonormality of the molecular orbitals discussed in Section 3.7. The equations, eq. (18.24), have to be rewritten in terms of the independent pairs of UP [3]. This does not affect the significance of the Z vector method.
18.2
The Z Vector Method and Second Derivative Properties for the CI Wavefunction
The second derivative of the electronic energy for the CI wavefunction was formulated in Chapter 6 and is given by MO
MO
E
a2Eelec 0a0b
Qii hg
+E
Gijkl(iiikirb +
2
mo
E
uz5Lixi,
ijkl MO
MO
+ 2
E
i
+ uzixt) +
ii
2
E qutayi,k, ijkl
ij
ci 2 E aciac, Oa Ob IJ
— IJEelec)
(18. 26)
The skeleton (core) first derivative Lagrangian matrices appearing in eq. (18.26) are defined by MO
X 4r i
E
MO
Qjmh m
+2E
mkt
(18.27)
THE Z VECTOR METHOD AND CI SECOND DERIVATIVES
359
and the Y matrix by MO Yijkl = Q jlhik +
2
MO
E
E
Gihnn (iklmn) + 4
mn
G jni tn,(iralkn)
.
(18.28)
mn
The Uab matrices, which are related to the second derivatives of the MO coefficients with respect to nuclear coordinates (see Chapter 3), are obtained straightforwardly by solving the following second-order CPHF equations:
A
Uab
(18.29)
For a closed-shell SCF wavefunction the A matrix is given in eq. (18.4) and the were found in Chapter 10 to be
BP
matrices
d.o.
F216 —
E — E 6142lki) —
(ik1j1)}
kl
all
+ E (upci F:, all
+
+E all d.o.
EE
Fa)
r4)E k
yzm ut,,, Aij,kl
kl 7Tt a 11 d.o.
+
i>
+
+ y:; Fibk +
+
lam ) Akj,Irn
kl rn all d.o.
+ EE (u:,u,bra + u Utm ) kl m
all d.o.
+
+ EE
(18.30)
I
k
In this equation the e b matrices, the skeleton (core) second derivative Fock matrices, and the A matrix and its skeleton (core) first derivative matrices are defined by =
s
all
Uibm
(yiam
— S
S — stn
(18.31)
(ikij O at) }
(18.32)
d.o. Fial
= h`e
E {2(iiikoab —
,
= 4(ijIkl) — (iklj 1) —
A clj,k/
4 (iilki)
a
(illikr
(18.33
)
(18.34)
360
CHAPTER 18.
THE Z VECTOR METHOD
Note that the A matrix in eq. (18.33) is a different quantity from the A matrix in eq. (18.4). The third term in eq. (18.26) is written in matrix form [2] as MO
E
2
2 XT uab
uex ii
(18.35)
and equation (18.29) is rewritten as uab
=
A lEttb
(18.36)
-
Combining eqs. (18.35) and (18.36), the third term of eq. (18.26) becomes MO
E iajb
2
=
Xi
=
2 XT -uab
(18.37)
2 xT A -i B ab
(18.38)
2 ZT B
ab
where the Z vector was defined by eqs. (18.12) and (18.13). evaluated through MO
2 where the elements of
(18.39) Thus, this term may be
MO
E
UXI =
2
E
(18.40)
Br matrices were defined in eq. (18.30).
In eq. (18.29), the simultaneous equations were solved for 3N(N +1)/2 degrees of freedom, while in eq. (18.39) these equations are solved for only one degree of freedom. This represents a substantial saving in computational effort. In the evaluation of second derivatives of the CI wavefunction, however, the Ua matrices must be explicitly obtained prior to application of eq. (18.26). The second derivative of the CI energy with respect to electric field perturbations, i.e., the electric polarizability, was given in Section 17.3 as of
(9 2 Eelec
=
(18.41)
a FOG mo
MO
E
- 2
-2
E
(ufix4 + ufixfj)
MO
-
2
Euifi ufiyiiki i3 kl CI
+
2
E aCi OF OG
—(111j - 61.7Eetec)
J
(18.42)
THE Z VECTOR METHOD AND CI SECOND DERIVATIVES
361
The skeleton (core) first derivative matrices appearing in this equation were given in Section 17.3 as
(18.43) The corresponding second-order CPHF equations to evaluate the elements of 1/ 12 are A U fg = Bfog
(18.44)
For a closed-shell SCF wavefunction the Bfog matrices were found in Section 17.5 to be d.o.
—
Bf 0,11 = —
E euf2(iiiki) —
(alit)}
kl all
+ E
+ ukfi Ffk +
(uLF,f, +
Ilk)
all
+E
+
ULUkfi)Ek
all d.o.
EE kfmUigmAij,k1 kl m all d.o.
+ EE
(u u
+ uf,
im ) Akj,lm
Ici m all d.o.
+ EE (uti urn, + ufc kl
,
U Aiki m
(18.45)
7rt
In this equation the eg matrices and the skeleton (core) first derivative Fock matrices are defined by
elf
all
=
E
+ uf,y,f77
,)
(18.46)
Tn
and
.
(18.47)
In a similar way as for the nuclear perturbations, these CPHF equations for the polarizability may be treated as
mo 2
E
=
2 XT U fg
(18.48)
2 X T A -1 Bfg
(18.49)
2 Z T Bfog
(18.50)
CHAPTER 18.
362
THE Z VECTOR METHOD
The Z vector in eq. (18.50) is again the same quantity defined in eq. (18.12). In eq. (18.44) the simultaneous equations were solved for six degrees (xx, xy, xz, yy, yz, and zz) of freedom, while in eq. (18.50) they are solved for only one degree of freedom. In the Z vector method the term due to the second-order changes of the MO coefficients may be reformulated in general as
mo
E
Iwo
E
Ilr.q X t.1 =
0,z3
-
(18.51)
where the superscript "pq" stands for any real second-order perturbations. The Z vector is obtained by solving the simultaneous equations given in eq. (18.24). The Z vector method is applicable whenever equations like (18.23), (18.24), and (18.51) can be formulated. If the Z vector is determined any second-order correction terms, given in the general form of eq. (18.51), with respect to different types (including mixed or crossed) of perturbations, may be evaluated simultaneously. However, the corresponding B r matrices must be available. The above approach is valid to any order of perturbation and for any property. In practice, the details are somewhat more complicated due to the linear dependence of the elements of the U" matrices, i.e., tryiq (18.52)
+
+
eft
=o
where all
(uirm uim
esFjq — ST39
srm sj„, _ st s)
(18.53)
m
The simultaneous equations, eq. (18.24), must be modified in terms of the independent pairs of the UPq matrices [4 ] . This does not diminish the importance of the Z vector method.
References 1. E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic Press, New York, 1976.
2. N.C. Handy and H.F. Schaefer, J. Chem. Phys. 81, 5031 (1984) 3. J.E. Rice, R.D. Amos, N.C. Handy, T.J. Lee, and H.F. Schaefer, J. Chem. Phys. 85, 963 (1986). 4. T.J. Lee, N.C. Handy, J.E. Rice, A.C. Scheiner, and H.F. Schaefer, J. Chem. Phys. 85, 3930 (1986).
Chapter 19
Applications of Analytic Derivatives In this chapter, some chemical applications of the analytic derivative techniques developed in the earlier chapters are illustrated. This chapter takes a somewhat historical viewpoint and focuses on a few of the key early studies of chemical reactions by derivative methods. In particular, differences in the results between SCF and correlated levels of theory obtained with derivative methodology will be highlighted. The application of analytic derivative techniques to ensure the precise determination of the equilibrium geometries of relatively weakly bound molecular complexes will be illustrated. The systematics of the changes in molecular properties including structures, dipole moments, and harmonic vibrational frequencies with basis set and theoretical method will be illustrated for two molecules. The evolution of derivative techniques has aided greatly in the production of a number of critical papers addressing the question of what level of theory is necessary to reach agreement with experiment for various molecular properties. In addition, the use of derivative methods to aid in the full (i.e., beyond the harmonic approximation) vibrational analysis of two triatoinic molecules will be considered. The immense practical advantages of the use of analytic derivatives in characterizing the nature of stationary points for larger molecules will be examined. Some of the quantitative results in this chapter have been superseded in very recent times by more elaborate methods of including electron correlation and perhaps most notably by the various coupled cluster (CC) approaches augmented by their own analytic derivative schemes. Nonetheless, the examples demonstrate clearly the importance of derivative approaches to real chemical problems and should give the reader an appreciation of the practical necessity of such methods. Table 19.1 summarizes some of the common applications of analytic gradients in chemistry.
363
CHAPTER 19. APPLICATIONS
364
Table 19.1 A summary of some common applications of analytic derivatives in chemistry.
This table is applicable to variational wavefunctions (i.e., SCF, CI, and MCSCF) and X is a nuclear coordinate or electric field perturbation.
Order Energy Derivative
Applications
Zeroth Relative energies Minima and transition states Electronic states
First
Second
Higher
01E
82 E
Tr2 C
Stationary point location Finite differences Force constants Infrared intensities Electric dipole moment Geometry optimization Force constants Harmonic vibrational frequencies Reaction path Hamiltonian Infrared intensities Electric polarizabilities
03 E
TX-5.
.94E
r§a71-
Cubic force constants Vibration-rotation interaction constants Raman intensities Electric hyperpolarizabilities Quartic force constants Vibrational anharmonic constants Electric higher-order hyperpolarizabilities
UNIMOLECULAR ISOMERIZATION
OF HCN
365
19.1 Unimolecular Isomerization of HCN Conceptually if not always experimentally or theoretically the simplest chemical reactions are isomerizations. A particularly important example of this reaction type is the methyl isocyanide, CH3NC, to methyl cyanide, CH3 CN, rearrangement. This reaction was a key test of unimolecular kinetics models such as RRKM. An early study of the analogous reaction of HCN was undertaken by Pearson, Schaefer, and Wahlgren without the use of analytic derivatives [1]. Through a careful but computationally tedious mapping of the potential energy surface at the SCF level supplemented by surface fits and CI level work on the stationary points, the conceptually and chemically significant intrinsic reaction coordinate or minimum energy path was determined. This important early study inspired the first chemical application of the configuration interaction (CI) analytic gradient method by Brooks et al. [2]. At the equilibrium geometries of both the linear reactant (HCN) and product (HNC) there are only two geometrical degrees of freedom, while there are three at the transition state. Thus the optimization of the equilibrium points could be done by using energy only methods. However, the ease of geometry optimization of the stationary points and particularly of the transition state, the precision of the various geometrical parameters determined, and the confidence in the results improved dramatically with the use of SCF and CI gradients. Figure 19.1 presents the three stationary point geometries at both the double zeta plus polarization (DZP) SCF and CISD levels of theory. Experimental structures for HCN and HNC also are shown. The lengthening of bonded and shortening of nonbonded distances in the transition state with the inclusion of electron correlation is evident. Such changes in transition state geometry have now been observed in many such reactions as derivative techniques have allowed the study of numerous chemical processes. The reaction barriers without correction for zero-point vibrational energy (ZPVE) changes were 40.0 and 36.3 kcal/mol with the SCF and CI theoretical models. Thus for this isomerization, the effects of electron correlation lower the reaction barrier by ca. 10 % and thus the SCF and CISD energetics agree reasonably well. The use of analytic derivatives, and more specifically SCF level first derivatives, was of great value in providing the input to one of the earliest reaction path Hamiltonian studies by Gray, Miller, Yamaguchi, and Schaefer [3 ] . This study of the HCN HNC reaction required the evaluation of force constants along the reaction coordinate to define the reaction path Hamiltonian of Miller, Handy, and Adams [4]. The labor in the repeated computation of force constants was reduced significantly by the availability of the analytic SCF gradient. The use of force constants from more reliable correlated wavefunctions in approximate kinetic models like the reaction path Hamiltonian is much more effi cient and accurate with analytic schemes than with any sort of numerical differencing approach. There have been extensive developments of methods to determine intrinsic reaction coordinates which depend on the availability of analytic higher derivatives of SCF and correlated methods. (For a recent article, see Page, Doubleday, and McIver [5].) The potential energy surface for the hydrogen cyanide-hydrogen isocyanide isomerization has remained of interest to the present day (see, e.g., Bentley, Bowman, Gazdy, Lee, and Dateo [6]).
1.062 DZP SCF 1.067 DZP CISD 1.065 (experiment)
H-C -N 1.137 1.162 1.153
1.155 1.168
9 .s'i
H
• 1`,
i
1.471 1.427
.
0.986 0.996 0.994
. .
C N 1.174 1.194
C N H 1.159 1.180 1.169
Figure 19.1 DZP SCF and CISD optimized structures for hydrogen cyanide, the isomerization transition state, and hydrogen isocyanide. Experimental results for the stable molecules also are shown.
FORMALDEHYDE DISSOCIATION
19.2
367
Formaldehyde Dissociation
The photochemically initiated dissociation of formaldehyde to molecular or radical products ranks as perhaps the most completely understood of all polyatomic unimolecular reactions. The use of analytic derivative methods has been critical in building up a reliable theoretical description of these reactions and especially of the transition state structures involved. Two key reviews on spectroscopic and photochemical aspects of formaldehyde were given by Clouthier and Ramsay [7] and by Moore and Weisshaar [8]. More recent experimental results on the molecular and radical dissociations may be found in Guyer, Polik, and Moore [9] and in Chuang, Foltz, and Moore [10]. Early pioneering ab initio theoretical studies of the molecular decomposition by Jaffe and Morokuma [11] and of the radical dissociations by Hayes and Morokuma [12] and Fink [13] are of particular note. Although these early papers made no use of analytic derivative methods, they did provide extraordinarily valuable information to supplement various orbital and state symmetry analyses of the reactions. An early study of both the molecular and radical processes on the ground state surface (closed-shell singlet) was carried out by Goddard and Schaefer [14] using fledgling SCF analytic first derivative technology. Another very early use (1981) of analytic CI gradient schemes was the thorough theoretical study by Goddard, Yamaguchi, and Schaefer [15] of the molecular dissociation of formaldehyde. It is with this study that the formaldehyde example considered here will commence. The DZP SCF and CISD geometries for formaldehyde, the molecular dissociation transition state, and the molecular products, H2 and CO, are shown in Figure 19.2. These structures are taken from Scuseria and Schaefer [16] although essentially comparable results on the transition state were given in [15]. The most notable feature of the transition state at both the SCF and correlated levels is its distortion to C s symmetry from C 2„ to avoid an otherwise orbitally forbidden process. Comparable geometrical results also are shown in Figure 19.2 with the same two theoretical methods but with a larger triple zeta plus double polarization (TZ2P) basis set [16]. It should be mentioned that, particularly in the region of transition state structures where bond breaking and formation occur, the condition of a zero analytic gradient makes the precise location of such stationary points possible. The simple energy only versus coordinate mapping of a potential energy surface to locate transition states would not only be highly computationally inefficient but also rather inaccurate. Geometrical uncertainty arises due to the flatness of the potential energy hypersurface with respect to certain degrees of freedom at the transition state. In addition, relatively standard sets of internal coordinates which reduce couplings between various modes can be chosen for the equilibrium structures of most molecules. However, such couplings can be important in transition state structures, making the availability of approximate Hessians derived from analytic derivatives crucial to the geometry optimization. The classical (no zero-point vibrational energy (ZPVE) correction) barrier height to the molecular dissociation of formaldehyde was predicted to be 101.9 kcal/mol at the TZ2P SCF level and 95.0 kcal/mol at the CISD level with the same basis set [16]. Analytic second SCF derivatives and finite differencing of analytic CI gradients allowed the reliable determination
CHAPTER 19. APPLICATIONS
368
H
1.094 DZP SCF 1.099 DZP CISD 1.091 TZ2P SCF 1.093 TZ2P CISD 1.099 (experiment)
ec
116.2 116.3 116.2 116.4 116.5
0 / 1.189 1.210 1.178 1.195 1.203
H
49.4 50.2 49.6 \\*1/4 50.5
H
_.. .
1.583 1.570 1.663 1.635 .,,,,,,.....
112.5 110.9 114.3 112.2
H(
/
1.102 1.096 1.099 1.094
0.735 0.744 0.733 0.741
c1.11 1.178 1.130 1.153
13
H
I + C 0
H
1.117 1.140 1.103 1.120
Figure 19.2 DZP and TZ2P SCF and CISD optimized geometries for formaldehyde, the molecular dissociation transition state, and the molecular products H2 and CO.
FORMALDEHYDE DISSOCIATION
369
of the harmonic vibrational frequencies not only for reactants and products but also for the transition state. These frequencies were then used to predict the effects of changes in ZPVE upon the barrier height. The ZPVE corrected TZ2P SCF barrier became 96.0 kcal/mol and the CISD result with this basis was 89.5 kcal/mol. It is clear that the ability to consistently predict harmonic vibrational frequencies and thus zero-point vibrational energies by analytic derivative approaches can have important energetic consequences when comparisons are drawn to experiment. It must be noted that still higher levels of electron correlation treatment (coupled cluster singles and doubles (CCSD) and coupled cluster including single, double, and linearized triple excitations (CCSDT-1)) and the associated gradient technology [16] finally result in a best prediction of an activation barrier of 81.4 kcal/mol in good agreement with the most reliable experimental value of 78.0 to 81.1 kcal/mol [9]. Consideration of the radical dissociation of formaldehyde by Yates, Yamaguchi, and Schaefer [17] involved the application of restricted open-shell SCF, CASSCF, and CISD analytic derivatives to the lowest triplet surface and to the formyl radical. Basis sets employed ranged from DZP through quadruple zeta plus triple polarization (QZ3P). Higher level correlation corrections (CISD Q) to the energies at lower level optimized geometries also were carried out to set the stage for a comparison with experiment [10]. The origin of the exit channel barrier in the triplet state radical dissociation of formaldehyde has been understood in terms of avoided crossings [12,13], but a high level theoretical description with rigorous geometry optimization and stationary point vibrational analyses via analytic derivative methods had been lacking. Similarly, the importance of the triplet dissociation mechanism had not been established firmly by experiment until the study by Moore and co-workers [10]. The T1 exit barrier height at 0 K was determined to lie in the range of 2.9 to 6.0 kcal/mol. This barrier is defined as the energy required for the hydrogen atom and forrnyl radical to approach from infinite separation and combine to yield triplet formaldehyde at its C s equilibrium geometry. Several key results of Yates et al. [17] will be discussed, but first we note a concluding remark from that paper. In order to estimate the energy difference between two stationary points to an accuracy of 2 kcal/mol, complete geometry optimizations and harmonic vibrational analyses of all species are required. In addition, these predictions must be made with large basis sets including multiple sets of polarization functions and at correlated levels including excitations beyond the simple CISD model. The determination of optimized structures and harmonic vibrational frequencies would be essentially impossible without at least the first analytic derivative of the various quantum chemical methods described in this book. Figure 19.3 contains the CISD optimized geometries for ground state formaldehyde, for the pyramidalized equilibrium structure of the lowest triplet state, for the strongly nonplanar transition state to radical dissociation, and for one of the products, the ground state formyl radical. Note that large basis sets have been used and the geometries precisely determined with analytic gradient techniques. The two predicted equilibrium structures for ground and triplet state formaldehyde are of interest since the ground state structure is in
1.0998 DZP CISD 1.0954 TZ2P CISD 1.0953 QZ3P CISD 1.099 (experiment)
• 116.27 116.40 116.43 116.5
\e-i t.., - 0 / 1.2116 1.1963 H 1.1975 1.203
117.86 117.76 118.14 40.30 39.60 38.M
1.0883 1.0824 4Zj
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-
CI
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82 Eetec
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APPENDIX BB
_
z (aaHa ij oi J
v.' aHij ab
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Bibliographical Survey on Analytic Energy Derivatives
1968
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456
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Index Born-Oppenheimer approximation, 10, 24, 25, 55
Activation energy (barrier), 369, 371, 373 Active space, 20, 111, 119, 175, 197 A matrix, 60, 129, 151, 181, 204, 296, 332, 359 first derivative, 133 Anharmonicity- (constants), 6, 218, 364, 379 Antisymmetrizer, 12, 111 Atomic orbital (AO), 12, 29, 309 Averaged Fock operator GRSCF, 150 MCSCF, 202 PEMCSCF, 180
Canonical orbitals, 128 Carbon dioxide (CO2), 380, 381 Classical barrier height, 367 Closed-shell SCF (CLSCF) wavefunction, 5, 13, 22, 53, 128, 293 Complete active space self-consistent-field (CASSCF) wavefunction, 4, 20, 119, 369 Configuration interaction (CI) wavefunction, 4, 16, 23, 83, 166, 218, 249, 355, 364, 385 Configuration interaction with singles and doubles (CISD), 18, 365, 367, 373, 376, 381, 383 Configuration state function (CSF), 4, 16, 18 Contracted Gaussian functions, 13 Core electrons, 18 orbitals, 18, 202 Correlated wavefunction, 3, 249, 256, 363 CASSCF, 4, 20, 119, 369 CI, 4, 16, 23, 83, 166, 218, 249, 355, 364, 385 GVB, 5, 98, 110, 270 MCSCF, 4, 18, 119,193, 258,364, 385 PEMCSCF, 5, 98, 110, 170, 270 TCSCF, 5, 98, 270 Correspondence equations (relationships), 6, 249, 256, 270, 271, 274, 278, 285, 293, 298 Coulomb integrals MO basis, 14 Coulomb operator, 14, 21, 54, 70
Bare generalized Lagrangian matrix PEMCSCF, 116, 174, 275 TCSCF, 105 Bare Lagrangian matrix CI, 90, 168, 252, 321, 340 first derivative CI, 90 first derivative MCSCF, 195, 346, 347 first derivative PEMCSCF, 173 MCSCF, 122, 194, 199, 261, 326, 343 PEMCSCF, 115, 173, 274 second derivative MCSCF, 210 TCSCF, 103 Bare T matrix PEMCSCF, 117, 174, 275 TCSCF, 105 Bare Y (y) matrix CI, 92, 170, 252, 341 first derivative MCSCF, 206 MCSCF, 125, 195, 262 Bare Z (z) matrix CI, 231 first derivative CI, 234 Basis set, 12, 376
463
464 Coupled cluster (CC) methods, 363 Coupled cluster with single, double and linearized triple excitations (CCSDT-1) wavefunction, 369, 373 Coupled cluster with singles and doubles and perturbative triples (CCSD(T)) wavefunction, 376, 385 Coupled cluster with singles and doubles (CCSD) wavefunction, 369, 373, 374, 376, 383, 385 Coupled perturbed configuration interaction (CPCI) equations, 6, 26, 94, 96, 166, 221, 248, 251, 253, 256, 322, 323, 330, 340 electric field perturbations first-order, 340 second-order, 341 mixed (cross) perturbations second-order, 342 multiple perturbations, 352 nuclear perturbations first-order, 166, 221, 340 second-order, 169, 221, 341 Coupled perturbed Hartree-Fock (CPHF) equations, 4, 26, 63, 79, 87, 96, 128, 144, 248, 250, 251, 253, 256, 290, 292, 300, 303, 315, 318, 323, 330, 332, 355 electric field perturbations CLSCF first-order, 333, 357 CLSCF second-order, 334, 361 GRSCF first-order, 336 GRSCF second-order, 338 mixed (cross) perturbations CLSCF second-order, 334 GRSCF second-order, 339 multiple perturbations CLSCF, 350 GRSCF, 350 nuclear perturbations CLSCF first-order, 128, 332, 356 CLSCF second-order, 134, 333, 359 GRSCF first-order, 144, 335 GRSCF second-order, 158, 336
INDEX Coupled perturbed multiconfiguration Hartree-Fock (CPMCHF) equations, 5, 26, 87, 126, 193, 260, 264, 270, 326, 332, 343, 355 electric field perturbations first-order, 344 second-order, 346 mixed (cross) perturbations second-order, 348 multiple perturbations, 352 nuclear perturbations first-order, 194, 343 second-order, 208, 345 Coupled perturbed paired excitation multiconfiguration Hartree-Fock (CPPEMCHF) equations, 6, 118, 172, 274, 278, 284 nuclear perturbations first-order, 172 second-order, 182 Coupled perturbed two-configuration Hartree-Fock (CPTCHF) equations, 109 see also CPPEMCHF Coupling constants see One-electron coupling constants see Two-electron coupling constants Cross perturbations see Mixed perturbations Deleted virtual approximation, 18 Density matrix CLSCF, 15, 22, 57 GRSCF, 73 Diagonality CLSCF Fock matrix, 54, 128, 304 GRSCF averaged Fock matrix, 150 MCSCF averaged rock matrix, 202 PEMCSCF averaged Fock matrix, 180 Diketene, 383 Dipole moment see Electric dipole moment Dipole moment derivative see Electric dipole moment derivative
INDEX Dipole moment integral AO basis, 310 MO basis, 310 Doublet state, 69 Double zeta plus heavy atom polarization (DZd) basis set, 383 Double zeta plus polarization (DZP) basis
set, 365, 367, 373, 376, 381 Eigenfunction, 11 Eigenvalue, 9, 18 Electric dipole moment, 312, 376 CI, 313, 357 CLSCF, 312 GRSCF, 313 MCSCF, 314 Electric dipole moment derivative, 326
CI, 329 CLSCF, 327 GRSCF, 328 MCSCF, 331 Electric field, 6, 29, 308, 364 Electric hyperpolarizability, 364 Electric polarizability, 314, 360, 364 CI, 320, 360 CLSCF, 315 GRSCF, 317 MCSCF, 323 Electronic energy, 10, 13 CI, 17, 23, 84, 219, 249 CLSCF, 13, 22, 53, 293 GRSCF, 68, 285 MCSCF, 19, 120, 258 PEMCSCF, 111, 114, 270 TCSCF, 99, 101 Electronic energy first derivative
CI, 84, 220, 250, 313, 355 CLSCF, 55, 295, 304, 312 GRSCF, 71, 286, 313 MCSCF, 121, 259, 314 PEMCSCF, 115, 272 TCSCF, 102 Electronic energy fourth derivative
CI, 224, 246, 253 CLSCF, 300, 306
465 GRSCF, 290 MCSCF, 264 PEMCSCF, 278 Electronic energy second derivative CI, 94, 221, 251, 322, 358 CLSCF, 60, 63, 296, 304, 315 GRSCF, 76, 79, 286, 318 MCSCF, 124, 259, 324 PEMCSCF, 116, 272 TCSCF, 106 Electronic energy third derivative
CI, 222, 233, 251 CLSCF, 298, 305 GRSCF, 288 MCSCF, 260 PEMCSCF, 274 Electronic Hamiltonian operator, 10, 84 Energy expectation value, 10, 13 nuclear repulsion, 10, 25, 51
orbital, 14, 15, 54, 129, 150, 180, 202, 304 total, 10, 25 see also Electronic energy Energy weighted density matrix
CI, 88 CLSCF, 57 GRSCF, 73 MCSCF, 123 Exchange integrals
MO, 14 Exchange operator, 14, 54, 70 Finite difference method, 3, 364 First derivative CI bare Lagrangian matrix, 90 CI bare Z matrix, 234 CI coefficients normalization
condition, 85, 168, 221, 323 CI Hamiltonian matrix, 89, 168, 220, 252, 323, 340 CI Lagrangian matrix, 321 CLSCF A matrix, 133 CLSCF Fock matrix, 58, 129, 298, 317 CLSCF orbital energy, 131, 305
466 GRSCF averaged Fock matrix, 151 GRSCF generalized Lagrangian matrix, 154 GRSCF Lagrangian matrix, 74, 145, 319 GRSCF r matrix, 155 MCSCF averaged Fock matrix, 204 MCSCF bare Lagrangian matrix, 195, 346, 347 MCSCF bare y matrix, 206 MCSCF Hamiltonian matrix, 121, 199, 346, 348 MCSCF Lagrangian matrix, 195, 261 MO orthonormality condition, 36, 38, 56, 88, 123, 129, 146, 178, 196, 200, 259, 311, 358 nuclear repulsion energy, 51 one-electron AO integral, 33, 310 one-electron MO integral, 39, 41 one-electron reduced density matrix CI, 321 MCSCF, 203 PEMCSCF, 181 orbital energy, 131, 305 overlap AO integral, 32 overlap MO integral, 34, 37 PEMCSCF averaged Fock matrix, 181 PEMCSCF bare Lagrangian matrix, 173 PEMCSCF Hamiltonian matrix, 115, 177, 276 PEMCSCF Lagrangian matrix, 174 TCSCF Hamiltonian matrix, 103 two-electron AO integral, 33 two-electron MO integral, 43, 47 see also Electronic energy first derivative Fock matrix CLSCF, 14, 15, 22, 54, 128, 293 first derivative, 58, 129, 298, 317 Fock operator CLSCF, 14, 54 MCSCF, 121
INDEX Formaldehyde (H2C0), 367, 371, 376 Formyl radical (HCO), 369 Fourth derivative CI Hamiltonian matrix, 235 MO orthonormality condition, 264 see also Electronic energy fourth derivative Frozen core approximation, 18 Full CI, 4, 18, 20
Gaussian-type orbital (GTO), 3, 12 Generalized Pock operator GRSCF, 16, 70, 144 MCSCF, 121 PEMCSCF, 114, 172 TCSCF, 101 Generalized Lagrangian matrix GRSCF, 75, 145, 287 PEMCSCF, 117, 174, 272 TCSCF, 106 Generalized valence bond (GVB) wavefunction, 5, 98, 110, 270 General restricted open-shell SCF (GRSCF) wavefunction, 5, 16, 68, 144, 285 Geometry optimization, 364 Glyoxal, 371 Hamiltonian matrix CI, 17, 84, 166, 219, 250 first derivative CI, 89, 168, 220, 252, 323, 340 first derivative MCSCF, 121, 199, 346, 348 first derivative PEMCSCF, 115, 177, 276 first derivative TCSCF, 103 fourth derivative CI, 235 MCSCF, 19, 120, 198, 258 PEMCSCF, 111, 177, 271 second derivative CI, 92, 170, 221, 252, 341, 342 second derivative MCSCF, 124, 213 second derivative PEMCSCF, 116, 187, 281
INDEX second derivative TCSCF, 104 TCSCF, 99 third derivative CI, 228, 254 Hamiltonian operator, 9, 29, 308 Harmonic vibrational frequencies, 3, 363, 376 Hartree-Fock approximation, 11 Hartree-Fock equations, 14 Hartree-Fock (HF) wavefunction, 11, 13 Hellmann-Feynman theorem, 312 Hessian, 3, 24, 367 Hessian index, 24 High-spin open-shell, 69, 99, 110, 175 Hydrogen cyanide (HCN), 365 Hydrogen fluoride (HF), 373 Hydrogen fluoride dimer [(HF)2], 373 Hydrogen isocyanide (HNC), 365 hyperpolarizability see Electric hyperpolarizability Imaginary frequency, 3, 371, 373 Independent and non-independent pairs CLSCF, 131, 315 GRSCF, 148, 150, 318, 329 MCSCF, 197, 206 PEMCSCF, 175, 180 Infrared (IR) intensities, 326, 364, 376 Intrinsic reaction coordinate, 365 Kinetic energy, 10 Koopmans' theorem, 129 Kronecker delta function, 14 Lagrange's method of undetermined multipliers, 167, 177, 199 Lagrangian matrix CI, 87, 90, 168, 250, 355 first derivative CI, 321 first derivative GRSCF, 74, 145, 319 first derivative MCSCF, 195, 261 first derivative PEMCSCF, 174 GRSCF, 16, 70, 144, 285 MCSCF, 20, 121, 194, 199, 258 PEMCSCF, 114, 173, 271 second derivative GRSCF, 156, 162
467 second derivative MCSCF, 211 second derivative PEMCSCF, 184 TCSCF, 102 Laplacian operator, 9 Linear combination of atomic orbitals-molecular orbital (LCAO-MO), 5, 9, 11, 15, 29 Linear dependency, 178, 195, 200, 362 Minimal basis set, 13 Minimum energy path, 365 Mixed (cross) perturbations, 308, 311, 326 Modified derivative Lagrangian matrix CI first, 252 CI second, 255 CLSCF first, 300 CLSCF second, 301, 303 GRSCF first, 289 GRSCF second, 290 MCSCF first, 261 MCSCF second, 267 PEMCSCF first, 276 PEMCSCF second, 279 Molecular orbital (MO), 12, 29 MO orthonormality condition, 14, 36, 56, 258, 311 CLSCF, 14, 56 MCSCF, 19, 258 first derivative, 36, 38, 56, 88, 123, 129, 146, 178, 196, 200, 259, 311, 358 fourth derivative, 264 second derivative, 34, 39, 62, 96, 108, 118, 126, 136, 160, 189, 213, 259, 311, 362 third derivative, 261 Multiconfiguration self-consistent-field (MCSCF) wavefunction, 4, 18, 119, 193, 258, 364, 385 Multiple perturbations, 349
Newton-Raphson method, 4, 24 Normalization condition of CI coefficients, 16, 83, 166, 219, 250 CLSCF MO coefficients, 16
468
INDEX
MCSCF CI coefficients, 19, 119, 199 PEMCSCF CI coefficients, 111, 177 TCSCF CI coefficients, 99 Normalization constant, 12 Nuclear perturbation, 4, 29, 364 Nuclear repulsion energy, 10, 25, 51 first derivative, 51 second derivative, 51 Occupation number, 68, 150, 180 Occupied orbital, 16, 22, 31, 68, 144, 148 One-electron coupling constant CI, 17, 84, 167, 219, 250, 321
GRSCF, 16, 68 MCSCF, 19, 120, 199, 258, 324 PEMCSCF, 111, 270 TCSCF, 99 One-electron integral AO basis, 15, 21, 33 first derivative AO basis, 33, 310 first derivative MO basis, 39, 41 MO basis, 14, 17, 20, 39 second derivative AO basis, 33, 310 second derivative MO basis, 39, 42 One-electron operator, 10, 14, 21, 54, 70,
121 One-electron reduced density matrix CI AO basis, 23, 87, 250 CI MO basis, 17, 23, 84, 167, 219, 250,
321 first derivative CI, 321
MCSCF, 203 PEMCSCF, 181 MCSCF AO basis, 123 MCSCF MO basis, 19, 120, 199, 258 PEMCSCF MO basis, 113, 270 TCSCF MO basis, 100 Open-shell SCF wavefunction, 16, 68, 285 Open-shell singlet, 68, 70 Orbital energies, 14, 15, 54, 129, 150, 180, 202, 304 first derivative CLSCF, 131, 305 Orbital exponent, 12 Orthogonality, 11
Orthonormality condition see MO orthonormality condition Overlap integral AO basis, 15, 32 first derivative AO basis, 32 first derivative MO basis, 34, 37 MO basis, 14 second derivative AO basis, 32 second derivative MO basis, 34, 39 Paired excitation MCSCF (PEMCSCF)
wavefunction, 5, 98, 110, 172, 270 Pauli exclusion principle, 12 Permutation operator, 14 Perturbations electric field, 6, 29, 308, 364 mixed (cross), 308, 311, 326 multiple, 349 nuclear, 4, 6, 29, 308
Polarizability see Electric polarizability Potential energy, 10 Potential energy (hyper) surface (PES),
3, 24 Primitive Gaussian functions, 13 Quadruple zeta plus triple polarization (QZ3P) basis set, 369, 374 Quartet state, 69 Raman intensities, 364 Reaction coordinate, 24, 365, 373 Reaction path Hamiltonian, 364, 365 Reduced density matrices see One-electron reduced density matrix see Two-electron reduced density matrix Reference wavefunction (or configuration), 5, 18, 87, 355 Restricted Hartree-Fock (RHF), 53, 68,
128, 144, 256 Rys polynomial, 3 Saddle point, 24
INDEX
Second derivative CI coe ffi cients normalization condition, 170 CI Hamiltonian matrix, 92, 170, 221, 252, 341, 342 CLSCF, Fock matrix, 136, 138 GRSCF, Lagrangian matrix, 156, 162 MCSCF bare Lagrangian matrix, 210 MCSCF Hamiltonian matrix, 124, 213 MCSCF Lagrangian matrix, 211 MO orthonormafity condition, 34, 39, 62, 96, 108, 118, 126, 136, 160, 189, 213, 259, 311, 362 nuclear repulsion energy, 51 one-electron AO integral, 33, 310 one-electron MO integral, 39, 42 overlap AO integral, 32 overlap MO integral, 34, 37, 39 PEMCSCF bare Lagrangian matrix, 183 PEMCSCF Hamiltonian matrix, 116, 187, 281 PEMCSCF Lagrangian matrix, 184 TCSCF Hamiltonian matrix, 104 two-electron AO integral, 34 two-electron MO integral, 43, 49 see also Electronic energy second derivative Secular equation, 15 Shell, 73, 147, 150, 175, 180 Simultaneous equations, 357 CI, 168, 169 CLSCF, 131 GRSCF, 148 MCSCF, 194, 201 PEMCSCF, 179 Skeleton (core) first derivative CI bare Lagrangian matrix, 91, 170, 252, 341, 342 CI bare Y matrix, 255 CI bare Z matrix, 235 CI Hamiltonian matrix, 90, 168, 252 CI Lagrangian matrix, 92, 170, 233,
469 251, 321, 330, 358, 361
CI Y matrix, 234, 253 CI Z matrix, 248, 256 CLSCF A matrix, 133, 298, 334, 359 CLSCF Fock matrix, 59, 129, 296, 316, 332, 334, 356, 361 GRSCF averaged Fock matrix, 151 GRSCF generalized Lagrangian matrix, 154, 288, 338 GRSCF Lagrangian matrix, 75, 145, 287, 319, 336 GRSCF T matrix, 154, 288 MCSCF averaged Fock matrix, 204 MCSCF bare Lagrangian matrix, 125, 195, 262, 346, 347 MCSCF bare y matrix, 207, 209, 267 MCSCF Hamiltonian matrix, 121, 199, 344 MCSCF Lagrangian matrix, 125, 195, 260, 324, 331, 344 MCSCF y matrix, 209, 263, 345, 347 MCSCF z matrix, 267 one-electron MO integral, 40, 310 overlap MO integral, 35 PEMCSCF averaged Fock matrix, 181 PEMCSCF bare generalized Lagrangian matrix, 184, 278 PEMCSCF bare Lagrangian matrix, 116, 174,275 PEMCSCF bare r matrix, 184, 278 PEMCSCF generalized Lagrangian matrix, 184, 274 PEMCSCF Hamiltonian matrix, 115, 177, 276 PEMCSCF Lagrangian matrix, 117, 174, 273 PEMCSCF T matrix, 184, 274 TCSCF bare Lagrangian matrix, 105 TCSCF Hamiltonian matrix, 103 TCSCF Lagrangian matrix, 105 two-electron MO integral, 46 Skeleton (core) fourth derivative CI Hamiltonian matrix, 237
470 Skeleton (core) second derivative CI bare Lagrangian matrix, 229, 255 CI Hamiltonian matrix, 93, 170, 252 CI Lagrangian matrix, 229, 233, 253 Cl Y matrix, 248, 256 CLSCF A matrix, 301 CLSCF Fock matrix, 133, 298, 334, 335, 359 GRSCF generalized Lagrangian matrix, 290 GRSCF Lagrangian matrix, 154, 288, 337, 339 GRSCF 7 matrix, 290 MCSCF bare Lagrangian matrix, 209, 266 MCSCF Hamiltonian matrix, 124, 214, 262, 346, 349 MCSCF Lagrangian matrix, 209, 263, 345, 349 MCSCF y matrix, 267 one-electron MO integral, 40, 310 overlap MO integral, 35 PEMCSCF bare Lagrangian matrix, 184, 278 PEMCSCF generalized Lagrangian matrix, 278 PEMCSCF Hamiltonian matrix, 116, 188, 276 PEMCSCF Lagrangian matrix, 184, 274 PEMCSCF T matrix, 278 TCSCF Hamiltonian matrix, 105 two-electron MO integral, 46 Skeleton (core) third derivative CI bare Lagrangian matrix, 237 CI Hamiltonian matrix, 229, 254 CI Lagrangian matrix, 237, 247, 256 CLSCF Fock matrix, 301 GRSCF Lagrangian matrix, 290 MCSCF Hamiltonian matrix, 266 MCSCF Lagrangian matrix, 267 PEMCSCF Hamiltonian matrix, 281 PEMCSCF Lagrangian matrix, 278 Slater determinant, 12
INDEX Slater-type orbital (STO), 12 Spatial orbital, 12 Spin function, 12 Spin-orbitals, 11 Split valence basis set, 13 Stationary point, 3, 24, 308, 363 s-Tetrazine, 371 Symmetric relationships one-electron AO integral, 21 MO integral, 21 reduced density matrix AO, 23 reduced density matrix MO, 17 T matrix, 147 two-electron AO integral, 21 coupling constant MO, 207 MO integral, 21 reduced density matrix AO, 23 reduced density matrix MO, 17 matrix GRSCF, 75, 145, 287 PEMCSCF, 117, 174, 273 TCSCF, 106 Third derivative CI Hamiltonian matrix, 228, 254 MO orthonormality condition, 261 see also Electronic energy third derivative Total energy, 10, 25 CI, 84 CLSCF, 54 GRSCF, 68 MCSCF, 120 PEMCSCF, 114 TCSCF, 101 Transition state, 3, 24, 364 Triplet state, 69, 369 Triple zeta plus double polarization (TZ2P) basis set, 367, 374, 376 Triple zeta plus polarization (TZP) basis set, 381 Two-configuration self-consistent field (TCSCF) wavefunction, 5, 98, 270 7
INDEX Two-electron coupling constant CI, 17, 84, 167, 219, 250 GRSCF, 16, 68 MCSCF, 19, 120, 200, 207, 258 PEMCSCF, 111, 270 TCSCF, 99 Two-electron integral AO basis, 15, 21, 33 first derivative AO basis, 33 first derivative MO basis, 43, 47 MO basis, 17, 21, 43 second derivative AO basis, 33 second derivative MO basis, 43, 49 Two-electron operator, 21, 34, 121 Two-electron reduced density matrix CI AO basis, 23, 87 Cl MO basis, 17, 23, 84, 167, 219, 250 MCSCF AO basis, 123 MCSCF MO basis, 19, 120, 200, 258 PEMCSCF MO basis, 113, 270 TCSCF MO basis, 101
U matrices, 27, 31, 309, 350, 356 Unitary transformation, 128, 150, 180, 202, 294 Unoccupied orbitals see Virtual orbitals Variational conditions, 11, 14, 18, 19 CI space, 18, 84, 166, 219, 250 CLSCF MO space, 14, 54, 128, 293, 304 GRSCF MO space, 16, 71, 144, 285, 319 MCSCF CI space, 20, 120, 198 MCSCF MO space, 20, 121, 194, 258, 325 PEMCSCF Cl space, 115, 177 PEMCSCF MO space, 114, 172, 271 TCSCF CI space, 102 TCSCF MO space, 102 Variation method, 10 Variation principle, 11 Virtual orbitals, 18, 31, 68, 71, 130, 144, 148, 202, 304
471 Water (H20), 376, 381 Wavefunction CASSCF, 4, 20, 119, 369 CI, 4, 16, 23, 83, 166, 218, 249, 355, 364, 385 CLSCF, 5, 13, 22, 53, 128, 293 GRSCF, 5, 16, 68, 144, 285 GVB, 5, 98, 110, 270 MCSCF, 4, 18, 119, 193, 258, 364, 385 PEMCSCF, 5, 98, 110, 172, 270 TCSCF, 5, 98, 270 Wigner's 2n-1-1 theorem, 25
Y (y) matrix CI, 92, 170, 251, 321, 359 MCSCF, 125, 195, 260, 325 Zero-point vibrational energy (ZPVE), 365, 369 Z (z) matrix CI, 231, 255 MCSCF, 267 Z vector method, 6, 26, 87, 96, 314, 323, 330, 355
