Density Functional and Ab Initio Study of Molecular Response

215 Pages·2014·1.54 MB·English

by  Degao Peng

Density Functional and Ab Initio Study of
Molecular Response
by
Degao Peng
Department of Chemistry
Duke University
Date:
Approved:
Weitao Yang, Supervisor
David N. Beratan
Patrick Charbonneau
Benjamin J. Wiley
Dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in the Department of Chemistry
in the Graduate School of Duke University
2014
Abstract
Density Functional and Ab Initio Study of Molecular
Response
by
Degao Peng
Department of Chemistry
Duke University
Date:
Approved:
Weitao Yang, Supervisor
David N. Beratan
Patrick Charbonneau
Benjamin J. Wiley
An abstract of a dissertation submitted in partial fulfillment of the requirements for
the degree of Doctor of Philosophy in the Department of Chemistry
in the Graduate School of Duke University
2014
Copyright (cid:13)c 2014 by Degao Peng
All rights reserved except the rights granted by the
Creative Commons Attribution-Noncommercial Licence
Abstract
Quantumchemistrymethodsnowadaysreachitsmaturitywithvariousrobustground
state correlation methods. However, many problems related to response do not have
satisfactory solutions. Chemical reactivity indexes are some static response to ex-
ternal fields and number of particles change. These chemical reactivity indexes have
important chemical significance, yet not all of them have analytical expressions for
direct evaluations. By solving coupled perturbed self-consistent field equations, ana-
lytical expressions are obtained and verified numerically. In the particle-particle (pp)
channel, the response to the pairing field describes N˘2 excitations, i.e. double ion-
ization potentials and double electron affinities. The linear response time-dependent
density-functional theory (DFT) with pairing fields is the response theory in the
density-functional theory (DFT) framework to describe N ˘ 2 excitations. Both
adiabatic and dynamic kernels are included in this response theory. The correla-
tion energy based on this response, the particle-particle random phase approxima-
tion (pp-RPA) correlation energy, is proven equivalent to the ladder approximation
of the well-established coupled-cluster doubles. These connections between the re-
sponse theory, ab initio methods, and Green’s function theory will be beneficial for
further development. Based on the particle-hole RPA and the pp-RPA, the theory
of the second particle-hole RPA and the second pp-RPA with restrictions capture
single and double excitations efficiently. We also present a novel method, variational
fractional spin DFT, to calculate singlet-triplet energy gaps for diradicals, which are
iv
usually studied through spin-flip response theories.
v
“I leave no trace of wings in the air, but I am glad I have had my flight.”
From Fireflies by Rabindranath Tagore (1928)
vi
Contents
Abstract iv
List of Tables xii
List of Figures xiii
List of Abbreviations and Symbols xiv
Acknowledgements xviii
1 Introduction 1
1.1 Overview of quantum chemistry . . . . . . . . . . . . . . . . . . . . . 1
1.2 Ab initio methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . 5
1.2.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Configuration interaction . . . . . . . . . . . . . . . . . . . . . 11
1.2.4 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.5 Coupled-cluster theory . . . . . . . . . . . . . . . . . . . . . . 16
1.2.6 Remarks on ab initio methods . . . . . . . . . . . . . . . . . . 18
1.3 Density-functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . 19
1.3.2 The Levy constrained-search functional . . . . . . . . . . . . . 21
1.3.3 Kohn-Sham formalism . . . . . . . . . . . . . . . . . . . . . . 22
1.3.4 Exchange-correlation energy . . . . . . . . . . . . . . . . . . . 25
vii
1.3.5 Challenges of DFT . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.6 Remarks on DFT . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Response calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.1 Static response . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.2 Time-dependent density-functional theory . . . . . . . . . . . 34
1.4.3 Equation-of-motion methods . . . . . . . . . . . . . . . . . . . 40
1.4.4 Summary of response calculations . . . . . . . . . . . . . . . . 42
1.5 Concluding remarks of the review . . . . . . . . . . . . . . . . . . . . 43
2 Fukui function and response function for nonlocal and fractional
systems 44
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 ErN,vs and its derivatives . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 The linearity condition and its extensions . . . . . . . . . . . . . . . . 50
2.4 Analytical expressions for derivatives in Kohn-Sham and generalized
Kohn-Sham framework . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.1 p`q “ 2 derivatives of a system with a fractional number of
electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.2 δ3E{δv3 and δ3E{δNδv2 for a system with an integer number
of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.3 δ3E{δN2δv and δ3E{δN3 for a system with an integer number
of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.4.4 Numerical verification . . . . . . . . . . . . . . . . . . . . . . 64
2.5 Extensions to nonlocal Fukui functions and linear-response functions 65
2.6 The constancy condition and its extensions . . . . . . . . . . . . . . . 72
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3 Linear-responsetime-dependentdensity-functionaltheory withpair-
ing fields 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
viii
3.2 DFT with pairing interactions . . . . . . . . . . . . . . . . . . . . . . 80
3.3 Adiabatic linear-response TDDFT-P for non-superconducting systems 86
3.4 Linear-response TDDFT-P with frequency-dependent pp kernels . . . 92
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Ladder-Coupled-Cluster Doubles and its equivalence to particle-
particle random phase approximation 97
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 The pp-RPA equation and its stability . . . . . . . . . . . . . . . . . 99
4.3 Proof of the equivalence of pp-RPA and ladder-CCD . . . . . . . . . 103
4.4 Numerical demonstrations . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5 Second random phase approximations 113
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.1 2ph-RPA formalism . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.2 2pp-RPA formalism . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.3 Excited state properties . . . . . . . . . . . . . . . . . . . . . 118
5.2.4 Orbital restrictions . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Computation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4.1 Sˆ2 expression tests . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4.2 H O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2
5.4.3 Be . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4.4 BH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4.5 CO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.6 Correlation energy offsets . . . . . . . . . . . . . . . . . . . . 129
ix
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Variational fractional spin density-functional theory for diradicals 133
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.1 Density-functional theory with fractional occupation numbers 135
6.2.2 Variational-fractional-spin DFT for diradicals . . . . . . . . . 136
6.2.3 Computational details . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3.1 Degenerate frontier orbitals . . . . . . . . . . . . . . . . . . . 141
6.3.2 Non-degenerate frontier orbitals . . . . . . . . . . . . . . . . . 141
6.3.3 Two types of diradicals . . . . . . . . . . . . . . . . . . . . . . 143
6.3.4 Carbene-like diradicals . . . . . . . . . . . . . . . . . . . . . . 146
6.3.5 Octacene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A Second Quantization 153
A.1 Operator and wavefunction representations . . . . . . . . . . . . . . . 153
A.2 Normal order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B Detailed derivations for Fukui and response functions 159
B.1 Detailed derivations of δ3E{δv3 and δ3E{δNδv2 . . . . . . . . . . . . 159
B.2 Detailed derivations of δ3E{δN2δv and δ3E{δN3 . . . . . . . . . . . . 164
C Mathematical Details of Non-Adiabatic Linear-Response TDDFT-
P 168
D Mathematical analysis of the pp-RPA equation 174
D.1 The zero signature of an eigenvector with an imaginary eigenvalue . . 174
D.2 The orthonormalization of eigenvectors with all real eigenvalues . . . 175
x