ParaGauss - a parallel implementation of the density functional method [Elektronische Ressource] : spin-orbit interaction in the Douglas-Kroll-Hess approach and a novel two-component treatment of spin-independent interaction terms / Alexei V. Matveev
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ParaGauss - a parallel implementation of the density functional method [Elektronische Ressource] : spin-orbit interaction in the Douglas-Kroll-Hess approach and a novel two-component treatment of spin-independent interaction terms / Alexei V. Matveev

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Publié le 01 janvier 2004
Nombre de lectures 25
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Institut fur¨ Physikalische und Theoretische Chemie
der Technischen Universit¨at Munc¨ hen
ParaGauss — A Parallel Implementation of the Density
Functional Method: Spin-Orbit Interaction in the
Douglas–Kroll–Hess Approach and a Novel Two-Component
Treatment of Spin-Independent Interaction Terms
Alexei V. Matveev
Vollst¨andiger Abdruck der von der Fakult¨at fur¨ Chemie zur Erlangung des
akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzende: Univ.-Prof. Dr. S. Weinkauf
Prufer¨ der Dissertation
1. Univ.-Prof. Dr. N. R¨osch
2. Univ.-Prof. Dr. M. Kleber
3. Univ.-Prof. Dr. W. Domcke
DieDissertationwurdeam24.09.2003beidertechnischenUniversit¨atMunc¨ hen
eingereicht und durch die Fakult¨at fur¨ Chemie am 30.10.2003 angenommen.Contents
1 Introduction 1
2 The Electron-Electron Interaction in the Douglas–Kroll–Hess Approach
to the Dirac–Kohn–Sham Problem 5
2.1 The Relativistic Kohn–Sham Problem. . . . . . . . . . . . . . . . . . . . . 7
2.1.1 The Four-Component Formalism . . . . . . . . . . . . . . . . . . . 7
2.1.2 The Douglas–Kroll Two-Component Formalism . . . . . . . . . . . 8
2.1.3 Scalar Relativistic Treatment and Spin-Orbit Interaction . . . . . . 15
2.1.4 Relativistic Transformation of the Coulomb Potential . . . . . . . . 15
2.1.5 Density-Fit Based Relativistic Expression for the Hartree Energy . 20
2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Primitive Integrals. Integrals Based on the Momentum Representation 27
2.2.2 Relativistic Transformations . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Relativistic Ttion of the Hartree Potential . . . . . . . . 38
2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.2 Spin-Orbit Splittings in the Hg Atom . . . . . . . . . . . . . . . . . 49
2.3.3 EffectsofRelativisticContributionstoee-Interactionontheg-Tensor
of NO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
2.3.4 Spin-Orbit Effects on Properties of Diatomic Molecules . . . . . . . 53
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 SymmetryTreatmentinRelativisticElectronicStructureCalculationsof
Molecules 61
3.1 Quaternion parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 The Eigenfunction Method . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Basic Concepts of the Eigenfunction Method:
Atomic Orbitals in C Symmetry . . . . . . . . . . . . . . . . . . 654v
3.2.2 Theory of the Eigenfunction Method . . . . . . . . . . . . . . . . . 69
iii
3.2.3 The Eigenfunction Method for Double Groups . . . . . . . . . . . 74
3.2.4 CSCO in orbital and spinor spaces . . . . . . . . . . . . . . . . . . 77
3.3 Symmetry Adapted Functions . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.1 Symmetrized Molecular Orbitals . . . . . . . . . . . . . . . . . . . 79
3.3.2 Pseudo 2D Representations . . . . . . . . . . . . . . . . . . . . . . 83
3.3.3 Symmetrized Molecular Spinors . . . . . . . . . . . . . . . . . . . . 84
3.3.4 Molecular Four-Spinors . . . . . . . . . . . . . . . . . 88
3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.1 Symmetrization Coefficients . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 Transformation from an Orbital Representation to
a Spinor Representation . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5 Application to Numerical Integration of the Exchange-Correlation Potential 99
3.6 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 Density Functional Study of Small Molecules and Transition Metal
Carbonyls Using Revised PBE Functionals 105
4.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.1 Small Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.2 Transition Metal Carbonyls . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Summary 121
A An Interface to Matrix Arithmetics 125
B Group-TheoreticalInformation:Generators,CanonicalSubgroupChains,
CSCO 129
Bibliography 133
Publications 146iii
List of abbreviations
BLYP Becke–Lee–Yang–Parr
BP Becke–Perdew
CG Clebsch–Gordan
CSCO Complete Set of Commuting Operators
DF Density-Functional
DFT Density-Functional Theory
DK Douglas–Kroll
DKH Douglas–Kroll–Hess
DKS Dirac–Kohn–Sham
ee Electron-Electron
EFM Eigenfunction Method
EPR Electronic Paramagnetic Resonance
FF Fitting Function
fpFW free-particle Foldy–Wouthuysen
FW Foldy–Wouthuysen
GGA Generalized Gradient Approximation
HF Hartree–Fock
irrep irreducible representation
KS Kohn–Sham
LCAO Linear Combination of Atomic Orbitals
LCGTO Linear Combination of Gaussian-Type Orbitals
LDA Local Density Approximation
NMR Nuclear Magnetic Resonance
PBE Perdew–Burke–Ernzerhof
PW91 Perdew–Wang, year 1991
QM Quantum Mechanics
revPBE Zhang and Yang revision of PBE
RPBE Hammer, Hansen, and Nørskov revision of PBE
SCF Self-Consistent Field
SO Spin-Orbit
SR Scalar Relativistic
VWN Vosko–Wilk–Nusair
xc exchange-correlation
ZORA Zeroth Order Regular ApproximationivChapter 1
Introduction
Electronic structure calculations evolved from illustrative numerical applications of the
Schr¨odinger equation to a scientific area of predictive strength [1, 2, 3, 4, 5, 6, 7]. Quan-
tum chemistry, the theoretical treatment of the electronic structure of molecular systems,
nowadays is one of the major subfields in the area of numerical calculations of the elec-
tronicstructureofmatter. Whetherspecificproblemscanbesolvedbyquantumchemistry
depends much on the availability of suitable methods which incorporate the relevant in-
teractions so that pertinent characteristics of the system can be modeled. Relativistic
quantum chemistry, which by now covers a vast variety of methods and approximations
[8], is a field that grew with the attempts to account for relativistic effects in the quantum
mechanics (QM) of molecular systems. Such effects are due to the fact that the speed of
light is finite, but large enough so that in most cases it can be assumed to be essentially
infinite. Relativistic effects are often thought of as unimportant for “normal” molecu-
lar system. Very many purely chemical applications of quantum chemistry are successful
without accounting for relativity. However, modern chemistry could hardly be imagined
without such spectroscopic techniques as electron paramagnetic resonance (EPR), nuclear
magnetic resonance (NMR) or other high-resolution methods where relativistic interac-
tions often play an important role. The chemistry of heavy elements cannot be described
theoretically without, in some way, accounting for relativistic effects.
A well known example of a relativistic effect is the spin-orbit (SO) interaction in atoms
and molecules. One of the observable effects of SO interaction is the fine structure of
atomic and molecular spectral lines [9, 10]. The term “spin-orbit” was originally used to
describe the coupling ζ(r)ls of the orbital angular momentum l of an electron and its
spin s in atoms. The eigenfunctions of the spherically symmetric SO Hamiltonian are
eigenfunctions of the total angular momentum j = l + s. The energy dependence on
1quantum numbers j = l± is able to describe the fine structure of atomic levels. The
2
radialfunctionζ(r)determinesthestrengthoftheSOcouplinginatomicshellslocalizedat
12 CHAPTER 1. INTRODUCTION
differentseparationsfromthenucleusandthusthemagnitudeofthefinestructureinatoms.
The expectation values of the electron magnetic moment μ = l+2s, responsible for the
interactionwithanexternalmagneticfield,hastobetakenwitheigenfunctionsofthetotal
angularmomentumj todetermineproperlytheg factor,anEPRparameterofanunpaired
electron. SO interaction plays an important role also in molecules although the orbital
angularmomentumisoftennotagoodquantumnumberthere. TheSOinteractionrenders
the g factor of an electron in a radical different from the exact g = 2 of a free electron
[11]. In molecular compounds where the orbital interplay involves the SO interaction,
structural parameters, e.g. the geometry, can be notably affected by the SO interaction.
The so-called scalar relativistic (SR) effects, the counterpart of the SO relativistic effects,
always make notable contributions to the physics and chemistry of the heavy elements and
theircompounds. However, theseeffectsareratherofquantitativethanqualitativenature.
OneofthecommonapproximationsassumesasimpleformfortheSOcouplingconstant
ζ(r) = dV/dr, where V is a effective electronic potential function. For a hydrogen-like
atom, the potential is V =−Z/r with Z being the nuclear charge; such a radial potential
is not that good an approximation for heavier atoms where the screening effects of the
electrondensityhastobeaccountedfor. Thescreeningeffectsoftheelectrondensityreduce
the effective nuclear charge Z, hence they also re

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