Relativistic Quantum Chemistry (eBook)

Markus Reiher obtained his PhD in Theoretical Chemistry in 1998, working in the group of Juergen Hinze at the University of Bielefeld on relativistic atomic structure theory. He completed his habilitation on transition-metal catalysis and vibrational spectroscopy at the University of Erlangen in the group of Bernd Artur Hess in 2002. During that time he had the opportunity to return to relativistic theories when working with Bernd Hess and Alex Wolf. From 2003 to 2005, Markus Reiher was Privatdozent at the University of Bonn and then moved to the University of Jena as Professor for Physical Chemistry in 2005. Since the beginning of 2006 he has been Professor for Theoretical Chemistry at ETH Zurich. Markus Reiher's research interests in molecular physics and chemistry are broad and diverse. Alexander Wolf studied physics at the University of Erlangen and at Imperial College, London. In 2004, he completed his PhD in Theoretical Chemistry in the group of Bernd Artur Hess in Erlangen. His thesis elaborated on the generalized Douglas-Kroll-Hess transformation and efficient decoupling schemes for the Dirac Hamiltonian. As a postdoc he continued to work on these topics in the group of Markus Reiher at the universities of Bonn (2004) and Jena (2005). Since 2006 he has been engaged in financial risk management for various consultancies and is currently working in the area of structuring and modeling of life insurance products. On a regular basis he has been using his spare time to delve into his old passion, relativistic quantum mechanics and quantum chemistry.

Preface

INTRODUCTION
Philosophy of this Book
Short Reader's Guide
Notational Conventions and Choice of Units

PART I: Fundamentals

ELEMENTS OF CLASSICAL MECHANICS AND ELECTRODYNAMICS
Elementary Newtonian Mechanics
Lagrangian Formulation
Hamiltonian Mechanics
Elementary Electrodynamics

CONCEPTS OF SPECIAL RELATIVITY
Einstein's Relativity Principle and Lorentz Transformations
Kinematic Effects in Special Relativity
Relativistic Dynamics
Covariant Electrodynamics
Interaction of Two Moving Charged Particles

BASICS OF QUANTUM MECHANICS
The Quantum Mechanical State
The Equation of Motion
Observables
Angular Momentum and Rotations
Pauli Antisymmetry Principle

PART II: Dirac's Theory of the Electron

RELATIVISTIC THEORY OF THE ELECTRON
Correspondence Principle and Klein-Gordon Equation
Derivation of the Dirac Equation for a Freely Moving Electron
Solution of the Free-Electron Dirac Equation
Dirac Electron in External Electromagnetic Potentials
Interpretation of Negative-Energy States: Dirac's Hole Theory

THE DIRAC HYDROGEN ATOM
Separation of Electronic Motion in a Nuclear Central Field
Schrödinger Hydrogen Atom
Total Angular Momentum
Separation of Angular Coordinates in the Dirac Hamiltonian
Radical Dirac Equation for Hydrogen-Like Atoms
The Nonrelativistic Limit
Choice of the Energy Reference and Matching Energy Scales
Wave Functions and Energy Eigenvalues in the Coulomb Potential
Finite Nuclear Size Effects
Momentum Space Representation

PART III: Four-Component Many-Electron Theory

QUANTUM ELECTRODYNAMICS
Elementary Quantities and Notation
Classical Hamiltonian Description
Second-Quantized Field-Theoretical Formulation
Implications for the Description of Atoms and Molecules

FIRST-QUANTIZED DIRAC-BASED MANY-ELECTRON THEORY
Two-Electron Systems and the Breit Equation
Quasi-Relativistic Many-Particle Hamiltonians
Born-Oppenheimer Approximation
Tensor Structure of the Many-Electron Hamiltonian and Wave Function
Approximations to the Many-Electron Wave Function
Second Quantization for the Many-Electron Hamiltonian
Derivation of Effective One-Particle Equations
Relativistic Density Functional Theory
Completion: The Coupled-Cluster Expansion

MANY-ELECTRON ATOMS
Transformation of the Many-Electron Hamiltonian to Polar Coordinates
Atomic Many-Electron Wave Function and jj-Coupling
One- and Two-Electron Integrals in Spherical Symmetry
Total Expectation Values
General Self-Consistent-Field Equations and Atomic Spinors
Analysis of Radial Functions and Potentials at Short and Long Distances
Numerical Discretization and Solution Techniques
Results for Total Energies and Radial Functions

GENERAL MOLECULES AND MOLECULAR AGGREGATES
Basis Set Expansion of Molecular Spinors
Dirac-Hartree-Fock Electronic Energy in Basis Set Representation
Molecular One- and Two-Electron Integrals
Dirac-Hartree-Fock-Roothaan Matrix Equations
Analytic Gradients
Post-Hartree-Fock Methods

PART IV: Two-Component Hamiltonians

DECOUPLING THE NEGATIVE-ENERGY STATES
Relations of Large and Small Components in One-Electron Equations
Closed-Form Unitary Transformations of the Dirac Hamiltonian
The Free-Particle Foldy-Wouthuysen Transformation
General Parametrization of Unitary Transformation
Fold-Wouthuysen Expansion in Powers of 1/c
The Infinite-Order Two-Component Two-Step Protocol
Toward Well-Defined Analytic Block-Diagonal Hamiltonians

DOUGLAS-KROLL-HESS THEORY
Sequential Unitary Decoupling Transformations
Explicit Form of the DKH Hamiltonians
Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH Method
Many-Electron DKH Hamiltonians
Computational Aspects of DKH Calculations

ELIMINATION TECHNIQUES
Naive Reduction: Pauli Elimination
Breit-Pauli Theory
The Cowan-Griffin and Wood-Boring Approaches
Elimination for Different Representations of Dirac Matrices
Regular Approximations

PART V: Chemistry with Relativistic Hamiltonians

SPECIAL COMPUTATIONAL TECHNIQUES
From the Modified Dirac Equation to Exact-Two-Component Methods
Locality of Relativistic Contributi Preface

INTRODUCTION
Philosophy of this Book
Short Reader's Guide
Notational Conventions and Choice of Units

PART I: Fundamentals

ELEMENTS OF CLASSICAL MECHANICS AND ELECTRODYNAMICS
Elementary Newtonian Mechanics
Lagrangian Formulation
Hamiltonian Mechanics
Elementary Electrodynamics

CONCEPTS OF SPECIAL RELATIVITY
Einstein's Relativity Principle and Lorentz Transformations
Kinematic Effects in Special Relativity
Relativistic Dynamics
Covariant Electrodynamics
Interaction of Two Moving Charged Particles

BASICS OF QUANTUM MECHANICS
The Quantum Mechanical State
The Equation of Motion
Observables
Angular Momentum and Rotations
Pauli Antisymmetry Principle

PART II: Dirac's Theory of the Electron

RELATIVISTIC THEORY OF THE ELECTRON
Correspondence Principle and Klein-Gordon Equation
Derivation of the Dirac Equation for a Freely Moving Electron
Solution of the Free-Electron Dirac Equation
Dirac Electron in External Electromagnetic Potentials
Interpretation of Negative-Energy States: Dirac's Hole Theory

THE DIRAC HYDROGEN ATOM
Separation of Electronic Motion in a Nuclear Central Field
Schrödinger Hydrogen Atom
Total Angular Momentum
Separation of Angular Coordinates in the Dirac Hamiltonian
Radical Dirac Equation for Hydrogen-Like Atoms
The Nonrelativistic Limit
Choice of the Energy Reference and Matching Energy Scales
Wave Functions and Energy Eigenvalues in the Coulomb Potential
Finite Nuclear Size Effects
Momentum Space Representation

PART III: Four-Component Many-Electron Theory

QUANTUM ELECTRODYNAMICS
Elementary Quantities and Notation
Classical Hamiltonian Description
Second-Quantized Field-Theoretical Formulation
Implications for the Description of Atoms and Molecules

FIRST-QUANTIZED DIRAC-BASED MANY-ELECTRON THEORY
Two-Electron Systems and the Breit Equation
Quasi-Relativistic Many-Particle Hamiltonians
Born-Oppenheimer Approximation
Tensor Structure of the Many-Electron Hamiltonian and Wave Function
Approximations to the Many-Electron Wave Function
Second Quantization for the Many-Electron Hamiltonian
Derivation of Effective One-Particle Equations
Relativistic Density Functional Theory
Completion: The Coupled-Cluster Expansion

MANY-ELECTRON ATOMS
Transformation of the Many-Electron Hamiltonian to Polar Coordinates
Atomic Many-Electron Wave Function and jj-Coupling
One- and Two-Electron Integrals in Spherical Symmetry
Total Expectation Values
General Self-Consistent-Field Equations and Atomic Spinors
Analysis of Radial Functions and Potentials at Short and Long Distances
Numerical Discretization and Solution Techniques
Results for Total Energies and Radial Functions

GENERAL MOLECULES AND MOLECULAR AGGREGATES
Basis Set Expansion of Molecular Spinors
Dirac-Hartree-Fock Electronic Energy in Basis Set Representation
Molecular One- and Two-Electron Integrals
Dirac-Hartree-Fock-Roothaan Matrix Equations
Analytic Gradients
Post-Hartree-Fock Methods

PART IV: Two-Component Hamiltonians

DECOUPLING THE NEGATIVE-ENERGY STATES
Relations of Large and Small Components in One-Electron Equations
Closed-Form Unitary Transformations of the Dirac Hamiltonian
The Free-Particle Foldy-Wouthuysen Transformation
General Parametrization of Unitary Transformation
Fold-Wouthuysen Expansion in Powers of 1/c
The Infinite-Order Two-Component Two-Step Protocol
Toward Well-Defined Analytic Block-Diagonal Hamiltonians

DOUGLAS-KROLL-HESS THEORY
Sequential Unitary Decoupling Transformations
Explicit Form of the DKH Hamiltonians
Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH Method
Many-Electron DKH Hamiltonians
Computational Aspects of DKH Calculations

ELIMINATION TECHNIQUES
Naive Reduction: Pauli Elimination
Breit-Pauli Theory
The Cowan-Griffin and Wood-Boring Approaches
Elimination for Different Representations of Dirac Matrices
Regular Approximations

PART V: Chemistry with Relativistic Hamiltonians

SPECIAL COMPUTATIONAL TECHNIQUES
From the Modified Dirac Equation to Exact-Two-Component Methods
Locality of Relativistic Contributi