# Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd Edition

# Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd Edition

ISBN: 978-3-527-66755-0

Dec 2014

750 pages

## Description

Einstein proposed his theory of special relativity in 1905. For a long time it was believed that this theory has no significant impact on chemistry. This view changed in the 1970s when it was realized that (nonrelativistic) Schrödinger quantum mechanics yields results on molecular properties that depart significantly from experimental results. Especially when heavy elements are involved, these quantitative deviations can be so large that qualitative chemical reasoning and understanding is affected. For this to grasp the appropriate many-electron theory has rapidly evolved. Nowadays relativistic approaches are routinely implemented and applied in standard quantum chemical software packages. As it is essential for chemists and physicists to understand relativistic effects in molecules, the first edition of "Relativistic Quantum Chemistry - The fundamental Theory of Molecular Science" had set out to provide a concise, comprehensive, and complete presentation of this theory.This second edition expands on some of the latest developments in this fascinating field. The text retains its clear and consistent style, allowing for a readily accessible overview of the complex topic. It is also self-contained, building on the fundamental equations and providing the mathematical background necessary. While some parts of the text have been restructured for the sake of clarity a significant amount of new content has also been added. This includes, for example, an in-depth discussion of the Brown-Ravenhall disease, of spin in current-density functional theory, and of exact two-component methods and its local variants.

A strength of the first edition of this textbook was its list of almost 1000 references to the original research literature, which has made it a valuable reference also for experts in the field. In the second edition, more than 100 additional key references have been added - most of them considering the recent developments in the field.

Thus, the book is a must-have for everyone entering the field, as well as for experienced researchers searching for a consistent review.

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 Contributions

Local Exact Decoupling

Efficient Calculation of Spin-Orbit Coupling Effects

Relativistic Effective Core Potentials

EXTERNAL ELECTROMAGNETIC FIELDS AND MOLECULAR PROPERTIES

Four-Component Perturbation and Response Theory

Reduction to Two-Component Form and Picture Change Artifacts

Douglas-Kroll-Hess Property Transformations

Magnetic Fields in Resonance Spectroscopies

Electric Field Gradient and Nuclear Quadrupole Moment

Parity Violation and Electro-Weak Chemistry

RELATIVISTIC EFFECTS IN CHEMISTRY

Effects in Atoms with Consequences for Chemical Bonding

Is Spin a Relativistic Effect?

Z-Dependence of Relativistic Effects: Perturbation Theory

Potential Energy Surfaces and Spectroscopic Parameters

Lanthanides and Actinides

Electron Density of Transition Metal Complexes

Relativistic Quantum Chemical Calculations in Practice

APPENDIX

Vector and Tensor Calculus

Kinetic Energy in Generalized Coordinates

Technical Proofs for Special Relativity

Relations for Pauli and Dirac Matrices

Fourier Transformations

Gordon Decomposition

Discretization and Quadrature Schemes

List of Abbreviations and Acronyms

List of Symbols

Units and Dimensions

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 Contributions

Local Exact Decoupling

Efficient Calculation of Spin-Orbit Coupling Effects

Relativistic Effective Core Potentials

EXTERNAL ELECTROMAGNETIC FIELDS AND MOLECULAR PROPERTIES

Four-Component Perturbation and Response Theory

Reduction to Two-Component Form and Picture Change Artifacts

Douglas-Kroll-Hess Property Transformations

Magnetic Fields in Resonance Spectroscopies

Electric Field Gradient and Nuclear Quadrupole Moment

Parity Violation and Electro-Weak Chemistry

RELATIVISTIC EFFECTS IN CHEMISTRY

Effects in Atoms with Consequences for Chemical Bonding

Is Spin a Relativistic Effect?

Z-Dependence of Relativistic Effects: Perturbation Theory

Potential Energy Surfaces and Spectroscopic Parameters

Lanthanides and Actinides

Electron Density of Transition Metal Complexes

Relativistic Quantum Chemical Calculations in Practice

APPENDIX

Vector and Tensor Calculus

Kinetic Energy in Generalized Coordinates

Technical Proofs for Special Relativity

Relations for Pauli and Dirac Matrices

Fourier Transformations

Gordon Decomposition

Discretization and Quadrature Schemes

List of Abbreviations and Acronyms

List of Symbols

Units and Dimensions

The new edition is updated to include the latest developments in the field. Some content of the chapters has been restructured for the sake of clarity.

A significant amount of completely new content including:

- extension of the discussion of continuum dissolution

- extended discussion of spin in relativistic density functional theory

- exact-decoupling procedures with a focus on X2C

- local approximations to exact-decoupling methods

- more than 100 new key references