Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science

Cover
Title Page
Contents
Preface
Preface to the Second Edition
Preface to the First Edition
1 Introduction
1.1 Philosophy of this Book
1.2 Short Reader’s Guide
1.3 Notational Conventions and Choice of Units
Part I: FUNDAMENTALS
2 Elements of Classical Mechanics and Electrodynamics
2.1 Elementary Newtonian Mechanics
2.1.1 Newton’s Laws of Motion
2.1.2 Galilean Transformations
2.1.2.1 Relativity Principle of Galilei
2.1.2.2 General Galilean Transformations and Boosts
2.1.2.3 Galilei Covariance of Newton’s Laws
2.1.2.4 Scalars, Vectors, and Tensors in Three-Dimensional Space
2.1.3 Basic Conservation Laws for One Particle in Three Dimensions
2.1.4 Collection of N Particles
2.2 Lagrangian Formulation
2.2.1 Generalized Coordinates and Constraints
2.2.2 Hamiltonian Principle and Euler–Lagrange Equations
2.2.2.1 Discrete System of Point Particles
2.2.2.2 Example: Planar Pendulum
2.2.2.3 Continuous Systems of Fields
2.2.3 Symmetries and Conservation Laws
2.2.3.1 Gauge Transformations of the Lagrangian
2.2.3.2 Energy and Momentum Conservation
2.2.3.3 General Space–Time Symmetries
2.3 Hamiltonian Mechanics
2.3.1 Hamiltonian Principle and Canonical Equations
2.3.1.1 System of Point Particles
2.3.1.2 Continuous System of Fields
2.3.2 Poisson Brackets and Conservation Laws
2.3.3 Canonical Transformations
2.4 Elementary Electrodynamics
2.4.1 Maxwell’s Equations
2.4.2 Energy and Momentum of the Electromagnetic Field
2.4.2.1 Energy and Poynting’s Theorem
2.4.2.2 Momentum and Maxwell’s Stress Tensor
2.4.2.3 Angular Momentum
2.4.3 Plane Electromagnetic Waves in Vacuum
2.4.4 Potentials and Gauge Symmetry
2.4.4.1 Lorenz Gauge
2.4.4.2 Coulomb Gauge
2.4.4.3 Retarded Potentials
2.4.5 Survey of Electro– and Magnetostatics
2.4.5.1 Electrostatics
2.4.5.2 Magnetostatics
2.4.6 One Classical Particle Subject to Electromagnetic Fields
2.4.7 Interaction of Two Moving Charged Particles
Further Reading
3 Concepts of Special Relativity
3.1 Einstein’s Relativity Principle and Lorentz Transformations
3.1.1 Deficiencies of Newtonian Mechanics
3.1.2 Relativity Principle of Einstein
3.1.3 Lorentz Transformations
3.1.3.1 Definition of General Lorentz Transformations
3.1.3.2 Classification of Lorentz Transformations
3.1.3.3 Inverse Lorentz Transformation
3.1.4 Scalars, Vectors, and Tensors in Minkowski Space
3.1.4.1 Contraand Covariant Components
3.1.4.2 Transformation Properties of Scalars, Vectors, and Tensors
3.2 Kinematic Effects in Special Relativity
3.2.1 Explicit Form of Special Lorentz Transformations
3.2.1.1 Lorentz Boost in One Direction
3.2.1.2 General Lorentz Boost
3.2.2 Length Contraction, Time Dilation, and Proper Time
3.2.2.1 Length Contraction
3.2.2.2 Time Dilation
3.2.2.3 Proper Time
3.2.3 Addition of Velocities
3.2.3.1 Parallel Velocities
3.2.3.2 General Velocities
3.3 Relativistic Dynamics
3.3.1 Elementary Relativistic Dynamics
3.3.1.1 Trajectories and Relativistic Velocity
3.3.1.2 Relativistic Momentum and Energy
3.3.1.3 Energy–Momentum Relation
3.3.2 Equation of Motion
3.3.2.1 Minkowski Force
3.3.2.2 Lorentz Force
3.3.3 Lagrangian and Hamiltonian Formulation
3.3.3.1 Relativistic Free Particle
3.3.3.2 Relativistic Particle Subject to External Electromagnetic Fields
3.4 Covariant Electrodynamics
3.4.1 Ingredients
3.4.1.1 Charge–Current Density
3.4.1.2 Gauge Field
3.4.1.3 Field Strength Tensor
3.4.2 Transformation of Electromagnetic Fields
3.4.3 Lagrangian Formulation and Equations of Motion
3.4.3.1 Lagrangian for the Electrodynamic Field
3.4.3.2 Minimal Coupling
3.4.3.3 Euler–Lagrange Equations
3.5 Interaction of Two Moving Charged Particles
3.5.1 Scalar and Vector Potentials of a Charge at Rest
3.5.2 Retardation from Lorentz Transformation
3.5.3 General Expression for the Interaction Energy
3.5.4 Interaction Energy at One Instant of Time
3.5.4.1 Taylor Expansion of Potential and Energy
3.5.4.2 Variables of Charge Two at Time of Charge One
3.5.4.3 Final Expansion of the Interaction Energy
3.5.4.4 Expansion of the Retardation Time
3.5.4.5 General Darwin Interaction Energy
3.5.5 Symmetrized Darwin Interaction Energy
Further Reading
4 Basics of Quantum Mechanics
4.1 The Quantum Mechanical State
4.4.1 Bracket Notation
4.1.2 Expansion in a Complete Basis Set
4.1.3 Born Interpretation
4.1.4 State Vectors in Hilbert Space
4.2 The Equation of Motion
4.2.1 Restrictions on the Fundamental Quantum Mechanical Equation
4.2.2 Time Evolution and Probabilistic Character
4.2.3 Stationary States
4.3 Observables
4.3.1 Expectation Values
4.3.2 Hermitean Operators
4.3.3 Unitary Transformations
4.3.4 Heisenberg Equation of Motion
4.3.5 Hamiltonian in Nonrelativistic Quantum Theory
4.3.6 Commutation Relations for Position and Momentum Operators
4.3.7 The Schrödinger Velocity Operator
4.3.8 Ehrenfest and Hellmann–Feynman Theorems
4.3.9 Current Density and Continuity Equation
4.4 Angular Momentum and Rotations
4.4.1 Classical Angular Momentum
4.4.2 Orbital Angular Momentum
4.4.3 Coupling of Angular Momenta
4.4.4 Spin
4.4.5 Coupling of Orbital and Spin Angular Momenta
4.5 Pauli Antisymmetry Principle
Further Reading
Part II: DIRAC'S THEORY OF THE ELECTRON
5 Relativistic Theory of the Electron
5.1 Correspondence Principle and Klein–Gordon Equation
5.1.1 Classical Energy Expression and First Hints from the Correspondence Principle
5.1.2 Solutions of the Klein–Gordon Equation
5.1.3 The Klein–Gordon Density Distribution
5.2 Derivation of the Dirac Equation for a Freely Moving Electron
5.2.1 Relation to the Klein–Gordon Equation
5.2.2 Explicit Expressions for the Dirac Parameters
5.2.3 Continuity Equation and Definition of the 4-Current
5.2.4 Lorentz Covariance of the Field-Free Dirac Equation
5.2.4.1 Covariant Form
5.2.4.2 Lorentz Transformation of the Dirac Spinor
5.2.4.3 Higher Level of Abstraction and Clifford Algebra
5.3 Solution of the Free-Electron Dirac Equation
5.3.1 Particle at Rest
5.3.2 Freely Moving Particle
5.3.3 The Dirac Velocity Operator
5.4 Dirac Electron in External Electromagnetic Potentials
5.4.1 Kinematic Momentum
5.4.2 Electromagnetic Interaction Energy Operator
5.4.3 Nonrelativistic Limit and Pauli Equation
5.5 Interpretation of Negative-Energy States: Dirac’s Hole Theory
Further Reading
6 The Dirac Hydrogen Atom
6.1 Separation of Electronic Motion in a Nuclear Central Field
6.2 Schrödinger Hydrogen Atom
6.3 Total Angular Momentum
6.4 Separation of Angular Coordinates in the Dirac Hamiltonian
6.4.1 Spin–Orbit Coupling
6.4.2 Relativistic Azimuthal Quantum Number Analog
6.4.3 Four-Dimensional Generalization
6.4.4 Ansatz for the Spinor
6.5 Radial Dirac Equation for Hydrogen-Like Atoms
6.5.1 Radial Functions and Orthonormality
6.5.2 Radial Eigenvalue Equations
6.5.3 Solution of the Coupled Dirac Radial Equations
6.5.4 Energy Eigenvalue, Quantization and the Principal Quantum Number
6.5.5 The Four-Component Ground State Wave Function
6.6 The Nonrelativistic Limit
6.7 Choice of the Energy Reference and Matching Energy Scales
6.8 Wave Functions and Energy Eigenvalues in the Coulomb Potential
6.8.1 Features of Dirac Radial Functions
6.8.2 Spectrum of Dirac Hydrogen-like Atoms with Coulombic Potential
6.8.3 Radial Density and Expectation Values
6.9 Finite Nuclear Size Effects
6.9.1 Consequences of the Nuclear Charge Distribution
6.9.2 Spinors in External Scalar Potentials of Varying Depth
6.10 Momentum Space Representation
Further Reading
Part III: FOUR-COMPONENT MANY-ELECTRON THEORY
7 Quantum Electrodynamics
7.1 Elementary Quantities and Notation
7.1.1 Lagrangian for Electromagnetic Interactions
7.1.2 Lorentz and Gauge Symmetry and Equations of Motion
7.2 Classical Hamiltonian Description
7.2.1 Exact Hamiltonian
7.2.2 The Electron–Electron Interaction
7.3 Second-Quantized Field-Theoretical Formulation
7.4 Implications for the Description of Atoms and Molecules
Further reading
8 First-Quantized Dirac-Based Many-Electron Theory
8.1 Two-Electron Systems and the Breit Equation
8.1.1 Dirac Equation Generalized for Two Bound-State Electrons
8.1.2 The Gaunt Operator for Unretarded Interactions
8.1.3 The Breit Operator for Retarded Interactions
8.1.4 Exact Retarded Electromagnetic Interaction Energy
8.1.5 Breit Interaction from Quantum Electrodynamics
8.2 Quasi-Relativistic Many-Particle Hamiltonians
8.2.1 Nonrelativistic Hamiltonian for a Molecular System
8.2.2 First-Quantized Relativistic Many-Particle Hamiltonian
8.2.3 Pathologies of the First-Quantized Formulation
8.2.3.1 Continuum Dissolution
8.2.3.2 Projection and No-Pair Hamiltonians
8.2.4 Local Model Potentials for One-Particle QED Corrections
8.3 Separation of Nuclear and Electronic Degrees of Freedom: The Born–Oppenheimer Approximation
8.4 Tensor Structure of the Many-Electron Hamiltonian and Wave Function
8.5 Approximations to the Many-Electron Wave Function
8.5.1 The Independent-Particle Model
8.5.2 Configuration Interaction
8.5.3 Detour: Explicitly Correlated Wave Functions
8.5.4 Orthonormality Constraints and Total Energy Expressions
8.6 Second Quantization for the Many-Electron Hamiltonian
8.6.1 Creation and Annihilation Operators
8.6.2 Reduction of Determinantal Matrix Elements to Matrix Elements Over Spinors
8.6.3 Many-Electron Hamiltonian and Energy
8.6.4 Fock Space and Occupation Number Vectors
8.6.5 Fermions and Bosons
8.7 Derivation of Effective One-Particle Equations
8.7.1 Avoiding Variational Collapse: The Minimax Principle
8.7.2 Variation of the Energy Expression
8.7.2.1 Variational Conditions
8.7.2.2 The CI Eigenvalue Problem
8.7.3 Self-Consistent Field Equations
8.7.4 Dirac–Hartree–Fock Equations
8.7.5 The Relativistic Self-Consistent Field
8.8 Relativistic Density Functional Theory
8.8.1 Electronic Charge and Current Densities for Many Electrons
8.8.2 Current-Density Functional Theory
8.8.3 The Four-Component Kohn–Sham Model
8.8.4 Electron Density and Spin Density in Relativistic DFT
8.8.5 Relativistic Spin-DFT
8.8.6 Noncollinear Approaches and Collinear Approximations
8.8.7 Relation to the Spin Density
8.9 Completion: The Coupled-Cluster Expansion
Further Reading
9 Many-Electron Atoms
9.1 Transformation of the Many-Electron Hamiltonian to Polar Coordinates
9.1.1 Comment on Units
9.1.2 Coulomb Interaction in Polar Coordinates
9.1.3 Breit Interaction in Polar Coordinates
9.1.4 Atomic Many-Electron Hamiltonian
9.2 Atomic Many-Electron Wave Function and jj-Coupling
9.3 One- and Two-Electron Integrals in Spherical Symmetry
9.3.1 One-Electron Integrals
9.3.2 Electron–Electron Coulomb Interaction
9.3.3 Electron–Electron Frequency-Independent Breit Interaction
9.3.4 Calculation of Potential Functions
9.3.4.1 First-Order Differential Equations
9.3.4.2 Derivation of the Radial Poisson Equation
9.3.4.3 Breit Potential Functions
9.4 Total Expectation Values
9.4.1 General Expression for the Electronic Energy
9.4.2 Breit Contribution to the Total Energy
9.4.3 Dirac–Hartree–Fock Total Energy of Closed-Shell Atoms
9.5 General Self-Consistent-Field Equations and Atomic Spinors
9.5.1 Dirac–Hartree–Fock Equations
9.5.2 Comparison of Atomic Hartree–Fock and Dirac–Hartree–Fock Theories
9.5.3 Relativistic and Nonrelativistic Electron Densities
9.6 Analysis of Radial Functions and Potentials at Short and Long Distances
9.6.1 Short-Range Behavior of Atomic Spinors
9.6.1.1 Cusp-Analogous Condition at the Nucleus
9.6.1.2 Coulomb Potential Functions
9.6.2 Origin Behavior of Interaction Potentials
9.6.3 Short-Range Electron–Electron Coulomb Interaction
9.6.4 Exchange Interaction at the Origin
9.6.5 Total Electron–Electron Interaction at the Nucleus
9.6.6 Asymptotic Behavior of the Interaction Potentials
9.7 Numerical Discretization and Solution Techniques
9.7.1 Variable Transformations
9.7.2 Explicit Transformation Functions
9.7.2.1 The Logarithmic Grid
9.7.2.2 The Rational Grid
9.7.3 Transformed Equations
9.7.3.1 SCF Equations
9.7.3.2 Regular Solution Functions for Point-Nucleus Case
9.7.3.3 Poisson Equations
9.7.4 Numerical Solution of Matrix Equations
9.7.5 Discretization and Solution of the SCF equations
9.7.6 Discretization and Solution of the Poisson Equations
9.7.7 Extrapolation Techniques and Other Technical Issues
9.8 Results for Total Energies and Radial Functions
9.8.1 Electronic Configurations and the Aufbau Principle
9.8.2 Radial Functions
9.8.3 Effect of the Breit Interaction on Energies and Spinors
9.8.4 Effect of the Nuclear Charge Distribution on Total Energies
Further Reading
10 General Molecules and Molecular Aggregates
10.1 Basis Set Expansion of Molecular Spinors
10.1.1 Kinetic Balance
10.1.2 Special Choices of Basis Functions
10.2 Dirac–Hartree–Fock Electronic Energy in Basis Set Representation
10.3 Molecular Oneand Two-Electron Integrals
10.4 Dirac–Hartree–Fock–Roothaan Matrix Equations
10.4.1 Two Possible Routes for the Derivation
10.4.2 Treatment of Negative-Energy States
10.4.3 Four-Component DFT
10.4.4 Symmetry
10.4.5 Kramers’ Time Reversal Symmetry
10.4.6 Double Groups
10.5 Analytic Gradients
10.6 Post-Hartree–Fock Methods
Further Reading
Part IV: TWO-COMPONENT HAMILTONIANS
11 Decoupling the Negative-Energy States
11.1 Relation of Large and Small Components in One-Electron Equations
11.1.1 Restriction on the Potential Energy Operator
11.1.2 The X-Operator Formalism
11.1.3 Free-Particle Solutions
11.2 Closed-Form Unitary Transformation of the Dirac Hamiltonian
11.3 The Free-Particle Foldy–Wouthuysen Transformation
11.4 General Parametrization of Unitary Transformations
11.4.1 Closed-Form Parametrizations
11.4.2 Exactly Unitary Series Expansions
11.4.3 Approximate Unitary and Truncated Optimum Transformations
11.5 Foldy–Wouthuysen Expansion in Powers of
11.5.1 The Lowest-Order Foldy–Wouthuysen Transformation
11.5.2 Second-Order Foldy–Wouthuysen Operator: Pauli Hamiltonian
11.5.3 Higher-Order Foldy–Wouthuysen Transformations and Their Pathologies
11.6 The Infinite-Order Two-Component Two-Step Protocol
11.7 Toward Well-Defined Analytic Block-Diagonal Hamiltonians
12 Douglas–Kroll–Hess Theory
12.1 Sequential Unitary Decoupling Transformations
12.2 Explicit Form of the DKH Hamiltonians
12.2.1 First Unitary Transformation
12.2.2 Second Unitary Transformation
12.2.3 Third Unitary Transformation
12.3 Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH Method
12.3.1 Convergence of DKH Energies and Variational Stability
12.3.2 Infinite-Order Protocol
12.3.3 Coefficient Dependence
12.3.4 Explicit Expressions of the Positive-Energy Hamiltonians
12.3.5 Additional Peculiarities of DKH Theory
12.3.5.1 Two-Component Electron Density Distribution
12.3.5.2 Transformation in the Presence of Off-Diagonal Potential Operators
12.3.5.3 Nonrelativistic Limit of Approximate Two-Component Hamiltonians
12.3.5.4 Rigorous Analytic Results
12.4 Many-Electron DKH Hamiltonians
12.4.1 DKH Transformation of One-Electron Terms
12.4.2 DKH Transformation of Two-Electron Terms
12.5 Computational Aspects of DKH Calculations
12.5.1 Exploiting a Resolution of the Identity
12.5.2 Advantages of Scalar-Relativistic DKH Hamiltonians
12.5.3 Efficient Approximations for the Transformation of Complicated Terms
12.5.3.1 Spin–Orbit Operators
12.5.3.2 Two-Electron Terms
12.5.3.3 One-Electron Basis Sets
12.5.4 DKH Gradients
13 Elimination Techniques
13.1 Naive Reduction: Pauli Elimination
13.2 Breit–Pauli Theory
13.2.1 Foldy–Wouthuysen Transformation of the Breit Equation
13.2.2 Transformation of the Two-Electron Interaction
13.2.2.1 All-Even Operators
13.2.2.2 Transformed Coulomb Contribution
13.2.2.3 Transformed Breit Contribution
13.2.3 The Breit–Pauli Hamiltonian
13.3 The Cowan–Griffin and Wood–Boring Approaches
13.4 Elimination for Different Representations of Dirac Matrices
13.5 Regular Approximations
Part V: CHEMISTRY WITH RELATIVISTIC HAMILTONIANS
14 Special Computational Techniques
14.1 From the Modified Dirac Equation to Exact-Two-Component Methods
14.1.1 Normalized Elimination of the Small Component
14.1.2 Exact-Decoupling Methods
14.1.2.1 The One-Step Solution: X2C
14.1.2.2 Two-Step Transformation: BSS
14.1.2.3 Expansion of the Transformation: DKH
14.1.3 Approximations in Many-Electron Calculations
14.1.3.1 The Cumbersome Two-Electron Terms
14.1.3.2 Scalar-Relativistic Approximations
14.1.4 Numerical Comparison
14.2 Locality of Relativistic Contributions
14.3 Local Exact Decoupling
14.3.1 Atomic Unitary Transformation
14.3.2 Local Decomposition of the X-Operator
14.3.3 Local Approximations to the Exact-Decoupling Transformation
14.3.4 Numerical Comparison
14.4 Efficient Calculation of Spin–Orbit Coupling Effects
14.5 Relativistic Effective Core Potentials
15 External Electromagnetic Fields and Molecular Properties
15.1 Four-Component Perturbation and Response Theory
15.1.1 Variational Treatment
15.1.2 Perturbation Theory
15.1.3 The Dirac-Like One-Electron Picture
15.1.4 Two Types of Properties
15.2 Reduction to Two-Component Form and Picture Change Artifacts
15.2.1 Origin of Picture Change Errors
15.2.2 Picture-Change-Free Transformed Properties
15.2.3 Free-Particle Foldy–Wouthuysen Transformation of Properties
15.2.4 Breit–Pauli Hamiltonian with External Electromagnetic Fields
15.3 Douglas–Kroll–Hess Property Transformation
15.3.1 The Variational DKH Scheme for Perturbing Potentials
15.3.2 Most General Electromagnetic Property
15.3.3 Perturbative Approach
15.3.3.1 Direct DKH Transformation of First-Order Energy
15.3.3.2 Explicit Expressions up to Third Order in the Unperturbed Potential
15.3.3.3 Alternative Transformation for First-Order Energy
15.3.4 Automated Generation of DKH Property Operators
15.3.5 Consequences for the Electron Density Distribution
15.3.6 DKH Perturbation Theory with Magnetic Fields
15.4 Magnetic Fields in Resonance Spectroscopies
15.4.1 The Notorious Diamagnetic Term
15.4.2 Gauge Origin and London Orbitals
15.4.3 Explicit Form of Perturbation Operators
15.4.4 Spin Hamiltonian
15.5 Electric Field Gradient and Nuclear Quadrupole Moment
15.6 Parity Violation and Electro-Weak Chemistry
16 Relativistic Effects in Chemistry
16.1 Effects in Atoms with Consequences for Chemical Bonding
16.2 Is Spin a Relativistic Effect?
16.3 Z-Dependence of Relativistic Effects: Perturbation Theory
16.4 Potential Energy Surfaces and Spectroscopic Parameters
16.4.1 Dihydrogen
16.4.2 Thallium Hydride
16.4.3 The Gold Dimer
16.4.4 Tin Oxide and Cesium Hydride
16.5 Lanthanides and Actinides
16.5.1 Lanthanide and Actinide Contraction
16.5.2 Electronic Spectra of Actinide Compounds
16.6 Electron Density of Transition Metal Complexes
16.7 Relativistic Quantum Chemical Calculations in Practice
Appendices
A Vector and Tensor Calculus
A.1 Three-Dimensional Expressions
A.1.1 Algebraic Vector and Tensor Operations
A.1.2 Differential Vector Operations
A.1.3 Integral Theorems and Distributions
A.1.4 Total Differentials and Time Derivatives
A.2 Four-Dimensional Expressions
A.2.1 Algebraic Vector and Tensor Operations
A.2.2 Differential Vector Operations
B Kinetic Energy in Generalized Coordinates
C Technical Proofs for Special Relativity
C.1 Invariance of Space-Time Interval
C.2 Uniqueness of Lorentz Transformations
C.3 Useful Trigonometric and Hyperbolic Formulae for Lorentz Transformations
D Relations for Pauli and Dirac Matrices
D.1 Pauli Spin Matrices
D.2 Dirac’s Relation
D.2.1 Momenta and Vector Fields
D.2.2 Four-Dimensional Generalization
E Fourier Transformations
E.1 Definition and General Properties
E.2 Fourier Transformation of the Coulomb Potential
F Gordon Decomposition
F.1 One-Electron Case
F.2 Many-Electron Case
G Discretization and Quadrature Schemes
G.1 Numerov Approach toward Second-Order Differential Equations
G.2 Numerov Approach for First-Order Differential Equations
G.3 Simpson’s Quadrature Formula
G.4 Bickley’s Central-Difference Formulae
H List of Abbreviations and Acronyms
I List of Symbols
J Units and Dimensions
References
Index