Relativistic Quantum Chemistry

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 xxi1 Introduction 11.1 Philosophy of this Book 11.2 Short Reader’s Guide 41.3 Notational Conventions and Choice of Units 6Part I — Fundamentals 92 Elements of Classical Mechanics and Electrodynamics 112.1 Elementary Newtonian Mechanics 112.1.1 Newton’s Laws of Motion 112.1.2 Galilean Transformations 142.1.2.1 Relativity Principle of Galilei 142.1.2.2 General Galilean Transformations and Boosts 162.1.2.3 Galilei Covariance of Newton’s Laws 172.1.2.4 Scalars, Vectors, Tensors in 3-Dimensional Space 172.1.3 Conservation Laws for One Particle in Three Dimensions 202.1.4 Collection of N Particles 212.2 Lagrangian Formulation 222.2.1 Generalized Coordinates and Constraints 222.2.2 Hamiltonian Principle and Euler–Lagrange Equations 242.2.2.1 Discrete System of Point Particles 242.2.2.2 Example: Planar Pendulum 262.2.2.3 Continuous Systems of Fields 272.2.3 Symmetries and Conservation Laws 282.2.3.1 Gauge Transformations of the Lagrangian 282.2.3.2 Energy and Momentum Conservation 292.2.3.3 General Space–Time Symmetries 302.3 Hamiltonian Mechanics 312.3.1 Hamiltonian Principle and Canonical Equations 312.3.1.1 System of Point Particles 312.3.1.2 Continuous System of Fields 322.3.2 Poisson Brackets and Conservation Laws 332.3.3 Canonical Transformations 342.4 Elementary Electrodynamics 352.4.1 Maxwell’s Equations 362.4.2 Energy and Momentum of the Electromagnetic Field 382.4.2.1 Energy and Poynting’s Theorem 382.4.2.2 Momentum and Maxwell’s Stress Tensor 392.4.2.3 Angular Momentum 402.4.3 Plane Electromagnetic Waves in Vacuum 402.4.4 Potentials and Gauge Symmetry 422.4.4.1 Lorenz Gauge 442.4.4.2 Coulomb Gauge 442.4.4.3 Retarded Potentials 452.4.5 Survey of Electro– and Magnetostatics 452.4.5.1 Electrostatics 452.4.5.2 Magnetostatics 472.4.6 One Classical Particle Subject to Electromagnetic Fields 472.4.7 Interaction of Two Moving Charged Particles 503 Concepts of Special Relativity 533.1 Einstein’s Relativity Principle and Lorentz Transformations 533.1.1 Deficiencies of Newtonian Mechanics 533.1.2 Relativity Principle of Einstein 553.1.3 Lorentz Transformations 583.1.3.1 Definition of General Lorentz Transformations 583.1.3.2 Classification of Lorentz Transformations 593.1.3.3 Inverse Lorentz Transformation 603.1.4 Scalars, Vectors, and Tensors in Minkowski Space 623.1.4.1 Contra- and Covariant Components 623.1.4.2 Properties of Scalars, Vectors, and Tensors 633.2 Kinematic Effects in Special Relativity 673.2.1 Explicit Form of Special Lorentz Transformations 673.2.1.1 Lorentz Boost in One Direction 673.2.1.2 General Lorentz Boost 703.2.2 Length Contraction, Time Dilation, and Proper Time 723.2.2.1 Length Contraction 723.2.2.2 Time Dilation 733.2.2.3 Proper Time 743.2.3 Addition of Velocities 753.2.3.1 Parallel Velocities 753.2.3.2 General Velocities 773.3 Relativistic Dynamics 783.3.1 Elementary Relativistic Dynamics 793.3.1.1 Trajectories and Relativistic Velocity 793.3.1.2 Relativistic Momentum and Energy 793.3.1.3 Energy–Momentum Relation 813.3.2 Equation of Motion 833.3.2.1 Minkowski Force 833.3.2.2 Lorentz Force 853.3.3 Lagrangian and Hamiltonian Formulation 863.3.3.1 Relativistic Free Particle 863.3.3.2 Particle in Electromagnetic Fields 893.4 Covariant Electrodynamics 903.4.1 Ingredients 913.4.1.1 Charge–Current Density 913.4.1.2 Gauge Field 913.4.1.3 Field Strength Tensor 923.4.2 Transformation of Electromagnetic Fields 953.4.3 Lagrangian Formulation and Equations of Motion 963.4.3.1 Lagrangian for the Electrodynamic Field 963.4.3.2 Minimal Coupling 973.4.3.3 Euler–Lagrange Equations 993.5 Interaction of Two Moving Charged Particles 1013.5.1 Scalar and Vector Potentials of a Charge at Rest 1023.5.2 Retardation from Lorentz Transformation 1043.5.3 General Expression for the Interaction Energy 1053.5.4 Interaction Energy at One Instant of Time 1053.5.4.1 Taylor Expansion of Potential and Energy 1063.5.4.2 Variables of Charge Two at Time of Charge One 1073.5.4.3 Final Expansion of the Interaction Energy 1083.5.4.4 Expansion of the Retardation Time 1103.5.4.5 General Darwin Interaction Energy 1103.5.5 Symmetrized Darwin Interaction Energy 1124 Basics of Quantum Mechanics 1174.1 The Quantum Mechanical State 1184.1.1 Bracket Notation 1184.1.2 Expansion in a Complete Basis Set 1194.1.3 Born Interpretation 1194.1.4 State Vectors in Hilbert Space 1214.2 The Equation of Motion 1224.2.1 Restrictions on the Fundamental Quantum Mechanical Equation 1224.2.2 Time Evolution and Probabilistic Character 1234.2.3 Stationary States 1234.3 Observables 1244.3.1 Expectation Values 1244.3.2 Hermitean Operators 1254.3.3 Unitary Transformations 1264.3.4 Heisenberg Equation of Motion 1274.3.5 Hamiltonian in Nonrelativistic Quantum Theory 1294.3.6 Commutation Relations for Position and Momentum Operators 1314.3.7 The Schrödinger Velocity Operator 1324.3.8 Ehrenfest and Hellmann–Feynman Theorems 1334.3.9 Current Density and Continuity Equation 1354.4 Angular Momentum and Rotations 1394.4.1 Classical Angular Momentum 1394.4.2 Orbital Angular Momentum 1404.4.3 Coupling of Angular Momenta 1454.4.4 Spin 1474.4.5 Coupling of Orbital and Spin Angular Momenta 1494.5 Pauli Antisymmetry Principle 155Part II — Dirac’s Theory of the Electron 1595 Relativistic Theory of the Electron 1615.1 Correspondence Principle and Klein–Gordon Equation 1615.1.1 Classical Energy Expression and First Hints from the Correspondence Principle 1615.1.2 Solutions of the Klein–Gordon Equation 1635.1.3 The Klein–Gordon Density Distribution 1645.2 Derivation of the Dirac Equation for a Freely Moving Electron 1665.2.1 Relation to the Klein–Gordon Equation 1665.2.2 Explicit Expressions for the Dirac Parameters 1675.2.3 Continuity Equation and Definition of the 4-Current 1695.2.4 Lorentz Covariance of the Field-Free Dirac Equation 1705.2.4.1 Covariant Form 1705.2.4.2 Lorentz Transformation of the Dirac Spinor 1715.2.4.3 Higher Level of Abstraction and Clifford Algebra 1725.3 Solution of the Free-Electron Dirac Equation 1735.3.1 Particle at Rest 1735.3.2 Freely Moving Particle 1755.3.3 The Dirac Velocity Operator 1795.4 Dirac Electron in External Electromagnetic Potentials 1815.4.1 Kinematic Momentum 1845.4.2 Electromagnetic Interaction Energy Operator 1845.4.3 Nonrelativistic Limit and Pauli Equation 1855.5 Interpretation of Negative-Energy States: Dirac’s Hole Theory 1876 The Dirac Hydrogen Atom 1936.1 Separation of Electronic Motion in a Nuclear Central Field 1936.2 Schrödinger Hydrogen Atom 1976.3 Total Angular Momentum 1996.4 Separation of Angular Coordinates in the Dirac Hamiltonian 2006.4.1 Spin–Orbit Coupling 2006.4.2 Relativistic Azimuthal Quantum Number Analog 2016.4.3 Four-Dimensional Generalization 2036.4.4 Ansatz for the Spinor 2046.5 Radial Dirac Equation for Hydrogen-Like Atoms 2046.5.1 Radial Functions and Orthonormality 2056.5.2 Radial Eigenvalue Equations 2066.5.3 Solution of the Coupled Dirac Radial Equations 2076.5.4 Energy Eigenvalue, Quantization and the Principal Quantum Number 2136.5.5 The Four-Component Ground State Wave Function 2156.6 The Nonrelativistic Limit 2166.7 Choice of the Energy Reference and Matching Energy Scales 2186.8 Wave Functions and Energy Eigenvalues in the Coulomb Potential 2196.8.1 Features of Dirac Radial Functions 2196.8.2 Spectrum of Dirac Hydrogen-like Atoms with Coulombic Potential 2216.8.3 Radial Density and Expectation Values 2236.9 Finite Nuclear Size Effects 2256.9.1 Consequences of the Nuclear Charge Distribution 2276.9.2 Spinors in External Scalar Potentials of Varying Depth 2296.10 Momentum Space Representation 233Part III — Four-Component Many-Electron Theory 2357 Quantum Electrodynamics 2377.1 Elementary Quantities and Notation 2377.1.1 Lagrangian for Electromagnetic Interactions 2377.1.2 Lorentz and Gauge Symmetry and Equations of Motion 2387.2 Classical Hamiltonian Description 2407.2.1 Exact Hamiltonian 2407.2.2 The Electron–Electron Interaction 2417.3 Second-Quantized Field-Theoretical Formulation 2437.4 Implications for the Description of Atoms and Molecules 2468 First-Quantized Dirac-Based Many-Electron Theory 2498.1 Two-Electron Systems and the Breit Equation 2508.1.1 Dirac Equation Generalized for Two Bound-State Electrons 2518.1.2 The Gaunt Operator for Unretarded Interactions 2538.1.3 The Breit Operator for Retarded Interactions 2568.1.4 Exact Retarded Electromagnetic Interaction Energy 2608.1.5 Breit Interaction from Quantum Electrodynamics 2668.2 Quasi-Relativistic Many-Particle Hamiltonians 2708.2.1 Nonrelativistic Hamiltonian for a Molecular System 2708.2.2 First-Quantized Relativistic Many-Particle Hamiltonian 2728.2.3 Pathologies of the First-Quantized Formulation 2748.2.3.1 Continuum Dissolution 2748.2.3.2 Projection and No-Pair Hamiltonians 2778.2.4 Local Model Potentials for One-Particle QED Corrections 2788.3 Born–Oppenheimer Approximation 2798.4 Tensor Structure of the Many-Electron Hamiltonian and Wave Function 2838.5 Approximations to the Many-Electron Wave Function 2858.5.1 The Independent-Particle Model 2868.5.2 Configuration Interaction 2878.5.3 Detour: Explicitly Correlated Wave Functions 2918.5.4 Orthonormality Constraints and Total Energy Expressions 2928.6 Second Quantization for the Many-Electron Hamiltonian 2968.6.1 Creation and Annihilation Operators 2968.6.2 Reduction of Determinantal Matrix Elements to Matrix Elements Over Spinors 2978.6.3 Many-Electron Hamiltonian and Energy 2998.6.4 Fock Space and Occupation Number Vectors 3008.6.5 Fermions and Bosons 3018.7 Derivation of Effective One-Particle Equations 3018.7.1 Avoiding Variational Collapse: The Minimax Principle 3028.7.2 Variation of the Energy Expression 3048.7.2.1 Variational Conditions 3048.7.2.2 The CI Eigenvalue Problem 3048.7.3 Self-Consistent Field Equations 3068.7.4 Dirac–Hartree–Fock Equations 3098.7.5 The Relativistic Self-Consistent Field 3128.8 Relativistic Density Functional Theory 3138.8.1 Electronic Charge and Current Densities for Many Electrons 3148.8.2 Current-Density Functional Theory 3178.8.3 The Four-Component Kohn–Sham Model 3188.8.4 Electron Density and Spin Density in Relativistic DFT 3208.8.5 Relativistic Spin-DFT 3228.8.6 Noncollinear Approaches and Collinear Approximations 3238.8.7 Relation to the Spin Density 3248.9 Completion: The Coupled-Cluster Expansion 3259 Many-Electron Atoms 3339.1 Transformation of the Many-Electron Hamiltonian to Polar Coordinates 3359.1.1 Comment on Units 3369.1.2 Coulomb Interaction in Polar Coordinates 3369.1.3 Breit Interaction in Polar Coordinates 3379.1.4 Atomic Many-Electron Hamiltonian 3419.2 Atomic Many-Electron Wave Function and jj-Coupling 3419.3 One- and Two-Electron Integrals in Spherical Symmetry 3449.3.1 One-Electron Integrals 3449.3.2 Electron–Electron Coulomb Interaction 3459.3.3 Electron–Electron Frequency-Independent Breit Interaction 3499.3.4 Calculation of Potential Functions 3519.3.4.1 First-Order Differential Equations 3529.3.4.2 Derivation of the Radial Poisson Equation 3539.3.4.3 Breit Potential Functions 3539.4 Total Expectation Values 3549.4.1 General Expression for the Electronic Energy 3549.4.2 Breit Contribution to the Total Energy 3569.4.3 Dirac–Hartree–Fock Total Energy of Closed-Shell Atoms 3579.5 General Self-Consistent-Field Equations and Atomic Spinors 3589.5.1 Dirac–Hartree–Fock Equations 3609.5.2 Comparison of Atomic Hartree–Fock and Dirac–Hartree–Fock Theories 3619.5.3 Relativistic and Nonrelativistic Electron Densities 3649.6 Analysis of Radial Functions and Potentials at Short and Long Distances 3669.6.1 Short-Range Behavior of Atomic Spinors 3679.6.1.1 Cusp-Analogous Condition at the Nucleus 3689.6.1.2 Coulomb Potential Functions 3699.6.2 Origin Behavior of Interaction Potentials 3709.6.3 Short-Range Electron–Electron Coulomb Interaction 3719.6.4 Exchange Interaction at the Origin 3729.6.5 Total Electron–Electron Interaction at the Nucleus 3769.6.6 Asymptotic Behavior of the Interaction Potentials 3789.7 Numerical Discretization and Solution Techniques 3799.7.1 Variable Transformations 3819.7.2 Explicit Transformation Functions 3829.7.2.1 The Logarithmic Grid 3829.7.2.2 The Rational Grid 3839.7.3 Transformed Equations 3839.7.3.1 SCF Equations 3849.7.3.2 Regular Solution Functions for Point-Nucleus Case 3849.7.3.3 Poisson Equations 3859.7.4 Numerical Solution of Matrix Equations 3869.7.5 Discretization and Solution of the SCF equations 3889.7.6 Discretization and Solution of the Poisson Equations 3919.7.7 Extrapolation Techniques and Other Technical Issues 3939.8 Results for Total Energies and Radial Functions 3959.8.1 Electronic Configurations and the Aufbau Principle 3979.8.2 Radial Functions 3979.8.3 Effect of the Breit Interaction on Energies and Spinors 3999.8.4 Effect of the Nuclear Charge Distribution on Total Energies 40010 General Molecules and Molecular Aggregates 40310.1 Basis Set Expansion of Molecular Spinors 40510.1.1 Kinetic Balance 40810.1.2 Special Choices of Basis Functions 40910.2 Dirac–Hartree–Fock Electronic Energy in Basis Set Representation 41310.3 Molecular One- and Two-Electron Integrals 41910.4 Dirac–Hartree–Fock–Roothaan Matrix Equations 41910.4.1 Two Possible Routes for the Derivation 42010.4.2 Treatment of Negative-Energy States 42110.4.3 Four-Component DFT 42210.4.4 Symmetry 42310.4.5 Kramers’ Time Reversal Symmetry 42310.4.6 Double Groups 42410.5 Analytic Gradients 42510.6 Post-Hartree–Fock Methods 428Part IV — Two-Component Hamiltonians 43311 Decoupling the Negative-Energy States 43511.1 Relation of Large and Small Components in One-Electron Equations 43511.1.1 Restriction on the Potential Energy Operator 43611.1.2 The X-Operator Formalism 43611.1.3 Free-Particle Solutions 43911.2 Closed-Form Unitary Transformation of the Dirac Hamiltonian 44011.3 The Free-Particle Foldy–Wouthuysen Transformation 44311.4 General Parametrization of Unitary Transformations 44711.4.1 Closed-Form Parametrizations 44811.4.2 Exactly Unitary Series Expansions 44911.4.3 Approximate Unitary and Truncated Optimum Transformations 45111.5 Foldy–Wouthuysen Expansion in Powers of 1/c 45411.5.1 The Lowest-Order Foldy–Wouthuysen Transformation 45411.5.2 Second-Order Foldy–Wouthuysen Operator: Pauli Hamiltonian 45811.5.3 Higher-Order Foldy–Wouthuysen Transformations and Their Pathologies 45911.6 The Infinite-Order Two-Component Two-Step Protocol 46211.7 Toward Well-Defined Analytic Block-Diagonal Hamiltonians 46512 Douglas–Kroll–Hess Theory 46912.1 Sequential Unitary Decoupling Transformations 46912.2 Explicit Form of the DKH Hamiltonians 47112.2.1 First Unitary Transformation 47112.2.2 Second Unitary Transformation 47212.2.3 Third Unitary Transformation 47512.3 Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH Method 47612.3.1 Convergence of DKH Energies and Variational Stability 47712.3.2 Infinite-Order Protocol 47912.3.3 Coefficient Dependence 48112.3.4 Explicit Expressions of the Positive-Energy Hamiltonians 48312.3.5 Additional Peculiarities of DKH Theory 48512.3.5.1 Two-Component Electron Density Distribution 48612.3.5.2 Off-Diagonal Potential Operators 48712.3.5.3 Nonrelativistic Limit 48712.3.5.4 Rigorous Analytic Results 48812.4 Many-Electron DKH Hamiltonians 48812.4.1 DKH Transformation of One-Electron Terms 48812.4.2 DKH Transformation of Two-Electron Terms 48912.5 Computational Aspects of DKH Calculations 49212.5.1 Exploiting a Resolution of the Identity 49412.5.2 Advantages of Scalar-Relativistic DKH Hamiltonians 49612.5.3 Approximations for Complicated Terms 49812.5.3.1 Spin–Orbit Operators 49812.5.3.2 Two-Electron Terms 49912.5.3.3 One-Electron Basis Sets 49912.5.4 DKH Gradients 50013 Elimination Techniques 50313.1 Naive Reduction: Pauli Elimination 50313.2 Breit–Pauli Theory 50713.2.1 Foldy–Wouthuysen Transformation of the Breit Equation 50813.2.2 Transformation of the Two-Electron Interaction 50913.2.2.1 All-Even Operators 51113.2.2.2 Transformed Coulomb Contribution 51213.2.2.3 Transformed Breit Contribution 51413.2.3 The Breit–Pauli Hamiltonian 51813.3 The Cowan–Griffin and Wood–Boring Approaches 52213.4 Elimination for Different Representations of Dirac Matrices 52313.5 Regular Approximations 524Part V — Chemistry with Relativistic Hamiltonians 52714 Special Computational Techniques 52914.1 From the Modified Dirac Equation to Exact-Two-Component Methods 53014.1.1 Normalized Elimination of the Small Component 53114.1.2 Exact-Decoupling Methods 53314.1.2.1 The One-Step Solution: X2C 53714.1.2.2 Two-Step Transformation: BSS 54214.1.2.3 Expansion of the Transformation: DKH 54314.1.3 Approximations in Many-Electron Calculations 54614.1.3.1 The Cumbersome Two-Electron Terms 54614.1.3.2 Scalar-Relativistic Approximations 54714.1.4 Numerical Comparison 54814.2 Locality of Relativistic Contributions 55114.3 Local Exact Decoupling 55314.3.1 Atomic Unitary Transformation 55414.3.2 Local Decomposition of the X-Operator 55514.3.3 Local Approximations to the Exact-Decoupling Transformation 55614.3.4 Numerical Comparison 55914.4 Efficient Calculation of Spin–Orbit Coupling Effects 56114.5 Relativistic Effective Core Potentials 56415 External Electromagnetic Fields and Molecular Properties 56715.1 Four-Component Perturbation and Response Theory 56915.1.1 Variational Treatment 57015.1.2 Perturbation Theory 57015.1.3 The Dirac-Like One-Electron Picture 57315.1.4 Two Types of Properties 57515.2 Reduction to Two-Component Form and Picture Change Artifacts 57615.2.1 Origin of Picture Change Errors 57715.2.2 Picture-Change-Free Transformed Properties 58015.2.3 Foldy–Wouthuysen Transformation of Properties 58015.2.4 Breit–Pauli Hamiltonian with Electromagnetic Fields 58115.3 Douglas–Kroll–Hess Property Transformation 58215.3.1 The Variational DKH Scheme for Perturbing Potentials 58315.3.2 Most General Electromagnetic Property 58415.3.3 Perturbative Approach 58715.3.3.1 Direct DKH Transformation of First-Order Energy 58715.3.3.2 Expressions of 3rd Order in Unperturbed Potential 58915.3.3.3 Alternative Transformation for First-Order Energy 59015.3.4 Automated Generation of DKH Property Operators 59215.3.5 Consequences for the Electron Density Distribution 59315.3.6 DKH Perturbation Theory with Magnetic Fields 59515.4 Magnetic Fields in Resonance Spectroscopies 59515.4.1 The Notorious Diamagnetic Term 59515.4.2 Gauge Origin and London Orbitals 59615.4.3 Explicit Form of Perturbation Operators 59715.4.4 Spin Hamiltonian 59815.5 Electric Field Gradient and Nuclear Quadrupole Moment 59915.6 Parity Violation and Electro-Weak Chemistry 60216 Relativistic Effects in Chemistry 60516.1 Effects in Atoms with Consequences for Chemical Bonding 60816.2 Is Spin a Relativistic Effect? 61216.3 Z-Dependence of Relativistic Effects: Perturbation Theory 61316.4 Potential Energy Surfaces and Spectroscopic Parameters 61416.4.1 Dihydrogen 61616.4.2 Thallium Hydride 61716.4.3 The Gold Dimer 61916.4.4 Tin Oxide and Cesium Hydride 62216.5 Lanthanides and Actinides 62216.5.1 Lanthanide and Actinide Contraction 62316.5.2 Electronic Spectra of Actinide Compounds 62316.6 Electron Density of Transition Metal Complexes 62516.7 Relativistic Quantum Chemical Calculations in Practice 629Appendix 631A Vector and Tensor Calculus 633A.1 Three-Dimensional Expressions 633A.1.1 Algebraic Vector and Tensor Operations 633A.1.2 Differential Vector Operations 634A.1.3 Integral Theorems and Distributions 635A.1.4 Total Differentials and Time Derivatives 637A.2 Four-Dimensional Expressions 638A.2.1 Algebraic Vector and Tensor Operations 638A.2.2 Differential Vector Operations 638B Kinetic Energy in Generalized Coordinates 641C Technical Proofs for Special Relativity 643C.1 Invariance of Space-Time Interval 643C.2 Uniqueness of Lorentz Transformations 644C.3 Useful Trigonometric and Hyperbolic Formulae for Lorentz Transformations 646D Relations for Pauli and dirac Matrices 649D.1 Pauli Spin Matrices 649D.2 Dirac’s Relation 650D.2.1 Momenta and Vector Fields 651D.2.2 Four-Dimensional Generalization 652E Fourier Transformations 653E.1 Definition and General Properties 653E.2 Fourier Transformation of the Coulomb Potential 654F Gordon Decomposition 657F. 1 One-Electron Case 657F. 2 Many-Electron Case 659G Discretization and Quadrature Schemes 661G.1 Numerov Approach toward Second-Order Differential Equations 661G.2 Numerov Approach for First-Order Differential Equations 663G.3 Simpson’s Quadrature Formula 665G.4 Bickley’s Central-Difference Formulae 665H List of Abbreviations and Acronyms 667I List of Symbols 669J Units and Dimensions 673References 675

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