Synopsis
I Angular-Momentum Theory and the Independent-Particle Model.- 1. Introduction.- 2. Angular-Momentum and Spherical Tensor Operators.- 2.1 Elementary Properties of Angular-Momentum and Spherical Tensor Operators.- 2.1.1 Angular-Momentum Operators.- 2.1.2 Spherical Tensor Operators.- 2.2 Rotations in Space.- 2.2.1 Relation Between Angular-Momentum Operators and Infinitesimal Rotations in Space.- 2.2.2 Transformation of Angular-Momentum States and Spherical Tensor Operators Under Infinitesimal Rotations.- 2.2.3 Transformation of Angular-Momentum States and Spherical Tensor Operators Under Finite Rotations.- 2.2.4 The Orbital Angular Momentum. Spherical Harmonics.- 2.2.5 Example of Rotation of Angular-Momentum Functions.- 2.3 Coupling of Angular-Momentum States and Spherical Tensor Operators.- 2.3.1 Coupling of States.- 2.3.2 Coupling of Tensor Operators.- 2.3.3 A Physical Example: The Coulomb Interaction.- 2.4 The Wigner-Eckart Theorem.- 2.4.1 Proof of the Theorem.- 2.4.2 A Physical Example: the Zeeman Effect.- 2.4.3 Reduced Matrix Elements of the C Tensor.- 3. Angular-Momentum Graphs.- 3.1 Representation of 3-j Symbols and Vector-Coupling Coefficients.- 3.1.1 Basic Conventions.- 3.1.2 Representation of the Vector-Coupling Coefficient.- 3.1.3 Representation of Coupled States.- 3.1.4 The Wigner-Eckart Theorem.- 3.1.5 Elimination of a Zero Line.- 3.2 Diagrams with Two or More Vertices.- 3.2.1 Summation Rules.- 3.2.2 Orthogonality Relations.- 3.3 A Physical Example: The Coulomb Interaction.- 3.3.1 Representation of a Single Matrix Element.- 3.3.2 Summation Over Filled Shells.- 3.4 Coupling of Three Angular Momenta. The 6-j Symbol.- 3.4.1 The 6-j Symbol.- 3.4.2 Equivalent Forms of the 6-j Symbol. The Hamilton Line.- 3.4.3 A Physical Example: The is Configuration.- 3.5 Coupling of Four Angular Momenta. The 9-j Symbol.- 4. Further Developments of Angular-Momentum Graphs. Applications to Physical Problems.- 4.1 The Theorems of Jucys, Levinson and Vanagas.- 4.1.1 The Basic Theorem.- 4.1.2 Diagrams Separable on Two Lines.- 4.1.3 Diagrams Separable on Three Lines.- 4.1.4 Diagrams Separable on Four Lines.- 4.2 Some Applications of the JLV Theorems.- 4.3 Matrix Elements of Tensor-Operator Products Between Coupled States.- 4.3.1 The General Formula.- 4.3.2 Special Cases.- 4.4 The Coulomb Interaction for Two-Electron Systems in LS Coupling.- 4.4.1 The Basic Formula.- 4.4.2 Antisymmetric Wave Functions.- 4.4.3 Two Equivalent Electrons in LS Coupling.- 4.4.4 Two Nonequivalent Electrons in LS Coupling.- 4.5 The Coulomb Interaction for Two-Electron Systems in j-j Coupling.- 5. The Independent-Particle Model.- 5.1 The Magnetic Interactions.- 5.2 Determinantal Wave Functions.- 5.3 Matrix Elements Between Slater Determinants.- 5.3.1 Matrix Elements of Single-Particle Operators.- 5.3.2 Matrix Elements of Two-Particle Operators.- 5.3.3 A New Notation.- 5.3.4 Feynman Diagrams.- 5.4 The Hartree-Fock Equations.- 5.5 Koopmans' Theorem.- 6. The Central-Field Model.- 6.1 Separation of the Single-Electron Equation for a Central Field.- 6.2 The Electron Configuration and the "Building-Up" Principle.- 6.2.1 The Meaning of a Configuration.- 6.2.2 The "Building-Up" Principle.- 6.3 Russell-Saunders Coupling.- 6.4 Angular-Momentum Properties of Determinantal States.- 6.5 LS Terms of a Given Configuration.- 6.6 Term Energies.- 6.6.1 The Single-Particle Operator.- 6.6.2 The Two-Particle Operator.- 6.6.3 Example: erm Energies of the 1s2 2s2 2p2 Configuration.- 6.6.4 A General Energy Expression.- 6.7 The Average Energy of a Configuration.- 6.7.1 Derivation of the General Formula.- 6.7.2 Example: Average of the 1s2 2s2 2p2 Configuration.- 7. The Hartree-Fock Model.- 7.1 Radial Equations for the Restricted Hartree-Fock Procedure.- 7.2 Koopmans' Theorem in Restricted Hartree-Fock.- 7.3 The Hartree-Fock Potential.- 7.4 Examples of Hartree-Fock Equations.- 7.4.1 A Closed-Shell System: 1s2 2s2.- 7.5 Examples of Hartree-Fock Calculations.- 7.5.1 The Carbon Atom.- 7.5.2 The
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