Atomic Many-Body Theory

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 Size of the Atom.- 7.5.3 The Magnitude of the Spin-Orbit Coupling Constant.- 7.5.4 The Ce2+ Ion.- 7.6 Properties of the Two-Electron Slater Integrals.- 7.7 Coupling Schemes for Two-Electron Systems.- 8. Many-Electron Wave Functions.- 8.1 Graphical Representation of the Fractional-Parentage Expansion.- 8.1.1 The (nl)3 Configuration.- 8.1.2 Classification of States.- 8.1.3 The Expansion of the nlN State.- 8.1.4 Graphical Representation of the Fractional-Parentage Coefficient.- 8.2 Matrix Elements of a Single-Particle Operator.- 8.2.1 General.- 8.2.2 Orbital Operator.- 8.2.3 Double-Tensor Operators.- 8.2.4 Standard Unit-Tensor Operators.- 8.2.5 A Physical Example: The Spin-Orbit Interaction.- 8.3 Matrix Elements of a Two-Particle Operator.- 8.4 More Than One Open Shell.- 8.4.1 Transition Probability Between the nlNand nlN-1n'l' Configurations.- 8.4.2 Coulomb Interaction Between the Configurations nlN and nlN-1n'l'.- 8.4.3 Transition Probability Between the Configurations nlNn'l' and nlN?1n'l'2.- II Perturbation Theory and the Treatment of Atomic Many-Body Effects.- 9. Perturbation Theory.- 9.1 Basic Problem.- 9.2 Nondegenerate Briilouin-Wigner Perturbation Theory.- 9.2.1 Basic Concepts.- 9.2.2 A Resolvent Expansion of the Wave Function.- 9.2.3 The Wave Operator.- 9.2.4 Determination of the Energy.- 9.2.5 The Feshbach Operator.- 9.3 The Green's Function.- 9.3.1 The Green's -Function Operator or the Propagator.- 9.3.2 The Green's Function in the Coordinate Space.- 9.4 General Rayleigh-Schrodinger Perturbation Theory.- 9.4.1 The Model Space.- 9.4.2 The Generalized Bloch Equation.- 9.4.3 The Effective Hamiltonian.- 9.5 The Rayleigh-Schrodinger Expansion for a Degenerate Model Space.- 10. First-Order Perturbation for Closed-Shell Atoms.- 10.1 The First-Order Wave Function.- 10.2 The First-Order Energy.- 10.3 Evaluation of First-Order Diagrams.- 11. Second Quantization and the Particle-Hole Formalism.- 11.1 Second Quantization.- 11.2 Operators in Normal Form.- 11.3 The Particle-Hole Formalism.- 11.4 Graphical Representation of Normal-Ordered Operators.- 11.5 Wick's Theorem.- 11.5.1 Statement of the Theorem.- 11.5.2 Proof of the Theorem.- 11.5.3 Wick's Theorem for Operator Products.- 11.6 The Wave Operator in Graphical Form.- 12. Application of Perturbation Theory to Closed-Shell Systems.- 12.1 The First-Order Contributions to the Wave Function and the Energy.- 12.2 The Second-Order Wave Operator.- 12.2.1 Construction and Evaluation of Diagrams.- 12.2.2 Equivalent Diagrams and Weight Factors.- 12.2.3 Choices of Single-Particle States.- 12.3 Perturbation Expansion of the Energy.- 12.3.1 The Correlation Energy.- 12.3.2 The Second-Order Energy.- 12.4 The Goldstone Evaluation Rules.- 12.5 The Linked-Diagram Expansion.- 12.5.1 Cancellation of Unlinked Diagrams in Third Order.- 12.5.2 The Linked-Diagram Theorem.- 12.6 Separation of Goldstone Diagrams into Radial and Spin-Angular Parts.- 12.6.1 Examples of Diagram Evaluations.- 12.6.2 Evaluation Rules for the Radial, Spin and Angular Factors.- 12.7 The Correlation Energy of the Beryllium Atom.- 12.8 Appendix. The Goldstone Phase Rule.- 13. Application of Perturbation Theory to Open-Shell Systems.- 13.1 The Particle-Hole Representation.- 13.1.1 Classification of Single-Particle States.- 13.1.2 The Particle-Hole Representation.- 13.1.3 Graphical Representation.- 13.2 The Effective Hamiltonian.- 13.2.1 The First-Order Effective Hamiltonian.- 13.2.2 The First-Order Wave Operator.- 13.2.3 The Second-Order Effective Hamiltonian.- 13.3 Higher-Order Perturbations. The Linked-Diagram Theorem.- 13.3.1 The Second-Order Wave Operator.- 13.3.2 The Linked-Diagram Expansion.- 13.3.3 The Third-Order Effective Hamiltonian.- 13.4 The LS Term Splitting of an (nl)N Configuration.- 13.4.1 The Slater and Trees Parameters.- 13.4.2 Evaluation of the First- and Second-Order Contributions to the Term Splitting.- 13.4.3 Application to the Pr3+ Ion.- 13.5 The Use of One-and Two-Particle Equations.- 13.5.1 The Single-Particle Equation.- 13.5.2 The Pair Equation.- 13.6 The Beryllium Atom Treated as an Open-Shell System.- 13.6.1 The Energy Matrix.- 13.6.2 The First-Order Results.- 13.6.3 The Second-Order Results.- 14. The Hyperfine Interaction.- 14.1 The Hyperfine Interaction.- 14.1.1 The Hyperfine Operator.- 14.1.2 The Hyperfine Splitting.- 14.1.3 The Hyperfine Interaction Constants.- 14.2 The Zeroth-Order Hyperfine Constants.- 14.2.1 The Effective Hyperfine Operator.- 14.2.2 The Zeroth-Order Hyperfine Constants in Second Quantization.- 14.2.3 The Spin-Angular Structure of the Hyperfine Operator.- 14.2.4 Single Open Shell.- 14.3 First-Order Core Polarization.- 14.3.1 Evaluation of the Polarization Diagrams.- 14.3.2 Contributions to the Effective Hyperfine Operators.- 14.3.3 Evaluation of the Radial Parts.- 14.3.4 Interpretation of the Core Polarization.- 14.4 Second-Order Polarization and Lowest-Order Correlation.- 14.4.1 The Second-Order Polarization.- 14.4.2 The Lowest-Order Correlation.- 14.5 All-Order Polarization.- 14.5.1 The Effective Hyperfine Interaction.- 14.5.2 The All-Order Radial Single-Particle Equation.- 14.5.3 Ground-State Correlations.- 14.5.4 Results of Some All-Order Calculations.- 14.6 Effective Two-Body Hyperfine Interactions.- 14.7 Relativistc Effects.- 14.7.1 Hyperfine Structure Calculations.- 14.7.2 Fine-Structure Calculations on Alkali-like Systems.- 15. The Pair-Correlation Problem and the Coupled-Cluster Approach.- 15.1 Introductory Comments.- 15.1.1 The Configuration-Interaction and the Linked-Diagram Procedures.- 15.1.2 The Separability Condition and the Coupled-Cluster Approach.- 15.2 Hierarchy of n-Particle Equations.- 15.2.1 General.- 15.2.2 The All-Order Single-Particle Equation.- 15.2.3 The All-Order Pair Equation.- 15.2.4 The All-Order Radial Equations.- 15.3 The Exponential Ansatz.- 15.3.1 Factorization of Closed-Shell Diagrams.- 15.3.2 Factorization of Open-Shell Diagrams.- 15.3.3 The Wave Operator in Exponential Form.- 15.4 The Coupled-Cluster Equations.- 15.4.1 General.- 15.4.2 The One- and Two-Particle Coupled-Cluster Equations.- 15.5 Comparison Between Different Pair-Correlation Approaches.- 15.5.1 Application to Helium.- 15.5.2 Application to Beryllium.- 15.5.3 Application to Neon.- 15.5.4 Examples of Open-Shell Coupled-Cluster Calculations.- Appendix A. Hartree Atomic Units.- Appendix B. States and Operators.- References.- Author Index.