Atomic Many-Body Theory

In the new edition only minor modifications have been made. Some print­ ing errors have been corrected and a few clarifications have been made. In recent years the activity in relativistic many-body theory has increased con­ siderably, but this field falls outside the scope of this book. A brief summary of the recent developments, however, has been included in the section on "relativistic effects" in Chap. 14. In addition, only a very limited number of references have been added, without any systematic updating of the material. Goteborg, December 1985 l. Lindgren· J. Morrison Preface to the First Edition This book has developed through a series of lectures on atomic theory given these last eight years at Chalmers University of Technology and several oth­ er research centers. These courses were intended to make the basic elements of atomic theory available to experimentalists working with the hyperfine structure and the optical properties of atoms and to provide some insight into recent developments in the 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-Schrödinger 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-Schrödinger 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.