Spin Eigenfunctions

1. Introduction.- 1.1. Electronic States with Definite Multiplicities.- 1.2. Basic Facts with Respect to the Spin.- 1.3. Spin Operators and Functions for One Electron.- 1.4. Addition Theorem of Angular Momenta.- References.- 2. Construction of Spin Eigenfunctions from the Products of One-Electron Spin Functions.- 2.1. The Resultant Spin Operator and the Dirac Identity.- 2.2. Eigenfunctions of S2.- 2.3. Construction of S2 Eigenfunctions by the Diagonalization of the S2 Matrix.- 2.4. Construction of S2 Eigenfunctions by the Orthogonalization Procedure.- 2.5. Dimension of the Spin Degeneracy.- 2.6. Genealogical Construction of Spin Eigenfunctions.- 2.7. Branching Diagram.- 2.8. Orthogonality of the Branching-Diagram Functions.- 2.9. Special Properties of the Branching-Diagram Functions.- 2.10. Ordering of the Primitive Spin Functions; the Path Diagram.- 2.11. Expression for X(N, S, S; 1).- 2.12. Expression for X(N, S, S; f).- 2.13. The Coefficient of a Primitive Spin Function in a Given Branching-Diagram Function.- References.- 3. Construction of Spin Eigenfunctions from the Products of Two-Electron Spin Eigenfunctions.- 3.1. Serber-Type Construction of Spin Eigenfunctions.- 3.2. Formulas for the Serber Construction.- 3.3. Geminal Spin Product Functions; Serber Path Diagram.- 3.4. Special Properties of the Serber Functions.- 3.5. The Coefficient of a Geminal Product Function in a Given Serber Function.- 3.6. The Algorithm of Carrington and Doggett.- 3.7. Construction of Serber-Type Functions by Direct Diagonalization of the S2 Matrix.- 3.7.1. S2 Matrix for an Even Number of Electrons.- 3.7.2. S2 Matrix for an Odd Number of Electrons.- 3.8. Construction of Spin Eigenfunctions from Those of Two Subsystems.- References.- 4. Construction of Spin Eigenfunctions by the Projection Operator Method.- 4.1. Projection Operator Method.- 4.1.1. Trial Function.- 4.1.2. The Projection Operator.- 4.2. The Projection of the First Primitive Function.- 4.3. The Projection of an Arbitrary Primitive Spin Function.- 4.4. The Choice of Spin Functions Whose Projections Are Linearly Independent.- 4.5. Relation between the Projected Functions and the Branching-Diagram Functions.- 4.6. Projected Functions for S > M; Sanibel Coefficients.- 4.7. Sasaki and Ohno's Derivation of the Sanibel Coefficients.- 4.8. Derivation of the Sanibel Coefficients from the Vector-Coupling Coefficients.- 4.9. Sanibel Coefficients by the Group Theoretical Projection Operator Method.- 4.10. The Construction of Serber-Type Functions by the Projection Operator Method.- 4.11. The Overlap Matrix of the Projected Spin Functions.- References.- 5. Spin-Paired Spin Eigenfunctions.- 5.1. Spin-Paired Spin Eigenfunctions.- 5.2. Extended Rumer Diagrams.- 5.3. Linear Independence of Extended Rumer Functions.- 5.4. The Relation between Rumer Functions and Branching-Diagram Functions.- 5.5. The Relation between Rumer Functions and Serber-Type Functions.- 5.6. Matrix Elements between the Spin-Paired Functions.- 5.6.1. Islands.- 5.6.2 O Chain.- 5.6.3 E Chain.- References.- 6. Basic Notions of the Theory of the Symmetric Group.- 6.1. Introduction.- 6.2. Permutations; Cyclic Structure.- 6.3. Young Frames; Young Tableaux.- 6.4. The Symmetric Group Algebra; Young Operators.- 6.4.1. Young Operators.- 6.4.2. Ordering of the Standard Tableaux.- 6.4.3. Yamanouchi Symbol.- 6.4.4. The Young Operator E?rs.- 6.4.5. Alternative Definition of the Young Operators.- 6.5. Representations of the Symmetric Group.- 6.5.1. Young's Orthogonal Representation.- 6.5.2. Young's Natural Representation.- 6.6. Matric Basis of the Symmetric Group Algebra.- 6.6.1. Calculation of the Characters of the Symmetric Group..- 6.6.2. Matsen's Method for the Construction of Matric Units..- 6.6.3. Salmon's Method for the Construction of Matric Units..- References.- 7. Representations of the Symmetric Group Generated by the Spin Eigenfunctions.- 7.1. Introduction.- 7.2. The Genealogical Spin Functions Generate a Representation of the Symmetric Group.- 7.3. Recursive Construction of the Representation Matrices: Yamanouchi-Kotani Method.- 7.3.1. Permutations That Do Not Affect the Last Number N.- 7.3.2. The Transposition (N - 1, N).- 7.4. Relation between the Yamanouchi-Kotani Representation and the Young Orthogonal Representation.- 7.4.1. Dimension of the Representation.- 7.4.2. One-to-One Correspondence between the Young Tableaux and Branching-Diagram Functions.- 7.4.3. Identity of the Young Orthogonal and the Yamanouchi-Kotani Representation.- 7.5. Construction of the Spin Functions from the Representation Matrices.- 7.6. Construction of the Branching-Diagram Functions by Use of the Matric Units.- 7.6.1. Conditions for Nonvanishing e?ii?.- 7.6.2. The Character Projection Operator.- 7.6.3. Construction of the Branching-Diagram Functions by Use of the Matric Units; Salmon's Procedure.- 7.7. Representation of the Symmetric Group Generated by the Serber-Type Spin Functions.- 7.7.1. Direct Method for the Calculation of the Representation Matrix.- 7.7.2. Recursive Calculation of the Representation Matrix.- 7.8. The Relation between the Serber and the Young-Yamanouchi Representations.- 7.8.1. The Transformation Matrix.- 7.8.2. Recursive Construction of the Transformation Matrix.- 7.9. Matric Basis of the Serber Representation.- 7.10. Representation Generated by the Spin-Coupled Functions.- 7.11. Relation between the Young-Yamanouchi and the Reduced Representations.- References.- 8. Representations of the Symmetric Group Generated by the Projected Spin Functions and Valence Bond Functions.- 8.1. Introduction.- 8.2. Representation Generated by the Projected Spin Functions.- 8.3. Construction of the Projected Spin Functions by the Use of the Young Operator.- 8.4. Construction of the Projected Spin Functions by the Character Projection Operator.- 8.5. Representation Generated by the Rumer Functions.- 8.6. Construction of the Spin-Paired Functions from the Alternative Young Operators.- 8.7. The Linear Independence of Vf's and Their Relation to the Genealogical Functions.- References.- 9. Combination of Spatial and Spin Functions; Calculation of the Matrix Elements of Operators.- 9.1. Introduction.- 9.2. Construction of Antisymmetric Wave Function.- 9.3. Separation of ?i into Spatial and Spin Functions.- 9.4. The Spatial Functions ?sji Generate a Representation of SN.- 9.5. Calculation of the Matrix Elements of the Hamiltonian.- 9.6. Computational Aspects of the Basic Formulas.- 9.7. The Form of the Spatial Function ?.- References.- 10. Calculation of the Matrix Elements of the Hamiltonian; Orthogonal Spin Functions.- 10.1. Introduction.- 10.2. Spatial Functions with a Number of Doubly Occupied Orbitals; Branching-Diagram Spin Functions.- 10.3. Calculation of the Energy Matrix.- 10.3.1. Alternative Method for the Calculation of the Invariant Part.- 10.3.2. Calculation of the Energy Matrix for the Case of Orthogonal Orbitals.- 10.4. Matrix Elements of the Hamiltonian for Serber-Type Spin Functions.- 10.4.1. Notation for the Spatial Functions.- 10.4.2. Geminal Spin Harmonics.- 10.4.3. Normalization Integral.- 10.4.4. The Lineup Permutation.- 10.4.5. The Wave Functions Form an Orthonormal Set.- 10.4.6. The Form of the Hamiltonian.- 10.4.7. Reduction of the Sum over the Permutations.- 10.4.8. Reduction of the Sum over Electron Pairs.- 10.4.9. Matrix Elements of the Hamiltonian.- 10.5. Calculation of the Matrix Elements of the Hamiltonian for Spin-Coupled Wave Functions.- 10.6. Calculation of the Energy for a Single Configuration.- 10.6.1. One-Electron Operators.- 10.6.2. Two-Electron Operators.- References.- 11. Calculation of the Matrix Elements of the Hamiltonian; Nonorthogonal Spin Functions.- 11.1. Introduction.- 11.2. A Single Configuration; Projected Spin Function.- 11.3. Different Orbitals for Different Spins.- 11.3.1. Alternant Molecular Orbitals.- 11.3.2. Calculation of the Normalization Integral.- 11.4. Many-Configuration Wave Function; Projection Operator Method.- 11.4.1. The Reference Permutation.- 11.4.2. Summation over the Subgroup Sv.- 11.4.3. The Spatial Integrals.- 11.4.4. Matrix Elements.- 11.5. Many-Configuration Wave Function; Bonded Functions.- 11.5.1. The Matching Rearrangement.- 11.5.2. The Effect of Double Occupancy.- 11.5.3. Matrix Elements of the Spin Functions.- 11.5.4. Matrix Elements of the Transpositions.- 11.5.5. Matrix Element of the Hamiltonian between Two Functions.- 11.5.6. Matrix Elements in Terms of Bonded Functions 224 References.- 12. Spin-Free Quantum Chemistry.- 12.1. Introduction.- 12.2. The Decomposition of the Antisymmetrizer.- 12.3. Spin-Free Hamiltonian.- 12.4. Construction of Spatial Functions ??ik.- 12.5. Invariance Group of the Primitive Ket.- 12.6. The Coset Representation Generated by the Invariance Group..- 12.6.1. Decomposition of the Vector Space V(?).- 12.6.2. Factorization of the Secular Equations.- 12.7. Construction of the Invariant Subspaces by Means of the Orthogonal Units.- 12.7.1. The Immanant Function.- 12.7.2. The Antisymmetric Space Spin Counterpart of the Immanant.- 12.8. Structure Projections.- 12.8.1. The Pair Diagram.- 12.8.2. The Pair Operators.- 12.8.3. Construction of Spin-Free Pair Functions.- 12.8.4. Pair Projections in the Function Space.- 12.8.5. Spin-Free Exclusion Principle.- 12.9. Matrix Elements of the Hamiltonian over the Structure Projections.- 12.10. Spin-Free Counterpart of the Projected Function.- 12.11. Gallup's Formulation of Spin-Free Quantum Chemistry.- 12.12. Calculation of Pauling Numbers.- References.- 13. Matrix Elements of the Hamiltonian and the Representation of the Unitary Group.- 13.1. Introduction.- 13.2. Formulation of the Hamiltonian.- 13.3. Basic Notions about the Unitary Group.- 13.4. Irreducible Representations of the Unitary Group.- 13.4.1. The Gel'fand-Tsetlin Representation of the Generator Algebra.- 13.4.2. Group-Theoretical Meaning of the Gel'fand Pattern..- 13.5. The Representation Matrices of E ij 's.- 13.5.1. The Diagonal Generators Eii.- 13.5.2. The Raising Generators Eij(i < j).- 13.5.3. The Lowering Generators Eij (i < j).- 13.5.4. Gel'fand-Tsetlin Formula for the Matrix of Ei,i+1.- 13.6. Weyl Tableaux.- 13.7. The Nth-Rank Tensor Space and Its Decomposition into Invariant Subspaces.- 13.8. Exclusion Principle and Gel'fand States.- 13.9. Matrix Elements of the Generators for Paldus Tableaux.- 13.9.1. Basis Generation.- 13.9.2. Matrix Elements of the Generators.- 13.10. Matrix Element of the Generators; Downward-Robb Algorithm.- 13.10.1. Basis Generation.- 13.10.2. Eigenvalues of the Diagonal Generators.- 13.10.3. Generators Eij.- 13.11. Graphical Representation of the Basis Functions; Relation to the Configuration Interaction Method.- References.- Appendix 1. Some Basic Algebraic Notions.- A.1.1. Introduction.- A.1.2. Frobenius or Group Algebra; Convolution Algebra.- A.1.2.1. Invariant Mean.- A.1.2.2. Frobenius or Group Algebra.- A.1.2.3. Convolution Algebra.- A.1.3. Some Algebraic Notions.- A.1.4. The Centrum of the Algebra.- A.1.5. Irreducible Representations; Schur's Lemma.- A.1.6. The Matric Basis.- A.1.7. Symmetry Adaptation.- A.1.8. Wigner-Eckart Theorem.- References.- Appendix 2. The Coset Representation.- A.2.1. Introduction.- A.2.2. The Character of an Element g in the Coset Representation..- Appendix 3. Double Coset.- A.3.1. The Double Coset Decomposition.- A.3.2. The Number of Elements in a Double Coset.- Appendix 4. The Method of Spinor Invariants.- A.4.1. Spinors and Their Transformation Properties.- A.4.2. The Method of Spinor Invariants.- A.4.3. Construction of the Genealogical Spin Functions by the Method of Spinor Invariants.- A.4.4. Normalization Factors.- A.4.5. Construction of the Serber Functions by the Method of Spinor Invariants.- A.4.6. Singlet Functions as Spinor Invariants.- References.- A.5.1. The Formalism of Second Quantization.- A.5.2. Representation of the Spin Operators in the Second-Quantization Formalism.- A.5.3. Review of the Papers That Use the Second-Quantization Formalism for the Construction of Spin Eigenfunctions.- A.5.3.1. Genealogical Construction.- A.5.3.2. Projection Operator Method.- A.5.3.3. Valence Bond Method.- A.5.3.4. The Occupation-Branching-Number Representation.- References.- Appendix 6. Table of Sanibel Coefficients.- Reference.- Author Index.