The aim of this book is to give a comprehensive treatment of the different methods for the construction of spin eigenfunctions and to show their interrelations. The ultimate goal is the construction of an antisymmetric many-electron wave function that has both spatial and spin parts and the calculation of the matrix elements of the Hamiltonian over the total wave function. The representations of the symmetric group playa central role both in the construction of spin functions and in the calculation of the matrix elements of the Hamiltonian, so this subject will be treated in detail. We shall restrict the treatment to spin-independent Hamiltonians; in this case the spin does not have a direct role in the energy expression, but the choice of spin functions influences the form of spatial functions through the antisymmetry principle; the spatial functions determine the energy of the system. We shall also present the "spin-free quantum chemistry" approach of Matsen and co-workers, in which one starts immediately with the construction of spatial functions that have the correct permutational symmetries. By presenting both the conventional and the spin-free approach, one gains a better understanding of certain aspects of the elec­ tronic correlation problem. The latest advance in the calculation of the matrix elements of the Hamiltonian is the use of the representations of the unitary group, so this will be the last subject. It is a pleasant task to thank all those who helped in writing this book.

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1. Introduction.- 2. Construction of Spin Eigenfunctions from the Products of One-Electron Spin Functions.- 3. Construction of Spin Eigenfunctions from the Products of Two-Electron Spin Eigenfunctions.- 4. Construction of Spin Eigenfunctions by the Projection Operator Method.- 5. Spin-Paired Spin Eigenfunctions.- 6. Basic Notions of the Theory of the Symmetric Group.- 7. Representations of the Symmetric Group Generated by the Spin Eigenfunctions.- 8. Representations of the Symmetric Group Generated by the Projected Spin Functions and Valence Bond Functions.- 9. Combination of Spatial and Spin Functions; Calculation of the Matrix Elements of Operators.- 10. Calculation of the Matrix Elements of the Hamiltonian; Orthogonal Spin Functions.- 11. Calculation of the Matrix Elements of the Hamiltonian; Nonorthogonal Spin Functions.- 12. Spin-Free Quantum Chemistry.- 13. Matrix Elements of the Hamiltonian and the Representation of the Unitary Group.- 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. Constructionof 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.