Spin Eigenfunctions

Construction and Use
Springer (Verlag)
  • erschienen am 6. Dezember 2012
  • |
  • XV, 370 Seiten
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978-1-4684-8526-4 (ISBN)
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.
  • Englisch
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  • USA
Springer US
978-1-4684-8526-4 (9781468485264)
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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
'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.

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