The author wrote a monograph 20 years ago on the construction of spin eigen functions; the monograph was published by Plenum. The aim of that mono graph was to present all aspects connected with the construction of spin eigen functions and its relation to the use of many-electron antisymmetric wavefunc tions. The present book is an introduction to these subjects, with an emphasis on the practical side. After the theoretical treatment, there will be many exam ples and exercises which will illustrate the different methods. The theory of the symmetric group and its representations generated by…mehr
The author wrote a monograph 20 years ago on the construction of spin eigen functions; the monograph was published by Plenum. The aim of that mono graph was to present all aspects connected with the construction of spin eigen functions and its relation to the use of many-electron antisymmetric wavefunc tions. The present book is an introduction to these subjects, with an emphasis on the practical side. After the theoretical treatment, there will be many exam ples and exercises which will illustrate the different methods. The theory of the symmetric group and its representations generated by the different spin eigen functions is an other subject, this is closely related to the quantum chemical applications. Finally we will survey the calculation of the matrix elements of the Hamiltonian, using the different constructions of the spin functions. The closing chapter will deal with a new method that gained much importance recently; the spin-coupled valence bond method. Since the publication of Spin Eigenfunctions, nearly 20 years ago there have been many interesting developments in the subject; there are quite a few new algorithms for the construction of spin eigenfunctions. Moreover the use of the spin-coupled valence bond method showed the importance of using different constructions for the spin functions. The subject matter of this book has been presented in a graduate course in the Technion. The author is obliged to the graduate students Averbukh Vitali, Gokhberg Kirill, and Narevicius Edvardas for many helpful comments.
1. The Spin Operator and Spin Functions.- 1.1 Introduction.- 1.2 Spin operators and Pauli matrices.- 1.3 Spin functions.- 1.4 The Dirac identity.- 1.5 Square brackets of spin functions.- 1.6 Graphical representation of the primitive spin functions.- 2. Genealogical Construction of Spin Eigenfunctions.- 2.1 Addition theorem of angular momenta.- 2.2 Addition theorem of spin forNelectrons.- 2.3 The spin degeneracy.- 2.4 Branching diagram symbol.- 2.5 Special properties of the branching diagram functions.- 2.6 The coefficient of a primitive spin function in a given branching diagram function.- 3. Serber Spin Functions.- 3.1 Serber spin functions.- 3.2 Geminal spin product functions. Serber path diagram.- 3.3 Serber branching diagram.- 3.4 Recursion formulas for the highest component.- 3.5 Special properties of the Serber functions.- 3.6 The coefficient of a geminal product in a given Serber function.- 3.7 The algorithm of Carrington and Doggett.- 4. Projected Spin Eigenfunctions.- 4.1 Projection operator.- 4.2 Expanded form of the projected function (Löwdin).- 4.3 Projection of other primitive functions.- 4.4 Relation between the projected functions and the branching diagram functions.- 4.5 Calculation of the overlap matrix of the projected functions.- 5. Spin-Paired Spin Eigenfunctions.- 5.1 Spin-paired spin functions.- 5.2 The Rumer algorithm for the selection of linearly independent spin-paired spin functions.- 5.3 Extended Rumer diagrams (Simonetta).- 5.4 The leading term (Simonetta).- 5.5 Linear independence of spin-paired eigenfunctions.- 5.6 Overlap integrals between Rumer functions..- 6. The Symmetric Group.- 6.1 Basic notions of the symmetric group.- 6.2 Classes of the symmetric group.- 6.3 Representations of the symmetric group.- 6.4 Young tableaux.- 6.5Young's orthogonal representation.- 6.6 The conjugate representation.- 6.7 The symmetric group algebra.- 6.8 The Young operator.- 7. Representations of SNGenerated by Spin Eigenfunctions.- 7.1 Representations of the symmetric group generated by the branching diagram functions.- 7.2 Yamanouchi-Kotani method for the representations.- 7.3 Branching diagram functions and Young tableaux.- 7.4 Representations of SNgenerated by the projected spin functions.- 7.5 Calculations of A(P) by the Rettrup-Pauncz algorithm.- 7.6 Correspondence between spin-paired functions and Young tableaux.- 7.7 Generation of projected spin functions by Young operators.- 8. Combination of Spatial and Spin Functions.- 8.1 Introduction.- 8.2 Antisymmetric wave function.- 8.3 Combination of spatial and spin functions.- 8.4 Representations of SNby the spatial functions ? jiS.- 9. Calculation of the Hamiltonian Matrix.- 9.1 Spin-free Hamiltonian.- 9.2 Branching diagram spin functions.- 9.3 The determinantal form of the wave function.- 9.4 Serber spin functions.- 9.5 Projected spin functions.- 9.6 Valence-bond spin function.- 9.7 Many-configuration wave functions.- 10. Spin-Coupled Functions.- 10.1 Introduction.- 10.2 Historical development.- 10.3 Spin-coupled wave functions.- 10.4 Spin-coupled valence-bond method.- 10.5 Core-valence separation.- 10.6 SPINS, computer program.- 11. Solutions to the Exercises.- 11.1 Chapter 1.- 11.2 Chapter 2.- 11.3 Chapter 3.- 11.4 Chapter 4.- 11.5 Chapter 5.- 11.6 Chapter 6.- 11.7 Chapter 7.- 11.8 Chapter 8.- 12. Index.
1. The Spin Operator and Spin Functions.- 1.1 Introduction.- 1.2 Spin operators and Pauli matrices.- 1.3 Spin functions.- 1.4 The Dirac identity.- 1.5 Square brackets of spin functions.- 1.6 Graphical representation of the primitive spin functions.- 2. Genealogical Construction of Spin Eigenfunctions.- 2.1 Addition theorem of angular momenta.- 2.2 Addition theorem of spin forNelectrons.- 2.3 The spin degeneracy.- 2.4 Branching diagram symbol.- 2.5 Special properties of the branching diagram functions.- 2.6 The coefficient of a primitive spin function in a given branching diagram function.- 3. Serber Spin Functions.- 3.1 Serber spin functions.- 3.2 Geminal spin product functions. Serber path diagram.- 3.3 Serber branching diagram.- 3.4 Recursion formulas for the highest component.- 3.5 Special properties of the Serber functions.- 3.6 The coefficient of a geminal product in a given Serber function.- 3.7 The algorithm of Carrington and Doggett.- 4. Projected Spin Eigenfunctions.- 4.1 Projection operator.- 4.2 Expanded form of the projected function (Löwdin).- 4.3 Projection of other primitive functions.- 4.4 Relation between the projected functions and the branching diagram functions.- 4.5 Calculation of the overlap matrix of the projected functions.- 5. Spin-Paired Spin Eigenfunctions.- 5.1 Spin-paired spin functions.- 5.2 The Rumer algorithm for the selection of linearly independent spin-paired spin functions.- 5.3 Extended Rumer diagrams (Simonetta).- 5.4 The leading term (Simonetta).- 5.5 Linear independence of spin-paired eigenfunctions.- 5.6 Overlap integrals between Rumer functions..- 6. The Symmetric Group.- 6.1 Basic notions of the symmetric group.- 6.2 Classes of the symmetric group.- 6.3 Representations of the symmetric group.- 6.4 Young tableaux.- 6.5Young's orthogonal representation.- 6.6 The conjugate representation.- 6.7 The symmetric group algebra.- 6.8 The Young operator.- 7. Representations of SNGenerated by Spin Eigenfunctions.- 7.1 Representations of the symmetric group generated by the branching diagram functions.- 7.2 Yamanouchi-Kotani method for the representations.- 7.3 Branching diagram functions and Young tableaux.- 7.4 Representations of SNgenerated by the projected spin functions.- 7.5 Calculations of A(P) by the Rettrup-Pauncz algorithm.- 7.6 Correspondence between spin-paired functions and Young tableaux.- 7.7 Generation of projected spin functions by Young operators.- 8. Combination of Spatial and Spin Functions.- 8.1 Introduction.- 8.2 Antisymmetric wave function.- 8.3 Combination of spatial and spin functions.- 8.4 Representations of SNby the spatial functions ? jiS.- 9. Calculation of the Hamiltonian Matrix.- 9.1 Spin-free Hamiltonian.- 9.2 Branching diagram spin functions.- 9.3 The determinantal form of the wave function.- 9.4 Serber spin functions.- 9.5 Projected spin functions.- 9.6 Valence-bond spin function.- 9.7 Many-configuration wave functions.- 10. Spin-Coupled Functions.- 10.1 Introduction.- 10.2 Historical development.- 10.3 Spin-coupled wave functions.- 10.4 Spin-coupled valence-bond method.- 10.5 Core-valence separation.- 10.6 SPINS, computer program.- 11. Solutions to the Exercises.- 11.1 Chapter 1.- 11.2 Chapter 2.- 11.3 Chapter 3.- 11.4 Chapter 4.- 11.5 Chapter 5.- 11.6 Chapter 6.- 11.7 Chapter 7.- 11.8 Chapter 8.- 12. Index.
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