| Abstract: | Techniques of diagrammatic spin algebra are employed to derive segment factorization formulas for spin-adapted matrix elements of one- and two-electron excitation operators. The spin-adapted basis is formed by the Yamanouchi–;Kotani geneological coupling method, and therefore constitutes an irreducible basis of the unitary group U(N), as prescribed by Gel'fand and Tsetlin. Several features distinguish this paper from similar work that has recently been published. First, intermediate steps in the derivation of each segment factor are fully documented. Comprehensive tables list the spin diagrams and phases that contribute to the possible segment factors. Second, a special effort has been made to distinguish between those parts of a segment factor that can be ascribed to a spin diagram and those parts which arise from the orbitals. The results of this paper should thus be useful for those who wish to extend diagrammatic spin algebra to evaluation of matrix elements for states built from nonorthogonal orbitals. Third, a novel graphical method has been introduced to keep track of phase changes that are induced by line up permutations of creation and annihilation operators. This technique may be useful for extension of our analysis to higher excitations. The necessary concepts of second quantization and diagrammatic spin algebra are developed in situ, so the present derivation should be accessible to those who have little prior knowledge of such methods. |
