Calculation of permutation matrices using graphical methods of spin algebras: Explicit expressions for the Serber-coupling case

Abstract

We present a general method, based on the diagrammatic techniques of spin algebras, for the calculation of permutation matrices of two-rowed (two-columned) irreducible representations of the symmetric group ${\mathcal{S}}_N$ relative to a basis (or bases) adapted to the subgroup(s) ${{\mathcal{S}}_N}_A\otimes {\mathcal{S}}_N\subset {\mathcal{S}}_N$ or to (a) subgroup chain(s) which may be generated by a recursive application of the above-given chain. These matrix elements are needed in spin-adapted configuration interaction calculations. This general technique is applied to the Serber coupling scheme, and general and explicit closed formulas are obtained for the matrix elements of an arbitrary transposition. An extension of this formalism to cases with an arbitrarily large frozen core is also outlined using the same technique. A computer program based on these derivations was written and its effectiveness compared with that of other approaches is discussed. An extension of these applications to more complex permutations than transpositions, as well as to other coupling schemes, is also briefly discussed.