Coefficients of fractional parentage in the L–S coupling scheme

An efficient procedure for the evaluation of the coefficients of fractional parentage (cfp’s) for L–S coupled wave functions is presented. The cfp’s are calculated separately for N particles, each with angular momentum l (s), coupled into a total angular momentum L (S). The N-particle states formed can belong to any permutational symmetry. The procedure for the evaluation of the L and the S cfp’s for arbitrary permutational symmetry is a generalization of the procedure proposed by Bayman and Lande [Nucl. Phys. 77, 1 (1966)] for symmetric and antisymmetric states. It involves the construction and diagonalization of the matrices representing the quadratic Casimir operators for the appropriate special unitary and symplectic (or orthogonal) groups. The cfp’s of the antisymmetric L–S coupled states are obtained in terms of products of cfp’s for L and S corresponding to conjugate representations of the symmetric group. This method is demonstrated to provide cfp’s for L–S states for systems with a considerably larger number of particles than is feasible using the procedures heretofore available.