Vector-coupling approach to orbital and spin-dependent tableau matrix elements in the theory of complex spectra

The power of the Young tableau scheme for labeling a complete spin-adapted basis set in the theory of complex spectra lies in one's ability to evaluate matrix elements of irreducible tensor operators directly in terms of the tableau labels and shapes. We show that the matrix-element rules stated by Harter for one-body operators can be easily derived from simple vector-coupling considerations. The graphical method of angular momentum analysis is used to derive closed-form expressions for the matrix elements of two-body operators. This study yields several interesting new relationships between spin-dependent operators and purely orbital operators.