Permutation Symmetry and theN-Electron Problem

The techniques of tensor algebra customarily applied to exploit spatial symmetry are applied to exploit permutational symmetry of the $N$-electron problem. In the approximation of no spin-orbit coupling, the results are nontrivial and give a further reduction of what is normally regarded as the reduced matrix element with respect to spatial symmetry alone. The required $3-j$ coefficients of the permutation group are evaluated in an appendix so that intermediate group-theoretical indices that have no direct physical significance are eliminated from the formulation. The spin integral for any operator can always be reduced to known integrals of the fundamental Pauli operators. Thus all matrix elements can be reduced to a corresponding spin-free form with known weighting coefficients. An explicit expression is given for the matrix element of an operator suitable for evaluating spin-own-orbit coupling or spin density at the nucleus. A recursion relation for the Clebsch-Gordan coefficients of bipartition representations of $S_N$ in terms of its subgroups and the $9-j$ sympols of $\mathrm{SU}\left(2\right)\mathrm{}$ is developed in the appendix. For one of the representations being the totally symmetric representation, the Clebsch-Gordan coefficient is known and the recursion relation (the group-orthogonality relation in this case) can be considered as giving nontrivial sum rules on the $9-j$ symbols of $\mathrm{SU}\left(2\right)\mathrm{}$.