The role of symmetry in representing reduced density operators and reduced transition density operators: General formulation with specific application to atomic systems

The p-particle reduced transition (density) matrix between a properly chosen given member of one degenerate manifold and a properly chosen given member of a (possibly) different manifold of eigenstates of a nonrelativistic atomic Hamiltonian contains sufficient information to construct the p-particle reduced transition (density) matrix between any member of the first manifold and any member of the second manifold. A synthetic procedure for this construction is presented in this article and is completely spelled out for p = 1. Application of these ideas to the more general symmetries of a molecular system or of an atom in a crystal field is outlined. This procedure is particularly useful when one has made a calculation of the reduced density matrix of the state belonging to one symmetry species within a degenerate manifold and desires the reduced density matrix of another state [or the reduced transition (density) matrix between two states] within the same manifold.