Simpler expression for evaluating the density matrix in the self-consistent Green's-function method

In a recent publication describing a self-consistent Green's function for use in studying defects in solids, one of the major steps was evaluating the density matrix ${\rho }_{\mathrm{mn}}$ which describes the valence charge density in a basis of localized orbitals. We have since discovered an algebraic identity which considerably simplifies that evaluation, and we here present the simplified form for ${\rho }_{\mathrm{mn}}$.