Direct analytical calculation of first-order density matrix elements through third order

A theory which permits the analytical calculation of atomic and molecular first-order density matrices in a computationally tractable manner is developed. Contour integral techniques are used to derive an equation relating the first-order density matrix γ to certain eigenvectors which arise in our earlier theory of molecular ionization potentials and electron affinities. By analytically evaluating the resulting contour integral, we obtain a closed expression giving γ in terms of Hartree-Fock orbital energies and two-electron integrals.