Possibilities for a density matrix theory

Two possible routes are considered to arrive at a one-particle reduced density matrix formulation of electronic structure theory. In the first scheme, an extended Fock matrix H is defined that has twice the dimension of the one-particle basis set. The corresponding Green’s function, defined as the upper left block of $\mathrm{(\omega }{1}-{H}{)}^{\text{−1}},$ yields the exact one-particle density matrix and energy. The poles of the Green’s function are precisely the ionization potentials and electron affinities of the extended Koopmans theorem. In the second scheme, a generalized Fock equation $[{F}\mathrm{(\rho ),\rho ]=}{X}$ is derived that is satisfied by the exact non-idempotent one-particle density matrix. The antisymmetric matrix X on the right-hand side is obtained from the irreducible part of the two-particle reduced density matrix, while F is the usual Fock matrix defined using the correlated one-matrix. The generalized Fock equation is a necessary condition but does not determine ρ uniquely. Alternatively, the one-matrix can be obtained from the irreducible part of the two-matrix directly, using a sum rule. The analysis leads to some additional desiderata and separability properties that may be imposed on traditional wave function based approaches. Possibilities for practical computational schemes are addressed briefly.