Hohenberg-Kohn theorem for nonlocal external potentials

The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel (x, x^'), but probably not of the particle density n (r) =s^ (rs, rs). The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists i. e. , if the functional derivative (x, {x^') } exists for a nondegenerate ground-state density kernel, then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals (0<〈₈||₈〉<1) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for N-representable density kernels, that any finite non-negative differentiable function is an N-representable particle density.