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 $\mu (x,x^{\prime })$, but probably not of the particle density $n\left(\overrightarrow{\mathrm{r}}\right)=\Sigma s^{}\mu (\overrightarrow{\mathrm{r}}s,\overrightarrow{\mathrm{r}}s)$. 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 Schrödinger-like form, and contains only local potentials. The latter may be obtained directly in Schrödinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\delta E}{\delta \mu (x,x^{\prime })}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<\mathrm{}〈{\phi }_i\left|\mu \right|{\phi }_i〉<1\mathrm{}$) 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.

• Received 10 June 1974

DOI:https://doi.org/10.1103/PhysRevB.12.2111