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Journal of Physics C Solid State Physics

An alternative interpretation of Theophilou's extension of the Hohenberg-Kohn theorem to excited states

Jacob Katriel

Published May 30, 1980• 29 citations
Publisher

Abstract

Theophilou's extension (ibid., vol.12, p.5419, 1979) of the Hohenberg-Kohn theorem to excited states is shown to follow from specialisation of the familiar Hohenberg-Kohn theorem to the ground state of an appropriately defined non-interacting super-Hamiltonian.
DOI
10.1088/0022-3719/13/15/001

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The starting point of this paper is a recent extension by Theophilou of the Hohenberg-Kohn-Sham (HKS) density-functional theory to ensembles of systems consisting of the M lowest eigenstates, equally weighted. As in the HKS theory the key quantities are the exchange-correlation energy, ExcM[n(r)]{E}_{\mathrm{xc}{}^{M}[\mathrm{n}(\mathrm{r})]}ExcM[n(r)]​, and potential, vxcM(r;[n(r′)]){v}_{\mathrm{xc}{}^{M}(\mathit{r};[\mathit{n}(\mathit{r}\mathcal{'})])}vxcM(r;[n(r′)])​. The present paper provides expressions for these quantities, valid for systems of slowly varying density. Even for such systems, however, there are essential nonlocal effects. Nevertheless both ExcM{E}_{\mathrm{xc}{}^{M}}ExcM​ and vxcM{v}_{\mathrm{xc}{}^{M}}vxcM​ can be calculated in terms of quantities characteristic of appropriate uniform thermal ensembles. This theory is the analog of the ground-state local-density approximation and allows calculation of excited-state energies and densities.
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 $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. 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 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 $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for NNN-representable density kernels, that any finite non-negative differentiable function is an NNN-representable particle density.