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Physical Review Letters

v -Representability and Density Functional Theory

Walter Kohn

Published Oct 24, 1983• 86 citations
Publisher

Abstract

It is shown that if $n(r)$ is the discrete density on a lattice (enclosed in a finite box) associated with a nondegenerate ground state in an external potential $v(r)$ (i.e., is "$v$-representable"), then the density $n(r)+\ensuremath{\mu}m(r)$, with $m(r)$ arbitrary (apart from trivial constraints) and $\ensuremath{\mu}$ small enough, is also associated with a nondegenerate ground state in an external potential ${v}^{\ensuremath{'}}(r)$ near $v(r)$; i.e., $n(r)+\ensuremath{\mu}m(r)$ is also $v$-representable. Implications for the Hohenberg-Kohn variational principle and the Kohn-Sham equations are discussed.

DOI
10.1103/physrevlett.51.1596

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It is shown that if n(r)n(r)n(r) is the discrete density on a lattice (enclosed in a finite box) associated with a nondegenerate ground state in an external potential v(r)v(r)v(r) (i.e., is "vvv-representable"), then the density n(r)+\ensuremathμm(r)n(r)+\ensuremath{\mu}m(r)n(r)+\ensuremathμm(r), with m(r)m(r)m(r) arbitrary (apart from trivial constraints) and \ensuremathμ\ensuremath{\mu}\ensuremathμ small enough, is also associated with a nondegenerate ground state in an external potential v\ensuremath′(r){v}^{\ensuremath{'}}(r)v\ensuremath′(r) near v(r)v(r)v(r); i.e., n(r)+\ensuremathμm(r)n(r)+\ensuremath{\mu}m(r)n(r)+\ensuremathμm(r) is also vvv-representable. Implications for the Hohenberg-Kohn variational principle and the Kohn-Sham equations are discussed.
In density-functional theory (DFT) of the ground state, a density distribution no{n}_{o}no​(r) is called v-representable (VR) if it is the ground-state density in some external potential. [It is known that not all ``reasonable'' n0{n}_{0}n0​(r) are VR.] In DFT of time-dependent linear response of a nondegenerate ground state, a similar question arises: Is a response density, n1{n}_{1}n1​(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$, VR, i.e., is it the response to some perturbing potential v1{v}_{1}v1​(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ [E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)]? In the present paper we show that (1) if the frequency \ensuremath{\omega}${\ensuremath{\omega}}_{\mathrm{min}}$ (the lowest excitation frequency), the answer is affirmative; and (2) if \ensuremath{\omega}>${\ensuremath{\omega}}_{\mathrm{min}}$, the answer is not necessarily affirmative, as demonstrated by counterexamples. [We exhibit ``reasonable'' functions n1{n}_{1}n1​(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ which, at isolated frequencies, are not VR.] Implications for time-dependent DFT of linear response are discussed.
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.
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A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.