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arXiv:cond-mat/0602067 · Journal of Physics B Atomic Molecular and Optical Physics

Exploring foundations of time-independent density functional theory for excited states

Prasanjit Samal, Manoj K. Harbola

Published Oct 2, 2006• 27 citations
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Abstract

Based on the work of Görling and that of Levy and Nagy, a density-functional formalism for many fermionic excited states is explored through a careful and rigorous analysis of the excited-state density to external potential mapping. It is shown that knowledge of the ground-state density is a must to fix the mapping from an excited-state density to an external potential. This is the excited-state counterpart of the Hohenberg–Kohn theorem, where instead of the ground-state density the density of the excited state gives the true many-body wavefunctions of the system. Further, the excited-state Kohn–Sham system is defined by comparing its non-interacting kinetic energy with the true kinetic energy. The theory is demonstrated by studying a large number of atomic systems.

DOI
10.1088/0953-4075/39/20/005

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Based on the work of Görling and that of Levy and Nagy, a density-functional formalism for many fermionic excited states is explored through a careful and rigorous analysis of the excited-state density to external potential mapping. It is shown that knowledge of the ground-state density is a must to fix the mapping from an excited-state density to an external potential. This is the excited-state counterpart of the Hohenberg–Kohn theorem, where instead of the ground-state density the density of the excited state gives the true many-body wavefunctions of the system. Further, the excited-state Kohn–Sham system is defined by comparing its non-interacting kinetic energy with the true kinetic energy. The theory is demonstrated by studying a large number of atomic systems.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.
The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of 4He^{4}\mathrm{He}4He atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.
The analysis in this paper shows that the Hohenberg-Kohn theorem is the constellation of two statements: (i) the mathematically rigorous Hohenberg-Kohn lemma, which demonstrates that the same ground-state density cannot correspond to two different potentials of an external field, and (ii) the hypothesis of the existence of the universal density functional. Based on the obtained explicit expression for the nonrel-ativistic particle energy in a local external field, we prove that the energy of the system of more than two non-interacting electrons cannot be a functional of the inhomogeneous density. This result is generalized to the system of interacting electrons. It means that the Hohenberg-Kohn lemma cannot provide justification of the universal density functional for fermions. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect. In the framework of the density matrix functional theory, the hypothesis of the existence of the universal density matrix functional corresponds to the cases of noninteracting particles and to interaction in the Hartree-Fock approximation.
The formalism of Kohn and Sham uses a specific (model) hamiltonian which highly simplifies the many-electron problem to that of noninteracting fermions. The theorem of Hohenberg and Kohn tells us that, for a given ground state density, this hamiltonian is unique. In principle, this density can be chosen as that of the real, interacting system. To obtain the energy, or other properties of the real system, approximations are needed. Working with non interacting fermions is an important simplification, but it may be easier to produce approximations with different choices of the model hamiltonian. The feature that the exact density is (ideally) reproduced can be kept in the newly defined fictitious systems. Using model hamiltonians having the same form as the physical one, that is, being built of one- and two-body operators, allows to approach the physical hamiltonian arbitrarily close, and thus a systematic reduction of the approximations.
The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.
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.