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Physical Review A

Electron densities in search of Hamiltonians

Mel Levy

Published Sep 1, 1982• 692 citations
Publisher

Abstract

By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}$ are ground-level densities for interacting or noninteracting Hamiltonians ${H}_{1}, {H}_{2},\dots{},{H}_{M}$ ($M$ arbitrarily large) with local potentials ${v}_{1}$,${v}_{2}$,$\dots{}$,${v}_{M}$, but given that we do not know which $\ensuremath{\rho}$ belongs with which $H$, the correct mapping is possible and is obtained by minimizing $\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]$ with respect to optimum permutations of the $\ensuremath{\rho}$'s among the $v$'s. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, $a$ density will not generally be associated with any ground-state wave function (is not wave function $v$ representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-$v$-representable as well as for $v$-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-$v$ representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-$v$ representability for some Kohn-Sham noninteracting systems.

DOI
10.1103/physreva.26.1200

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By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that \ensuremathρ\ensuremathα,\ensuremathρ\ensuremathβ,…,\ensuremathρ\ensuremathω{\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}\ensuremathρ\ensuremathα​,\ensuremathρ\ensuremathβ​,…,\ensuremathρ\ensuremathω​ are ground-level densities for interacting or noninteracting Hamiltonians H1,H2,…,HM{H}_{1}, {H}_{2},\dots{},{H}_{M}H1​,H2​,…,HM​ (MMM arbitrarily large) with local potentials v1{v}_{1}v1​,v2{v}_{2}v2​,…\dots{}…,vM{v}_{M}vM​, but given that we do not know which \ensuremathρ\ensuremath{\rho}\ensuremathρ belongs with which HHH, the correct mapping is possible and is obtained by minimizing \ensuremath∫dr\ensuremath→[v1(r\ensuremath→)\ensuremathρ\ensuremathα(r\ensuremath→)+v2(r\ensuremath→)\ensuremathρ\ensuremathβ(r\ensuremath→)+\ensuremath⋯vM(r\ensuremath→)\ensuremathρ\ensuremathω(r\ensuremath→)]\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]\ensuremath∫dr\ensuremath→[v1​(r\ensuremath→)\ensuremathρ\ensuremathα​(r\ensuremath→)+v2​(r\ensuremath→)\ensuremathρ\ensuremathβ​(r\ensuremath→)+\ensuremath⋯vM​(r\ensuremath→)\ensuremathρ\ensuremathω​(r\ensuremath→)] with respect to optimum permutations of the \ensuremathρ\ensuremath{\rho}\ensuremathρ's among the vvv's. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, aaa density will not generally be associated with any ground-state wave function (is not wave function vvv representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-vvv-representable as well as for vvv-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-vvv representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-vvv representability for some Kohn-Sham noninteracting systems.
On the non-interacting vvv-representability conjecture within density functional theory
Density functional theory together with the Kohn-Sham (KS) scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary system (one-body with a local potential that is a density functional). However, to fully achieve establishing the KS scheme, a general proof of the validity of the so-called vvv-representability conjecture is still required. In this article, we propose such a proof. We reduce the demonstration to proving (1) that the KS potential is differentiable for all non-interacting vvv-representable densities and (2) that its derivative is bounded. Then, we demonstrate points (1) and (2) applying static linear response to the non-interacting system.
Exact density-functional theory is recast here as two parallel exact ensemble variational hierarchies: an interacting hierarchy rooted in Lieb's ensemble formulation and a noninteracting hierarchy rooted in exact noninteracting ensemble theory. In optimization terms, NNN-representability is primal feasibility, Legendre-Fenchel duality equates the primal and dual values, vvv-representability is dual attainment, and the Hohenberg-Kohn theorem gives uniqueness, modulo constants, of an attained local potential. The Kohn-Sham construction couples the interacting density-space optimality condition to a compatible noninteracting dual realization on their common NNN-representable density domain. State-class restrictions yield the Levy-Lieb and single-determinant branches, while fractional particle number and fractional occupations lead naturally to piecewise linearity, one-sided chemical potentials, Janak-type relations, and the derivative discontinuity. This organization locates the exactness of Kohn-Sham theory in the preservation of the interacting density-space optimization together with its compatible noninteracting realization, without implying a general many-body spectral interpretation of Kohn-Sham eigenvalues.
Aiming to combine density functional theory (DFT) and wave-function theory, we study a mapping from the many-body interacting system to an effectively interacting Kohn-Sham system instead of a noninteracting Kohn-Sham system. Because a ground state of effectively interacting systems requires having a solution for the correlated many-body wave functions, this provides a natural framework to many-body wave-function theories such as the configuration interaction and the coupled-cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems---namely, the one-dimensional helium atom, the hydrogen molecule, and the heteronuclear diatomic molecule---we investigate properties of many-body wave functions and exact exchange-correlation potentials of effectively interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact noninteracting Kohn-Sham system, namely, a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit, can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively interacting Kohn-Sham scheme.