Physical Review A

v-representability for noninteracting systems

Jiqiang Chen, Malcolm J. Stott

Published Jan 1, 1993• 9 citations

Abstract

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.

DOI: 10.1103/physreva.47.153

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Physical Review A

v-representability for noninteracting systems

Jiqiang Chen, Malcolm J. Stott

Published Jan 1, 1993• 9 citations

Publisher

Abstract

DOI: 10.1103/physreva.47.153

Related papers

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Jiqiang Chen, Malcolm J. Stott

We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if ${\mathit{E}}_{1}$ (\ensuremath{\lambda}), ${\mathit{E}}_{2}$ (\ensuremath{\lambda}), and ${\mathit{E}}_{3}$ (\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if ${\mathit{E}}_{1}$(\ensuremath{\lambda}), ${\mathit{E}}_{2}$(\ensuremath{\lambda}), and ${\mathit{E}}_{3}$(\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.

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F. Aryasetiawan, Malcolm J. Stott

A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of $v$ representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state $v$ representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of $v$ representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state $v$ representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.

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On the non-interacting $v$ -representability conjecture within density functional theoryOn the non-interacting $v$-representability conjecture within density functional theory

J. Messud

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 $v$ -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 $v$ -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.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 $v$-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 $v$-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.

Feb 6, 20210 citations

We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if

{\mathit{E}}_{1}

(\ensuremath{\lambda}),

{\mathit{E}}_{2}

(\ensuremath{\lambda}), and

{\mathit{E}}_{3}

(\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.

v

-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.

v

representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state

v

representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.

On the non-interacting

v

-representability conjecture within density functional theory

v

-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

v

-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.