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Physical review. B, Condensed matter

Density-functional theory for two noninteracting spinless fermions

F. Aryasetiawan, Malcolm J. Stott

Published Sep 15, 1986• 48 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.34.4401

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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 vvv representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form e\ensuremath−ar{e}^{\ensuremath{-}ar}e\ensuremath−ar for two-particle systems are not ground-state vvv 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.
The v-representability of the particle density for a system of noninteracting fermions is considered. It is shown that any reasonable spherical density for two spinless fermions is either the nondegenerate ground-state density or a linear combination of degenerate ground-state densities for some effective local potential, and is therefore ensemble v-representable. The results of model calculations are given that illustrate various possible situations for two-particle systems. An extension of this result for systems of more particles is made and 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 E1{\mathit{E}}_{1}E1​(\ensuremath{\lambda}), E2{\mathit{E}}_{2}E2​(\ensuremath{\lambda}), and E3{\mathit{E}}_{3}E3​(\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.
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.
Sets of orbitals are considered, in each of which all the orbitals correspond either individually or pairwise to a single density. We address the questions of whether such orbitals can themselves be a set of solutions of a single, one-particle Schrödinger equation, or whether they can be linearly equivalent to such a set. This is possible for one or two particles, but not in general for more than two particles. Equations relating density, phase and potential in the two-particle case are obtained and discussed. The special form in which the phase is an integral of the density is not in general possible, but a particular density in one dimension for which such is the case is presented. We conclude that equidensity orbitals are rarely of physical significance.
A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.
The N-particle configurational probability density P(N) of a noninteracting Fermi gas may be approximated as P(N) ≃ Πi<je−u(rij) / ∫d 3NrΠi<je −u(rij), where u(r) is an effective pair potential. Our reason for considering these potentials is that they have been successfully used in computing properties of interacting Fermi systems. We define the best effective potential to be the one whose corresponding two-body distribution function is closest to the exact two-body distribution function. A method for calculating the best u(r) is not known. Lado has calculated u(r) by using the Born-Green-Yvon integrodifferential equation relating u(r) to the two- and three-body distribution functions. Effective potentials may also be calculated using the hypernetted chain and Percus-Yevick integral equations. It is found that potentials computed by the above three methods are in fairly close agreement. The principal purpose of this paper is to present tables of effective potentials approximating both the spin-summed system and parallel spin subsystem probability densities for a system of noninteracting spin-1/2 fermions.