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Physical Review A

Exact single-particle kinetic energy functional for general two-level and modeln-level one-dimensional systems: Dependence only on electron density and its gradient

Pawel Michal Kozlowski, N. H. March

Published Apr 1, 1989• 7 citations
Publisher

Abstract

From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ${t}_{s}$[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ${t}_{s}$[\ensuremath{\rho}] is explicitly obtained. Again ${t}_{s}$[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.

DOI
10.1103/physreva.39.4270

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Journal of Mathematical Physics•Aug 1, 2001•1 citation
From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ts{t}_{s}ts​[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ts{t}_{s}ts​[\ensuremath{\rho}] is explicitly obtained. Again ts{t}_{s}ts​[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.
When the kinetic energy density in k (defined positive definite) of a system of one-dimensional, non-interacting fermions is approximated by an ordinary function of the density rho , and of the lowest n derivatives of rho , in k >or=0 can satisfy the differential virial theorem for arbitrary density distributions only if in k actually depends on rho and rho ' only. Thus the case n>1 is ruled out, and one is left with the result in k = kappa rho 3 + rho ' 2 /(8 rho ) ( kappa >or=0) already obtained in previous work.
We consider the density functional theory from the point of view of one-dimensional two-particle systems, where the particles are either spin-1/2 fermions or spin-0 bosons. The basic equations are derived and applied to the system of two coupled harmonic oscillators. An exact density functional solution is constructed from the exactly known single-particle density for this system. We demonstrate how this solution may be applied to other two-particle systems to calculate the exact single-particle density and energy.
The differential virial theorem of March and Young for N fermions moving in a common one-dimensional potential energy V(x) is here combined with the Euler equation of density functional theory expressing the constancy of the chemical potential throughout the entire inhomogeneous particle density. The functional derivative of the single-particle kinetic energy is thereby expressed directly in terms of the kinetic energy density; a line integral being involved in establishing the connection. This result is then used to establish a formula measuring departures from simple homogeneity of the kinetic energy functional: a matter of current interest in density functional theory. Finally, the general theory of the functional derivative of the single-particle kinetic energy with respect to the particle density is exemplified for the case of harmonic confinement of fermions in one dimension.