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Europhysics Letters (EPL)

Single-Particle Kinetic Energy for N Fermions in One-Dimension: General and Model Results

Pawel Michal Kozlowski, Norman Henry March

Published Apr 1, 1990• 3 citations
Publisher

Abstract

It is shown by combining the Euler equation of density functional theory with the differential form of the virial theorem that the single-particle kinetic-energy density T s [ρ], on the minimum, and for N fermions, is a function solely of ρ, ρ' and x . This formal result is illustrated by taking the example of a harmonic oscillator with N levels filled. Model forms of T s [ρ] and δ T s [ρ]/δρ then follow.

DOI
10.1209/0295-5075/11/7/006

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It is shown by combining the Euler equation of density functional theory with the differential form of the virial theorem that the single-particle kinetic-energy density T s [ρ], on the minimum, and for N fermions, is a function solely of ρ, ρ' and x . This formal result is illustrated by taking the example of a harmonic oscillator with N levels filled. Model forms of T s [ρ] and δ T s [ρ]/δρ then follow.
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.
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.
In earlier work, expressions have been constructed for the single-particle kinetic-energy functional ${T}_{s}[\ensuremath{\rho}]$ for independent fermions subject to harmonic confinement in low dimensions, with $\ensuremath{\rho}$ the particle density. Here, the differential equation for $\ensuremath{\rho}$ is first obtained in d dimensions for an arbitrary number of closed shells. Then, by using the known Euler-Lagrange equation, the functional derivative $\ensuremath{\delta}{T}_{s}/\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r})$ is constructed. ${T}_{s}[\ensuremath{\rho}]$ itself is proved to take the form of a linear combination of three pieces: (i) a von Weizs\"acker inhomogeneity kinetic energy, but with the original coefficient reduced by a dimensionality factor 1/d,1/d,1/d, (ii) a Thomas-Fermi kinetic energy, and (iii) a truly nonlocal contribution that, however, is shown to involve only the density $\ensuremath{\rho}$ itself and its first derivative. Thus, for this model, which is currently highly relevant to the interpretation of experiments on the evaporative cooling of dilute, and hence almost noninteracting, fermions, a complete density-functional theory now exists.
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.