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Journal of Physics A Mathematical and General

On possible gradient approximations to the one-dimensional kinetic energy density functional compatible with the differential virial theorem

Jiushu Shao, R. Baltin

Published Dec 21, 1990• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/23/24/033

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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.
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.
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
The virial theorem is used to derive an exact hierarchy of coupled equations for the kinetic energy functional T[\ensuremath{\rho}] of an inhomogeneous system of noninteracting fermions. The hierarchy links the nth functional derivative of T[\ensuremath{\rho}] with respect to the particle density \ensuremath{\rho}(r) to the (n+1)th functional derivative and to \ensuremath{\rho}(r) for n=1,2,. . . . These functional derivatives are the analogs of the so-called ``direct correlation functions'' in the theory of classical fluids, and the hierarchy obtained resembles the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the distribution functions associated with such fluids. Some general properties of the two- and three-point correlation functions are described, in particular, their relations to the linear and quadratic response functions that describe the response of the inhomogeneous system to a change in the potential. The hierarchy is then used to obtain several equivalent expressions for T[\ensuremath{\rho}], one of which is an infinite series of successively higher-order terms, of which the nth term represents the contribution from the n-point correlation function and the densities associated with those points. The equations are then examined in the homogeneous (uniform) gas limit and the first and second equations are verified explicitly with the aid of the associated response functions. The analysis indicates that the infinite series expression converges rapidly for the three-dimensional gas; it also reveals that for both the one-dimensional and two-dimensional gas, the series terminates after the first few terms, suggesting that the structure of T[\ensuremath{\rho}] is much simpler in one and two dimensions than in three dimensions.