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Modern Physics Letters B

EXACT KINETIC ENERGY FUNCTIONAL OF NONINTERACTING FERMIONS

Jiushu Shao

Published Aug 10, 1993• 13 citations
Publisher

Abstract

The exact expression of kinetic energy as a functional of the density for a system of noninteracting fermions, which has been searched for many years, is derived based on the direct method.
DOI
10.1142/s0217984993001193

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It is known in density-functional theory that the noninteracting kinetic-energy density functional ${T}_{s}[\ensuremath{\rho}]$ is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional ${T}_{N}[\ensuremath{\rho}],$ that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}).$ This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of ${T}_{N}[\ensuremath{\rho}],$ the full ${T}_{N}[\ensuremath{\rho}]$ in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define ${T}_{s}[\ensuremath{\rho}],$ the Lagrange multiplier that forces the orbitals to yield $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r})$ is not other than the first derivative of ${T}_{s}[\ensuremath{\rho}],$ $\ensuremath{\delta}{T}_{s}[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}),$ which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schr\"odinger equation can be derived from the basics of density-functional theory as well.