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The Journal of Chemical Physics

Homogeneity and isotropy of space requisites for kinetic-energy functional derivatives

Daniel P. Joubert

Published Dec 1, 1994• 4 citations
Publisher

Abstract

Homogeneity and isotropy of space lead to two hierarchies of equations for the independent Fermion kinetic-energy functional Ts[n]. The hierarchies link the mth functional derivative of Ts[n] to the (m+1)th functional derivative. The constraints imposed on Ts[n] are different for nonlocal functionals but equivalent for local functionals. As a consequence of the first hierarchy, Ts[n] can be expressed in terms of any of its functional derivatives.

DOI
10.1063/1.467935

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Homogeneity and isotropy of space lead to two hierarchies of equations for the independent Fermion kinetic-energy functional Ts[n]. The hierarchies link the mth functional derivative of Ts[n] to the (m+1)th functional derivative. The constraints imposed on Ts[n] are different for nonlocal functionals but equivalent for local functionals. As a consequence of the first hierarchy, Ts[n] can be expressed in terms of any of its functional derivatives.
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Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
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.