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International Journal of Quantum Chemistry

Many‐fermion systems: Differential equations and kinetic energy functionals for different confining potentials

I. A. Howard, N. H. March

Published Nov 18, 2002• 2 citations
Publisher

Abstract

Abstract Determination of the kinetic energy functional is one of the central problems in the density functional theory of a many‐fermion system. We consider several many‐fermion systems to be characterized by simple confining potentials: the harmonic oscillator potential in one and three dimensions, the Coulomb potential, and the potential V(x) = −Uosech2(x/L). We discuss for each of these cases the differential equation that determines the fermion density in the given potential, and the corresponding kinetic energy density. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

DOI
10.1002/qua.10418

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Abstract Determination of the kinetic energy functional is one of the central problems in the density functional theory of a many‐fermion system. We consider several many‐fermion systems to be characterized by simple confining potentials: the harmonic oscillator potential in one and three dimensions, the Coulomb potential, and the potential V(x) = −Uosech2(x/L). We discuss for each of these cases the differential equation that determines the fermion density in the given potential, and the corresponding kinetic energy density. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
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.
Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 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.