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Kinetic energy functional for Fermi vapors in spherical harmonic confinement

Anna Minguzzi, A N. H. March B, M. P. Tosi A

Published Jan 1, 2001• 13 citations
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Abstract

Two equations are constructed which reflect, for fermions moving independently in a spherical harmonic potential, a differential virial theorem and a relation between the turning points of kinetic energy and particle densities. These equations are used to derive a differential equation for the particle density and a non-local kinetic energy functional.

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Exact and asymptotic local virial theorems for finite fermionic systems

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We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper (Brack and Murthy 2003 J. Phys. A: Math. Gen. 36 1111), some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper, we show that these local virial theorems (LVTs) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although formally they are only valid asymptotically for large particle numbers N, we show that practically they are surprisingly accurate also for moderate values of N.

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

Pawel Michal Kozlowski, Norman Henry March

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Europhysics Letters (EPL)•Apr 1, 1990•3 citations
Two equations are constructed which reflect, for fermions moving independently in a spherical harmonic potential, a differential virial theorem and a relation between the turning points of kinetic energy and particle densities. These equations are used to derive a differential equation for the particle density and a non-local kinetic energy functional.
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.
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.
We investigate the particle and kinetic-energy densities for a system of NNN fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers NNN, we show that they practically are surprisingly accurate also for moderate values of NNN.
We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper (Brack and Murthy 2003 J. Phys. A: Math. Gen. 36 1111), some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper, we show that these local virial theorems (LVTs) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although formally they are only valid asymptotically for large particle numbers N, we show that practically they are surprisingly accurate also for moderate values of N.