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Journal of Physics A Mathematical and General

The Pauli potential in one-dimensional density functional theory: general result for two-level systems and specific example forNharmonically confined Fermions

I. A. Howard, N. H. March

Published Jun 5, 2003• 3 citations
Publisher

Abstract

There is ongoing interest in the kinetic energy functional T s [ρ] in density functional theory. The present study lies in this area and concerns the Pauli potential V P [ρ]. A differential equation is obtained here for V P ( x ) in one dimension for a general two-level system. Also, as a specific example, such a functional of ρ( x ), the ground-state Fermion density, is given for the case of N Fermions which are harmonically confined.

DOI
10.1088/0305-4470/36/24/103

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There is ongoing interest in the kinetic energy functional T s [ρ] in density functional theory. The present study lies in this area and concerns the Pauli potential V P [ρ]. A differential equation is obtained here for V P ( x ) in one dimension for a general two-level system. Also, as a specific example, such a functional of ρ( x ), the ground-state Fermion density, is given for the case of N Fermions which are harmonically confined.
Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential V(r)V(\mathbf{r})V(r), Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.
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.
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.