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Journal of Physics A Mathematical and General

Density and near-diagonal Dirac density matrix for closed shells of isotropic harmonically confined fermions in three dimensions

I. A. Howard, Norman Henry March

Published Sep 3, 2001• 11 citations
Publisher

Abstract

In early work, Lawes and March obtained a differential equation for an arbitrary number of independent harmonically confined fermions in one dimension, and very recently this result has been generalized to apply to three-dimensional (3D) isotropic harmonic confinement. Here, an exact solution of this 3D equation for the fermion particle density ρ( r ) is constructed, and the near-diagonal form of the Dirac density matrix is also obtained.

DOI
10.1088/0305-4470/34/36/101

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The determination of the Dirac density matrix of the d-dimensional harmonic oscillator for an arbitrary number of closed shells

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In 1959, March and Young ( Nucl. Phys. 12 237) rewrote the equation of motion for the Dirac density matrix γ( x , x 0 ) in terms of sum and difference variables. Here, γ( , 0 ) for the d -dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables ∣ + 0 ∣/2 and ∣ − 0 ∣/2, a generalized partial differential equation embracing the March–Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix γ( , 0 ) and the diagonal density ϱ( r ) = γ( , 0 )∣ 0 = , this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for d = 1 through 3. Some comments are finally made on the d -dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrödinger equations (the so-called Slater–Kohn–Sham equations).

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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 $1/d,$ (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.

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It is shown explicitly that the fermion particle density ϱ( r ) in d dimensions for isotropic harmonic confinement and odd d is determined by the one-dimensional wavefunctions of the highest occupied state. Knowledge of the Dirac density matrices for d = 1 and 2 suffices to completely determine the general d -dimensional matrices.

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Single-particle kinetic energy in density-functional theory: Harmonic confinement in two and three dimensions

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Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential $V(\mathbf{r})$, 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.

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Analytical expression for the first-order density matrix of ad-dimensional harmonically confined Fermi gas at finite temperature

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We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-$N$ limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.

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Comment on “Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement”

Sébastien Ragot

In a recent paper A. Akbari, N. H. March, and A. Rubio [Phys. Rev. A 76, 032510 (2007)] investigated the one-electron reduced density matrix and the momentum density of several two-electron model atoms, including Hooke's atom. The method used by the authors for deriving an integral form of the momentum density is in fact the most suited for deriving a closed-form expression of the exact reciprocal form factor, in a special case. This function is important inasmuch as it reflects the off-diagonal side of the exact reduced density matrix.

arXiv:0802.2561•Physical Review A•Jul 22, 2008•4 citations
In early work, Lawes and March obtained a differential equation for an arbitrary number of independent harmonically confined fermions in one dimension, and very recently this result has been generalized to apply to three-dimensional (3D) isotropic harmonic confinement. Here, an exact solution of this 3D equation for the fermion particle density ρ( r ) is constructed, and the near-diagonal form of the Dirac density matrix is also obtained.
In 1959, March and Young ( Nucl. Phys. 12 237) rewrote the equation of motion for the Dirac density matrix γ( x , x 0 ) in terms of sum and difference variables. Here, γ( , 0 ) for the d -dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables ∣ + 0 ∣/2 and ∣ − 0 ∣/2, a generalized partial differential equation embracing the March–Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix γ( , 0 ) and the diagonal density ϱ( r ) = γ( , 0 )∣ 0 = , this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for d = 1 through 3. Some comments are finally made on the d -dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrödinger equations (the so-called Slater–Kohn–Sham equations).
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.
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.
We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-NNN limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.
In a recent paper A. Akbari, N. H. March, and A. Rubio [Phys. Rev. A 76, 032510 (2007)] investigated the one-electron reduced density matrix and the momentum density of several two-electron model atoms, including Hooke's atom. The method used by the authors for deriving an integral form of the momentum density is in fact the most suited for deriving a closed-form expression of the exact reciprocal form factor, in a special case. This function is important inasmuch as it reflects the off-diagonal side of the exact reduced density matrix.