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The Journal of Chemical Physics

The effect of discretization on a path integral expression for the one-electron density

Gary G. Hoffman

Published Mar 1, 1990• 3 citations

Abstract

Using an integral expression which is based on the use of discretized path integrals, the one-particle density matrix is evaluated for individual fermions in a one- and three-dimensional harmonic well. In this regard, the present paper improves and extends work previously done by Handler. By deriving a closed form expression for the path integral with finite discretization, the density is expressed as a one-dimensional integral that can be evaluated numerically. In addition, the trace of this density matrix, which corresponds to the number of particles present as a function of the Fermi energy, is evaluated in closed form. For finite discretization, the function is continuous in the Fermi energy, but approaches the correct discontinuous limit as the discretization becomes large. Oscillations are observed at the discontinuities analogous to the Gibbs phenomenon of Fourier analysis. Calculations of the density show a marked improvement in the classically allowed region when the discretization is only one level beyond the simple Thomas–Fermi theory. The density also penetrates into the classically forbidden region as the discretization is increased, but the presence of negative oscillations requires a higher level of discretization for a proper description in this region.

DOI: 10.1063/1.457892

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Using an integral expression which is based on the use of discretized path integrals, the one-particle density matrix is evaluated for individual fermions in a one- and three-dimensional harmonic well. In this regard, the present paper improves and extends work previously done by Handler. By deriving a closed form expression for the path integral with finite discretization, the density is expressed as a one-dimensional integral that can be evaluated numerically. In addition, the trace of this density matrix, which corresponds to the number of particles present as a function of the Fermi energy, is evaluated in closed form. For finite discretization, the function is continuous in the Fermi energy, but approaches the correct discontinuous limit as the discretization becomes large. Oscillations are observed at the discontinuities analogous to the Gibbs phenomenon of Fourier analysis. Calculations of the density show a marked improvement in the classically allowed region when the discretization is only one level beyond the simple Thomas–Fermi theory. The density also penetrates into the classically forbidden region as the discretization is increased, but the presence of negative oscillations requires a higher level of discretization for a proper description in this region.

A new approach has been used to determine directly the reduced single particle density matrix for noninteracting fermions in a one-dimensional potential. A time evolution operator closely related to the classical limit of the Feynman path integral propagator is derived which has superior behavior in turning point and nonclassical regions. This is used to determine the reduced density matrix in terms of the phase and time of oscillation at the Fermi energy. The density so obtained has the correct quantum oscillations and nonclassical tails and is a distinct improvement over the Thomas-Fermi density. Two simple examples are presented and implications for electronic structure calculations are discussed.

One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution

{n}_{0}(z,N).

For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near

{2k}_{F},

with

{k}_{F}

being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.

Numerical evaluation of functional integrals usually involves a finite ( L -slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L -slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as L (β) = [ 1 (β/ L )] L , if the density matrix 1 (β) in the static approximation is known. We investigate the convergence of the partition function Z L (β) ≡ Tr L (β), the internal energy and the density of states g L ( E ) (the inverse Laplace transform of Z L ), as L → ∞ . For the simple harmonic oscillator, g L ( E ) is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-½ dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, Z L is an even function of the coupling constant for L < 3: ferromagnetic and antiferromagnetic coupling can be distinguished only for L ⩾3, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as L → ∞ .

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.

Using the discretized path integral formalism we develop a numerically convenient and accurate method for evaluating the finite time propagator (and density matrix) of a given system. The eigenfunctions and energies of large number of states of the system are also found. We demonstrate the accuracy of the method by computing eigenvalues, eigenfunctions, position and flux autocorrelation functions and complex time propagators for a large number of systems and comparing these results with those available from other sources. In all the cases we find very good agreement. The method thus provides a very simple and accurate procedure for the study of static and dynamic properties of a quantum system.