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Physical Review A

Wigner-Kirkwoodħexpansion of the linear-response function for inhomogeneous Fermi systems

Guy Chanfray, Peter Schuck

Published Nov 1, 1988• 8 citations
Publisher

Abstract

For the treatment of inhomogeneous Fermi systems we derive a semiclassical expansion of the linear-response function in powers of \ensuremath{\Elzxh} (up to ${\ensuremath{\Elzxh}}^{2}$) for the interacting (random-phase approximation) and noninteracting (Lindhardt function) case. We find that the series is rapidly converging and that the semiclassical result compares very well with exact model calculations for cases where the response does not exhibit narrow resonances (e.g., the quasielastic peak region); otherwise, it still represents the average value.

DOI
10.1103/physreva.38.4832

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For the treatment of inhomogeneous Fermi systems we derive a semiclassical expansion of the linear-response function in powers of \ensuremath{\Elzxh} (up to \ensuremath\Elzxh2{\ensuremath{\Elzxh}}^{2}\ensuremath\Elzxh2) for the interacting (random-phase approximation) and noninteracting (Lindhardt function) case. We find that the series is rapidly converging and that the semiclassical result compares very well with exact model calculations for cases where the response does not exhibit narrow resonances (e.g., the quasielastic peak region); otherwise, it still represents the average value.
Explicit kinetic-energy density functionals for noninteracting electron systems that are based on reproducing the linear response function in the homogeneous limit are the most accurate approximations to the exact Kohn-Sham functional. In this work we study several such models for one-dimensional systems. In this case, the geometrical simplicity allows the calculation of the static linear response resulting from these functionals for inhomogeneous systems. Likewise, the discussion of the energies and density profiles obtained is easier and clearer than that for more realistic problems. Special attention is paid to the application of explicit kinetic energy functionals to systems with odd and noninteger number of particles.
This paper presents a theory of the linear density response to an external time-dependent scalar potential for a Fermi system whose unperturbed density varies slowly. Simple local-density-functional response theory is valid except in a region of small qqq and $\ensuremath{\omega}$ (wave number and frequency of the perturbation). For this region we have worked out a generalization, to the case of inhomogeneous systems of Landau Fermi-liquid theory.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.
The semiclassical $\ensuremath{\hbar}$ expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of B. Grammaticos and A. Voros [Ann. Phys. (N.Y.) 123, 359 (1979)], originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virtue of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\ensuremath{\hbar}$ expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a finite, second-order gradient correction to the Hartree-Fock energy, which takes the form $\ensuremath{\varepsilon}{(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}/\sqrt{\ensuremath{\rho}}$, with $\ensuremath{\varepsilon}$ being small ($|\ensuremath{\varepsilon}|\ensuremath{\ll}1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.
We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory [Y. R. Shimizu, J. D. Garrett, R. A. Broglia, M. Gallardo, and E. Vigezzi, Rev. Mod. Phys. 61, 131 (1989)]. It is demonstrated that our formula is equivalent to a contour integral representation recently proposed [F. Donau, D. Almehed, and R. G. Nazmitdinov, Phys. Rev. Lett. 83, 280 (1999)] being numerically more efficient for realistic calculations. Examples are presented for pairing correlations in rapidly rotating nuclei.