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Canadian Journal of Physics

Nonlocal exchange contribution to the free energy of inhomogeneous many–fermion systems. I: Formulation of the gradient expansion

Ewa Dunlap, D. J. W. Geldart

Published Jan 1, 1994• 9 citations
Publisher

Abstract

Nonlocal exchange contributions to the free energy of a weakly inhomogeneous many-fermion system at finite temperature are obtained to first order in the interparticle interaction. Explicit representations are given for the second-order gradient coefficient Bx (n(r); T). These results apply for arbitrary velocity-independent interparticle interaction and arbitrary temperature.

DOI
10.1139/p94-001

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Nonlocal exchange contributions to the free energy of a weakly inhomogeneous many-fermion system at finite temperature are obtained to first order in the interparticle interaction. Explicit representations are given for the second-order gradient coefficient Bx (n(r); T). These results apply for arbitrary velocity-independent interparticle interaction and arbitrary temperature.
The coefficient Bx(T), which determines the leading gradient contribution to the nonlocal free energy of an inhomogeneous many-fermion system, has been evaluated exactly to first order in the interparticle interaction, in the case of pure Coulomb interactions. The result is valid for all temperatures and is expressed in terms of Fermi–Dirac integrals having known properties. Explicit analytical results are given for the degenerate and nondegenerate limits and numerical results are given for the intermediate regime of temperature and density. The limiting value of Bx(T) at T → 0 differs from previous evaluations of this quantity based on ground-state perturbation theory. The reasons for this discrepancy are discussed.
The exchange contributions to the ground-state energy of an inhomogeneous many-fermion system of arbitrary dimensionality are calculated. Explicit results are given for both the local-density approximation and the lowest-order gradient corrections for a class of interparticle interactions, including the Coulomb potential. For wider application, both the local-density approximation and the gradient corrections are decomposed into their wave-vector components. These exchange contributions can then be determined for arbitrary interparticle interaction by a simple quadrature.
We present a general scheme based on nonlinear response theory to calculate the expansion of correlation functions such as the pair-correlation function or the exchange-correlation hole of an inhomogeneous many-particle system in terms of density derivatives of arbitrary order. We further derive a consistency condition that is necessary for the existence of the gradient expansion. This condition is used to carry out an infinite summation of terms involving response functions up to infinite order from which it follows that the coefficient functions of the gradient expansion can be expressed in terms of the local density profile rather than the background density around which the expansion is carried out. We apply the method to the calculation of the gradient expansion of the one-particle density matrix to second order in the density gradients and recover in an alternative manner the result of Gross and Dreizler [Gross and Dreizler, Z. Phys. A 302, 103 (1981)], which was derived using the Kirzhnits method. The nonlinear response method is more general and avoids the turning point problem of the Kirzhnits expansion. We further give a description of the exchange hole in momentum space and confirm the wave vector analysis of Langreth and Perdew [Langreth and Perdew, Phys. Rev. B 21, 5469 (1980)] for this case. This is used to derive that the second-order gradient expansion of the system averaged exchange hole satisfies the hole sum rule and to calculate the gradient coefficient of the exchange energy without the need to regularize divergent integrals.
A simple alternative derivation of conventional cluster expansion formulas of Iwamoto and Yamada and others for the expectation value ≪H > of the energy of a many-Fermion system is offered. The assumed trial function is of the general form FΦp/(FΦp, FΦp)1/2, where Φp is a Slater determinant of appropriate single-particle functions and the factor F is attached to account for the possibility of strong interparticle correlations. The proposed scheme is applicable to finite as well as infinite systems and is easily generalized to the calculation of non-diagonal as well as diagonal matrix elements, with respect to the members of the set {FΦp/(FΦp, FΦp)1/2}. Explicit results for ≪H > are displayed to second cluster order, contributions being classified according to their assigned order of magnitude in a correlation parameter ξ=|Nω/Ω| of the sort introduced by Iwamoto and Yamada and Clark and Westhaus.
By performing Wang-Landau simulations in Fourier space, we study coarse grained free energies of generalized lattice spin models. The temperature dependence of the lowest order Landau-Ginzburg free energy expansion coefficients of the s4{s}^{4}s4 metamodel for phase transitions of the Ising universality class for a given coarse-graining length is calculated for the first time and compared to the standard assumptions made in the literature. In particular, the assumption that the so-called gradient coefficient is constant with temperature is in excellent agreement with the simulations. Our results also indicate that the coarse grained potentials follow finite size scaling. Finally, a coarse grained free energy with a temperature-dependent coarse-graining length equal to the system's correlation length is computed. The resulting potentials are compared to their mean-field counterparts.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.