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Canadian Journal of Physics

Nonlocal exchange contribution to the free energy of inhomogeneous many-fermion systems. II: Second-order gradient coefficient for a Coulomb system

M. L. Glasser, D. J. W. Geldart, Ewa Dunlap

Published Jan 1, 1994• 9 citations
Publisher

Abstract

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.

DOI
10.1139/p94-002

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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 consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion TTT matrix, which we determine to second order in the boson-fermion sss-wave scattering length. We also discuss how to incorporate the usual boson-boson TTT matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to aBFkF.{a}_{\mathrm{BF}}{k}_{\mathrm{F}}.aBF​kF​. The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of 6Li{}^{6}\mathrm{Li}6Li and 7Li{}^{7}\mathrm{Li}7Li and of 3He{}^{3}\mathrm{He}3He and 4He.{}^{4}\mathrm{He}.4He. Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length aBF.{a}_{\mathrm{BF}}.aBF​.
We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose-Einstein condensation. 1
We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first-order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose–Einstein condensation.
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.