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Physical review. B, Condensed matter

Nonlocal exchange energy of many-fermion systems of arbitrary dimensionality and interparticle interaction

Godfrey A. Gumbs, D. J. W. Geldart

Published Nov 15, 1986• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.34.6847

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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.
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.
A density-functional theory is developed for fermions in one dimension, interacting via a $\ensuremath{\delta}$ function. Such systems provide a natural testing ground for questions of principle, as the local-density approximation should be highly accurate since for this interaction type the exchange contribution to the local-density approximation is intrinsically self-interaction-free. The exact-exchange contribution to the total energy is a local functional of the density. A local-density approximation for correlation is obtained using perturbation theory and Bethe ansatz results for the one-dimensional contact-interacting uniform Fermi gas. The ground-state energies are calculated for two finite systems, the analogs of helium and of Hooke's atom. The local-density approximation is shown to be excellent as expected.
We introduce density-functional theory for inhomogeneous Bose-Fermi mixtures, derive the associated Kohn-Sham equations, and determine the exchange-correlation energy in local-density approximation. We solve numerically the Kohn-Sham system, and determine the boson and fermion density distributions and the ground-state energy of a trapped, dilute mixture beyond mean-field approximation. The importance of the corrections due to exchange correlation is discussed by a comparison with current experiments; in particular, we investigate the effect of the repulsive potential-energy contribution due to exchange correlation on the stability of the mixture against collapse.
We explicitly calculate the four leading-order terms of the formal asymptotic series for large |ε| (where ε denotes the external energy parameter) of the single-particle Green function G σ(ε), σ ϵ {−s, −s + 1, …, s}, and the three leading-order terms of that of the self-energy operator ∑σ(ε), pertaining to systems of spin-s fermions (s is a half-integer) in d-dimensional spatial space, interacting through an arbitrary two-body potential v(r-r'). These contributions, which are expressed in terms at most of a three-body static ground-state correlation function, are amenable to accurate numerical calculation through employing correlated many-particle ground-state wavefunctions such as determined within, for example, the quantum Monte Carlo framework. Such calculations will provide indisputably reliable information with regard to the spectral function of the single-particle excitations at high energies of correlated systems, as well as energy moments of this spectral function, and thus help to assess the reliability of theoretical approaches that are applied in studying such systems. We give especial attention to d = 3 and v ≡ v C, the long-range Coulomb potential, for which case we explicitly calculate the five leading-order terms of the regularized large-|ε| asymptotic series of ∑σ(ε). Our considerations reveal some interesting aspects which are very specific to the behaviour of v C(r-r') at both small and large values of ||r-r'||. In particular, we show that in these systems an inhomogeneity, even on the atomic scale, in the particle spin polarization density gives rise to a pronounced effect, directly discernible in the inverse photoemission spectra; we show this effect to be absent in models with v bounded at origin and those in which v ≡ v C but Umklapp processes are neglected, unless the ground state possesses long-range magnetic order. Our analyses shed light on the importance of the non-local part of the self-energy operator and disclose that some of the strictly non-local contributions to this operator transform into local contributions upon replacing v ≢ v C by v C, implying that a local approximation to ∑σ(ε), which for v ≢ v C proves accurate, is necessarily less accurate for v ≡ v C. These findings establish a fundamental limitation of the so-called ‘dynamical mean-field’ approximation to ∑σ(ε), which is strictly local, specifically in applications where v ≡ v C. We further explicitly establish some of the shortcomings of ∑σ(ε) as calculated within the framework of the many-body perturbation theory. In this context we demonstrate the empirically well-known inadequacy of the dynamically screened exchange self-energy operator in particular for describing the photoemission and inverse photoemission spectra of interacting systems at intermediate and large transfer energies and put forward a workable scheme that rids this self-energy of its fundamental defects. We present ample explicit analyses of our results in terms of uncorrelated many-body wavefunctions.
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.