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The Journal of Chemical Physics

Alternative expressions for the Fermi hole curvature

John F. Dobson

Published Jun 1, 1993• 104 citations
Publisher

Abstract

The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero separation r′=r for a pair of Fermions in a many-Fermion system. It has been used by a number of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, it is shown that a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, it is shown that a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-Fermion state and is not limited to the single-determinant case investigated previously.

DOI
10.1063/1.464444

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The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero separation r′=r for a pair of Fermions in a many-Fermion system. It has been used by a number of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, it is shown that a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, it is shown that a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-Fermion state and is not limited to the single-determinant case investigated previously.
Using two different interpretations of the concept of the Fermi hole curvature, the author derives a class of correlation energy expressions for the inhomogeneous interacting electron gas. Their properties include freedom from spurious orbital self-interaction, invariance under unitary transformations among occupied orbitals, correct values in the homogeneous limit, correctly normalized correlation hole, inclusion of kinetic energy (KE) as well as potential energy of correlation, and non-vanishing values for fully spin-polarized systems (in contrast with some similar schemes developed for chemical applications). Minimization of the energy with respect to the orbitals leads to a Euler-Lagrange equation resembling the Hartree-Fock one-electron effective Schrodinger equation, with the addition of a term resembling the KE operator for an inhomogeneous effective mass. For current-carrying states there is a further term involving an effective dynamically induced vector potential. Despite these complications the effective one-electron Hamiltonian is Hermitian, so that the canonical orbitals are orthogonal, in contrast with those of the commonest self-interaction correction scheme.
A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.
The metallic phase of strongly correlated electrons is discussed within a novel statistical approach that correctly reproduces the principal features of the Mott insulating state when localization is reached at half-filled band (n=1). The entropy in the normal phase decomposes into a spinless-fermion part representing holes, and a part describing localized moments. The explicit form of the gap ${\ensuremath{\Delta}}_{\mathrm{k}}$ is derived; this solution is shown to disappear when n=1. Most of the results cannot be obtained by using the Fermi-Dirac distribution for single-particle states.
The equations which determine the one-particle energy and effective two-body interaction in an interacting Fermi gas are constructed within the approximation which sums up all pair creation-annihilation processes. The equation corresponds to the familiar equation for the KKK matrix which represents the interaction between particles (or holes) and sums up the particle-particle (or hole-hole) scattering processes. The method of the equation of motion is used in this paper. Our result for the one-particle energy is shown to lead to the result previously obtained by Quinn and Ferrell, and by Rockmore for the case of the electron gas with Coulomb interactions, when we replace screened potentials by bare potential in the self-consistent energy equation. For nuclear matter, it is shown that the presence of an attractive interaction in the equation of motion for number density causes an "enhancement" of exchange forces, whereas in the electron gas repulsive Coulomb interactions lead to "screening" of the exchange force. The strength of the isospin density interaction pseudopotential is enhanced by a factor of two when one solves the self-consistent equation; and a simple estimate shows that the Goldhaber-Teller mode lies about 15% higher than the value pFqm\frac{{p}_{F}q}{m}mpF​q​ previously estimated by Glassgold et al. (qqq: momentum of the oscillation, pF{p}_{F}pF​: Fermi momentum).
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.
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.