Analysis of the density-gradient-expansion approximation for the exchange-correlation energy of density-functional theory

Abstract

To understand the density-gradient expansion approximation for the exchange-correlation energy of density-functional theory from a fundamental viewpoint, we have performed an analysis of the corresponding expansion of the Fermi–coulomb hole charge distribution. The Fermi–Coulomb hole represents the correlations between electrons resulting from the Pauli exclusion principle and Coulomb's law. The analysis is performed in the exchange-only approximation by considering the expansion for the Fermi hole to terms of O(▽3) as applied to atoms. Our study shows that the expansions to O(▽), O(▽2), and O(▽3) all severely violate the constraint of positivity, becoming progressively worse with increasing orders of ▽. Further, the expansion to O(▽2) also severely violates the constraint of charge neutrality. (Terms of O(▽) and O(▽3) do not contribute to this constraint or to the exchange energy.) Thus the description of the physics of Pauli correlations in atoms as given by this approximation is highly unphysical. In spite of this, the exchange energy to O(▽2) is superior to the local density approximation because the expansion hole better approximates the exact Fermi hole in the interior of atoms from which arise the principal contributions to the energy. However, the improvement is not substantial, as the oscillations in the expansion Fermi hole occur within the atom itself. For asymptotic positions of the electron, the expansion holes to each order neither approximate the local density approximation nor the exact Fermi hole. Thus we understand why the expansion cannot lead to accurate highest occupied eigenvalues. The oscillations of the expansion Fermi hole also demonstrate why the Slater potential and electric field that result from these hole charge distributions are singular. On the other hand, we show that the expansion approximation is mathematically consistent in that the coefficient of the gradient correction term for screened Coulomb interaction to O(▽2) as obtained from the approximate Fermi hole is the same as that derived from linear response theory. We conclude with remarks on the Coulomb hole as obtained within this gradient expansion approximation scheme.