The gradient expansion approximation for exchange: A physical prespective

Abstract

In recent work, we have provided a rigorous physical interpretation for the exchange energy and potential (or functional derivative) as obtained within the local-density approximation via the Harbola–Sahni formulation of many-electron theory. In this article, we analyze the gradient-expansion approximation (GEA) for these properties from the same physical perspective. The source charge distribution in this approximation is the GEA Fermi hole to O(▽3). This charge distribution is unphysical, so that the resulting force field and work done cannot be defined in a physically meaningful manner, and the exchange energy is singular. Thus, when viewed from the perspective of a source charge, the existence of the gradient expansions for the potential and energy is questionable. We next discuss the conventional method of employing a screened-Coulomb interaction to eliminate the singularities due to the GEA source charge, and show that it leads to inconsistent results. These inconsistencies are also intrinsic to a proof of the inequivalence of the Harbola–Sahni and Kohn–Sham exchange potentials within the GEA. Thus, although the inequivalence of these potentials has been established by other analyses, this proof is shown not to be rigorous. Finally, we demonstrate that when the physics of the GEA exchange source charge is corrected by the satisfaction of sum rules, the modified charge distribution then leads to a well-behaved local exchange potential and exchange energy density, and to a finite exchange energy. The consequences of our analysis on the gradient expansions for the correlation and exchange-correlation potential and energy are also noted. © 1992 John Wiley & Sons, Inc.