Exchange potentials in density-functional theory

The Harbola-Sahni exchange potential is the work needed to move an electron against the electric field of its hole charge distribution. We prove that it is not the exact exchange potential of density-functional theory, by showing that it yields the wrong second-order gradient expansion in the slowly varying limit. But we also discover that it yields the correct local-density approximation. Thus the Harbola-Sahni potential is a more physically correct version of the Slater potential, one that is better suited for molecular and solid-state applications. As a step in our derivation, we present the third-order gradient expansion of the exchange hole density, and discuss its structure. We also describe a new version of the Harbola-Sahni potential which corrects its path dependence. The exact exchange potential for an atom is given by the optimized potential model (OPM) of Talman and Shadwick. By using enhanced numerics, we confirm that the OPM potential satisfies the Levy-Perdew virial relation and exhibits correct -1/r behavior at large r. Numerical calculations also show that the intershell maxima in the exact exchange potential are needed to lower the total energy. These ``bumps'' are missing from the Harbola-Sahni and Slater potentials.