Formal expressions and corresponding expansions for the exact Kohn-Sham exchange potential

Abstract

Formal expressions and their corresponding expansions in terms of Kohn-Sham (KS) orbitals are deduced for the exchange potential ${v}_{x}(\mathbf{r})$. After an alternative derivation of the basic optimized effective potential integrodifferential equations is given through a Hartree-Fock adiabatic connection perturbation theory, we present an exact infinite expansion for ${v}_{x}(\mathbf{r})$ that is particularly simple in structure. It contains the very same occupied-virtual quantities that appear in the well-known optimized effective potential integral equation, but in this new expression ${v}_{x}(\mathbf{r})$ is isolated on one side of the equation. An orbital-energy modified Slater potential is its leading term which gives encouraging numerical results. Along different lines, while the earlier Krieger-Li-Iafrate approximation truncates completely the necessary first-order perturbation orbitals, we observe that the improved localized Hartree-Fock (LHF) potential, or common energy denominator potential (CEDA), or effective local potential (ELP), incorporates the part of each first-order orbital that consists of the occupied KS orbitals. With this in mind, the exact correction to the LHF, CEDA, or ELP potential (they are all equivalent) is deduced and displayed in terms of the virtual portions of the first-order orbitals. We close by observing that the newly derived exact formal expressions and corresponding expansions apply as well for obtaining the correlation potential from an orbital-dependent correlation energy functional.