Systematic approximations to the optimized effective potential: Application to orbital-density-functional theory

The integral equation originally derived by Sharp and Horton for the optimized effective potential (OEP) is exactly transformed into an equivalent form from which it is manifestly clear that the OEP, ${\mathit{V}}_{\mathit{x}\mathrm{\sigma }}^0$(r), is an implicit functional of only {${\mathit{n}}_{\mathit{i}\mathrm{\sigma }}$}, the orbital densities of the occupied states {${\mathrm{\psi }}_{\mathit{i}\mathrm{\sigma }}$}, and the corresponding single-particle exchange potentials {${\mathit{v}}_{\mathit{i}\mathrm{\sigma }}$}. Furthermore, the transformed OEP has exactly the same form as one recently developed by the authors [Phys. Rev. A 45, 101 (1992)] from a more heuristic approach, the only difference being that in the present work a term proportional to the gradient of ${\mathit{n}}_{\mathit{i}\mathrm{\sigma }}$ is added to each ${\mathit{v}}_{\mathit{i}\mathrm{\sigma }}$ whose average value when taken over the iσ state is zero. This result leads to the natural development of an iterative approximation for ${\mathit{V}}_{\mathit{x}\mathrm{\sigma }}^0$, with the zeroth approximation being given by our previous result. The application of this technique to the calculation of the total energy and highest-energy single-particle eigenvalue for selected atoms is presented. In addition, we note that our results are applicable to the calculation of the OEP for any assumed exchange-correlation functional ${\mathit{E}}_{\mathrm{xc}}$[{${\mathrm{\psi }}_{\mathit{i}\mathrm{\sigma }}$}], where ${\mathit{v}}_{\mathit{i}\mathrm{\sigma }}$ is taken as the appropriate functional derivative of ${\mathit{E}}_{\mathrm{xc}}$. In the case that ${\mathit{E}}_{\mathrm{xc}}$ is a functional of {${\mathit{n}}_{\mathit{i}\mathrm{\sigma }}$} only, as in the case of the local-density approximation with self-interaction correction, the resulting ${\mathit{V}}_{\mathrm{xc}\mathrm{\sigma }}^0$ is a functional of the {${\mathit{n}}_{\mathit{i}\mathrm{\sigma }}$} only.