Theorem for exact local exchange potential

Consider a local exchange potential v${\mathrm{¯}}_{\mathit{x}}$(r) which depends on a set of orbitals {${\mathrm{\phi }}_{\mathit{i}}$}. It is proven that v${\mathrm{¯}}_{\mathit{x}}$ is the exact Kohn-Sham density-functional exchange potential, when {${\mathrm{\phi }}_{\mathit{i}}$} is optimum, if and only if ${\mathit{v}}_{\mathit{x}}$ satisfies three specified conditions. Exchange potentials are presented which satisfy two of the conditions, one of which is satisfied neither by the original Slater potential nor by the local-density approximation.