Correlated optimized effective-potential treatment of the derivative discontinuity and of the highest occupied Kohn-Sham eigenvalue: A Janak-type theorem for the optimized effective-potential model

A Janak theorem is derived for the correlated optimized effective-potential model of the Kohn-Sham exchange-correlation potential $v_{\mathrm{xc}}.$ It is used to evaluate the derivative discontinuity (DD) and to show that the highest occupied Kohn-Sham eigenvalue, ${\epsilon }_{\mathrm{H}}\cong -I,$ the negative of the ionization potential, when relaxation and correlation effects are included. This reconciles an apparent inconsistency between the ensemble theory and fractional occupation number approaches to noninteger particle number in density-functional theory. For finite systems, ${\epsilon }_{\mathrm{H}}=-I$ implies that $v_{\mathrm{xc}}^{\infty }=0\mathrm{}$ independent of particle number, and that the DD vanishes asymptotically as $1/r.$ The difference in behavior of the DD in the bulk and asymptotic regions means that the DD affects the shape of $v_{\mathrm{xc}},$ even at fixed, integer particle number.