Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy

Abstract

Recent arguments for and against the equivalence of the highest occupied orbital eigenvalue of the Kohn-Sham theory and ionization energy are discussed. It is shown that for physically realistic systems with a nonintegral number of electrons, which are described by the thermal average of two systems, each with an integer number of electrons, an equivalent Kohn-Sham system exists. This is done by writing explicit expressions for the exchange-correlation potential constructed to give the mixed-state density, and then relating it to the mixed-state exchange-correlation energy functional by employing the virial theorem sum rule of Levy and Perdew [Phys. Rev. A 32, 2010 (1985)]. Further, the functional derivative of the mixed-state exchange-correlation energy functional is obtained in terms of this potential. This is then used to show, without recourse to Janak's theorem [Phys. Rev. B 18, 7165 (1978)], that ${\ensuremath{\varepsilon}}_{\mathrm{max}}(N)=\ensuremath{-}I(Z),$ where Z is an integer and $(Z\ensuremath{-}1)lNlZ.$ Thus the original arguments about the equivalence of the highest occupied Kohn-Sham orbital eigenenergy and the ionization energy which were based on Janak's theorem are valid, and the two quantities are equal.