Comment on “Significance of the highest occupied Kohn-Sham eigenvalue”

With more explanation than usual and without appeal to Janak's theorem, we review the statement and proof of the ionization potential theorems for the exact Kohn-Sham density-functional theory of a many-electron system: (1) For any average electron number $N$ between the integers $Z\ensuremath{-}1$ and $Z,$ and thus for $N\ensuremath{\rightarrow}Z$ from below, the highest occupied or partly occupied Kohn-Sham orbital energy is minus the ionization energy of the $Z$-electron system. (2) For $Z\ensuremath{-}1<N<Z,$ the exact Kohn-Sham effective potential ${v}_{s}(\mathbf{r})$ tends to zero as $|\mathbf{r}|\ensuremath{\rightarrow}\ensuremath{\infty}.$ We then argue that an objection to these theorems. [L. Kleinman, Phys. Rev. B 56, 12 042 (1997)] overlooks a crucial step in the proof of theorem (2): The asymptotic exponential decay of the exact electron density of the $Z$-electron system is controlled by the exact ionization energy, but the decay of an approximate density is not controlled by the approximate ionization energy. We review relevant evidence from the numerical construction of the exact Kohn-Sham potential. In particular, we point out a model two-electron problem for which the ionization potential theorems are exactly confirmed. Finally, we comment on related issues: the self-interaction correction, the discontinuity of the exact Kohn-Sham potential as $N$ passes through the integer $Z,$ and the generalized sum rule on the exchange-correlation hole.