Asymptotic behavior of the electron density and the Kohn-Sham potential in case of a Kohn-Sham HOMO nodal plane

It is known that the asymptotic decay of the electron density $n(\br)$ outside a molecule is informative about its first ionization potential $I_0$, $n(|\br|\to\infty) \sim \text{exp}(-2\sqrt{2I_0}\,r)$. This dictates the orbital energy of the highest occupied Kohn-Sham (KS) molecular orbital (HOMO) to be $ε_H=-I_0$, if the KS potential goes to zero at infinity. However, when the Kohn-Sham HOMO has a nodal plane, the KS density in that plane will decay as $\exp{(-2\sqrt{-2ε_{H-1}}\,r)}$. Conflicting proposals exist for the KS potential: from exact exchange calculations it has been found that the KS potential approaches a {\em positive} constant in the plane, but from the assumption of isotropic decay of the exact (interacting) density it has been concluded this constant needs to be {\em negative}. Here we show that either 1) the exact density decays differently (according to the second ionization potential $I_1$) in the HOMO nodal plane than elsewhere, and the KS potential has a regular asymptotic behavior (going to zero everywhere) provided that $ε_{H-1}=-I_1$; or 2) the density does decay like $\text{exp}(-2\sqrt{2I_0}\,r)$ everywhere but the KS potential exhibits strongly irregular if not divergent behavior around (at) the nodal plane.11 pages, 5 figures