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

TL;DR

This paper investigates the asymptotic decay of electron density and the behavior of the Kohn-Sham potential in molecules with a HOMO nodal plane, revealing conditions under which the potential remains regular or becomes irregular.

Contribution

It clarifies the asymptotic behavior of the Kohn-Sham potential and electron density in the presence of a HOMO nodal plane, resolving conflicting proposals and identifying conditions for regularity.

Findings

Density decays according to the second ionization potential in the HOMO nodal plane if the potential is regular.

The Kohn-Sham potential approaches zero at infinity when the second orbital energy equals the negative of the second ionization potential.

Irregular or divergent behavior of the KS potential occurs if the density decay is uniform but the potential is not regular.

Abstract

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 $\epsilon_H=-I_0$, 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 $\exp{(-2\sqrt{-2\epsilon_{H-1}}\,r)}$. 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 $\epsilon_{H-1}=-I_1$; 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