Asymptotic behaviour of the electron density and the Kohn–Sham potential in case of a Kohn–Sham HOMO nodal plane

ABSTRACT

It is known that the asymptotic decay of the electron density $n\left(\mathbf{r}\right)$ outside a molecule is informative about its first ionisation potential I₀, $n\left(\right|\mathbf{r}|\to \infty )\sim \text{exp}(-2\sqrt{2I_0}r)$. This dictates the orbital energy of the highest occupied Kohn–Sham (KS) molecular orbital (HOMO) to be ε_H = −I₀, if the KS potential goes to zero at infinity. However, when the KS HOMO has a nodal plane, the KS density in that plane will decay as $exp(-2\sqrt{-2{ϵ}_{H-1}}r)$. Conflicting proposals exist for the KS potential: from exact exchange calculations it has been found that the KS potential approaches a 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 negative. Here, we show that either (1) the exact density decays differently (according to the second ionisation potential I₁) in the HOMO nodal plane than elsewhere, and the KS potential has a regular asymptotic behaviour (going to zero everywhere) provided that ε_{H − 1} = −I₁; 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 behaviour around (at) the nodal plane.

GRAPHICAL ABSTRACT

Keywords:

• DFT
• Kohn–Sham potential
• HOMO nodal plane
• asymptotic density

Acknowledgments

It is our pleasure to dedicate this paper to Andreas Savin, who always enjoyed to discover strange features in the Kohn–Sham potential and has been a pioneer in asking fundamental questions in exact DFT.

We thank the Netherlands Science Foundation NWO for a visitors grant for Tamás Gál and a Vidi grant for Paola Gori-Giorgi, and the WCU (World Class University) program of the Korea Science and Engineering Foundation (Project No. R32-2008-000-10180-0) for support. Paola Gori-Giorgi acknowledges useful discussions with A. Görling and S. Kümmel.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The Netherlands Science Foundation NWO [grant number 700.59.428]; The WCU (World Class University) program: Korea Science and Engineering Foundation [Project Number R32-2008-000-10180-0].