Breakdown of the ionization potential theorem of density functional theory in mesoscopic systems

TL;DR

This paper demonstrates that the ionization potential theorem of density functional theory fails in mesoscopic systems due to their different asymptotic electron density behavior, leading to the proposal of an alternative IP band structure concept.

Contribution

It reveals the breakdown of the IP-theorem in mesoscopic systems and introduces the concept of an observable IP band structure as an alternative to the KS eigenvalues.

Findings

IP-theorem does not hold for mesoscopic systems like Q2D electron gas.

The asymptotic electron density behavior differs in mesoscopic systems, causing the violation.

A practical method to determine the IP band structure from DFT is proposed.

Abstract

The IP-theorem of the Kohn-Sham (KS) density functional theory (DFT) states that the energy of the highest occupied molecular orbital (HOMO) $\epsilon_{HOMO}$ equals the negative of the first ionization potential (IP), thus ascribing a physical meaning to one of the eigenvalues of the KS hamiltonian. We scrutinize the fact that the validity of the IP-theorem relies critically on the electron density $n({\bf r})$, far from the system, to be determined by HOMO only, behaving as $n({\bf r}) \underset{r\to\infty}{\sim} e^{- 2 \sqrt{-2 \epsilon_{HOMO}} r}$. While this behavior always holds for finite systems, it does not hold for mesoscopic ones, such as quasi-two-dimensional (Q2D) electron gas or Q2D crystals. We show that this leads to the violation of the IP-theorem for the latter class of systems. This finding has a strong bearing on the role of the KS valence band with respect to the work-function problem in the mesoscopic case. Based on our results, we introduce a concept of the IP band structure as an observable alternative to its unphysical KS counterpart. A practical method of the determination of IP band structure in terms of DFT quantities is provided.