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

The ionization potential (IP)-theorem of Kohn-Sham (KS) density functional theory (DFT) states that the energy of the highest occupied molecular orbital (HOMO) ϵ_HOMO equals the negative of the first 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(r), far from the system, to be determined by HOMO only, behaving as n(r)∼r→∞e^-2-2ϵ. 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 the IP band structure in terms of DFT quantities is provided.