Exact exchange-correlation potential of an ionic Hubbard model with a free surface

In Kohn-Sham density functional theory (DFT) the interacting electron problem is mapped into a noninteracting problem in an effective potential v(KS). It is known that the charge gap of the interacting system is different from the gap of the effective problem due to a jump Δxc in v(KS) when an electron is added but its magnitude and its role in the ubiquitous discrepancy between the experimental gaps and approximate DFT computations is poorly understood. Here we compute the exact v(KS) of a strongly interacting one-dimensional lattice model which can be driven from an ionic to a Mott insulating state. Presence of a "vacuum" region allows to determine the absolute value of v(KS). We show that in the ionic regime Δxc is determined by nearest-neighbor interaction, while in the Mott regime Δxc is determined by on-site Hubbard interaction.