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 vKS. It is known that the charge gap of the interacting system is different from the gap of the effective problem due to a jump ?xc in vKS 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 vKS 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 vKS. We show that in the ionic regime ?xc is determined by nearest-neighbor interaction, while in the Mott regime ?xc is determined by on-site Hubbard interaction.