Application of the Lany-Zunger polaron correction for calculating surface charge trapping

Defect calculations, using density functional theory in a local or semi-local approximation, in transition metal oxides are severely handicapped by the electron self-interaction error. The underestimation of the band gap may cause incorrect occupation of defect states and wrong formation energies, and the underestimated localization of the states disfavors the formation of small polarons. These problems can be avoided by using higher level approximations (GW or a correctly chosen hybrid functional), but those methods are computationally too expensive to be used for calculating surface defects in a periodic slab model. Lany and Zunger have suggested a convenient (low-cost) solution for solving the band-gap and charge delocalization problem, by applying a correction scheme to the standard local or semi-local approximations. Most importantly, the linearity of the total energy as a function of the fractional occupation numbers, is restored, leading to the fulfillment of the generalized Koopmans' theorem. The method works well in the bulk but, as we show here, it is not accurate on the surface due to the different screening environment. We also show that by making the atom- and angular-momentum dependent parameters of the Lany-Zunger polaron-correction also coordination dependent, it is possible to correctly describe charge trapping in small polaron states on the anatase (1 0 1) and rutile (1 1 0) surfaces at a low computational cost.