Effect of on-site Coulomb repulsion termUon the band-gap states of the reduced rutile (110)TiO2surface

Abstract

We present a study concerning the effect of the on site $d\text{\ensuremath{-}}d$ Coulomb interaction energy $U$ on the band-gap states of nonstoichiometric rutile (110) $\mathrm{Ti}{\mathrm{O}}_{2}$ surface. As well known, the excess electrons resulting from the formation of oxygen vacancies localize on the $\mathrm{Ti}\phantom{\rule{0.2em}{0ex}}3d$ orbitals forming band-gap states. Local density approximation (LDA) does not give a correct description of these band-gap states, either with or without gradient corrections. The failure of LDA is often attributed to an inadequate treatment of electron correlation in systems with localized orbitals and is commonly corrected with an empirical local Coulomb repulsion term, i.e., the $\mathrm{LDA}+U$ method. This study provides a completely general strategy to estimate the $U$ value in this kind of systems, illustrated here for reduced (110) $\mathrm{Ti}{\mathrm{O}}_{2}$ surface, well characterized from experiments. From ab initio embedded cluster configuration interaction calculations, combined with the effective Hamiltonian theory, a value of $U$ of $5.5\ifmmode\pm\else\textpm\fi{}0.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ is obtained, in good agreement with those reported for this system from x-ray photoemission spectroscopy experiments $(U=4.5\ifmmode\pm\else\textpm\fi{}0.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV})$. It is observed that when the ab initio estimate of $U$ is injected into the periodic $\mathrm{LDA}+U$ calculations, a correct description of the gap states is obtained from the periodic $\mathrm{LDA}+U$ calculations. Additionally, the results indicate that the position of these states on the band gap strongly depends on the level at which lattice relaxation is taken into account, with significant differences between the density of states curves at the $\mathrm{LDA}+U$ level obtained using the optimal generalized gradient approximation or $\mathrm{LDA}+U$ geometries. These results suggest that this combined strategy could be a useful tool for those systems where electron correlation plays a key role, and no experimental data are available for the on-site Coulomb repulsion.