Do Small Hole Polarons Form in Bulk Rutile TiO$_2$?

Hole transport and localization through small polarons is essential to the performance of TiO$_2$ in photocatalysis applications. The existence of small hole polaron in bulk rutile TiO$_2$ has been however controversial with contradicting evidences from theory and experiments. Here, we use first principles computations and more specifically a Koopmans' compliant hybrid functional and charge correction to study small hole polarons in bulk rutile. We find that a fraction of exchange exists satisfying Koopmans' compliance for the polaron state and reproducing the band gap provided that charge correction is used. We clearly show that first principles computations indicate that the small hole polaron is unstable in bulk for rutile and stable for anatase TiO$_2$.