Calculation of charge transition levels of oxygen vacancies in rutile TiO$_2$ using the GW method*

Titanium dioxide (TiO$_2$) is a semiconductor displaying photovoltaic and photocatalytic properties and is widely used in numerous technological applications, such as solar cells, hydrolysis of water, photocatalysis, etc. Defects are important for optical properties of TiO$_2$, e.g, they play a crucial role in photocatalytic reactions. Gaining a deep understanding of the influence of intrinsic defects, such as vacancies and interstitials, on the properties of this material is highly desirable. In spite of many theoretical and experimental investigations of defects in TiO$_2$, controversies still remain. In this work, the charge transition levels of oxygen vacancies are studied in the rutile form of TiO$_2$. For this purpose the GW method is employed for the calculation of the band gap and the position of the defect levels in the gap. The effects of lattice relaxations for systems with various charge states are taken into account. This is done with the help of hybrid functionals, since LDA or PBE mean fields do not describe defect levels properly, which may result in incorrect lattice relaxations. To account for spurious effects due to periodic images of the charged defects, appropriate electrostatic correction techniques are used.