  We perform first-principles calculations of the quasiparticle defect states,
charge transition levels, and formation energies of oxygen vacancies in rutile
titanium dioxide. The calculations are done within the recently developed
combined DFT+GW formalism, including the necessary electrostatic corrections
for the supercells with charged defects. We find the oxygen vacancy to be a
negative U defect, where U is the defect electron addition energy. For the
values of Fermi level below 2.8 eV (relative to the valence band maximum) we
find the +2 charge state of the vacancy to be the most stable, while above 2.8
eV we find that the neutral charge state is the most stable.
