  We studied the oxygen vacancies($V_{O}$) in rutile TiO2 by using $G_0W_0$
approximation on top of GGA+$U$ as a method of choice to improve the gap. Since
there is no extensive agreement regarding the characteristic of electron
localization for $ TiO_2$, we examine combined $G_0W_0$@GGA+$U$ scheme in which
both are conceptually one step toward better enumeration of the non locality of
exchange-correlation potential . Our $G_0W_0$@GGA+$U$ results realize and
confirm the weak nature of electron correlation in rutile $TiO_2$ and shows
that the $U$-dependence of the energy gap in perfect bulk is slightly stronger
than in defected sample. In addition, we studied the $U$- dependency of $V_O$
defect states and found that different charged vacancies shows different
$U$-dependence . While the application of $G_0W_0$ correction would improve the
quasiparticle gap and formation energies, however the $V_O$ states, in contrast
to experiment, remains entangled with the conduction band. Finally, we used
PBE0 and HSE06 hybrid functionals and found that these exchange-correlation
functional particularly HSE06 that reproduced the real gap and provide desire
description of the screening, properly disentangle the neutral and singly
ionized $V_O$ from the conduction band . According to hybrid functional
calculations, all vacancies are stabilized and $V_O$ with 2+ charge state is
the most stable vacancy in the whole Fermi-level range inside the gap and hence
$V_O$ acts as a shallow donor.
