  We perform full-potential screened-hybrid density-functional theory (DFT)
calculations to compare the thermodynamic stability of neutral and charged
states of the surface oxygen vacancy at the rutile TiO$_2$(110) surface.
Solid-state (QM/MM) embedded-cluster calculations are employed to account for
the strong TiO$_2$ polarization response to the charged defect states. Similar
to the situation for the bulk O vacancy, the +2 charge state $V_{\rm O}^{2+}$
is found to be energetically by far most stable. Only for Fermi-level positions
very close to the conduction band, small polarons may at best be trapped by the
charged vacancy. The large decrease of the $V_{\rm O}^{2+}$ formation energy
with decreasing Fermi-level position indicates strongly enhanced surface O
vacancy concentrations for $p$-doped samples.
