  Cu$_2$O has appealing properties as an electrode for photo-electrochemical
water splitting, yet its practical performance is severely limited by
inefficient charge extraction at the interface. Using hybrid DFT calculations,
we investigate carrier capture processes by oxygen vacancies (V$_\mathrm{O}$)
in the experimentally observed ($\sqrt{3} \times \sqrt{3}$)R30$^{\circ}$
reconstruction of the dominant (111) surface. Our results show that these
V$_\mathrm{O}$ are doubly ionized and that associated defects states strongly
suppress electron transport. In particular, the excited electronic state of a
singly charged V$_\mathrm{O}$ plays a crucial role in the non-radiative
electron capture process with a capture coefficient of about 10$^{-9}$~cm$^3$/s
and a lifetime of 0.04~ps, explaining the experimentally observed ultrafast
carrier relaxation. These results highlight that engineering the surface
V$_\mathrm{O}$ chemistry will be a crucial step in optimizing Cu$_2$O for
photoelectrode applications.
