  This work explores the Zn vacancy in ZnO using hybrid density functional
theory calculations. The Zn vacancy is predicted to be an exceedingly deep
polaronic acceptor that can bind a localized hole on each of the four
nearest-neighbor O ions. The hole localization is accompanied by a distinct
outward relaxation of the O ions, which leads to lower symmetry and reduced
formation energy. Notably, we find that initial symmetry-breaking is required
to capture this effect, which might explain the absence of polaronic hole
localization in some previous hybrid density functional studies. We present a
simple model to rationalize our findings with regard to the approximately
equidistant thermodynamic charge-state transition levels. Furthermore, by
employing a one-dimensional configuration coordinate model with parameters
obtained from the hybrid density functional theory calculations, luminescence
lineshapes were calculated. The results show that the isolated Zn vacancy is
unlikely to be the origin of the commonly observed luminescence in the visible
part of the emission spectrum from \textit{n}-type material, but rather the
luminescence in the infrared region.
