An analytical result was obtained for the value of the dynamical charge necessary for calculations of O vacancy kinetics in dielectric perovskite-type crystals. It was shown, by using the Berry phase analysis, that this charge equalled the nominal charge of the vacancy: for example, for the doubly charged state (no electrons associated with the vacancy), it was 2e where e was the electron charge; a neutral vacancy (two electrons in the vacancy) had zero average charge.

On the Average Charge of the Oxygen Vacancy in Perovskites Necessary for Kinetics Calculations. S.A.Prosandeev: Journal of Physics - Condensed Matter, 2002, 14[49], L745-8