On the average charge of the oxygen vacancy in perovskites necessary for   kinetic calculations
  An analytical result has been obtained for the value of the dynamical charge necessary for calculations of oxygen vacancy kinetics in dielectric perovskite-type crystals. It is shown by using the Berry phase analysis that this charge equals the nominal charge of the vacancy: for example, for the double charged state, it is 2; a neutral vacancy has the zero charge. 
S.A. Prosandeev