Positron annihilation at vacancies in semiconductors is studied. The authors are interested in the formation of singlet and triplet positron-electron states, and especially, in their influence on the positron annihilation characteristics. These states are formed by a positron trapped by the vacancy and an unpaired (paramagnetic) electron in a deep level. They provide examples by determining the two ensuing positron lifetime components for vacancies in Si and GaAs. For the practical calculations it is shown how the self-consistent electron structures obtained by the pseudopotential-plane wave methods can be used. Finally, the implications are discussed from the experimental point of view.