A slightly modified version of the $X\ensuremath{\alpha}$ scattered-wave method, generalized to include relativistic Darwin and mass-velocity corrections to the energy, has been applied to clusters of up to 27 atoms (through and including third-nearest neighbors about some arbitrarily chosen lattice site) representing bulk PbS. Calculations have been carried out both on clusters representing the ideal solid (no defects) and on clusters containing either a lead or sulfur vacancy at the center. The effects of the vacancy are deduced via a direct comparison of the ideal and cluster results, avoiding the convergence problems associated with perturbative techniques. The effect of electronic relaxation has been accounted for by using the cluster wave functions to calculate a new potential, using the $X\ensuremath{\alpha}$ local-exchange approximation, and iterating until the self-consistent-field limit has been reached. The results of this investigation are discussed and compared to the PbTe work of Parada and of Pratt, based on the non-self-consistent Koster-Slater method. In the case of a lead vacancy, the two approaches are shown to yield a reasonably consistent picture: each lead vacancy produces two holes in the valence band, forming a $p$-type sample, and no impurity states are introduced into the fundamental-gap region. There seems to be no localization of charge about the vacancy site, and the scattering is expected to be much weaker than that of an ionized acceptor. The two pictures diverge somewhat, however, in the case of a chalcogen vacancy. While Parada finds no states to be introduced into the fundamental-gap region by the creation of a tellurium vacancy, the present calculation indicates the presence of a bound state below the conduction-band edge due to a sulfur vacancy. The two electrons associated with the sulfur vacancy seem to be localized to within a lattice constant or so of the defect site. It is expected that scattering by the sulfur vacancy should be much stronger than that predicted for a tellurium vacancy by Parada and Pratt.