A full perturbation treatment of the Dirac density matrix is developed as a basis for self-consistent calculations in free-electron metals containing localized defects. The present perturbation series is shown to sum to the well-known result for the case of slowly varying potentials. To first order in perturbation theory, exact self-consistent results for the radial density of displaced charge and the Hartree potential in the presence of point singularities have been obtained over a density range sufficient to cover all metals under normal conditions. The basic limitations of Mott's first-order method, based on the assumption of slowly varying potentials, are shown to be completely removed and the self-consistent density and potential display long-range oscillations. Finally, the application of the present approach to Bloch wave functions rather than plane waves is briefly considered. Friedel's generalized first-order method, applicable to a band structure, may be obtamed from the present theory for sufficiently slowly varying potentials. Unfortunately, such an assumption is seen by comparison with the free-electron findings to lead to serious errors.