Most density-functional studies of defects in semiconductors invariably use (i) a supercell that imitates the host crystal, as well as (ii) a local treatment of the exchange-correlation potential. The first approximation has had an enormous success in many materials, provided that the size of cell is large enough to minimize long-range interactions between the infinite lattice of defects. For instance, these may arise from strain fields or from the overlap between shallow defect states. The second approximation, when combined with the periodic boundary conditions, leads to an essentially metallic density of states in Ge, which can compromise any investigation of electronic transitions involving gap levels. Here, we report on two approaches to surmount these difficulties, namely (i) to open the gap by reducing the host to a Ge cluster of atoms whose states are confined by a surface potential and (ii) to use supercells, but choosing carefully the Brillouin zone sampling scheme, taking k-points that minimize the admixture between defect-related gap states and the host density of states. These methods are utilized in the calculation of the electronic structure of the vacancy, divacancy, and vacancy-donor pairs in Ge.