Comparison between the adspace and cluster-extended Green's-function techniques for localized defects

The adspace technique of Williams, Feibelman, and Lang [Phys. Rev. B 26, 5433 (1982)] and the cluster-extended Green's-function method of Baraff, Schlüter, and Allan [Phys. Rev. B 27, 1010 (1983)] are two solutions to the problem of including extra orbital flexibility into the Green's function $G^0$ of the perfect crystal which is used for the calculation of localized defects. We compare the two methods theoretically and find that, for a small extra computational effort, the cluster-extended technique, which is an approximation, can be converted into the adspace method, which is exact. Numerical comparison of total defect energies illustrate the effects of the approximation and the superiority of the adspace method.