Comparison of the embedding and Dyson-equation methods in the Green’s-function calculation of a defect in solids

We discuss the relationship between the Dyson-equation method of Wachutka et al. [J. Phys. Condens. Matter 4, 2831 (1992)] and the embedding method of Inglesfield [J. Phys. C 14, 3795 (1981)] in the Green’s-function calculation of a defect in solids. We will show that if the Green’s function is expanded using the same basis set, the Green’s-function matrix of the embedding method, $G^E,$ is related to that of the Dyson-equation method, $G^D,$ by a simple Dyson-type equation $G^E{=G}^D{+G}^D\delta {\mathrm{hG}}^E,$ where the matrix $\delta h$ is related to the incompleteness of the basis set. With the increasing number of basis functions, the Green’s functions calculated with the two methods converge to each other rapidly in the interior of the perturbed volume, while they differ persistently on the boundary surface because the Dyson-equation method fails to incorporate the boundary condition of the Green’s function. Reflecting this behavior, $\delta h$ tends to vanish rather slowly with increasing number of basis functions. To demonstrate this, we perform a numerical calculation using a simplified one-dimensional model.