Two contributions to the Green's-function formulation of electronic-structure calculations are described. We show that a seemingly minor change in the definition of the relevant perturbed and unperturbed Green's functions eliminates several problems that have hampered development of the formalism. Defining the Green's functions in terms of matrices, rather than differential operators, substantially simplifies the description of many systems, including displaced atoms, transition-metal impurities in non-transition-metal hosts, and chemisorbed atoms. In particular, our methods provide a natural solution to the so-called cluster-embedding problem. The second development we describe pertains to the summation over the occupied states of a system that is required in order to construct quantities such as the total energy and the electron density. We show that these summations, which are an essential part of self-consistent calculations, can be represented as contour integrals and that certain displacements of the contour lead to a significant simplification of the calculations. © 1982 The American Physical Society.