Perturbed crystals in thekqrepresentation. I. The impurity problem

A method is developed for defining localized states and effective Hamiltonians in perturbed crystals. The method is based on the localization ideas in the $\mathrm{kq}$ representation for perfect lattices. An equation is derived defining localized states for perturbations caused by an impurity, a magnetic and electric fields. In this paper the impurity problem is considered in detail. A correction term is obtained to the one-band Koster-Slater effective Hamiltonian. It is shown to be significant for bound states and scattering cross sections of a localized impurity.