Perturbation approach to the impurity problem in lattice dynamics

Using the localisation ideas in the kq representation for perfect lattices, an equation is postulated defining perturbed localised modes for the impurity problem in lattice dynamics. The equation is solved by perturbation theory in the unit-cell scheme, up to second order. Multibranch and one-branch effective equations of motion are derived. The formalism is illustrated through the example of a molecular diatomic chain with a strongly localised impurity and in the tight-binding limit. A first-order correction is found for the displacement function of the optical localised mode.