Treatment of the Metallic Impurity Problem Using Green's Functions

The calculation of the charge density or spin density at an impurity in a metal is conveniently performed using a suitably weighted density of electron states. Working in the Wannier function representation, we derive this function making use of two Green's functions, one defined for the pure metal and the other for the metal containing the impurity. The former is directly related to the usual density of states of the pure metal, the latter to the weighted density of states we require. The attractive feature of the derivation is that there is a simple relationship between the two Green's functions, as was shown by Waller for the analogous problem in lattice dynamics. The weighted density-of-states function has a continuous part coming from the band and may also have isolated parts coming from bound states which have been pulled out of the band by the perturbation. The contribution of each to a number of physical properties is discussed for the simple case of a single band of conduction electrons and a well-localized impurity potential.