The condition for the equivalence of localised state and Bloch state expansions in generalized coherent potential approximations

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Abstract
For generalized coherent potential approximations a theorem is derived which gives the necessary and sufficient condition for the equivalence of the localized state (or locator) expansion and the band state (or propagator) expansions. This condition is that the self-energy or coherent potential (and hence the locator) be cluster-diagonal. In the ordinary CPA this is satisfied sine the coherent potential is site-diagonal, but in most of the attempted generalizations of the CPA to clusters now in the literature this condition is not met. An exception is the molecular CPA model of Tsukada which is discussed as an example. Furthermore it is pointed out that when the self-energy is cluster-diagonal the multiple-scattering approach of Nickel and Krumhansl and that of Cyrot-Lackmann and Ducastelle and the diagrammatic technique all merge to a single result.