Contribution to the Theory of Frenkel Excitons in Disordered Molecular Crystals *

In this paper, we discuss the Frenkel excitons in disordered molecular crystals. Most of our results can be generalized to cover electrons and phonons in other disordered systems as well. In particular, the relationship among the exact Green's function, the CPA (coherent potential approximation) Green's function, and the method of moments is discussed. It is emphasized that, despite its complicated form, the exact Green's function not only yields correct moments for the spectral density function and the overall density-of-states function but also leads to the CPA Green's function if certain terms are modified. Both the exact and the CPA self-energies are expanded with the aid of generating functions. This enables us to demonstrate both systemmatically and analytically how the latter is derived from the former and what exactly the approximations are. A better understanding of the CPA is thus obtained. Specifically, it is shown that the exact self-energy is wave vector and branch independent in the first five terms of its expansion in powers of Z-l (reciprocal energy). A comparison with a similar expansion of the CPA selfenergy indicates that the CPA self-energy is exact to the Z-5 power. Consequently, the spectral density function and the over-all density-of-states functions determined from the CPA method have the correct seven and eight lower moments, respectively. Analytical expressions for these lower moments are also given for future applications.