  We study the dynamics of a single Frenkel exciton in a disordered molecular
chain. The coherent-potential approximation (CPA) is applied to the situation
when the single-molecule excitation energies as well as the transition dipole
moments, both their absolute values and orientations, are random. Such model is
believed to be relevant for the description of the linear optical properties of
one-dimensional $J$ aggregates. We calculate the exciton density of states, the
linear absorption spectra and the exciton coherence length which reveals itself
in the linear optics. A detailed analysis of the low-disorder limit of the
theory is presented. In particular, we derive asymptotic formulas relating the
absorption linewidth and the exciton coherence length to the strength of
disorder. Such expressions account simultaneously for all the above types of
disorder and reduce to well-established form when no disorder in the transition
dipoles is present. The theory is applied to the case of purely orientational
disorder and is shown to agree well with exact numerical diagonalization.
