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