The partition function of a dimer with one electron, interacting with two local Einstein phonons, is reduced to the pure phonon partition function. It is investigated analytically in the coherent-state path-integral representation in order to recognize the competition between electron hopping and electron-phonon interaction. The non-polynomial term in the phonon action, promoted by electron hopping, is deduced. The low-temperature resonance decoupling between electron and phonon systems destroys the low-temperature tendency of electron localization in the case of small hopping and large electron-phonon coupling. The comparison with polaronic theory is completed.