The theory of the transfer of electronic energy between two molecules, or between two parts of a "double molecule", is discussed for the case in which vibrational quanta are present. The results are obtained for two limiting cases of weak and strong electronic interaction. Some of the phenomena occurring in the intermediate region are discussed qualitatively. With the aid of these results the spectra of biphenyl, diphenylmethane, and dibenzyl have been analyzed, and the electronic energy exchange parameter has been determined for each. These data are compared with similar data on durene crystal and the paracyclophanes. The electronic energy exchange parameter is given approximately correctly by considering the interaction of electronic transition dipoles except for very close and for very distant molecules.