The Franck-Condon Principle and Its Application to Crystals

The semiclassical Franck-Condon principle is shown to be related to the more rigorous (``exact'') quantum-mechanical perturbation formula in the following ways: (1) the Franck-Condon formula can be derived from the ``exact'' formula by using a mean value approximation or by neglecting certain commutators; (2) if the electric dipole moments are treated as approximately independent of position, the Franck-Condon and the ``exact'' absorption (or emission) spectrum have the same zeroth, first, and second moments, i.e., the same integrated spectrum, mean absorption frequency, and breadth; (3) the errors in higher moments than the second become relatively unimportant at high temperatures. If the electron-nuclear interaction is sufficiently strong the errors are unimportant even at absolute zero. The use of a quasi-molecular description in a many particle problem is found to be possible only if the masses or stiffnesses are allowed to be temperature dependent. A detailed analysis is made of the case in which the energy difference between the two electronic states is a linear function of the vibrational coordinates—and the latter are describable by normal modes. ``Exact'' formulas for the absorption and emission spectrum are obtained.