The Franck-Condon Principle and Its Application to Crystals

TLDR

The study analyzes the case where the electronic energy gap depends linearly on vibrational coordinates expressed as normal modes. The Franck–Condon principle reproduces the exact quantum absorption spectrum up to second moments, with higher‑order errors becoming negligible at high temperatures or strong electron–nuclear coupling, and exact absorption/emission formulas are derived, though a quasi‑molecular description requires temperature‑dependent masses or stiffnesses.

Abstract

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.