Radiationless Transitions in Polyatomic Molecules. I. Calculation of Franck—Condon Factors

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Abstract
Radiationless transitions between two electronic states are studied for a system consisting of a polyatomic molecule in a medium where vibrational relaxation is rapid. The transition rate is then governed by a vibronic matrix element and a vibrational overlap factor. Only the latter, known as the Franck—Condon factor and denoted by F, is investigated in detail. For harmonic oscillators F derives from shifts in equilibrium distance (displacements) and shifts in frequency (distortions). It is shown that for radiationless transitions involving large energy gaps (E), F is dominated by distortions, whereas for optical transitions it is dominated by displacements of the oscillators. These distortions lead to an approximately exponential decrease of F with increasing E. An isotope rule for F is derived which is valid for both displaced and distorted oscillators provided E is not too small. Since all formulas are of the form F (E), where E is the principal variable, comparison with experiment is only possible for a class of related molecules. Within this class E should vary a great deal but all other parameters should either be constant or vary in a simple, systematic manner. The dependence of some of these parameters on the number of oscillators involved in the transition is examined. Finally the theory is extended to molecular crystals and the corresponding liquids. It is concluded that in these systems the rate constant of an intramolecular radiationless transition must be equal to or smaller than that in dilute solutions, except when excited dimers are present in the initial state.