Theory of electron transfer reactions. II. Exact quantum number scaling relationship for semiclassical time-dependent dynamics

We consider the quantum number scaling properties of the exact solutions to the dynamical equations describing electron transfer between two heavy particles of arbitrary size. The only assumptions are that the Born–Oppenheimer approximation holds for the separated reactants and products, and that the relative translational motion of the heavy structured particles is classical. We derive an exact scaling relationship between the various rovibrational state transition amplitudes in the reaction. As an example of the utility of this result, we investigate the implications of the Franck–Condon approximation on the quantum number variation of electron transfer cross sections, without recourse to dynamical calculations.