Theory of nonradiative decays in the non-Condon scheme

Without invoking the so-called Condon approximation, a theory is developed for the nonradiative decay rate constant in molecules. The electronic coupling matrix elements obtained by using adiabatic electronic wavefunctions are expressed in terms of commutators [p, η], where p is the nuclear momentum operator and η is a function of nuclear coordinates. Evaluating commutator [p, η] at a fixed nuclear configuration, we obtain the rate constant within a Condon-like approximation. Assuming harmonic and displaced potential energy surfaces, the rate constant of the nonradiative decays where the promoting mode has a displacement is derived in the low temperature limit. The dependences of the non-Condon correction factors on the vibronic coupling constant ηgamma, the displacement Δ, and the energy gap E are examined by performing model calculations. The correction factors decrease with increasing γ owing to the resonance effect near the potential surface crossing. The calculations also show that the rate constant in the non-Condon scheme exceeds that obtained invoking the Condon approximation by about two orders of magnitude. Finally, a simple upper bound to the decay rate constant is derived by applying a Condon-like approximation to the electronic coupling matrix elements. It is shown that in the weak coupling situation the evaluation of the electronic coupling matrix elements by the use of a Q-centroid method gives an upper bound to the rate constant.