The energy gap law for radiationless transitions in large molecules

In this paper we present a unified treatment of non-radiative decay processes in large molecules which involve either electronic relaxation between two electronic states or unimolecular rearrangement reactions in excited electronic states. The present treatment is analogous to the formalism previously applied for the line shape problem in nuclear recoil and in the optical spectra of solids. We were able to derive theoretical expressions for the non-radiative decay probability so that an arbitrary number of different molecular vibrations can be incorporated in the vibrational overlap factors. The general expressions obtained herein can be reduced to analytical form for two limiting cases, which we call the strong coupling case (which corresponds to a substantial horizontal displacement of the potential energy surfaces of the two electronic states) and the weak coupling limit (whereupon the relative horizontal displacement of the two potential energy surfaces is small). Quantitative criteria for the applicability of these two coupling limits are provided. In the strong coupling limit the transition probability is determined by a gaussian function of the energy parameter (ΔE - EM ), where ΔE is the energy gap between the origins of the two electronic states and 2EM is the Stokes shift. This limit exhibits a generalized Arrhenius type temperature behaviour whereupon the transition probability depends exponentially on the energy barrier for the intersection of the two potential surfaces. At low temperatures the transition probability is determined by the mean vibrational frequency and is thus expected to reveal only a moderately weak deuterium isotope effect. The weak coupling limit reveals an exponential (or rather superexponential) dependence of the transition probability on the energy gap ΔE. In this limit the transition probability is dominated by the highest vibrational frequency (e.g. the C-H or C-D vibrations) and thus will reveal a marked isotope effect. When semi-empirical estimates of the pre-exponential factors are provided, the approximate theoretical expression for the weak coupling limit is found to be consistent with the available experimental data on electronic relaxation in large organic molecules.