Multiphonon Processes in the Nonradiative Decay of Large Molecules

Radiationless transition rates for polyatomic molecules are investigated in the simplest case of the statistical limit, i.e., large energy gaps between the two electronic states. By analogy with the theory of optical line shapes in solids, the “golden rule” rate expression for the nonradiative decay, which is usually written as a double sum over initial and final vibronic states, is equivalently represented in closed form as a single Fourier integral. For the case in which the vibrations are assumed to be harmonic, but may have different frequencies and equilibrium positions in the two electronic states, general closed-form analytic expressions, which include all of the vibrational modes, are obtained for the transition rates. The energy gap law is again obtained in the weak coupling statistical limit along with a proper description of the propensity rules for the promoting mode. For the case of the aromatic hydrobarbons, the effective energy gap is found to be in agreement with Siebrand's analysis, and explicit account is taken of the competition between the C–H modes and the C–C skeletal modes (which have large oscillator displacements) or the out-of-plane modes (which have large frequency changes) for the electronic energy which is involved in the relaxation process. The deuterium isotope effect is again found to be dominated by the modes of highest frequency (C–H or C–D), but corrections due to the C–C modes, which may be measurable, are explicitly included. The theory is also extended to include the role of the host medium as an inert heat bath by considering the intermolecular phonons as potential accepting modes. The approximations employed are compared with those usually made in Boltzmann statistical mechanics, thereby providing greater understanding as to the role of the Franck–Condon principle in the determination of the nonradiative decay rates.