The restricted quantum exchange theory of intramolecular T1 and T2 relaxation rates

Abstract

We present a theory for intramolecular vibrational relaxation in polyatomic molecules. The theory postulates the existence of a restriction on the magnitudes of matrix elements connecting zero-order states which favors coupling between vibrational modes. Specifically, the matrix elements are assumed to depend on two parameters, one determining the overall magnitude of coupling, and the other the rate of falloff with increasing quantum exchange. We use this ’’restricted quantum exchange’’ (RQE) hypothesis to derive analytic expressions for T1 (energy-transfer) and T2 (coherence-loss) relaxation rates which depend only on the two coupling parameters, the average molecular frequency, the number of modes, the energy in the molecule, and (for T1) the size of the energy transfer. In the derivation we obtain analytic expressions for the number of pairs of states on an energy shell related to each other by exchange of a fixed number of quantua M. The analysis has been carried out for arbitrary M, allowing us formally to include the effects of all higher order couplings and to show rigorously when they can be eliminated. The resulting T1 and T2 relaxation rates and associated linewidths exhibit saturation with increasing vibrational energy, a property which has been shown to be essential to obtaining reasonable cross sections for multiphoton excitation. We propose on the basis of restricted quantum exchange a simple explanation for the observed narrowing of the linewidths of benzene with increasing vibrational quantum number.