First passage times and the kinetics of unimolecular dissociation

Approximate solutions for multistep master equations describing the time evolution of product formation in multiphoton or thermal unimolecular reactions are investigated. In particular, a method based on fitting the first few moments of the passage time distribution associated with the given stochastic process to proposed simple expressions for the product yield function is studied. It is shown that reasonable agreement with the exact numerical solution of the corresponding master equation is obtained with a two parameter fit (using two passage time moments) and an excellent agreement is obtained with a three parameter fit (using three passage time moments). In no case studied does a need arise for more than a three-moment description and the quality of available experimental results makes the simpler two-moment description sufficient in most cases. Analytical solutions for the first and second passage time moments are obtained for simple discrete and continuous master equation models. Expressions for the incubation time and the reaction rate are obtained in terms of these solutions. The validity of discretizing a continuous master equation (which is an important simplifying step in evaluating the time evolution associated with multiphoton dissociations in the presence of collisions, or with thermal unimolecular reactions involving large molecules) is studied using both the approximate two-moment solutions and exact numerical solutions. It is concluded that a proper discretization of a continuous master equation may be carried out provided ε≪k_BT*, where ε is the discretization energy step, k_B the Boltzmann constant, and T* the effective (density of states weighted) temperature. A larger discretization step can be used if only the incubation time is required. Using the approximately discretized master equation, we next calculate the effect of collisions on the incubation time and the rate of multiphoton dissociation using a model constructed to correspond to the unimolecular dissociation of tetramethyldioxethane. Incubation times are found to be less sensitive to collisions then the reaction rates. Finally, we investigate the applicability of the passage time moments method to describe the time evolution of product formation in a system whose dynamics is determined by a quantum mechanical Liouville equation. Again the two-moment description provides a reasonable and the three-moment approximation a good approximation to the exact solution. The three-moment approximations, however, cannot be used when the pressure (i.e., the dephasing rate) is too low.