Reformulation and Solution of the Master Equation for Multiple-Well Chemical Reactions

Abstract

We consider an alternative formulation of the master equation for complex-forming chemical reactions with multiple wells and bimolecular products. Within this formulation the dynamical phase space consists of only the microscopic populations of the various isomers making up the reactive complex, while the bimolecular reactants and products are treated equally as sources and sinks. This reformulation yields compact expressions for the phenomenological rate coefficients describing all chemical processes, i.e., internal isomerization reactions, bimolecular-to-bimolecular reactions, isomer-to-bimolecular reactions, and bimolecular-to-isomer reactions. The applicability of the detailed balance condition is discussed and confirmed. We also consider the situation where some of the chemical eigenvalues approach the energy relaxation time scale and show how to modify the phenomenological rate coefficients so that they retain their validity.

Adventures on the C3H5O potential energy surface: OH + propyne, OH + allene and related reactions

“…To obtain a consistent set of phenomenological rate coefficients for the important pathways, the time-dependent 1-D master equation methodology of Miller and Klippenstein [25][26][27] was used as implemented in the newly developed PAPER (Predictive Automated Phenomenological Elementary Rates) code of Georgievskii et al [28]. Below 400 K we used the low-eigenvalue method to obtain numerically stable results, while above 400 K the results of the direct-diagonalization method are presented.…”

“…We used the VaReCof code [32] to sample the long-range interaction potential, KinBot [18] to develop the network of stationary points, and PAPER [28] to solve the 1-D master equation.…”

“…Both addition barriers for propyne feature two hindered rotors, a methyl and an OH one. We replaced the separable 1-D hindered rotor model for these saddle points by a 2-D coupled rotor model as implemented in PAPER [28], but this resulted in less than 5% change in the rate coefficient for the central addition (where the two rotors are close) and no change in the terminal addition rate coefficient. Note also that the good agreement after the change in the barrier height between the lowand the high-pressure experiments signals that the empirical collisional parameters used are adequate.…”

Bimolecular recombination reactions: K-adiabatic and K-active forms of the bimolecular master equations and analytic solutions

“…7,10 In so revealing features of the kinetics, in such situations, tracing the evolution of the time dependence of the populations may be important. In the context of unimolecular theory, 1-3 the two typical approaches to this problem are the lowest-eigenvalue matrix approach 1,[7][8][9][10] and the stochastic approach, 6,[11][12][13][14] introduced earlier in this section. In the present paper, recurrence relations [43][44][45] are also given for the time dependent populations of the K-active and K-adiabatic master equation cases, for single and multiple well and reaction channels.…”

“…6 In the atmosphere or in other chemical systems, the buffer gas may react with the vibrationally excited species, as well as deactivate it, and such processes may also be investigated by master equations. 6 The usual treatments of master equations for studying a unimolecular or bimolecular process utilize either an eigenvalue method 1,[7][8][9][10] or a stochastic based approach, 6,[11][12][13][14] for obtaining a unimolecular dissociation rate constant k uni or bimolecular recombination rate constant k rec using detailed balance. Presently, both methods are available for the K-active case, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product.…”