A master equation model for bimolecular reaction via multi-well isomerizing intermediates

Communication: Thermal unimolecular decomposition of syn-CH3CHOO: A kinetic study

“…Using the calculated PES properties, anharmonic constants, and rovibrational parameters obtained with mHEAT (see above), a simplified two-dimensional-(E,J)-grained master equation was solved using an eigenvalue-eigenvector matrix technique [51][52][53][54][55][56][57][58][59][60][61] to obtain the time evolution of various intermediates as well as phenomenological rate constants as functions of pressure (P = 10 −3 to 10 4 Torr) and temperature (T = 200-400 K) that cover atmospheric conditions. Figure 2 displays the time evolution of various species as a function of pressure at 300 K, while Figure 3 shows the time evolution of various species as a function of temperature at 1 atm.…”

Fast, scalable master equation solution algorithms. III. Direct time propagation accelerated by a diffusion approximation preconditioned iterative solver

“…The modeling of the 1 CH 2 ϩC 2 H 2 system in this work is similar to that used in previous ME studies. 6,7,44 However, in this work a better treatment of the propargyl formation step is used.…”

“…While solving the diffusion version of the ME is clearly an approximation to the solution of the full ME, this approximation has proved useful. 6,7,16,[32][33][34][35] One of the keys to the scalability of the methods presented in the current work is the inversion of the diffusion approximation matrix. As described elsewhere, 6,7 rearranging the ordering of the energy grains within the state space p to bring grains of the same energy together results in a banded matrix with the bandwidth equal to the number of isomers being modeled, p. A banded matrix of this type can be factorized with computational effort scaling at O(np 2 ), a vast improvement over the O(n 3 ) standard solve as usually p Ӷn.…”

“…Critically the factorization of such an arrowhead matrix does not lead to fill-in outside of the arrowhead structure so that bimolecular channels do not alter the basic scaling of solving linear systems involving the diffusion approximation ME. 6,7 It should be pointed out that bimolecular reactions under second-order conditions can be modeled with a ME, but this is complicated considerably by the fact that the ODE is no longer linear. Equation ͑3͒ changes form to become dp dt…”

“…Determining transient behavior is particularly important in multi-well systems describing isomerization between a number of isomers, which are increasingly being modeled with ME methods. [6][7][8][9][10][11][12][13] The matrices arising from multi-well MEs are significantly larger than unimolecular ME matrices. The rank of the matrix increases approximately linearly with the number of isomers being modelled, leading to very large matrices to model moderately sized systems.…”