Application of inverse iteration to 2-dimensional master equations

Abstract

Recent developments in unimolecular theory have placed great emphasis on the role played by angular momentum in determining the details of the dependence of the rate coefficient on pressure and temperature. The natural way to investigate these dependencies is through the master equation formulation, where the rate coefficient is recovered as the eigenvalue of the smallest magnitude of the spatial operator. Except for very simple cases, the master equation must be solved with numerical methods. For the 2‐dimensional master equation this leads to large sparse matrices and correspondingly lengthy computational times in order to determine the eigenvalue of the least magnitude. A reformulation of the problem in terms of a diffusion equation approximates the final matrix with a narrow banded matrix that can easily be factored using a variation of Gaussian elimination. The 2‐dimensional master equation can then be solved with inverse iteration, which rapidly converges to the desired eigenpair. This method can be up to 10 times faster than conventional iterative algorithms for finding the desired eigenpair. © 1997 John Wiley & Sons, Inc. J Comput Chem 18:1004–1010, 1997

Modeling Collisional Transitions in Thermal Unimolecular Reactions: Successive Trajectories and Two-Dimensional Master Equation for Trifluoromethane Decomposition in an Argon Bath

“…A straightforward choice is the two conserved quantities of the molecule: the total internal (rovibrational) energy, E , and total angular momentum, J . This leads to the two-dimensional formulation of the master equation as a function of E and J . − …”

Pressure-Dependent Rate Constant Caused by Tunneling Effects: OH + HNO3 as an Example

“…An E , J -resolved two-dimensional master equation − that describes the time evolution of one well (PRC) and multiple products (as shown in Figure ) is given by where J max is the maximum angular momentum; E max is the maximum internal energy; C 1 ( E i , J i , t ) represents the population density of PRC in state ( E i , J i ) and time t; ω LJ (in s –1 ) is the Lennard–Jones collisional frequency; and k 1→ l ( E i , J i ) (in s –1 ) is the ( E i , J i )-resolved microcanonical rate coefficient from PRC to products. For the dissociations of PRC via TS1 and TS2, Miller’s semiclassical TST (SCTST) theory, − which includes coupled vibrations and multidimensional tunneling, is used to compute the microcanonical rate constants without angular momentum effects, k ( E , J = 0); the J effects are then included using the J -shifting approximation, ,, assuming an active K-rotor model for both reactant and TS. − For the barrierless dissociation of PRC back to OH + HNO 3 , variational RRKM theory , is used to characterize a kinetic bottleneck as well as to compute k ( E , J ) for a loose TS.…”

“…It should be emphasized that other approaches for kinetics calculations could be used to generate the same qualitative results. For example, there are several master-equation approaches from other groups (refs − ), which can be used in a manner similar to that above. In addition, the PolyRate software package (ref ) can be used to compute thermal rate constants at the high-pressure limit, where Boltzmann thermal equilibrium distribution can be assumed.…”

Thermal Decomposition of CH3O: A Curious Case of Pressure-Dependent Tunneling Effects

“…An E , J -resolved two-dimensional master equation − that describes the time evolution for a thermally activated reaction system of CH 3 O (as shown in Figure and Scheme ) is expressed as …”