The Recombination of Propargyl Radicals: Solving the Master Equation

Abstract

We have investigated theoretically the recombination reaction between two propargyl (C3H3) radicals using previously published BAC-MP4 calculations (supplemented by DFT-B3LYP results) to characterize the potential energy surface, RRKM theory to compute microcanonical rate coefficients, and solutions to the time-dependent, multiple-well master equation to predict thermal rate coefficients and product distributions as a function of temperature and pressure. The thermal rate coefficient k(T,p) drops off precipitously at high temperature, regardless of the pressure. Below 500 K, k(T,p) ≈ k∞(T), the high-pressure limit rate coefficient for initial complex formation, independent of p. For 500 K < T < 2000 K, the rate coefficient increases with increasing pressure, as one would normally expect. At 2000 K, the “coalescence temperature” for this reaction, k(T,p) = k0(T), the zero-pressure rate coefficient, and only bimolecular products (phenyl + H) are predicted, no matter how high we make the pressure. The latter effect is a consequence of all the intermediate complexes reaching their “stabilization limits,” a concept discussed extensively in the text. Below 800 K, many C6H6 isomers are formed as products, and the pressure and temperature dependence of the branching fractions is easily understood from conventional reasoning. Above 800 K, the product distributions begin to be dominated by isomers reaching their stabilization limits and disappearing as important products. Above 1200 K, the only significant products are fulvene, benzene, and phenyl + H. Beyond 1700 K fulvene disappears, and for T > 2000 K the only products are phenyl + H. We discuss our results in terms of the eigenvalues and eigenvectors of G, the transition matrix of the master equation. A “good” rate coefficient exists only when the rate is controlled by a single eigenvalue of G. A jump of the k(T,p) curve for any pressure from one eigenvalue to another is triggered by the reaching of critical stabilization limits, producing “avoided crossings” of the eigenvalue curves. It is at such avoided crossings that biexponential reactant decays occur.