Microscopic rate coefficients in reactions with flexible transition states: Analysis of the transitional-mode sum of states

Expressions are derived for the energy- and angular-momentum-resolved transitional-mode sum of states, WTM(E,J), for flexible transition states in unimolecular, recombination, or bimolecular collision–complex-forming reactions. The expressions are derived classically by evaluation of the phase-space volume integral. The phase-space integral is so arranged that the total available momentum-space volume, Φ(E,J,q), for a frozen configuration q is first evaluated. Accurate analytic expressions for Φ(E,J,q) are obtained for all relevant pairwise combinations of atom, linear, spherical-top, symmetric-top, and asymmetric-top fragments in flexible transition states. The analytic expressions for Φ(E,J,q) indicate clearly the conditions under which the common method of imposing angular momentum conservation, which assumes that J≊L (L being the orbital angular momentum), will fail. WTM(E,J) is then obtained by integration of Φ(E,J,q) over configuration space. Exact evaluation of the integral over configuration space involves at most a five-dimensional numerical integral. Accurate analytical expressions for WTM(E,J) are derived for model potentials which admit sufficient flexibility for the fitting of more-accurate potentials. These expressions enable the calculation of accurate microscopic rate coefficients k(E,J) by microcanonical variational Rice–Ramsperger–Kassel–Marcus (RRKM) theory at little more computational expense than a standard RRKM calculation.