Bimolecular Recombination Reactions: K-Adiabatic and K-Active Forms of RRKM Theory, Nonstatistical Aspects, Low-Pressure Rates, and Time-Dependent Survival Probabilities with Application to Ozone. 2

Abstract

We consider for bimolecular recombination reactions the K-adiabatic versus the K-active forms of RRKM theory, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. When that product is approximately a prolate symmetric top, with two moments of inertia of the product substantially larger than the third, K becomes a dynamically slowly varying quantity and the K-adiabatic form of RRKM theory is the appropriate version to use. Using classical trajectory results for the rate constant for ozone formation in the low-pressure region as an example, excellent agreement for the recombination rate constant k rec with the K-adiabatic RRKM theory is observed. Use of a two transition state (inner, outer TS) formalism also obviates any need for assessing recrossings in the exit channel. In contrast, the K-active form of RRKM theory for this system disagrees with the trajectory results by a factor of about 2.5. In this study we also consider the distribution of the (E, J) resolved time-dependent survival probabilities P(E, J, t) of the intermediate O 3 * formed from O + O 2 . It is calculated using classical trajectories. The initial conditions for classical trajectories were selected using action-angle variables and a total J representation for (E, J) resolved systems, as described in Part I. 1 The difference between K-active and K-adiabatic treatments is reflected also in a difference of the K-active RRKM survival probability P(E, J, t) from its trajectory-based value and from its often non-single-exponential decay. It is shown analytically that k rec (K-active) ≥ k rec (K-adiabatic), independent of the details of the TS (e.g., variational or fixed RRKM theory, 1-TS or 2-TS). Nonstatistical effects for O 3 * formation include a small initial recrossing of the transition state, a slow (several picoseconds) equipartitioning of energy among the two O−O bonds of the newly formed O 3 *, and a small nondissociation (a quasi-periodicity) of some trajectories originating in O 3 * (∼10%) and so, by microscopic reversibility, are not accessible from O + O 2 . An apparently new feature of the present results is the comparison of classical trajectories with K-adiabatic and K-active theories for rate constants of bimolecular recombinations. The quantum mechanical counterpart of classical K-adiabatic RRKM theory is also given, and its comparison with the experimental k rec for O 3 is given elsewhere.

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

“…The ozone molecule is an example of such a system where its K has been investigated to exhibit adiabaticity, 15,[18][19][20] e.g., the role of K in the intermolecular energy transfer with a bath gas, 18 K-diffusion in the metastable ozone, 19,20 and its role in the recombination rate constant for O + O 2 → O 3 whereby a K-active form of Rice-Ramsperger-Kassel-Marcus (RRKM) theory yielded a k rec for O 3 to be greater than a factor of ∼2 compared to the K-adiabatic k rec . 15 Comparison of various alternative combinations for the treatment of angular momentum in RRKM theory 16,17 has also been used to investigate Cl + H 2 O 2 association 21 and explored elsewhere.…”

“…15 Comparison of various alternative combinations for the treatment of angular momentum in RRKM theory 16,17 has also been used to investigate Cl + H 2 O 2 association 21 and explored elsewhere. 17 Correspondingly, a K-adiabatic 16,17 formulation of master equation would be relevant and is derived in the present work, along with the K-active 16,17 case for the treatment of angular momenta.…”

“…23 Another, ancillary, consideration on selecting to determine k rec directly or from the detailed balance condition is the phase space accessibility for a unimolecular versus a bimolecular process. 15 For example, the volume for the density of states ρ for the ozone molecule is ∼10% smaller when ρ is determined from a unimolecular process versus a bimolecular process at 298 K, where its physical basis lies upon the presence of quasi-periodic trajectories persisting for a unimolecular process and manifesting in the survival probability of the ozone and for its k rec . 15 The recombination rate constant may also be determined directly, upon using the populations acquired from the analytic solution of the master equation.…”

“…15 For example, the volume for the density of states ρ for the ozone molecule is ∼10% smaller when ρ is determined from a unimolecular process versus a bimolecular process at 298 K, where its physical basis lies upon the presence of quasi-periodic trajectories persisting for a unimolecular process and manifesting in the survival probability of the ozone and for its k rec . 15 The recombination rate constant may also be determined directly, upon using the populations acquired from the analytic solution of the master equation. 27 In previous works, approximate analytic solutions, distribution functions for populations, for a master equation for a unimolecular process for a strong collision model, 1 and solutions for the low pressure limit unimolecular master equation based on various energy transfer models have been derived for their dissociation rate constants.…”

“…(16) are now the K-active counterpart of those defined earlier for Eq. (15). In the usual treatment, the term k r i appearing in the master equation is rewritten as k d i and the equilibrium constant K eq i for A + BC i using detailed balance condition.…”