Four Isotope-Labeled Recombination Pathways of Ozone Formation

A theoretical approach is developed for the description of all possible recombination pathways in the ozone forming reaction, without neglecting any process a priori, and without decoupling the individual pathways one from another. These pathways become physically distinct when a rare isotope of oxygen is introduced, such as 18O, which represents a sensitive probe of the ozone forming reaction. Each isotopologue of O3 contains two types of physically distinct entrance channels and two types of physically distinct product wells, creating four recombination pathways. Calculations are done for singly and doubly substituted isotopologues of ozone, eight rate coefficients total. Two pathways for the formation of asymmetric ozone isotopomer exhibit rather different rate coefficients, indicating large isotope effect driven by ΔZPE-difference. Rate coefficient for the formation of symmetric isotopomer of ozone (third pathway) is found to be in between of those two, while the rate of insertion pathway is smaller by two orders of magnitude. These trends are in good agreement with experiments, for both singly and doubly substituted ozone. The total formation rates for asymmetric isotopomers are found to be somewhat larger than those for symmetric isotopomers, but not as much as in the experiment. Overall, the distribution of lifetimes is found to be very similar for the metastable states in symmetric and asymmetric ozone isotopomers.

In the doubly substituted case the processes of formation, decay and stabilization of scattering resonances (metastable intermediates) of ozone are represented by the following reaction scheme: Including these processes in the rate of change expression for the concentrations of intermediate (S2) Figure S1. Two distinct channels of ozone formation (left and right) and the metastable ozone states (middle) in the case of double isotopic substitution. Here "6" denotes 16  In the steady state conditions, applicable to ozone formation reaction, we have: As before, the rate coefficients for formation of resonances through two channels are converted into the corresponding rate coefficients of decay using two equilibrium constants, but here, in the case of double isotopic substitutions, the expressions are different: Namely, here the −ΔZPE/ factor shows up in Channel 2 (in contrast with the singly substituted case, where it appears in Channel 1). The value of ΔZPE itself in the doubly substituted case is also slightly different than in the singly substituted case. Here we have: Using these formulae, we obtain the following expressions for two product-specific rates of recombination process: The following four pathway-specific dynamical partition functions can be introduced to simplify these expressions: Note that here the ΔZPE shift appears in the expressions for ̃A and ̃S , while in the case of single isotopic substitution it appeared in ̃B and ̃I (see Equations 41-44 of the main text).
The insertion pathway is shown schematically in Fig. S2 The isotope exchange process, and the corresponding equilibrium constant, in the case of double substitution are: where the factor −ΔZPE/ is used in Channel 2. The forward direction of the isotope exchange here is defined as from Channel 1 to Channel 2, opposite to the singly substituted case. The value of this equilibrium constant is expected to be on the order of 2 due to symmetry of the homonuclear diatomic reagent in the Channel 1, in which every other rotational state is forbidden by symmetry.
Using this expression, we can rewrite formula for the rates in the way where all pathways are referenced relative to the lowest energy channel, as follows: (S15) Figure S2. Schematic of the global PES of ozone that possesses a three-fold symmetry with respect to the entrance channels and the product wells. Reagents are indicated by black numbers, the product ozone molecules by white numbers, for the case of double isotopic substitution. The insertion pathway is shown schematically by red arrows.
A relationship between the pathway-specific rate coefficients and the product-specific rate coefficients then looks as follows: Note that here, in the doubly substituted case, the ex sticks to the rate coefficients of pathways A and S that originate in the higher energy channel, the Channel 2. Interestingly, the equations for product-specific dynamical partition functions ̃s ym and ̃a sym in the doubly-substituted case appear to be entirely identical to those derived for the singly-substituted case, simply because these formula do not contain any −ΔZPE/ factors (they analytically cancel, as it was discussed in Sec. II of the main text). Therefore, they will not be repeated here.

B. Rate expressions for channel-specific processes
Since tot = A + B + S + I , we can regroup the four contributions into two rates that correspond to two entrance channels: First consider the case of single isotopic substitution. Using the definitions of Eqs. (36)(37)(38)(39)(45)(46)(47)(48) of the main text, one immediately obtains expressions for the corresponding rate coefficients: (S16) sym = S ex + I = stab̃s ym ch1 (S17) Two dynamical partition functions for these processes can be defined as a simple sum: Using This result is also quite interesting. These expressions indicate that the splitting of tot onto sym and asym has no influence on ̃c h1 vs. ̃c h2 since the same overall stabilization probability enters both expressions. In this case the difference between ̃c h1 and ̃c h2 comes entirely from splitting of Γ tot onto Γ ch1 and Γ ch2 , for individual resonances. In this procedure, Γ ch1 comes with the ΔZPE shift of energy reference, as emphasized above.
In the doubly substituted case: Again, the difference from the singly substituted case is that here the −ΔZPE/ factor comes with the upper Channel 2.

C. Rate expressions for low-pressure and high-pressure limits
Low-pressure limit is relevant to the atmospheric chemistry, to some laboratory studies, but is also important methodologically. In the limit of zero pressure of bath gas,

D. Technical details of calculations
The following changes were made in this work, relative to the earlier work of Teplukhin et al. 14 : a) The value of temperature was changed from 296 K used by Teplukhin to = 293 K used This was corrected in the updated version of the code, and resulted in a significant change of B and ex , but no change in the isotope effects where these two moieties enter together and the effect of the diatomic mass cancels.
The characteristic values of resonance widths for three kinds of states were computed as average using the following formula: where ̃= (2 + 1) − is a contribution of each resonance into the dynamical partition function ̃, used here as a weighting factor in the averaging. The values of ̃c ov , ̃v dw and ̃f ree are obtained by substitution of cov , vdw and free into this formula in place of .
Note that Γ is the total resonance width. The average values of resonance width in symmetric and asymmetric isotopomers were computed in the same way, but using sym and asym in place of . Zero-pressure limit was assumed for calculations of all average values, which corresponds to = 1 for each resonance, regardless of its width (as in ̃° above).
Location of the effective transition state that separates the covalently bound ozone molecule from the weekly-bound van der Waals complex is accurately determined by inspecting the dependence of adiabatic energy (of 2D solution for hyper-angles θ and φ) as a function of hyperradius ρ. Such dependence indicates a well-defined barrier near ρ † ~ 5.5 Bohr, sensitive to the rotational excitation (quantum numbers J and K), recombination pathway under consideration, and the number of isotopic substitutions. First, the covalent well probability cov is obtained by integrating the vibrational wave function of scattering resonance through the range 0 < ρ < ρ † . The contributions of symmetric and asymmetric ozone molecules, sym and asym , are determined using hyper-angle φ (see Fig. 4 in the main text). One third of the range in the vicinity of symmetry axis (2π/3 < φ < 4π/3) corresponds to symmetric ozone isotopomer, while the remaining two thirds of the φ-range correspond to asymmetric ozone molecules.
A complex absorbing potential (CAP) in the form suggested by Manolopoulos [J. Chem.
Phys. 117, 9552 (2002)] was used to impose the boundary conditions. The CAP was defined with the minimum absorption energy min = 7 cm -1 and spans the range of ~ 6 Bohr going inward from the end of the -grid. The optimized DVR-grid for coordinate covered the range of 3 ≤ ≤ 15 Bohr and consisted of 88 points. It is nearly impossible to converge every individual state above the dissociation threshold. Therefore, our convergence parameters were adjusted to ensure convergence of the overall recombination rate coefficient to within few percent. Convergence of the individual states depends on their properties. Broad resonances with Γ ~ 10 cm -1 are converged to within 1 cm -1 or better (both energy and width). Narrower resonances are converged much better. Convergence of broader resonances is not important, since they make negligibly small contributions to the dynamical partition function (their weights are close to 1, but their probabilities cov are close to 0).