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Reaction Cross Sections

The quantum cross sections are shown in Fig. 4, and compared with the quasiclassical results of Varandas.[25] The quantum curves are from the sg and mw filtered probabilities obtained using quadratic interpolation/linear extrapolation. These two curves are very similar, although the mw curve, while smoother than the sg curve, does not show proper threshold behavior since it ought to decay to zero as E$_{tr}$ $\rightarrow$ 0.7 eV. The two quasiclassical curves differ from one another and these differences are instructive. Zero point energy effects present significant difficulties for quasiclassical trajectory studies of this reaction, particularly in the threshold region. Varandas[20,21,25] has developed a suite of methods to circumvent this problem. Results obtained with two of these are displayed here. In the IEQMT method, a trajectory is not considered in the statistical ensemble unless it has enough internal energy to satisfy the zero-point energy requirement for the final product in the case of a reactive trajectory or the reformed reagents, in the nonreactive case. The VEQMT method is similar except that it uses only the final vibrational energy of product (or reactant) to determine whether to include a trajectory. The considerable difference in these curves indicates the importance of zero-point energy constraints for this reaction. While the classical results are produced for a 300K ensemble, they are expected to be similar to results from ground state O$_2$;[21] rotational excitation of the reactants does not enhance reactivity.[21,32]

The VEQMT cross sections are much closer to the QM results than the IEQMT cross-sections, indicating that the more stringent requirement on zero-point energy conservation is necessary for accuracy. The VEQMT cross sections are somewhat lower than the QM ones near threshold, then rise to be somewhat higher than the QM as E$_{tr}$ increases. Quasiclassical methods based on the discarding of ``unphysical trajectories'' can be accurate only if they are unbiased in discarding reactive or nonreactive trajectories. As explained in Ref. res:vara1993, at low energies, discarded trajectories will more likely be reactive trajectories formed with insufficient internal energy. At higher energies however, most reactive trajectories will meet the internal energies requirements. For nonreactive trajectories, the reactants are more likely to reform with an excess of translational energy. Therefore, we may expect that at higher energies, these QC methods will overestimate the QM cross sections.

Fig. 5 displays the computed QM cross sections with the most recent experimental cross sections of Wolfrum and Volpp.[17,50] Note that these cross sections, determined from O rather than OH detection, do not have the pronounced maximum that was reported in earlier work.[7,8,9,17] Nevertheless, the new experimental results are higher than the computed QM results at almost every energy. Note in particular that the experimental results show a much faster rise at threshold and a smaller energy dependence than the theoretical ones. If one accepts the accuracy of the experimental cross sections, then the most likely reason for the discrepancy between our values and the experimental ones is that the DMBE IV PES is not accurate enough to give reliable cross sections.

Recent high level ab initio calculations by Harding et al. [84] on the H + O$_2$ interaction, indicate that the DMBE IV PES is too attractive in the entrance channel, perhaps making the probability of forming the metastable HO$_2$ radical too large. Similar difficulty with the DMBE IV PES can be seen in Ref.kend1995. How these inaccuracies affect the overall reaction rate is not clear, however, since, especially at energies above 1.2 eV, both complex-forming and direct processes contribute to the reaction cross section.[1,2]

It is also possible that electronically nonadiabatic interactions play a significant role in this reaction, although one would expect that these would be more important at higher energies. The lowest lying conical intersection between the $^2A''$ PES's is the C$_{2v}$ intersection between the $\tilde{X} ^2A''$ and $2 ^2A''$ surfaces at 1.89 eV above the energy of the reactants.[49] Except for possible geometric phase effects, nonadiabatic interactions between these two electronic states are not expected to be important at the energies considered in this work. It is possible that interactions between states of different symmetry could pay some role. The $\tilde{X} ^2A''$ and $1 ^2A'$ surfaces are members of a Renner-Teller pair at collinear geometries; the Renner-Teller intersection occurs at $\approx$ 0.5 eV above the energy of the reactants. This $1 ^2A'$ surface is attractive and correlates to the same product asymptote as the $\tilde{X} ^2A''$ surface, although it correlates to excited reactants: H + O$_2$ ($^1\Delta_g$) which is 0.982 eV higher in energy that the H + O$_2$ ($^3\Sigma_g^-$).[49] Because reaction on the 1A' state is slightly exoergic, one might expect that if it does play a role via the nonadiabatic Renner-Teller interaction, that it would facilitate reaction. Of course, in order to reach these linear configurations, considerable energy must be transferred from the reaction coordinate to the bending modes. Thus, if the Renner-Teller interaction plays a role at all, it is not likely to involve direct processes but rather reactions that proceed via HO$_2$ collision complexes.

There is some indication that nonadiabatic effects may play a role, since only about 65-70% of the O atoms are formed in $j' = 2$ with the rest being formed in $j' = 1$, where $j'$ in this case refers to the total angular momentum of the O atom, including spin angular momentum. Only O($^3P_{j' =
2}$) correlates with the ground state of the reactants. It is very likely that spin-orbit transitions occur in the exit channel, which would have no effect on reactivity. However, until a proper calculation which includes the relevant surfaces and couplings is performed, it is impossible to say anything conclusive about the role of nonadiabatic interactions in contributing to the reaction cross section.

It is still an open question as to whether the geometric phase induced by the conical intersection on the ground state PES will have a significant effect on the reactive cross section. Kendrick and Pack[38,39] found that, at thermal energies below the reaction threshold, the geometric phase had a significant effect on both the lifetimes and positions of the $J=0$ resonances as well as state-to-state inelastic transition probabilities. It is quite possible that at the higher energies involved in reaction, there will be significant geometric phase effects on the individual reaction probabilities, $P_{\nu_i,j_i,\Omega_i}^{J,p}$. It is less clear, however, how they will affect the total reaction cross section, a highly averaged quantity in which resonance structure does not seem to be evident.[50] In a detailed study of the geometric phase on the H + D$_2$ reaction, Kendrick concludes that due to cancellations included by symmetry, the effects of the geometric phase cancel out when even and odd $J$ values are added together, thus having no effect at all on either the differential or integral cross sections.[85] He notes that it is possible that this will be true of the H + O$_2$ reaction as well. In a somewhat earlier study on H + D$_2$ that did not include the geometric phase, De Miranda et al.[86] also achieved excellent agreement with experimental differential cross sections.

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Next: Conclusions Up: Results and Discussion Previous: Reaction Probabilities
Anthony J. H. M. Meijer 2000-10-05