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Next: Conclusions Up: Results and Discussion Previous: CC calculations for J=10

   
Helicity conserving calculations

We performed HC calculations in the R-embedding and in the r-embedding for J=1, J=2, J=5, and J=10. Together with the CC results (for which the R-embedding and the r-embedding give identical results) they are plotted in Fig. 4a-d. For J=1 in Fig. 4a we see that there is good agreement between the CC results (solid line) and the HC-r results (dotted line).

  
Figure 4: Total reaction probability as a function of J for 3 calculation methods. All panels: solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (a): J=1.
\begin{figure}
\centering\epsfig{file=J1CS.ps,width=10cm}\end{figure}

The HC-R results show more deviation from the CC than the HC-r results. The same observation can be made for Fig. 4b (J=2), although the agreement between the three calculations is not as good as for J=1, especially at higher energies.
 
Figure: Total reaction probability as a function of J for 3 calculation methods. All panels: solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (b): J=2.
\begin{figure}
\centering\epsfig{file=J2CS.ps,width=10cm}
\end{figure}

This trend continues for J=5 in Fig. 4c and for J=10 in Fig. 4d.
 
Figure: Total reaction probability as a function of J for 3 calculation methods. All panels: solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (c): J=5.
\begin{figure}
\centering\epsfig{file=J5CS.ps,width=10cm}
\end{figure}

[htb]
 
Figure: Total reaction probability as a function of J for 3 calculation methods. Solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (d): J=10.
\begin{figure}\centering\epsfig{file=J10CS.ps,width=10cm}
\end{figure}

In Fig. 4d we see that for J=10 the agreement between HC-r and CC calculations is poor (deviation 30 %) at high energies and reasonable at lower energies. The agreement between HC-R and CC calculations is poor over the entire energy range between E=0.8 eV and E=1.8 eV, although the average value for the reaction probability agrees reasonably well at low energies.

In Figs. 5a-c we plot the HC-R reaction probabilities, the HC-rreaction probabilities, and the CC reaction probabilities as a function of initial $\Omega _i$ state for J=10.

  
Figure: Total reaction probability for J=10 for different $\Omega _i$ for 3 calculation methods. All panels: solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (a): $\Omega _i=0^+$.
\begin{figure}
\centering\epsfig{file=J10CSomega0p.ps,width=10cm}\end{figure}


 
Figure: Total reaction probability for J=10 for different $\Omega _i$ for 3 calculation methods. All panels: solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (b): $\Omega _i=1^+$.
\begin{figure}
\centering\epsfig{file=J10CSomega1p.ps,width=10cm}
\end{figure}


 
Figure: Total reaction probability for J=10 for different $\Omega _i$ for 3 calculation methods. All panels: solid line: CC calculations, dashed line: HC calculation in R-embedding, dotted line: HC calculation in r-embedding. (c): $\Omega _i=1^-$.
\begin{figure}
\centering\epsfig{file=J10CSomega1m.ps,width=10cm}
\end{figure}

From comparing these results we can learn why the HC calculations give different results from the CC calculations.

In Figs. 5a-c we see for all three HC-R results a large number of resonances at low energies, which is indicative of a long-lived reaction complex. For high energies we see that only for $\Omega _i=0^+$ the reaction probability significantly increases and the resonance structure almost disappears. This last feature for the $\Omega _i=0^+$ calculations is indicative of a ``direct'' reaction mechanism. The idea of two different reaction mechanisms for the H + O2 ( $^3\Sigma_g^+$) has been brought up in the literature before.[11,12,13,14,23,24,43] We used it in combination with a classical model in paper I to explain the CC results for J=1. The same classical model can be used here to explain the HC-R results. If a calculation is started in the $\Omega _i=0^+$ state (and for a HC-R calculation it stays in this state), the reaction geometry can be called ``planar''. This means that the molecule is rotating in the plane of the three atoms. In this case it is possible for the H atom to collide with one of the O atoms directly, leading possibly to reaction if the total energy is sufficient. At lower energies a reaction complex will be formed. Assuming also that the direct reaction has a higher reaction probability than the complex forming reaction, we get a curve with a small reaction probability and a large number of resonances at low energies and with a high reaction probability and a smaller number of resonances at higher energies, as in Fig. 5a. For calculations starting in the $\Omega _i=1^+$ or $\Omega _i=1^-$ states, the direct reaction path is less likely, because initially the molecule is not rotating in the plane of the 3 atoms, making it harder for the atom to hit one of the O atoms directly. Thus, the complex forming reaction is favored. This results in a curve with a low reaction probability and a large number of resonances throughout, as can be seen in Figs. 5b and 5c. The resonance structure at low energies in Figs. 5a-c also suggests that our explanation for the disappearance of the resonance structure at low energies in Fig. 1 for J=10 in the CC-R calculations is correct. Since the HC-Rcalculations use only one $\Omega $ state, there cannot be any overlap between different $\Omega $ states. Consequently, the resonance structure does not get washed out.

The reaction model used above and the difference in reactivity between the two reaction geometries are corroborated by experimental results. The difference in reactivity between the two reaction geometries has been found experimentally by Bronikowski et al,[13] who studied the relative populations of the $\Lambda$ doublet states of the ground state of OH($^2\Pi$), which correlate to the two different reaction geometries in the interaction region. The reaction model used above is consistent with measurements of differential cross sections (DCSs) by Fei et al.[14] At low energies they find an almost isotropic DCS, indicative of a reaction complex. At higher energies, they find a DCS with a large maximum in the forward scattering region and a smaller maximum in the backward scattering region. Both of these maxima are consistent with a direct mechanism.

From Figs. 5a-c we can also see that the main cause for the differences between the HC-R and the CC results for J=10 lies in the HC-R results for J=10, $\Omega _i=0^+$ at high energies (Fig. 5a). We can also use our reaction model to explain this. Both calculations start in the same initial state, $\Omega _i=0^+$, which corresponds to a planar reaction geometry. For the HC-R calculation $\Omega $ is conserved, meaning that in this case also the reaction geometry is conserved. In the CC calculation the $\Omega $ selection is lost through Coriolis coupling, allowing the wave packet to sample other (less reactive) reaction geometries as well. This results in a lower reaction probability, especially at higher energies.

If we compare the HC-r results to the HC-R results in Figs. 5a-c, we that the largest differences occur for the J=10, $\Omega _i=0^+$ calculation in Fig. 5a. Notable are the absence of sharp resonances at low energies and the lower reaction probabilities at higher energies for the HC-r calculation. This can be explained in the following way. Initially, the wave packet in the HC-r calculation is in the $\Omega _i=0^+$ state through a coherent superposition of K states. Although there is no Coriolis coupling between the K states, this coherence is nevertheless lost during the calculation. The wave packet evolves differently in each K state, because each K state has a different Hamiltonian. This decoherence of the wave packet means that other reaction geometries (components of other $\Omega $ states) become accessible to the wave packet during the propagation, resulting in a lower reaction probability at higher energies. The decoherence is not fast enough, however. The HC-rreaction probability still overestimates the correct (CC) reaction probability. The fact that each K state evolves with its own Hamiltonian also means that the reaction probability for each K state has a different dependence on energy. This results in overlapping and subsequent washing out of resonances at low energies as in the CC calculations.

We conclude from the HC calculations that neither $\Omega $ nor K are good quantum numbers. Therefore, we think that neither the R-embedding nor the r-embedding are optimal coordinate embeddings for this reaction. This corroborates the observation from Ref. [30] that no optimal frame for this reaction will be found, because of the floppiness of the HO2 complex.


next up previous
Next: Conclusions Up: Results and Discussion Previous: CC calculations for J=10
Anthony J. H. M. Meijer
1998-10-14