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CC calculations for J=10

The total reaction probability for J=0, J=1, J=2, J=5, and J=10 is plotted in Fig. 1, where the J=0, J=1, J=2, and J=5 results were taken from paper I.

  
Figure: Total reaction probability as a function of J. J=0, J=1, J=2, and J=5 results taken from Ref. [43].
\begin{figure}
\centering\epsfig{file=Jall.ps,width=10cm}\end{figure}

A number of things are immediately clear from the figure. First, the increase in the reaction probability around E=1.25 eV that is so clear for J=0 has completely disappeared at J=10. Instead, the reaction probability shows a linear increase as a function of energy. Second, compared to low J, the resonance structure for J=10 has nearly disappeared. In paper I we suggested that the washing out of the resonance structure is due to the fact that for higher J states there are more $\Omega $ states one has to sum over to get the reaction probability. The overlap between the resonances for each of these states would then result in the washing out of the resonances for the total reaction probability. Thus, we examined the reaction probabilities for the constituent $\Omega $ states for each $\Omega _i$ separately. Representative examples for J=10, $\Omega _i=0^+$ ( $\Omega_i=0$, parity even) are plotted in Fig. 2.
  
Figure: Total reaction probability for J=10, $\Omega _i=0^+$ as a function of $\Omega $ for a number of representative values of $\Omega $.
\begin{figure}
\centering\epsfig{file=J10O0omega.ps,width=10cm}\end{figure}

From this figure we see that the curves indeed show more resonance structure than the total reaction probability for J=10, $\Omega _i=0^+$, which is plotted in Fig. 3. This validates our explanation of the disappearance of the resonance structure with increasing J. Fig. 2 also shows that the resonances in the reaction probabilities for the individual $\Omega $ states stay sharp for the J=10, $\Omega _i=0^+$ calculation. This means that an alternative explanation for the washing out of the resonance structure suggested in paper I is not very likely; namely that the life time of the complex decreases with increasing J, resulting in broadened resonance peaks. Note also that the averaging over $\Omega _i$, needed to get the overall reaction probability, introduces a second overlapping of resonances and subsequent washing out of the resonance structure. This becomes clear upon comparing the reaction probabilities for the individual $\Omega _i$ in Fig. 3 with the total reaction probability for J=10.

In Fig. 3 we plot, as mentioned, the reaction probability as a function of $\Omega _i$ together with the total reaction probability for J=10.

  
Figure: Total reaction probability for J=10 as a function of $\Omega _i$.
\begin{figure}
\centering\epsfig{file=J10omega.ps,width=10cm}\end{figure}

We see a number of trends continue from our earlier J=1, J=2, and J=5 results (Figs. 4a-4c in paper I). First of all, the reaction probability for $\Omega _i=0^+$ is larger than the reaction probabilities for the $\Omega _i=1^+$ and the $\Omega _i=1^-$ calculations. As was the case for J=2 and J=5 (this is not so clear for J=1), the $\Omega _i=0^+$ curve also exhibits the largest amount of structure. Second, the gap between the reaction probabilities for $\Omega _i=1^+$ and $\Omega _i=1^-$ continues to grow. For J=1 the two lines in Fig. 4a of paper I are practically equivalent, whereas for J=5 (Fig. 4c, paper I) the reaction probability for $\Omega _i=1^+$ is clearly larger than the reaction probability for $\Omega _i=1^-$. This is still more obvious for J=10, where even at low energies, the two reaction probabilities have a different magnitude. The only difference between the two calculations for a specific J is the fact that for $\Omega _i=1^-$, the $\Omega=0$ state is forbidden. That the omission of the $\Omega=0$ state has more effect on the dynamics with increasing Jis entirely reasonable, since Coriolis coupling becomes more important with increasing J (the Coriolis terms of the Hamiltonian have a $\sqrt{J(J+1)-\Omega(\Omega\pm1)}$ dependence).

In paper I we investigated the possibility of decreasing the size of the calculation by leaving out $\Omega $ states with a negligible contribution to the reaction probability. We consider an $\Omega $ state to be negligible if its contribution to the reaction probability at a certain energy is 3-4orders of magnitude lower than the highest contribution and if this is the case for all energies between 0.8 and 1.8 eV. We concluded that for J=5and lower, no states could be left out. We expected at that time that for higher J it might be possible to leave out $\Omega $ states.

In Table I we give the relative contribution to the total reaction probability as a function of the final $\Omega $ state, $\Omega _f$, for J=10, $\Omega _i=0^+$ for a number of selected energies. The last row in the table gives the ratio of the largest element of a column with the smallest element of the column (which in this case is the $\Omega_f=10$ state at all energies). For comparison, we give the corresponding results for J=5, $\Omega _i=0^+$ in Table II. As is clear from comparing Tables I and II, the $\Omega=J$ state indeed becomes less important in going from J=5 to J=10, because the ratio of the highest population with the lowest population ($\Omega=J$) increases for each energy point given in Tables I and II. However, it is also clear that that ratio for J=10 is nowhere near the 3-4 orders of magnitude for any energy. Looking at the distribution of the reaction probability over the final $\Omega $ states, we also conclude that Coriolis coupling is more important for J=10 than for J=5 in the R-embedding, because for J=5the largest three contributions to the reaction probability accounted for more than 60 %, whereas for J=10 the reaction probability is much more smeared out. In Tables III and IV we tabulated the relative contributions of the final $\Omega $ states to the total reaction probability for the J=10, $\Omega _i=1^+$ and the J=10, $\Omega _i=1^-$ calculations, respectively. Comparing Table I to Table III, we see that qualitatively not much changes. In both cases the lower $\Omega $ states are the most important and the highest ratio between the smallest and the largest element is approximately the same ($\sim$2 orders of magnitude). However, if we look at Table IV, we see that for the J=10, $\Omega _i=1^-$calculation the situation is reversed. Here, the higher $\Omega $ states are more important than the lower $\Omega $ states, especially at higher energies. It seems unlikely that any state can be left out, since the contribution from all of them is similar. Thus, we conclude (tentatively) that an approximate calculation in which a number of $\Omega $ states is left out to make the calculation smaller, will not lead to accurate results. However, we will have to perform the approximate calculations to be absolutely sure about this.[95] Preliminary J=10, $\Omega _i=0^+$ calculations show that leaving out the $\Omega=10$ has indeed a sizable effect and cannot be done without significant loss of accuracy. We cannot examine the relative importance of Coriolis coupling as a function of J for calculations done in the r-embedding as straightforwardly, since in that embedding the calculations start in a coherent superposition of all K states.

The series of reaction probabilities from Fig. 1 also allows us to predict the maximum value of J that will have a non-negligible reaction probability. We did the following analysis of our data for each ``bin'' of 0.2 eV between 0.8 eV and 1.8 eV for J>0. For each bin and each J we calculated the average reaction probability. Then, for each bin, we tried a number of different fitting schemes to fit the data as a function of J. Using the fits, we extrapolated to Pr(E)=0. This is of course a very crude way to get an estimate for the maximum J still contributing, but we think that it gives a good idea of the size of that J value.

The best fit for all curves was obtained using a quadratic fitting functional (aJ2+bJ+c form). A slightly worse fit was obtained using an exponential quadratic fitting functional ( $\exp[aJ^2+bJ+c]$ form). Unfortunately, not all fitted curves for the quadratic fit had roots on the real axis. However, for the ones that did, PJr(E) became zero between J=16 and J=26. This is remarkably similar to the J values of 15-30 we estimated in paper I based on the maximum impact parameter found in classical trajectory calculations on the H + O2 system.[12,23,24]


next up previous
Next: Helicity conserving calculations Up: Results and Discussion Previous: Results and Discussion
Anthony J. H. M. Meijer
1998-10-14