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Next: Conclusions Up: Results and Discussion Previous: J=0


In this section we present our results for J=1, 2, and 5. We will do that in order of increasing detail, i.e., we start with the reaction probabilities summed over the possible initial states and we end with the reaction probability per $\Omega $ substate for a given initial state $\Omega _i$.

In Fig. 1 we plot the reaction probability for J=0 and showed that the envelope is basically converged at T=30 000 a.u. In Fig. 2 we show a similar plot for PJ=1(E) as a function of the propagation time T. [Note, that in order to use this reaction probability in the calculation of a cross section, one has to multiply it by (2J+1)]. This figure also shows that the envelope of the reaction probability is basically converged at T=30 000 a.u. Therefore, we stopped the calculations for J=2 and J=5 at T=30 000 a.u, since we are mainly interested in the envelope of the reaction probability and not in the exact position and widths of the resonances. For these calculations also more than 97 % of the wave function has already left the grid at T=30 000 a.u. and Pr (E) + Pnr (E) for the region between 0.85 eV and 1.8 eV never deviates more than 10-3 from 1. Between 0.8 eV and 0.85 eV the deviation is less than 10-2.

In Fig. 3 we show the total reaction probability PJ(E) for J=0, 1, 2, and 5, averaged over all (3) possible initial states with the molecule initially in the vibrational state $\nu=0$ and in the rotational state j=1. A number of things are immediately clear from this figure. The division into two parts of the reaction probability persists at J>0. For energies lower than 1.25 eV the reaction probability is on average independent of energy. Interestingly, a number of the resonances for J=0 persist in the J>0 calculations. This indicates that it might be possible to measure them in an experimental setup. At energies larger than 1.25 eV the reaction probability increases rapidly. However, the increase is less pronounced for J>0 than it is for J=0. Moreover, at J=5 the reaction probability increases in a linear fashion instead of with a jump as is the case for J=0, 1, and 2. We are currently performing calculations for J=10, 15, and 30 to see if this trend continues for higher J and to see if the reaction probability will go to zero with higher total angular momentum[84]. This last aspect is prompted by the fact that classical trajectory calculations suggest that the reaction has a very small maximum impact parameter between 1 and 2 a.u. (depending on energy)[25,26,27], which means that the maximum total angular momentum for ji=1 lies somewhere between 15 and 30 (again depending on energy).

In the 3 panels of Fig. 4 we show the reaction probability for different initial states for J=1, 2, and 5, respectively. The notation $\Omega_i=1^+$ means, that we started the calculation with the wave packet in the $\Omega=1$ substate and that the spectroscopic parity [ (-1)J+p with p the parity of the wave function] was even. Starting in the $\Omega=1$ substate with odd spectroscopic parity is designated by $\Omega_i=1^-$.

In Fig. 4a we plot the reaction probability for J=1, PJ=1(E). As is clear from this figure, the initial $\Omega $ substate has a distinct influence on the reaction probability. The $\Omega _i=0^+$ calculation shows a much higher reaction probability than the $\Omega_i=1^+$ and $\Omega_i=1^-$ calculations. This results in a lower total reaction probability than for J=0. The difference can be explained to a certain extent using the following classical model. Initially, there are marked differences in reaction geometry for the $J=0,\ \Omega_i=0^+$ and the $J=1,\ \Omega_i=0^+$ calculations on the one hand and $J=1,\ \Omega_i=1^+$ and $J=1,\ \Omega_i=1^-$ calculations on the other hand. In the former case initially the configuration is ``planar'' (which for J=0 of course remains that way). This means that the molecule is rotating in the plane spanned by the three atoms. One can imagine that in this configuration it becomes easier for the H-atom to ``hit'' one of the two O-atoms, leading to a direct reaction. For $J=1,\ \Omega_i=1^+$ and $J=1,\ \Omega_i=1^-$ the system is initially in a ``perpendicular'' configuration. This means that the molecule rotates perpendicular to the instantaneous plane of the three atoms. If it retains this geometry, it is very hard for the H-atom to hit one of the O-atoms directly and create products. In fact, it is more likely that a reaction geometry like this will create an intermediate complex. Further state-resolved reaction probabilities and energy-dependent wave functions are needed to resolve this issue. However, the mechanism given above is consistent with the findings of Bronikowski et al. who concluded on the basis of the populations of the two $\Lambda$ doublet states of the product OH that the planar configuration was more reactive than the perpendicular configuration at E=1.6 eV[16].

Looking at Fig. 4b and Fig. 4c, we see that the trends for J=1 continue at J=2 and J=5. The $\Omega _i=0^+$ initial state shows a higher reaction probability than the other two initial states. Comparing $J=0,\ \Omega_i=0^+$ to $J=1,\ \Omega_i=0^+$, $J=2,\
\Omega_i=0^+$, and $J=5,\ \Omega_i=0^+$ in Fig. 5, we see that the reaction probability goes steadily down. The same holds when comparing the other initial states for different J. This might be due to the fact that the centrifugal barrier[85] (i.e., the J(J+1) term in the Hamiltonian) becomes larger for higher J. This suggests that one might get a good guess for the reaction probability for a specific initial state for higher total angular momentum by using an approximate method (see e.g., Refs. [86,87,88]), starting from the J=1 calculations. We plan to investigate this issue further[84].

Looking at Figs. 4 and 5 it is clear that for higher total angular momentum the resonances get broader and less pronounced. This is the case not only for the reaction probability, averaged over the initial states, but also for the reaction probabilities for the separate initial states. This effect might be caused by the fact that for higher J there are more states available for the wave function to dissipate into. This also suggests that another frame for doing the calculations in, e.g., the frame in which the O2 bond is the z-axis[89] may not be a better frame in the sense of conservation of the projection of J onto that z-axis (this is in agreement with earlier observations by Pack et al.[34]). We are currently looking into this issue[84]. On the other hand, given the strong dependence of the reaction probability on the initial state, it is clear that $\Omega $ does not scramble as quickly as one might expect. Knowing that the Coriolis interaction goes to zero as R-2 when R becomes large, the strong dependence suggests that the Coriolis interaction becomes non-negligible only in the interaction region. At higher J it might even be possible to leave out a few $\Omega $ substates, if the scrambling is slow enough. In order to investigate this we put the reaction probability per $\Omega $ substate for $J=5,\ \Omega_i$ in Table II. For $\Omega _i=0^+$ there is a no clear trend for the reaction probability as a function of $\Omega $. For $J=5,\ \Omega_i=1^+$ there seems to be a weak trend for decrease of the reaction probability with increasing $\Omega $. For $J=5,\ \Omega_i=1^-$, the trend seems to be the reverse. Clearly, J=5 is too low to leave out any $\Omega $ states. It will be interesting to see what happens at higher J[84].

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Next: Conclusions Up: Results and Discussion Previous: J=0
Anthony J. H. M. Meijer