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Hamiltonian, Propagation, and Analysis

We use the following two Hamiltonians in our calculations for the R-embedding and for the r-embedding, respectively.


  
$\displaystyle \hat{H}_R = -\frac{\hbar^2}{2\mu_R}\frac{\partial^2}{\partial R^2...
...{J}-\hat{j})^2}{2\mu_R R^2} + \frac{\hat{j}^2}{2\mu_r r^2} + V
(R,r,\vartheta),$     (8)
$\displaystyle \hat{H}_r = -\frac{\hbar^2}{2\mu_R}\frac{\partial^2}{\partial R^2...
...{J}-\hat{l})^2}{2\mu_r r^2} + \frac{\hat{l}^2}{2\mu_R R^2} + V
(R,r,\vartheta),$     (9)

where $\mu_R$ and $\mu_r$ are the reduced mass of the H + O2 collision complex and the reduced mass of O2, respectively. $V(R,r,\vartheta)$ is the intermolecular potential. In our case, we use the DMBE IV surface.[47]

We use the time-dependent Schrödinger equation to propagate the wave function in time. Using the basis set expansion, given in Eqs. (3) and (4), we get the equations of motion for the coupled channels (CC) method. For the R-embedding they are given in Ref. [43]. The equations of motion can be derived for the r-embedding in an analogous fashion. The equations of motion in both embeddings are diagonal in the projection quantum numbers $\Omega $ and Kapart from two Coriolis terms in each set, which couple $\Omega $ to $\Omega\pm1$ or K to $K\pm1$.

This tridiagonal form for the equations of motion forms the basis of the parallel method we use to facilitate our calculations. We assign blocks of the wave function corresponding to a particular $\Omega $ or K to a specific processor on a parallel machine.[69,70,71] The only communication between the processors arises from calculating the Coriolis terms from the equations of motion and is small compared to e.g., the calculation of the vibrational terms in the equations of motion.[43,69] In our calculations we use MPI[87,88,89] to perform the communication between the processors. See Refs. [43] and [69] for more details.

The time-evolution of the wave function is accomplished by using a symplectic integrator of the (m=6, n=4) type.[90] This means we take short and discrete time steps $\Delta t$, each of which requires 6 evaluations of $\hat{H}_R \Psi_R$ or $\hat{H}_r \Psi_r$. At the edges of the grid the wave function is absorbed using an exponential damping function.[91] This procedure is equivalent to using an imaginary potential to absorb the wave function.[92,93]

During the propagation of the wave function we calculate the derivative of the wave function along a cut through the Potential Energy surface in the O + OH exit channel. Together with the wave function at this cut it is stored on disk. After the propagation they are restored and Fourier transformed to calculate the reactive flux as a function of energy[29] and from there the reaction probability Pr (E).[29,43]

In all calculations we take ji=1 and $\nu_i=0$, where ji is the initial rotational angular momentum quantum number of O2 and $\nu_i$ its initial vibrational angular momentum. The quantum number $\Omega $ is not determinable experimentally, so we have to do a calculation for each $\Omega _i$ initially possible and average over all these calculations.[94] To make comparison possible for the r-embedding we start in the same initial state as in the R-embedding. For this we use Eq. (6), where we fill in the known parameters, which leads us to


 
$\displaystyle A_{\lambda\nu l_i K_i} (t_0)$ = $\displaystyle (-1)^{K_i+\Omega_i}
\left[\frac{\left(1+\delta_{K0}\right)}{\left...
...)}\right]^\frac{1}{2}C_{\lambda \nu j_i \Omega_i} (t_0)
\sqrt{(2l_i+1)(2j_i+1)}$  
    $\displaystyle \left(\begin{array}{ccc}l_i & J & j_i \\  0 & \Omega_0 & -\Omega_...
... & J & j_i \\  -K_i & K_i & 0\end{array}\right) \left[2+2(-1)^{p+l_i+1}\right],$ (10)

where the 1 in the exponent of the last term arises because ji is always odd for the H + O2 $(^3\Sigma^-_g)$ system. In Eq. (10), liruns between $\left\vert J-j_i\right\vert$ and J+ji and K is always larger than or equal to $K_{\rm
min}$, where $K_{\rm min} = 0$ or 1, depending on the spectroscopic parity. Thus, the quantities li and Ki are not fixed in Eq. (10), in contrast to ji, $\nu_i$, and $\Omega _i$. For each set of (J,ji) states in both embeddings we have to perform 2$\times$min(J,ji)+1 calculations to get a complete set of initial conditions. Each calculation itself runs on J+1 (even spectroscopic parity) or J (odd spectroscopic parity) processors.[69] For more details regarding parameters for the calculations, basis set sizes, etc. we refer to Ref. [43].

For our helicity conserving (HC) calculations,[73] we use the same method as described above, apart from the fact that we ignore the Coriolis coupling between the $\Omega $ states or the K states. For the R-embedding this means that throughout the calculations the wave function remains in a pure $\Omega $ state. This makes the HC calculation J or J+1 times cheaper than the corresponding CC calculations, where we have J or J+1 $\Omega $states to deal with. Moreover, it is clear from studying the equations of motion for this system that leaving out Coriolis coupling makes the reaction probability for even parity equal to the reaction probability for odd parity, if $\Omega _i$ is larger than zero, because the only difference between the two CC calculations is the (non)-existence of an $\Omega=0$state, which (for $\Omega_i >0$) is immaterial for the HC-R calculations. This means we only have to do min(J,ji)+1 calculations instead of 2$\times$min(J,ji)+1 to get the complete set of initial conditions. (This means of course that the calculations for $\Omega_i >0$ have a weight of two in the calculation of the total reaction probability).

The HC calculations in the r-embedding (HC-r calculations) are not much cheaper than the corresponding CC calculations (CC-r calculations). For the HC-r and CC-r calculations we also start in a pure $\Omega $ and j state to be able to compare the results with the HC-R and CC-Rcalculations. Therefore, for the r-embedding we start in a coherent superposition of K states and l states through the transformation Eq. 10. This has two effects. First, a calculation with even parity does not give the same result as a calculation starting with odd parity, because they transform to different initial states in the r-embedding (see Sec. IIIB). This means that all 2$\times$min(J,ji)+1 calculations have to be performed. Second, all Kstates contribute in the transformed wave function, meaning that the basis for the HC-r calculations is as large as the basis for the CC-rcalculations. Thus, in going from the HC-r calculations to the CC-rcalculations only the work associated with the Coriolis terms is added, which is negligible, as stated earlier.


next up previous
Next: Results and Discussion Up: Theory and Computational details Previous: Coordinates, Basis set, and
Anthony J. H. M. Meijer
1998-10-14