Quantum Dynamics of the Complex Forming OH + CO H + CO Reaction

Evelyn M. Goldfield and Stephen K. Gray

Chemistry Division
Argonne National Laboratory

The detailed quantum dynamics of a scattering event involving four atoms represents an extremely difficult computational problem that is reasonably well suited to parallel computer architectures. We have determined the quantum dynamics of a model for a particular four-atom chemical reaction that is very important in combustion and atmospheric chemistry. Our calculation would have required over four months to accomplish on a single RS/6000 workstation with sufficient memory. Moreover, the required memory is about 1028 MB. Using just 32 processors of Argonne's SP, our calculation was completed in one week.

In order to identify the relevant scattering information such as reaction probabilities or cross sections, it is necessary to determine (in a time-dependent picture) wave packets that satisfy the time-dependent Schroedinger equation. Assuming the electronic motion is described by a potential function that varies with the nuclear positions, a wave packet is a complex function of all the nuclear coordinates and time. It is still not possible to determine full-dimensional solutions to the Schroedinger equation for the chemical reaction OH + CO H + CO because the wave packet would involve (after separating out the center-of-mass motion) 4 x 3 - 3 = 9 nuclear coordinates and time. If, for example, 20 points were used to describe each nuclear degree of freedom, the wave packet would involve complex numbers at each time. Some simpler four-atom chemical reactions involving mostly hydrogen atoms could conceivably be studied in full dimension because large de Broglie wavelengths and relatively few points per degree of freedom are involved. The above reaction is challenging because three of the atoms are relatively heavy, and the chemical forces (the potential) are quite strong, demanding rather fine grids per degree of freedom.

Of course, in addition to the memory difficulties when detailed four-atom quantum scattering events are to be described, is the actual computational burden: The "propagation" of a wave packet according to the time-dependent Schroedinger equation may be broken down into a series of time steps with the time-dependent propagator exp(-i H dt) acting on the current wave packet at time t to generate one at time t + dt. Here, H is the Hamiltonian operator which involves derivative operators with respect to all the coordinates, and a multiplication by the potential. The propagator is evaluated by carrying out a number of acts of H on the current wave packet, and then constructing an approximation to the exponential operator (similar in spirit to a Taylor series, but we actually use a better series representation, the Chebyshev series).

We considered just a planar model of the OH + CO reaction, and further restricted the CO internuclear distance to be frozen, and examined total angular momentum J = 0 states. These approximations/limitations result in just a four degree of freedom (4-DOF) model. Nonetheless, this is still an immensely challenging model -- the wave packet involves up to 340 MB and several copies of it are required to perform the details of our propagation. Fast Fourier transforms (FFTs) are used to evaluate the effect of the derivative terms in the Hamiltonian operator. The wave packet was distributed among the processors used, with each processor containing only a portion of the wave packet at any given time. Our problem, unfortunately, is not ``embarassingly parallel'' and information must be passed from processor to processor. A transpose operation was found to be quite convenient for our particular purposes.

Despite our model being a reduced dimension one, we were able to use certain approximate theories to extend our results to full-dimension and make an estimate of the observable rate constant. We compared our results with previous (even more) reduced dimension results, and certain classical simulations results. We have found that HOCO intermediate complexes play a crucial role in the determination of H + CO products. Our results have also pointed to a need for refining the potential model used to describe the OH + CO reaction. We are currently considering extending our model to 5-DOF and perhaps even to 6-DOF (which would be full dimensional with zero total angular momentum).

The figure displays a snapshot of our wave packet at a certain time during the chemical reaction. The R axis is the OH to CO separation, and the axis is the OH separation. The wave packet starts at t = 0 as a Gaussian-like function in the large R (small ) limit. It moves into the interaction region (small R), where HOCO complexes form. (There are two other coordinates, of course. The figure displays the absolute square of the wave packet averaged over those two other coordinates.) The figure shows the wave packet at such a time. One sees a large density in the interaction (HOCO) region, and a small trickle of density moving out to the large limit, that is, to the limit of H + CO products.

>