Hamiltonian, Propagation, and Analysis

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

where and are the reduced mass of the H + O

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
and *K*apart from two Coriolis terms in each set, which couple
to
or *K* to .

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
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 ,
each of which requires 6
evaluations of
or
.
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 *P*_{r} (*E*).[29,43]

In all calculations we take *j*_{i}=1 and ,
where *j*_{i} is the initial
rotational angular momentum quantum number of O_{2} and
its initial
vibrational angular momentum. The quantum number
is not determinable
experimentally, so we have to do a calculation for each
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

where the 1 in the exponent of the last term arises because

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
states or the *K* states. For the
*R*-embedding this means that throughout the calculations the wave function remains in a
pure
state. This makes the HC calculation *J* or *J*+1 times cheaper
than the corresponding CC calculations, where we have *J* or *J*+1 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
is larger than zero, because the only difference
between the two CC calculations is the (non)-existence of an state, which (for
)
is immaterial for the HC-*R* calculations.
This means we only have to do min(*J*,*j*_{i})+1 calculations instead of
2min(*J*,*j*_{i})+1 to get the complete set of initial conditions.
(This means of course that the calculations for
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
and
*j* state to be able to compare the results with the HC-*R* and CC-*R*calculations. 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
2min(*J*,*j*_{i})+1 calculations have to be performed. Second, all *K*states contribute in the transformed wave function, meaning that the
basis for the HC-*r* calculations is as large as the basis for the CC-*r*calculations. Thus, in going from the HC-*r* calculations to the CC-*r*calculations only the work associated with the Coriolis terms is added,
which is negligible, as stated earlier.