Hamiltonian and Propagation

The triatomic Hamiltonian in Jacobi coordinates for the H + O_{2} system in
the BF frame is given
by[33,49,50,61,62,63,67,68]

where and are the reduced mass of the H + O

(8) |

Using the expansion of the wave function, given in
Eq. (2), and taking the inner product with the basis
functions we get a set of equations of motion for the coefficients
,
which are simplified using the
orthogonality properties of the Wigner D-matrices. Furthermore,
consistent with the DVR approach the integrals in *R* and *r* involving
multiplicative operators are evaluated within the quadrature
approximation, resulting in diagonal matrix representations for these
operators. This results in the following set of equations of motion of
the coefficient
.

and can be evaluated analytically[55,56,57,73]. is the reduced potential in DVR-point and is defined as

This expression is evaluated using a Gauss-Legendre quadrature.

The last two terms in the equations of motion are the Coriolis terms and represent the Coriolis coupling in the system. From its form the sparsity of the Coriolis coupling matrix is immediately clear. There is only coupling between the substates and , and the substates and . The coefficients of the form are given as

The time-evolution of the wave function is accomplished by using a
symplectic integrator of the (*m*=6,*n*=4) type, introduced by Gray and
Manolopoulos in 1996 (see Ref. [74]). This means
that we take short and discrete time steps ,
each of which
requires 6 evaluations of
.

At the end of each time step we absorb the wave function at the edges of the
grid according to
with
for
and
for
[61]. A similar definition holds for .
The
damping coefficients *B*_{R} and *B*_{r} are determined by trial and error, where
they must be sufficiently strong to dampen the wave function as it approaches
the edge of the grid, but not so strong as to cause artificial reflections.
This procedure is equivalent to using an imaginary potential to absorb the
wave function[75,76]

The propagation of the wave function for *J*>0 is facilitated by the
Coriolis-coupled parallel method, introduced recently by Goldfield and
Gray[49]. This method is most efficient on parallel
computers with fast connections between the processors, e.g., the IBM
SP2 or the Cray T3E. In this method we distribute the portions of the
wave function corresponding to different
substates over
different processors. Communication between processors is needed only
when the Coriolis terms in Eq. (7) are calculated. The
overhead generated is negligible compared to the computational work
associated with the kinetic/potential terms in Eq. (7).
This method requires the same amount of CPU-time per processor as a
*J*=0 calculation, allowing the calculation to finish in approximately
the same amount of wall-time. We use the Message-Passing Interface
(MPI) library[77,78,79] to perform the communication
between the processors. We wish to emphasize here that implementing
this method is *extremely* simple. About 15 lines of Fortran code
were needed to convert a serial program into a parallel program. For
more details we refer to Ref. [49].