The detailed formulation of reactive scattering based on hyperspherical coordinates and local variational hyperspherical surface functions (LHSF) is discussed elsewhere [Kuppermann:86a], [Hipes:87a], [Cuccaro:89a]. We present a very brief review to facilitate the explanation of the parallel algorithms.
For a triatomic system, we label the three atoms 
,
and 
. Let (
) be any cyclic
permutation of the indices (
). We define the
coordinates, the mass-scaled [Delves:59a;62a]
internuclear vector 
from 
to 
, and the mass-scaled
position vector 
of 
with respect to
the center of mass of 
diatom. The symmetrized
hyperspherical coordinates [Kuppermann:75a] are the hyper-radius
, and a set of five angles
, 
, 
,
and 
, denoted collectively as
. The first two of these are in the range 0 to
and are, respectively, 
and
the angle between 
and 
. The
angles 
, 
are the polar angles of 
in a space-fixed frame and 
is the tumbling
angle of the 
, 
half-plane around
its edge 
. The Hamiltonian 
is
the sum of a radial kinetic energy operator term in 
, and the
surface Hamiltonian 
, which contains all differential
operators in 
and the electronically adiabatic potential
. The surface
Hamiltonian 
depends on 
parametrically and is
therefore the ``frozen'' hyperradius part of 
.
The scattering wave function 
is labelled by
the total angular momentum J, its projection M on the
laboratory-fixed Z axis, the inversion parity 
with respect to
the center of mass of the system, and the irreducible representation
of the permutation group of the system (
for 
) to
which the electronuclear wave function, excluding the nuclear spin part,
belongs [Lepetit:90a;90b]. It can be expanded in terms of the
LHSF 
defined below, and calculated at the values
of 
:
The index i is introduced to permit consideration of a set of many linearly independent solutions of the Schrödinger equation corresponding to distinct initial conditions which are needed to obtain the appropriate scattering matrices.
The LHSF 
and
associated energies 
are,
respectively, the eigenfunctions and eigenvalues
of the surface Hamiltonian 
. They are obtained using
a variational approach [Cuccaro:89a]. The variational basis set
consists of products of Wigner rotation matrices
,
associated Legendre functions of 
and functions of
which depend parametically on 
and are
obtained from the numerical solution of one-dimensional
eigenvalue-eigenfunction differential equations in 
,
involving a potential related to
.
The variational method leads to an eigenvalue problem with coefficient and
overlap matrices 
and
and whose elements are five-dimensional
integrals involving the variational basis functions.
The coefficients 
defined by
Equation 8.12 satisfy a coupled set of second-order
differential equations involving an interaction matrix
whose elements are defined
by
The configuration space 
is divided into a set of Q
hyperspherical shells 
within each of which we choose a value 
used in
expansion 8.12.
When changing from the LHSF set at 
to the one at
, neither 
nor its derivative with
respect to 
should change. This imposes continuity conditions on the
and their 
-derivatives at 
,
involving the overlap matrix 
between the LHSF evaluated at 
and
The five-dimensional integrals required to evaluate the elements of
, 
, 
, and
are performed analytically over 
,
, and 
and by two-dimensional numerical
quadratures over 
and 
. These
quadratures account for 90% of the total time needed to calculate the LHSF
and the matrices 
and
.
The system of second-order ordinary differential equations in the 
is integrated
as an initial value problem from small values of 
to large values
using Manolopoulos' logarithmic derivative propagator
[Manolopoulos:86a]. Matrix inversions account for more than 90%
of the time used by this propagator. All aspects of the physics can be
extracted from the solutions at large 
by a constant 
projection [Hipes:87a], [Hood:86a], [Kuppermann:86a].
