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].