Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory

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Abstract
The theory of reactive (rearrangement) scattering for three atoms in three physical dimensions using adiabatically adjusting, principal axes hyperspherical (APH) coordinates is given. The relationships of the APH coordinates to Delves and Jacobi coordinates are given, and the kinetic energy operator is shown to be relatively simple. Procedures for solving the equations via either an exact coupled channel (CC) method or an optimum centrifugal sudden (CSAPH) approximation are given as well as procedures for applying scattering boundary conditions. Surface functions of two angles are obtained using a finite element method with an optimized, nonuniform mesh, and the CC equations are solved using the efficient VIVAS method. Sample CC results are given for the H3 system. The present approach has the advantages that all arrangements are treated fully equivalently; it is a principal axis system, so that both axes and internal coordinates swing smoothly with the reactions; it is directly applicable to both symmetric and unsymmetric systems and mass combinations and all total angular momenta; it gives convenient mappings for visualization of potential energy surfaces and wave functions; only regular radial solutions are required; all coordinate matching is by simple projection; and the expensive parts of the calculation are energy independent, so that, once they are done, the scattering matrices can be rapidly generated at the large numbers of energies needed to map out reactive thresholds and resonances. Accurate reactive scattering calculations are now possible for many chemically interesting reactions that were previously intractable.