Numerical Initial Value Problems in Ordinary Differential Equations

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.

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

We show how the linear reference potential method for solution of the close‐coupled equations, which arise in inelastic scattering theory, can be reformulated in terms of an ‘‘imbedding‐type’’ propagator. Explicit expressions are given for the blocks of the propagator matrix in terms of Airy functions. By representing these functions in terms of moduli and phases, in both classically allowed and classically forbidden regions, one can evaluate the propagator without any numerical difficulty. The resulting algorithm is tested on a highly pathological problem—the rotationally inelastic scattering of a polar molecule by a spherical ion at extremely low kinetic energy—and found to be completely stable.We show how the linear reference potential method for solution of the close‐coupled equations, which arise in inelastic scattering theory, can be reformulated in terms of an ‘‘imbedding‐type’’ propagator. Explicit expressions are given for the blocks of the propagator matrix in terms of Airy functions. By representing these functions in terms of moduli and phases, in both classically allowed and classically forbidden regions, one can evaluate the propagator without any numerical difficulty. The resulting algorithm is tested on a highly pathological problem—the rotationally inelastic scattering of a polar molecule by a spherical ion at extremely low kinetic energy—and found to be completely stable.

A stable linear reference potential algorithm for solution of the quantum close‐coupled equations in molecular scattering theory

We present the full close‐coupling formulation of the collision between a diatomic molecule in a 2Π state in the Hund’s case (a) limit and a structureless target. Due to the possibility of transitions between spin‐orbit and/or Λ‐doubling states the scattering displays an additional degree of complexity not seen in rotationally inelastic collisions of 1Σ+ molecules. The well‐known coupled states and infinite‐order‐sudden (IOS) approximation techniques can be applied in a straightforward manner. The factorization and scaling relations between the various cross sections, which are valid in the energy sudden limit, are explored. For transitions within either spin‐orbit manifold (Ω = 1/2, Ω = 3/2), these scaling relations allow both matrices of cross sections, for parity conserving the parity violating transitions, to be expressed in terms of the cross sections for parity conserving transitions out of the Ω = 1/2, J = 1/2 level. Under conditions in which either a Born or sudden formulation of the collision dyn...

Rotationally inelastic collisions between a diatomic molecule in a2Π electronic state and a structureless target

The interaction potential energy surfaces for CN(X 2∑+, A 2∏)+He have been computed from ab initio MCSCF and MCSCF‐CI wave functions using an extensive basis set. In the presence of the He atom the two degenerate components of the CN 2∏ state split into wave functions of A’ and A‘ symmetry, and the symmetry of the 2∑+ state reduces to A’. The two adiabatic potentials for the A’ states are transformed to a diabatic basis, which yields a fourth potential energy surface V1, describing the collision‐induced electrostatic coupling between the two A’ states. The degree of mixing of the two diabatic A’ states has been determined by integration of the relevant nonadiabtic coupling matrix elements and, in a simpler method, from the coefficients of the MCSCF configurations. Both procedures yield virtually identical results. The nonadiabatic coupling matrix elements are strongly peaked near the CN bond distance at which the X 2∑+ and A 2∏ states cross in the isolated molecule. The diabatic coupling potential V1, however, is only weakly dependent on the CN bond distance, and decreases exponentially with the CN–He separation. Near the classical turning points for room temperature collisions the magnitude of V1 is approximately 50 cm−1. The V1 potential shows a bimodal character as a function of the collision angle θ. These results are discussed in connection with recent experiments of Dagdigian and co‐workers.

Adiabatic and diabatic potential energy surfaces for collisions of CN(X 2Σ+, A 2Π) with He

https://escholarship.org/content/qt3m53q8kg/qt3m53q8kg.pdf

B.R Johnson

Papers

Comparison of numerical methods for solving the second-order differential equations of molecular scattering theory