Theory of Three-Dimensional Reactive Collisions Using Natural Collision Coordinates

Abstract

Starting from the classical kinetic energy expressed in natural collision coordinates, the Hamiltonian operator is derived for three dimensional reactions of the form $\mathrm{AB}+\mathrm{C}$ →near linear intermediate→ $\mathrm{A}+\mathrm{BC}$ . After writing the wavefunction as a sum of products of symmetric top wavefunctions (which provide for orientation of the plane of the three particles) times internal functions (which determine the size and shape of the three particle triangle), a system of coupled equations is obtained for the internal functions. Coupling mixes states of different K (component of angular momentum along rotating z axis). At large distances, $\text{K± 1}$ coupling is important even though the bending potential vanishes. At small distances, $\text{K ± 2}$ coupling is most important but even this should be small for a near linear intermediate. After expanding the internal functions in a perturbed stationary state basis, coupled differential equations (containing only local potentials) are obtained for the translational functions. The rotational part of the basis is then treated in more detail. At large atom—molecule distances, the free rotator states are obtained. At smaller distances, where internal rotation is slightly hindered, rotator functions are obtained as expansions in free rotator states. For the near linear intermediate, bending states are obtained using continued fraction techniques. The asymptotic form of the total wavefunction is then examined. Formulas are obtained for the nonreactive and reactive scattering amplitudes, and for the degeneracy averaged total reaction cross section. The latter has a form which in the classical limit becomes integration of the reaction probability over the impact parameter.