Upper bounds to the impact parameter and cross section for atom–diatom exchange reactions

Publisher Website
Google Scholar
Abstract
We establish upper bounds to the impact parameter and cross section for any A+BC exchange reaction in which the reagents are in a given initial state. The approach we take is to use the centrifugal forces generated in a collision to place a bound on the values of the total angular momentum for which reaction can occur. The bounds on the impact parameter and cross section then follow directly from the restrictions imposed by energy and angular momentum conservation. Our approach is related to theories based on the properties of periodic trajectories in that the system configuration which determines the angular momentum bound is also that of a quasibound ABC rigid rotor periodic trajectory. The equation which defines the configuration of this trajectory is similar in form to a generating function recently derived by Child and Pollak. Furthermore, an analysis of the symmetric stretch periodic trajectories in the H+H2 reaction suggests that the rigid rotor trajectory is the maximum angular momentum member of a family of periodic trajectories which exist at energies below and above the dissociation threshold. Our approach is also related to variational transition state theory. However, rather than vary the location of a diving surface, we keep the surface fixed in the reagents’ region of the system phase space and vary instead its boundary. We compare the bounds we place on the impact parameter and cross section to the quasiclassical trajectory data of Karplus, Porter, and Sharma for the H+H2 exchange reaction and to that of Persky for the reactions of Cl with H2, D2, and HD. The cross section ratios show a near-linear dependence on the fraction of the total system energy which is partitioned initially into relative translational energy of the reagents, whereas the impact parameter ratios smoothly increase from ∼0.2 near threshold to a maximum of ∼0.9.