Franck–Condon theory of chemical dynamics. VI. Angular distributions of reaction products

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Abstract
We calculate planar and three‐dimensional angular distributions for the products of atom–diatom chemical reactions by means of the Franck–Condon (FC) model. The wave functions on the reactant and product quasiadiabatic surfaces are expanded in partial wave series. A local uncoupling of the different degrees of freedom, as justified earlier, is assumed and consequently the individual members of the partial wave series can be separated into products of angular factors and rovibration–translation factors. To evaluate these factors, we consider the limit of weak and strong potential, and weak and strong kinematic couplings. The center of mass differential cross section is obtained by means of the T matrix formalism, where the T matrix is approximated by a generalized Franck–Condon overlap of the reactantlike and productlike wave functions. We use several further satisfactory approximations, e.g., linearization of the potential in the region of maximal overlap, and semiclassical approximation to the oscillator wave functions, beyond those of the FC model to obtain an analytic expression for the T matrix. For assumed LEPS surfaces of the systems H+H2 →H2+H, H2+F→HF+H and H+Cl2→HCl+Cl, we calculate angular distributions of reaction products in the various coupling limits for ranges of final states. The angular distributions in the strong potential coupling limits have a Gaussian shape peaked about the backscattering angle (π) (the hard sphere deflection angle for the chosen critical configuration) for each of the three reactions studied. In all three cases the 3D angular distribution is narrower than the planar (2D) angular distribution. Our calculations show no difference between the angular distributions of the weak and strong kinematic coupling limits. The angular distribution of the 2D weak potential coupling case are broader than those of the strong potential coupling. For H+H2 we find our results in the strong potential limit to be in qualitative agreement with exact quantum mechanical calculations. The angular distribution for a given product state broadens as the initial relative kinetic energy is increased, in agreement with classical trajectory calculation (F+H2). The angular distribution is also predicted to broaden as the final relative velocity increases, in agreement with experiment (H+Cl2, F+H2). Finally we introduce several simplifying approximations to our analytical model and find that, for exothermic reactions like F+H2, the radial contribution to the T matrix is dominated by certain features of the potential: the barrier width, the slope of the potential on the reactant side, and force constants in the region of maximum overlap. Our analysis provides a basis for the formulation of reduced variables which may be of use in comparing reactions. Finally we discuss some sufficient conditions for the separability of product velocity and angular distributions.