Franck–Condon factors in studies of dynamics of chemical reactions. II. Vibration–rotation distributions in atom–diatom reactions

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Abstract
We study vibration–rotation distributions in two and three dimensional atom–diatom chemical reactions using the generalized Franck–Condon overlap theory of chemical dynamics presented in paper I. We begin by reducing the Franck–Condon overlap form of the transition amplitude to a partial wave series involving overlap expressions for the partial wave transition matrices. These quantities are then approximately separated into products of angular coordinate factors and vibration–translation factors. To evaluate the angular coordinate factor, we examine the limits of strong and weak potential and kinematic coupling, where simple dynamical approximations (such as sudden approximations) may be used, and we are thus able to do the integrations analytically. The vibration–translation factor is evaluated by reducing it to an overlap similar in form to one previously analyzed for collinear reactions (Ref. 1), with some differences arising from centrifugal forces. The resulting formulas for product internal state distributions are then applied to the 2D and 3D H+H2 reaction, and to the 3D exoergic reactions F+H2(D2) and H(D)+Cl2. For H+H2, we find that centrifugal effects may be ignored, and that the angular coordinate part of the Franck–Condon overlap provides an accurate qualitative description of the rotational distributions. In addition, we find that on reaction the projection quantum number distribution shows a marked propensity for the condition mj=mj′=0 for the z component of the rotational angular momentum. Centrifugal effects are important for the exoergic reactions considered, and the simple formulas developed for describing vibration–rotation distributions for such reactions give a qualitatively (and often quantitatively) accurate description of the shapes of these distributions and their dependence on dynamical and kinematic parameters such as mass ratios, potential anisotropy, and energy release behavior. In addition, a qualitative understanding of the Levine–Bernstein information theoretic procedure for relating 1D and 3D vibrational distributions is provided, and possible sources of errors in this method are identified.