Half-collision description of final state distributions of the photodissociation of polyatomic molecules

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Abstract
The full three-dimensional quantum mechanical scattering equations, describing direct photodissociation and weak predissociation from initially selected levels, are analyzed within a formulation which permits the use of the different nuclear coordinate systems appropriate to the bound and dissociative surfaces. The coupled two surface scattering equations satisfy the physical boundary conditions of regularity at the origin and purely outgoing flux on the dissociative surface (with incoming photon flux.) These equations are transformed, both in integral and differential equation forms, into single surface half-collision equations wherein the initial bound state wave function, multiplied by the appropriate coupling operator, is propagated on the dissociative surface with the physical boundary conditions. These driven equations are shown to yield transition amplitudes which are equivalent to the transition amplitudes obtained from the Gell-Mann and Goldberger (GMG) scattering formulation which employs plane wave plus purely incoming wave eigenfunctions to evaluate the transition amplitudes. Given the direct transition amplitudes evaluated for the full three-dimensional case by Morse et al., the scattering equations may be integrated along the reaction coordinate, and the full state-to-state photodissociation amplitudes are obtained from the asymptotic limit of the driven single surface equations. Although the driven equation formulation is applicable to molecules of arbitrary size, the theory is presented specifically for the case of photodissociation of a triatomic molecule, the case for which a full three-dimensional calculation is most feasible. The GMG formulation is utilized to enable the application of standard scattering approximations to the single surface driven half-collision equations. The cases of the coupled states and the infinite order sudden approximations are treated in detail along with a discussion of some of the conditions of their applicability.