Differential cross sections for chemically reactive systems

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Abstract
A quantum mechanical theory is developed for the differential and total elastic and reactive scattering cross sections for an electronically adiabatic bimolecular exchange reaction, with the restriction that the three atoms are constrained to move on a straight line, but with the whole system free to rotate in three dimensions. The introduction of a set of natural collision coordinates, together with a vibrationally adiabatic approximation, is used to reduce the scattering problem to the solution of one-dimensional Schrödinger equations. Semi-classical expressions for the elastic and reactive phase shifts are derived. The partial wave summations that occur in the theory are evaluated by semi-classical techniques and elastic and reactive differential cross sections are calculated for three different kinds of potential surface. A feature of the calculations is the appearance of a new kind of rainbow, which is named a ‘cubic’ rainbow since it arises when the deflection function varies cubically with impact parameter. The classical and semi-classical theory of cubic rainbows is developed.

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