Open-Channel Projectors for Rearrangement Processes in Molecular Collisions

A general form for the projector onto open channels for rearrangement processes in molecular collisions is given. This form is investigated for two special cases of the three-atom process $A+BC\to AB+C$ on a single potential-energy surface: (a) the rearrangement to and from the ground vibrational and rotational states of $\mathrm{BC}$ and $\mathrm{AB}$ for arbitrary total angular momentum and (b) the semiclassical limit of this form. The projector for special case (a) can be found exactly under either of two limiting conditions, and for case (b) we find the projector to be a local operator, such that in the semiclassical limit the process $A+BC\to AB+C$ occurs only when the distance from $A$ to the center of mass of $\mathrm{BC}$ equals the distance of $C$ from the center of mass of $\mathrm{AB}$.