Half and full collision matrix methods for scattering: Application to multichannel curve crossing

We present methods of numerically solving the multichannel Schrödinger equation by propagating exact first-order coupled equations for a specially designed half collision matrix X(r). The method requires choosing a convenient set of reference potentials with which to generate a pair of reference radial functions for each channel. It is easy to tailor the choice of basis to the nature of the exact multichannel interaction matrix in a given region of space, either to enhance the numerical efficiency of propagating the exact X(r), or, to facilitate the introduction of useful approximations. In particular, we define a classical half collision matrix Z(r) by neglecting rapidly oscillating terms in the propagation of X(r). This random phase approximation can only be justified in classically accessible regions of r, and results in a set of first order equations which imposes the unitarity of Z(r)Z†(r) = 1 throughout the propagation. Fortunately, such regions often contain the dominant couplings, particularly for heavy collision partners, and the unitarity constraint generally can be utilized out to r = ∞. We only begin to explore the consequences of this useful result in this paper. Because of the divergence of the reference functions, both the exact X(r) and the approximate Z(r) half collision amplitudes can only be propagated with confidence in classical regions of r. However, using the same basis, we derive a stable algorithm for propagating the full collision matrix Σ(r) = X(X*)^-1 which asymptotically defines the scattering matrix. Since Σ(r) is always unitary, the first-order equations for Σ(r) can be propagated in all regions of r. The various methods are demonstrated by applying them to a multichannel curve crossing in atom-diatom vibrational energy transfer collisions. Exact results are obtained from both the complete propagation of Σ(r) and from the partial propagation of X(r) in classical allowed regions. Excellent results are obtained from the unitarized classical half collision matrix Z(r) when reasonable reference potentials are employed.