Solution of the coupled-channel Schroedinger equation using constant, linear and quadratic reference potentials

Systems of coupled second order differential equations arise in many problems in molecular quantum physics. Approximate potential methods that solve these differential equations are investigated. A series method for the evaluation of the propagators is developed for the multichannel Schroedinger equation where the reference potential is expressed as a polynomial. The case for a linear approximate potential is treated in detail and the results compared with the Magnus and Bessel methods which approximate the potential as a constant in each interval. It is shown that the most efficient way to fit the reference potential is to require it to interpolate the potential at the zeros of the Psub(m+l) Legendre polynomial. Explicit expressions for the perturbative corrections to the Magnus propagator are presented through the leading terms in B2 (B is the linear term in the reference potential). These expressions are valid for any number of channels. This perturbative technique leads to the most efficient approximate potential method available. Its efficiency is shown by comparison with the log derivative method. (U.K.)