NUMERICALLY STABLE SOLUTION OF COUPLED-CHANNEL EQUATIONS - THE LOCAL TRANSMISSION MATRIX

The Schrodinger function and its first derivative are expressed in terms of a local reflection matrix (LORE) and an inverse local transmission matrix (INTRA). In the asymptotic regions of a transmission problem (with an activation barrier of finite height) these two matrices approach zero (one) on one side of the barrier and the physical reflection (transmission) matrix on the opposite side. The Schrodinger equation then can be transformed to two first-order differential equations. The first one is a nonlinear differential equation for the LORE alone. The second one involves both INTRA and LORE. The (numerically unstable) boundary value problem then can be turned into a (stable) initial value problem. In a transmission problem the LORE can be used to determine fully state-resolved reflection probabilities but only certain partially state-resolved transmission probabilities using unitarity. The INTRA-LORE combination yields fully state-resolved reflection and transmission quantities. Unitarity then can be used as an additional check of numerical consistency. The computing time for the combined procedure is negligibly increased as compared to the time for the LORE alone. With this method a two-dimensional potential energy surface for the desorption of hydrogen from Pd(100) is discussed under the viewpoint of nearly thermal translational distribution.