Solution of vibrational excitation problems with constant step size Magnus propagators: Convergence, perturbation analysis, and approximate decoupling

Constant step size Magnus propagators are used to integrate the Secrest–Johnson vibrational excitation problem with parameters given by Stechel, Walker, and Light. Calculations are done for a six channel basis for E=6 and 8 and for a 30 channel basis for E=60. For all the calculations the error scales as the fourth power of the step size for small steps. This indicates a significant cancellation between the errors in the Magnus propagation and the assumed diagonalization of the potential over each interval. The error grows rapidly when the step size is such that at least one channel is propagated by near half a wavelength. The errors in the E=6 calculation are compared with those for the log derivative, renormalized Numerov, R matrix propagation methods, and a quadratic approximate potential method. The constant step size Magnus method is superior to the other methods for this vibrational excitation problem. A perturbation analysis is presented to show why accurate calculations are possible with large nonclassical steps, since we have used steps much larger than those usually recommended for use with the Magnus propagator. Finally, a perturbative calculation of the transform that diagonlizes the potential matrix is described. The perturbative transform is rapidly calculated, and gives excellent results for diagonalizations at large distances.