Schrödinger's radial equation: Solution by extrapolation

Abstract

Combining an appropriate finite difference method with iterative extrapolation to the limit results in a simple, highly accurate, numerical method for solving a one-dimensional Schrödinger's equation appropriate for a diatomic molecule. This numerical procedure has several distinct advantages over the more conventional methods such as Numerov's method or the method of finite differences without extrapolation. The advantages are the following: (i) initial guesses for the term values are not needed; (ii) the algorithm is easy to implement, has a firm mathematical foundation and provides error estimates; (iii) the method is relatively less sensitive to round-off error since a small number of mesh points is used and, hence, can be implemented on small computers; (iv) the method is faster for equivalent accuracy. We demonstrate the advantages of the present algorithm by solving Schrödinger's equation for (a) a Morse potential function appropriate for HCl and (b) a numerically derived Rydberg-Klein-Rees potential function for the X 1 Σ + state of CO. A direct comparison of the results for the X 1 Σ + state of CO is made with results obtained using Numerov's method.