Exponential and Bessel fitting methods for the numerical solution of the Schrödinger equation (Q1115133)

A new method is developed for the numerical integration of the one dimensional radial Schrödinger equation. This method involves using different integration formulae in different parts of the range of integration rather than using the same integration formula throughout. Two new integration formulae are derived, one which integrates Bessel and Neumann functions exactly and another which exactly integrates certain exponential functions. It is shown that, for large r, these new formulae are much more accurate than standard integration methods for the Schrödinger equation. The benefit of using this new approach is demonstrated by considering some numerical examples based on the Lennard- Jones potential.