Using rectangular collocation with finite difference derivatives to solve electronic Schrödinger equation

We show that a rectangular collocation method, equivalent to evaluating all matrix elements with a quadrature-like scheme and using more points than basis functions, is an effective approach for solving the electronic Schrödinger equation (ESE). We test the ideas by computing several solutions of the ESE for the H atom and the H₂ ⁺ cation and several solutions of the Kohn-Sham equation for CO and H₂O. In all cases, we achieve millihartree accuracy. Two key advantages of the collocation method we use are (1) collocation points need not have a particular distribution or spacing and can be chosen to reduce the required number of points - they need not converge any quadrature; (2) the better the basis is, the less sensitive the results are to the choice of the point set. The ideas of this paper make it possible to use any basis functions and thus open the door to using basis functions that are not Gaussians or plane waves. We use basis functions that are similar to Slater-type orbitals. They are rarely used with the variational method, but present no problems when used with collocation.