Spectrally accurate numerical solution of the single-particle Schrödinger equation

Abstract

We have formulated a three-dimensional fully numerical (i.e., chemical basis-set free) method and applied it to the solution of the single-particle Schr\odinger equation. The numerical method combines the rapid ``exponential'' convergence rates of spectral methods with the geometric flexibility of finite-element methods and can be viewed as an extension of the spectral element method. Singularities associated with multicenter systems are efficiently integrated by a Duffy transformation and the discrete operator is formulated by a variational statement. The method is applicable to molecular modeling for quantum chemical calculations on polyatomic systems. The complete system is shown to be efficiently inverted by the preconditioned conjugate gradient method and exponential convergence rates in numerical approximations are demonstrated for suitable benchmark problems including the hydrogenlike orbitals of nitrogen.