A fourth order real-space algorithm for solving local Schrödinger equations

We describe a rapidly converging algorithm for solving the Schrödinger equation with local potentials in real space. The algorithm is based on evolving the Schrödinger equation in imaginary time by factorizing the evolution operator e^-εH to fourth order with purely positive coefficients. The states |ψj〉 and the associated energies extracted from the normalisation factor e^-εEj converge as Ο(ε⁴). Our algorithm is at least a factor of 100 more efficient than existing second order split operator methods. We apply the new scheme to a spherical jellium cluster with 20 electrons. We show that the low-lying eigen-states converge very rapidly and that the algorithm does not lose any of its effectiveness for very steep potentials.