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

We describe a rapidly converging algorithm for solving the Schrodinger equa tion with local potentials in real space. The algorithm is based on solving the Schrodinger equation in imaginary time by factorizing the evolution op erator e(-epsilonH) to fourth order with purely positive coefficients. The wave functions \ psi (j)> and the associated energies extracted from the no rmalization factor e(j)(-epsilonE) converge as O(epsilon (4)). The energies computed directly from the expectation value, < psi (j)\H \ psi (j)>, conv erge as O(epsilon (8)). When compared to the existing second-order split op erator method, our algorithm is at least a factor of 100 more efficient. We examine and compare four distinct fourth-order factorizations for solving the sech(2)(ax) potential in one dimension and conclude that all four algor ithms converge well at large time steps, but one is more efficient. We also solve the Schrodinger equation in three dimensions for the lowest four eig enstates of the spherical analog of the same potential. We conclude that th e algorithm is equally efficient in solving for the low-lying bound-state s pectrum in three dimensions. In the case of a spherical jellium cluster wit h 20 electrons, our fourth-order algorithm allows the use of very large tim e steps, thus greatly speeding up the rate of convergence. This rapid conve rgence makes our scheme particularly useful for solving the Kohn-Sham equat ion of density-functional theory and the Gross-Pitaevskii equation for dilu te Bose-Einstein condensates in arbitrary geometries. (C) 2001 American Ins titute of Physics.