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

abstract

  • We describe a rapidly converging algorithm for solving the Schrdinger equation with local potentials in real space. The algorithm is based on solving the Schrdinger equation in imaginary time by factorizing the evolution operator eH to fourth order with purely positive coefficients. The wave functions |j and the associated energies extracted from the normalization factor eEj converge as O(4). The energies computed directly from the expectation value, j|H|j, converge as O(8). When compared to the existing second-order split operator method, our algorithm is at least a factor of 100 more efficient. We examine and compare four distinct fourth-order factorizations for solving the sech2(ax) potential in one dimension and conclude that all four algorithms converge well at large time steps, but one is more efficient. We also solve the Schrdinger equation in three dimensions for the lowest four eigenstates of the spherical analog of the same potential. We conclude that the algorithm is equally efficient in solving for the low-lying bound-state spectrum in three dimensions. In the case of a spherical jellium cluster with 20 electrons, our fourth-order algorithm allows the use of very large time steps, thus greatly speeding up the rate of convergence. This rapid convergence makes our scheme particularly useful for solving the KohnSham equation of density-functional theory and the GrossPitaevskii equation for dilute BoseEinstein condensates in arbitrary geometries.

published proceedings

  • JOURNAL OF CHEMICAL PHYSICS

author list (cited authors)

  • Auer, J., Krotscheck, E., & Chin, S. A.

complete list of authors

  • Auer, J||Krotscheck, E||Chin, SA