A multigrid method is presented for the real space numerical solution of the Poisson equation arising in ab-initio quantum computations. The specific algorithm presented is designed to compute the total electrostatic potential in real space density functional theory electronic structure calculations. Using a high-order finite difference approximation for the Laplacian and representing the nuclei as distributed charges, accurate electrostatic potentials due to both nuclei and electrons are obtained simultaneously in contrast to separate calculations for each component. Computations are presented for finite and periodic model problems in order to illustrate the accuracy of the approximation in addition to the speed and linear scaling properties of the algorithm.
A multigrid method for real-space solution of the Kohn᎐Sham equations is presented. By using this multiscale approach, the problem of critical slowing down typical of iterative real-space solvers is overcome. The method scales linearly in computer time with the number of electrons if the orbitals are localized. Here, we describe details of our multigrid method, present preliminary many-electron numerical results illustrating the efficiency of the solver, and discuss its strengths and limitations.
In the distributed nucleus approximation we represent the singular nucleus as smeared over a small portion of a Cartesian grid. Delocalizing the nucleus allows us to solve the Poisson equation for the overall electrostatic potential using a linear scaling multigrid algorithm.This work is done in the context of minimizing the Kohn-Sham energy functional directly in real space with a multiscale approach. The efficacy of the approximation is illustrated by locating the ground state density of simple one electron atoms and molecules and more complicated multiorbital systems.
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