Efficient Real Space Solution of the Kohn-Sham Equations with Multiscale Techniques

TL;DR

This paper introduces a multigrid algorithm for efficiently solving the Kohn-Sham equations in real space, achieving rapid convergence with only a few self-consistency iterations.

Contribution

The paper develops a multiscale, real space finite difference method combined with a Full Multigrid algorithm for fast, self-consistent solutions of Kohn-Sham equations.

Findings

Rapid convergence to ground state electron distribution

Only two or three self-consistency iterations needed

High efficiency in solving Kohn-Sham equations

Abstract

We present a multigrid algorithm for self consistent solution of the Kohn-Sham equations in real space. The entire problem is discretized on a real space mesh with a high order finite difference representation. The resulting self consistent equations are solved on a heirarchy of grids of increasing resolution with a nonlinear Full Approximation Scheme, Full Multigrid algorithm. The self consistency is effected by updates of the Poisson equation and the exchange correlation potential at the end of each eigenfunction correction cycle. The algorithm leads to highly efficient solution of the equations, whereby the ground state electron distribution is obtained in only two or three self consistency iterations on the finest scale.