Journal of Computational Physics 149, 75–94 (1999) Article ID jcph.1998.6138, available online at http://www.idealibrary.com on Finite Difference Schemes and Block Rayleigh Quotient Iteration for Electronic Structure Calculations on Composite Grids Jean-Luc Fattebert 1 Department of Mathematics, Ecole Polytechnique F´ ed´ erale de Lausanne, 1015 Lausanne, Switzerland E-mail: fatteber@nemo.physics.ncsu.edu Received December 10, 1997; revised August 10, 1998 We present an original numerical method to discretize the Kohn–Sham equations by a finite difference scheme in real-space when computing the electronic struc- ture of a molecule. The singular atomic potentials are replaced by pseudopotentials and the discretization of the 3D problem is done on a composite mesh refined in part of the domain. A “Mehrstellenverfahren” finite difference scheme is used to approximate the Laplacian on the regular parts of the grid. The nonlinearity of the potential operator in the Kohn–Sham equations is treated by a fixed point algorithm. At each step an iterative scheme is applied to determine the searched solutions of the eigenvalue problem for a given fixed potential. The eigensolver is a block gen- eralization of the Rayleigh quotient iteration which uses Petrov–Galerkin approx- imations. The algorithm is adapted to a multigrid resolution of the linear systems obtained in the inverse iterations. Numerical tests of the different algorithms are presented on problems coming from the electronic structure calculation of some molecules. c  1999 Academic Press Key Words: Rayleigh quotient iteration; multigrid method; finite differences; mesh refinement; electronic structure calculations; Kohn–Sham equations. 1. INTRODUCTION Over the last years real-space methods have appeared in the domain of 3D ab initio electronic structure calculations for numerically solving the Kohn–Sham (KS) equations [1–10]. The main reason to replace the classical plane wave expansion of the orbitals by these new discretization schemes is their local feature, allowing natural local mesh refinements [7, 10] and efficient subdomains decompositions on massively parallel supercomputers [4]. Moreover, this approach is more natural for nonperiodic systems. The computation 1 Present address: Department of Physics, North Carolina State University, Raleigh, NC 27695-8202. 75 0021-9991/99 $30.00 Copyright c  1999 by Academic Press All rights of reproduction in any form reserved.