Finite difference schemes and block Rayleigh quotient iteration for electronic structure calculations on composite grids
We present an original numerical method to discretize the Kohn-Sham equations by a finite difference scheme in real-space when computing the electronic structure 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 generalization of the Rayleigh quotient iteration which uses Petrov-Galerkin approximations. 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.
finite differences rayleigh quotient iteration electronic structure calculations kohn-sham equations mesh refinement multigrid method