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.