Efficient multiscale algorithms for solution of self-consistent eigenvalue problems in real space

Real-space multiscale methods provide efficient algorithms for large-scale electronic structure calculations. In this paper, we present multigrid strategies for solving self-consistent problems in density functional theory. The full approximation scheme (FAS) formulation of the multigrid method allows for transfer of the expensive orthogonalization and Ritz projection operations to coarse levels. In addition, the effective potential may be updated on coarse levels during multiscale processing of the eigenfunctions. We investigate modifications of a previously proposed algorithm which are necessary to yield robust convergence rates. With these modifications, rapid convergence is observed without orthonormalization or Ritz projection for the full occupied subspace on the fine level. Calculations comparing the various algorithms are performed on three many-electron examples: benzene, benzenedithiol, and the amino acid glycine. The modified algorithm is also illustrated on several larger test cases. Recently developed relativistic separable dual-space Gaussian pseudopotentials are utilized to remove the core electrons.