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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 ͑FAS͒ 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 manyelectron 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.

Iterative multiscale methods for electronic structure calculations offer several advantages for large-scale problems. Here we examine a nonlinear full approximation scheme (FAS) multigrid method for solving fixed potential and self-consistent eigenvalue problems. In principle, the expensive orthogonalization and Ritz projection operations can be moved to coarse levels, thus substantially reducing the overall computational expense. Results of the nonlinear multiscale approach are presented for simple fixed potential problems and for self-consistent pseudopotential calculations on large molecules. It is shown that, while excellent efficiencies can be obtained for problems with small numbers of states or well-defined eigenvalue cluster structure, the algorithm in its original form stalls for large-molecule problems with tens of occupied levels. Work is in progress to attempt to alleviate those difficulties.

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