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Davidson iteration scheme

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The preconditioned residual vector is calculated for each band resulting in a basis-set:

Within this subspace the NBANDS lowest eigenfunctions are calculated solving the eigenvalue problem

The NBANDS lowest eigenfunctions are used in the next step.

Implemented Davidson-block iteration scheme

The implemented scheme selects a subset of all bands from . The following steps are then performed on this subset:

  • Optimize this subset by adding the orthogonalized preconditioned residual vectors to the presently

considered subspace

  • Apply Rayleigh-Ritz optimization in the space spanned by these vectors (``sub-space rotation in a dim. space) to determine the lowest vectors .
  • Add additional preconditioned residuals calculated from the yet optimized bands

  • Sub-space rotation in a dim. space.
  • Continue iteration by adding a fourth set of preconditioned vectors if required. If the iteration is finished, store the optimized wavefunction back in the set

.

  • Continue with next sub-block .
  • After each band has been optimized a Raighly Ritz optimization in the space

is performed.

This method is approximately a factor of 1.5-2 slower than RMM-DIIS, but always stable. It is available in parallel for any data distribution.