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# Davidson iteration scheme

The preconditioned residual vector is calculated for each band resulting in a $2*N_{bands}$ basis-set:

$b_{i,i=1,2*{N_{\rm {bands}}}}=\{\phi _{n}/p_{n}|n=1,N_{\rm {bands}}\}.$ Within this subspace the NBANDS lowest eigenfunctions are calculated solving the eigenvalue problem

$\langle b_{i}|{\bf {H}}-\epsilon _{j}{\bf {S}}|b_{j}\rangle =0.$ 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 $\{\phi _{n}|n=1,..,N_{\rm {bands}}\}\Rightarrow \{\phi _{k}^{1}|k=1,..,n_{1}\}$ . The following steps are then performed on this subset:

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

considered subspace
$\left\{\phi _{k}^{1}\,/\,g_{k}^{1}=\left(1-\sum _{n=1}^{N_{\rm {bands}}}|\phi _{n}\rangle \langle \phi _{n}|{\bf {S}}\right){\bf {K}}\left({\bf {H}}-\epsilon _{\rm {app}}{\bf {S}}\right)\phi _{k}^{1}\,|\,k=1,..,n_{1}\right\}.$ • Apply Rayleigh-Ritz optimization in the space spanned by these vectors (sub-space rotation in a $2n_{1}$ dim. space) to determine the $n_{1}$ lowest vectors $\{\phi _{k}^{2}|k=1,n_{1}\}$ .

$\left\{\phi _{k}^{2}\,/\,g_{k}^{1}\,/\,g_{k}^{2}=\left(1-\sum _{n=1}^{N_{\rm {bands}}}|\phi _{n}\rangle \langle \phi _{n}|{\bf {S}}\right){\bf {K}}\left({\bf {H}}-\epsilon _{\rm {app}}{\bf {S}}\right)\phi _{k}^{2}\,|\,k=1,..,n_{1}\right\}.$ • Sub-space rotation in a $3n_{1}$ 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

$\{\phi _{k}|k=1,..,N_{\rm {bands}}\}$ .

• Continue with next sub-block $\{\phi _{k}^{1}|k=n_{1}+1,..,2n_{1}\}$ .
• After each band has been optimized a Raighly Ritz optimization in the space

$\{\phi _{k}|k=1,..,N_{\rm {bands}}\}$ 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.