Implemented Davidson-block iteration scheme

• selects a subset of all bands from $\{ \phi_n\vert n=1,..,N_{\rm bands}\}\Rightarrow \{ \phi^1_k\vert k=1,..,n_1\}$
  • Optimize this subset by adding the orthogonalized preconditioned residual vectors to the presently considered subspace
    
    $$\left \{ \phi^1_k \, / \, g^1_k = \left (1- \sum_{n=1}^{N_{... ...pp} {\bf S} \right ) \phi^1_k \, \vert \, k=1,..,n_1 \right \}$$
    
    

  • apply Raighly Ritz optimization in the space spanned by these vectors (``sub-space'' rotation in a $2 n_1$ dim. space) to determine $n_1$ lowest vectors $\{ \phi^2_k \vert k=1,n_1 \}$

  • Add additional preconditioned residuals calculated from the yet optimized bands
    
    $$\left \{ \phi^2_k \,/ \, g^1_k \, / \, g^2_k = \left (1- \s... ...app} {\bf S} \right) \phi^2_k \, \vert \, k=1,..,n_1 \right \}$$
    
    
    and ``sub-space'' rotation in a $3 n_1$ 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 $\{ \phi_k\vert k=1,..,N_{\rm bands}\}$.

  • Continue with next sub-block $\{ \phi^1_k\vert k=n_1+1,..,2 n_1\}$

  • After each band has been optimized a Raighly Ritz optimization in the space $\{ \phi_k\vert k=1,..,N_{\rm bands}\}$ is performed

• Approximately a factor of 1.5-2 slower than RMM-DIIS, but always stable.
• Available in parallel for any data distribution.