Blocked-Davidson algorithm

The workflow of the blocked-Davidson iterative matrix diagonalization scheme implemented in VASP is as follows:

Take a subset (block) of [math]\displaystyle{ n_1 }[/math] orbitals out of the total set of NBANDS orbitals:

[math]\displaystyle{ \{ \psi_n| n=1,..,N_{\rm bands}\}\Rightarrow \{ \psi^1_k| k=1,..,n_1\} }[/math].

Extend the subspace spanned by [math]\displaystyle{ \{\psi^1\} }[/math] by adding the preconditioned residual vectors of [math]\displaystyle{ \{\psi^1\} }[/math]:

[math]\displaystyle{ \left \{ \psi^1_k \, / \, g^1_k = \left (1- \sum_{n=1}^{N_{\rm bands}} | \psi_n \rangle \langle\psi_n | {\bf S} \right) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right ) \psi^1_k \, | \, k=1,..,n_1 \right \}. }[/math]

Rayleigh-Ritz optimization ("subspace rotation") within the [math]\displaystyle{ 2n_1 }[/math] dimensional space spanned by [math]\displaystyle{ \{\psi^1/g^1\} }[/math], to determine the [math]\displaystyle{ n_1 }[/math] lowest eigenvectors:

[math]\displaystyle{ {\rm diag}\{\psi^1/g^1\} \Rightarrow \{ \psi^2_k| k=1,..,n_1\} }[/math]

Extend the subspace with residuals of [math]\displaystyle{ \{\psi^2\} }[/math]:

[math]\displaystyle{ \left \{ \psi^2_k \,/ \, g^1_k \, / \, g^2_k = \left (1- \sum_{n=1}^{N_{\rm bands}} | \psi_n \rangle \langle\psi_n | {\bf S} \right ) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right) \psi^2_k \, | \, k=1,..,n_1 \right \}. }[/math]

Rayleigh-Ritz optimization ("subspace rotation") within the [math]\displaystyle{ 3n_1 }[/math] dimensional space spanned by [math]\displaystyle{ \{\psi^1/g^1/g^2\} }[/math]:

[math]\displaystyle{ {\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k| k=1,..,n_1\} }[/math]

If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace, up to a depth [math]\displaystyle{ d }[/math]:

[math]\displaystyle{ \{\psi^1/g^1/g^2/../g^d\}\lt /math * Continue iteration by adding a fourth set of preconditioned vectors if required, etc etc. * When the iteration is finished, store the optimized block of orbitals back in the set: :\lt math\gt \{ \psi^d_k| k=1,..,n_1\} \Rightarrow \{ \psi_k| k=1,..,N_{\rm bands}\} }[/math].

Continue with the next block [math]\displaystyle{ \{ \psi^1_k| k=n_1+1,..,2 n_1\} }[/math].
After each band has been optimized a Rayleigh-Ritz optimization in the complete subspace [math]\displaystyle{ \{ \phi_k| k=1,..,N_{\rm bands}\} }[/math] 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.