The workflow of the blocked-Davidson iterative matrix diagonalization scheme implemented in VASP is as follows:[1][2]
- Take a subset (block) of
orbitals out of the total set of NBANDS orbitals:
.
- Extend the subspace spanned by
by adding the preconditioned residual vectors of
:

- Rayleigh-Ritz optimization ("subspace rotation") within the
-dimensional space spanned by
, to determine the
lowest eigenvectors:

- Extend the subspace with the residuals of
:

- Rayleigh-Ritz optimization ("subspace rotation") within the
-dimensional space spanned by
:

- 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:

- Per default VASP will not iterate deeper than
, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.
- When the iteration is finished, store the optimized block of orbitals back into the set:
.
- Move on to the next block
.
- When LDIAG=.TRUE. (default), a Rayleigh-Ritz optimization in the complete subspace
is performed after all orbitals have been optimized.
The blocksize
used in the blocked-Davidson algorithm can be set by means of the NSIM tag.
In principle
NSIM, but for technical reasons it needs to be dividable by an integer N:

where
is the "number of band groups per k-point group":

(see the section on parallelization basics).
As mentioned before, the optimization of a block of orbitals is stopped when either the maximum iteration depth (NRMM), or a certain convergence threshold has been reached. The latter may be fine-tuned by means of the EBREAK, DEPER, and WEIMIN tags. Note: we do not recommend you to do so! Rather rely on the defaults instead.
The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the RMM-DIIS, but more robust.
References