Algorithms used in VASP to calculate the electronic groundstate

*The following section discusses the minimization
algorithms implemented in VASP. We generally have one outer loop
in which the charge density is optimized, and one inner loop
in which the wavefunctions are optimized.
Have at least a look at the flowchart.*

Most of the algorithms implemented in VASP use an iterative matrix-diagonalization scheme: the used algorithms are based on the conjugate gradient scheme [20,21], block Davidson scheme [22,23], or a residual minimization scheme - direct inversion in the iterative subspace (RMM-DIIS) [19,26]). For the mixing of the charge density an efficient Broyden/Pulay mixing scheme[24,25,26] is used. Fig. 4 shows a typical flow-chart of VASP. Input charge density and wavefunctions are independent quantities (at start-up these quantities are set according to INIWAV and ICHARG). Within each selfconsistency loop the charge density is used to set up the Hamiltonian, then the wavefunctions are optimized iteratively so that they get closer to the exact wavefunctions of this Hamiltonian. From the optimized wavefunctions a new charge density is calculated, which is then mixed with the old input-charge density. A brief flow chart is given in Fig. 4.

1cm (0,0)[bc] (0,0.5) (-6,-1)3.0 (0,0.5) ( 6,-1)3.0 (0,-.5) (-6, 1)3.0 (0,-.5) ( 6, 1)3.0 (-3,0.1) (0,0)[br]no

(13.0,1.0)[bl] (0,0.0) ( 1, 6)0.166 (0,0.0) ( 1, 0)12.8 (12.8,0.0) ( 1, 6)0.166 (.166,1.0) ( 1, 0)12.8

The conjugate gradient and the residual minimization scheme do not recalculate
the exact Kohn-Sham eigenfunctions but an arbitrary linear combination
of the NBANDS lowest eigenfunctions. Therefore it is in addition necessary to
diagonalize the Hamiltonian in the subspace spanned by the
trial-wavefunctions, and to transform the wavefunctions accordingly
(i.e. perform a unitary transformation of the wavefunctions,
so that the Hamiltonian is diagonal in the subspace spanned by transformed wavefunctions).
This step is usually called sub-space diagonalization (although a more
appropriate name would be, using the Rayleigh Ritz variational scheme in
a sub space spanned by the wavefunctions):

The sub-space diagonalization can be performed before or after the conjugate gradient or residual minimization scheme. Tests we have done indicate that the first choice is preferable during selfconsistent calculations.

In general all iterative algorithms work very similar: The core quantity is the residual vector

with | (7.1) |

This residual vector is added to the wavefunction , the algorithms differ in the way this is exactly done.

- Preconditioning
- Simple Davidson iteration scheme
- Single band, steepest descent scheme
- Efficient single band eigenvalue-minimization
- Conjugate gradient optimization
- Implemented Davidson-block iteration scheme
- Residual minimization scheme, direct inversion in the iterative subspace (RMM-DIIS)