RMM-DIIS: Difference between revisions

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: and its corresponding residual vector <math>R(\psi^M_m)</math> are added to the iterative subspace, that is subsequently inverted to yield <math>\bar{\psi}^M</math>.
: and its corresponding residual vector <math>R(\psi^M_m)</math> are added to the iterative subspace, that is subsequently inverted to yield <math>\bar{\psi}^M</math>.
: The algorithm keeps iterating until the norm of the residual has dropped below a certain threshold, or the maximum number of iterations per orbital has been reached ({{TAG|NRMM}}).
: The algorithm keeps iterating until the norm of the residual has dropped below a certain threshold, or the maximum number of iterations per orbital has been reached ({{TAG|NRMM}}).
* Replace <math>\psi^0_m</math> by <math>\bar{\psi}^M</math>, and move on to start work on the next orbital, ''e.g.'' <math>\psi^0_{m+1}</math>.
* Replace <math>\psi^0_m</math> by <math>\bar{\psi}^M</math> and move on to start work on the next orbital, ''e.g.'' <math>\psi^0_{m+1}</math>.


== References ==
== References ==

Revision as of 10:26, 20 October 2023

The implementation of the Residual Minimization Method with Direct Inversion in the Iterative Subspace (RMM-DIIS) in VASP[1][2] is based on the original work of Pulay:[3]

  • The procedure starts with the evaluation of the preconditioned residual vector for some selected orbital :
where is the preconditioning function, and the residual is computed as:
with
  • Then a Jacobi-like trial step is taken in the direction of the vector:
and a new residual vector is determined:
  • Next a linear combination of the initial orbital and the trial orbital
is sought, such that the norm of the residual vector is minimized. Assuming linearity in the residual vector:
this requires the minimization of:
with respect to .
This step is usually called direct inversion of the iterative subspace (DIIS).
  • The next trial step () starts from , along the direction . In each iteration is increased by 1, and a new trial orbital:
and its corresponding residual vector are added to the iterative subspace, that is subsequently inverted to yield .
The algorithm keeps iterating until the norm of the residual has dropped below a certain threshold, or the maximum number of iterations per orbital has been reached (NRMM).
  • Replace by and move on to start work on the next orbital, e.g. .

References