RMM-DIIS: Difference between revisions

Revision as of 10:20, 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 a selected orbital $\psi^0_m$ :

{\displaystyle K \vert R^0_m \rangle = K \vert R(\psi^0_m) \rangle }

where

K

is the preconditioning function, and the residual is computed as:

{\displaystyle \vert R(\psi) \rangle = (H-\epsilon_{\rm app}) \vert \psi \rangle }

with

{\displaystyle \epsilon_{\rm app} = \frac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert S \vert \psi \rangle} }

Then a Jacobi-like trial step is taken in the direction of the vector:

{\displaystyle \vert \psi^1_m \rangle = \vert \psi^0_m \rangle + \lambda K \vert R^0_m \rangle }

and a new residual vector is determined:

{\displaystyle \vert R^1_m \rangle = \vert R(\psi^1_m) \rangle }

Next a linear combination of the initial orbital $\psi^0_m$ and the trial orbital $\psi^1_m$

\vert \bar{\psi}^M \rangle = \sum^M_{i=0} \alpha_i \vert \psi^i_m \rangle, \,\, M=1

is sought, such that the norm of the residual vector is minimized. Assuming linearity in the residual vector:

{\displaystyle \vert \bar{R}^M \rangle = \vert R(\bar{\psi}^M) \rangle = \sum^M_{i=0} \alpha_i \vert R^i_m \rangle }

this requires the minimization of:

{\displaystyle \frac{\sum_{ij} \alpha_i^* \alpha_j \langle R^i_m \vert R^j_m \rangle}{\sum_{ij}\alpha_i^* \alpha_j \langle \psi^i_m \vert S \vert \psi^j_m \rangle} }

with respect to

{\{\alpha_i | i=0,..,M\}}

.

This step is usually called direct inversion of the iterative subspace (DIIS).

The next trial step ( $M=2$ ) starts from $\bar{\psi}^1$ , along the direction $K \bar{R}^1$ . In each iteration $M$ is increased by 1, and a new trial orbital:

{\displaystyle \vert \psi^M_m \rangle = \vert \bar{\psi}^{M-1} \rangle + \lambda K \vert \bar{R}^{M-1} \rangle }

and its corresponding residual vector

R(\psi^M_m)

are added to the iterative subspace, that is subsequently inverted to yield

\bar{\psi}^M

.

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).

Move on to the next orbital $\psi^0_{m+1}$ .

References

[kresse:cms:1996-1] G. Kresse and J. Furthmüller, Comp. Mater. Sci. 6, 15 (1996)

[kresse:prb:96-2] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).

[pulay:cpl:1980-3] P. Pulay, Chem. Phys. Lett. 73, 393 (1980).

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-The schemes like {{TAG|Davidson iteration scheme}} and {{TAG|Conjugate gradient optimization}}, try to optimize the expectation value
+The implementation of the Residual Minimization Method with Direct Inversion in the Iterative Subspace (RMM-DIIS) in {{VASP}}{{cite|kresse:cms:1996}}{{cite|kresse:prb:96}} is based on the original work of Pulay:{{cite|pulay:cpl:1980}}{{cite}}
-of the Hamiltonian for each wavefunction using an increasing trial basis-set.
-Instead of minimizing the expectation value it is also possible to
-minimize the norm of the residual vector. This leads to a similar iteration scheme as
-described in {{TAG|Efficient single band eigenvalue-minimization}}, but a different eigenvalue problem
-has to be solved (see Refs. {{cite|wood:jpa:1985}}{{cite|pulay:cpl:1980}}).
-There is a significant difference
-between optimizing the eigenvalue and the norm of the residual vector.
-The norm of the
-residual vector is given by
+* The procedure starts with the evaluation of the preconditioned residual vector for a selected orbital <math>\psi^0_m</math>:
+::<math>
+K \vert R^0_m \rangle = K \vert R(\psi^0_m) \rangle
+</math>
+:where <math>K</math> is the [[preconditioning]] function, and the residual is computed as:
+::<math>\vert R(\psi) \rangle = (H-\epsilon_{\rm app}) \vert \psi \rangle
+</math>
+:with
 ::<math>
- \langle R_n   | R_n \rangle =\langle \phi_n | (H - \epsilon)^+  (H - \epsilon) | \phi_n \rangle,
+\epsilon_{\rm app} = \frac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert S \vert \psi \rangle}
+</math>
+* Then a Jacobi-like trial step is taken in the direction of the vector:
+::<math> \vert \psi^1_m \rangle = \vert \psi^0_m \rangle + \lambda K \vert R^0_m \rangle
+</math>
+: and a new residual vector is determined:
+::<math>\vert R^1_m \rangle = \vert R(\psi^1_m) \rangle
 </math>
+* Next a linear combination of the initial orbital <math>\psi^0_m</math> and the trial orbital <math>\psi^1_m</math>
+::<math>\vert \bar{\psi}^M \rangle = \sum^M_{i=0} \alpha_i \vert \psi^i_m \rangle, \,\, M=1 </math>
+:is sought, such that the norm of the residual vector is minimized. Assuming linearity in the residual vector:
+::<math>\vert \bar{R}^M \rangle = \vert R(\bar{\psi}^M) \rangle = \sum^M_{i=0} \alpha_i \vert R^i_m \rangle
+</math>
+: this requires the minimization of:
+::<math>\frac{\sum_{ij} \alpha_i^* \alpha_j \langle R^i_m \vert R^j_m \rangle}{\sum_{ij}\alpha_i^* \alpha_j \langle \psi^i_m \vert S \vert \psi^j_m \rangle}
+</math>
+: with respect to <math>{\{\alpha_i | i=0,..,M\}}</math>.
+: This step is usually called ''direct inversion of the iterative subspace'' (DIIS).
+* The next trial step (<math>M=2</math>) starts from <math>\bar{\psi}^1</math>, along the direction <math>K \bar{R}^1</math>. In each iteration <math>M</math> is increased by 1, and a new trial orbital:
+::<math>\vert \psi^M_m \rangle = \vert \bar{\psi}^{M-1} \rangle + \lambda K \vert \bar{R}^{M-1} \rangle
+</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}}).
+* Move on to the next orbital <math>\psi^0_{m+1}</math>.
-and possesses a quadratic unrestricted minimum at the each
-eigenfunction <math>\phi_n</math>. If you have a good starting guess for
-the eigenfunction it is possible to use this algorithm without
-the knowledge of other wavefunctions, and therefore
-without the explicit orthogonalization of the preconditioned residual vector
-(eq. for <math>g_n</math> in {{TAG|Single band steepest descent scheme}}).
-In this case, after a sweep over all bands a Gram-Schmidt orthogonalization is  necessary
-to obtain a new orthogonal trial-basis set.
-Without the  explicit orthogonalization to the current set of trial wavefunctions
-all other algorithms tend to converge to the lowest band, no matter
-from which band they started.
-More information regarding related input variables and potential problems is available in the [[IALGO#RMM-DIIS]] section.
 == References ==
 <references/>
 ----
 [[Category:Electronic minimization]][[Category:Theory]]