RMM-DIIS

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

• The procedure starts with the evaluation of the preconditioned residual vector for some selected orbital $\psi _{m}^{0}$:

$K\vert R_{m}^{0}\rangle =K\vert R(\psi _{m}^{0})\rangle$

where $K$ is the preconditioning function, and the residual is computed as:

$\vert R(\psi )\rangle =(H-\epsilon _{\rm {app}})\vert \psi \rangle$

with

$\epsilon _{\rm {app}}={\frac {\langle \psi \vert H\vert \psi \rangle }{\langle \psi \vert S\vert \psi \rangle }}$

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

$\vert \psi _{m}^{1}\rangle =\vert \psi _{m}^{0}\rangle +\lambda K\vert R_{m}^{0}\rangle$

and a new residual vector is determined:

$\vert R_{m}^{1}\rangle =\vert R(\psi _{m}^{1})\rangle$

• Next a linear combination of the initial orbital $\psi _{m}^{0}$ and the trial orbital $\psi _{m}^{1}$

$\vert {\bar {\psi }}^{M}\rangle =\sum _{i=0}^{M}\alpha _{i}\vert \psi _{m}^{i}\rangle ,\,\,M=1$

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

$\vert {\bar {R}}^{M}\rangle =\vert R({\bar {\psi }}^{M})\rangle =\sum _{i=0}^{M}\alpha _{i}\vert R_{m}^{i}\rangle$

this requires the minimization of:

${\frac {\sum _{ij}\alpha _{i}^{*}\alpha _{j}\langle R_{m}^{i}\vert R_{m}^{j}\rangle }{\sum _{ij}\alpha _{i}^{*}\alpha _{j}\langle \psi _{m}^{i}\vert S\vert \psi _{m}^{j}\rangle }}$

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:

$\vert \psi _{m}^{M}\rangle =\vert {\bar {\psi }}^{M-1}\rangle +\lambda K\vert {\bar {R}}^{M-1}\rangle$

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 ${\bar {R}}^{M}$ has dropped below a certain threshold, or the maximum number of iterations per orbital has been reached (NRMM).

• Replace $\psi _{m}^{0}$ by ${\bar {\psi }}^{M}$ and move on to start work on the next orbital, e.g. $\psi _{m+1}^{0}$.

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