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 $\psi^0_m$:

$K \vert R^0_m \rangle = K \vert R(\psi^0_m) \rangle$

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

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

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:

$\vert \psi^1_m \rangle = \vert \psi^0_m \rangle + \lambda K \vert R^0_m \rangle$

and a new residual vector is determined:

$\vert R^1_m \rangle = \vert R(\psi^1_m) \rangle$

• 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:

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

this requires the minimization of:

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

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:

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

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).

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

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