DIIS Convergence Acceleration

Direct Inversion in the Iterative Subspace (DIIS) accelerates SCF convergence by extrapolating the Fock matrix:

$${\mathbf{F}}_{\text{new}}=\sum _i{c}_i{\mathbf{F}}_i$$

where coefficients $c_i$ minimize the error vector norm subject to ${\sum }_i{c}_i=1$.

Error Vector

The error vector is the commutator:

$$\mathbf{e}=\mathbf{FPS}-\mathbf{SPF}$$

At convergence, $\mathbf{FPS}=\mathbf{SPF}$, so the commutator is zero.

DIIS Matrix Construction

Build the $M\times M$ overlap matrix of error vectors:

$$B_{ij}=⟨{\mathbf{e}}_i,{\mathbf{e}}_j⟩=\mathrm{Tr}[{\mathbf{e}}_i^{†}{\mathbf{e}}_j]$$

Constrained Optimization

Find coefficients $c_i$ that minimize the error norm subject to ${\sum }_i{c}_i=1$:

$$\text{Minimize: }{\mathbf{c}}^{†}\mathbf{Bc}$$ $$\text{Subject: }{1}^{†}\mathbf{c}=1$$

This is solved via the Lagrangian system:

$$\left[\begin{array}{cc}\mathbf{B}& -1\\ -1^{†}& 0\end{array}\right]\left[\begin{array}{c}\mathbf{c}\\ \lambda \end{array}\right]=\left[\begin{array}{c}0\\ -1\end{array}\right]$$

SCF Keywords

Control DIIS in the SCF section:

SCF
  Diis     6      # subspace size (default 6)
  NoDiis   false  # disable DIIS
  ...
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