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# 9.1.7 GDIIS

(July 14, 2022)

Direct inversion in the iterative subspace (DIIS) was originally developed by Pulay for accelerating SCF convergence. 996 Pulay P.
J. Comput. Chem.
(1982), 3, pp. 556.
Subsequently, Csaszar and Pulay used a similar scheme for geometry optimization, which they termed GDIIS. 245 Csaszar P., Pulay P.
J. Mol. Struct. (Theochem)
(1984), 114, pp. 31.
The method is somewhat different from the usual quasi-Newton type approach and is included in Optimize as an alternative to the EF algorithm. Tests indicate that its performance is similar to EF, at least for small systems, however, there is rarely an advantage in using GDIIS in preference to EF.

In GDIIS geometries generated in previous optimization cycles, $\mathbf{x}_{i}$, are linearly combined to find the “best” geometry for the current cycle

 $\mathbf{x}_{n}=\sum\limits_{i=1}^{m}c_{i}\mathbf{x}_{i}$ (9.33)

where the problem is to find the best values for the coefficients $c_{i}$.

If we express each geometry by its deviation from the sought-after final geometry, $\mathbf{x}_{f}$, i.e., $\mathbf{x}_{f}=\mathbf{x}_{i}+\mathbf{e}_{i}$, where $\mathbf{e}_{i}$ is an error vector, then it is obvious that if the conditions

 $\mathbf{r}=\sum c_{i}\mathbf{e}_{i}$ (9.34)

and

 $\sum{c_{i}}=1$ (9.35)

are satisfied, then the relation

 $\sum c_{i}\mathbf{x}_{i}=\mathbf{x}_{f}$ (9.36)

also holds.

The true error vectors $\mathbf{e}_{i}$ are, of course, unknown. However, in the case of a nearly quadratic energy function they can be approximated by

 $\mathbf{e}_{i}=-\mathbf{H}^{-1}\mathbf{g}_{i}$ (9.37)

where $\mathbf{g}_{i}$ is the gradient vector corresponding to the geometry $\mathbf{x}_{i}$ and $\mathbf{H}$ is an approximation to the Hessian matrix. Minimization of the norm of the residuum vector $\mathbf{r}$, Eq. (9.34), together with the constraint equation, Eq. (9.35), leads to a system of $m+l$ linear equations

 $\left({{\begin{array}[]{c c c c}{B_{11}}&\cdots&{B_{1m}}&1\\ \vdots&\ddots&\vdots&\vdots\\ {B_{m1}}&\cdots&{B_{mm}}&1\\ 1&\cdots&1&0\\ \end{array}}}\right)\left({{\begin{array}[]{c}{c_{1}}\\ \vdots\\ {c_{m}}\\ {-\lambda}\\ \end{array}}}\right)=\left({{\begin{array}[]{c}0\\ \vdots\\ 0\\ 1\\ \end{array}}}\right)$ (9.38)

where $B_{ij}=\langle\mathbf{e}_{i}|\mathbf{e}_{j}\rangle$ is the scalar product of the error vectors $\mathbf{e}_{i}$ and $\mathbf{e}_{j}$, and $\lambda$ is a Lagrange multiplier.

The coefficients $c_{i}$ determined from Eq. (9.38) are used to calculate an intermediate interpolated geometry

 $\mathbf{x}_{m+1}^{\prime}=\sum c_{i}\mathbf{x}_{i}$ (9.39)

 $\mathbf{g}_{m+1}^{\prime}=\sum c_{i}\mathbf{g}_{i}$ (9.40)
 $\mathbf{x}_{m+1}=\mathbf{x}_{m+1}^{\prime}-\mathbf{H}^{-1}\mathbf{g}_{m+1}^{% \prime}\;.$ (9.41)