Direct inversion in the iterative subspace DIIS

When three-point interpolation fails to yield a convergent calculation, you can request a second accelerator for any SCFcalculation via the Semi-empirical Options dialog box and the Ab Initio Options dialog box. This alternative method. Direct Inversion in the Iterative Subspace (DIIS), was developed by Peter Pulay [P. Pulay, Chem. Phys. Lett., 73, 393 (1980) J. Comp. Chem., 3, 556(1982)]. DIIS relies on the fact that the eigenvectors of the density and Fock matrices are identical at self-consistency. At SCF convergence, the following condition exists... 

Direct Inversion in the Iterative Subspace (DIIS). This procedure was developed by Pulay and is an extrapolation procedure. It has proved to be very efficient in fiDrciug convergence, and in reducing the number of iterations at the same time. It is------new-one of the most commonly-used methods for helping SC-F convergence. The... 

The effective Fock matrix (20) is in our implementation [51] the quantity which is averaged in the optimization based on the direct inversion in the iterative subspace (DIIS) method [52],... 

Hamilton T P and Pulay P 1986 Direct inversion in the iterative subspace (DIIS) optimization of open-shell, excited-state and small multiconfigurational SCF wavefunctions J. Chem. Phys. 84 5728... 

To avoid variational collapse, it is probably advisable to use an SCF convergence algorithm that is based on direct minimization rather than extrapolation methods such as direct inversion in the iterative subspace (DIIS) " and related methods, which are the default convergence algorithms in most quantum chemistry programs. Direct minimization, while often very slow to reach convergence, is more likely to converge to the desired local minimum in the space of MO coefficients. 

Figure 14.11 An example of a function and the associated gradient norm known of these is perhaps the GDIIS (Geometry Direct Inversion in the Iterative Subspace) which is directly analogous to the DIIS for electronic wave functions... 

GDIIS. - Another approach based on the Hessian and devised for structure optimization is the GDIIS,25 i.e., geometry optimization using direct inversion in the iterative subspace. GDIIS is a special version, devised for structure optimization, of DIIS. It has, e.g., been described by Farkas and Schlegel.26... 

A more sophisticated method that is often very successful is Pulay s direct inversion of the iterative subspace (DIIS) [Pulay 1980]. Here, the energy is assumed to vary as a quadratic function of the basis set coefficients. In DUS the coefficients for the next iteration are calculated from their values in the previous steps. In essence, one is predicting where the minimum in the energy will lie from a knowledge of the points that have been visited and by assuming that the energy surface adopts a parabolic shape. 

Newton-Raphson methods can be combined with extrapolation procedures, and the best known of these is perhaps the Geometry Direct Inversion in the Iterative Subspace (GDIIS), which is directly analogous to the DIIS for electronic wave functions described in Section 3.8.1. In the GDIIS method, the NR step is not taken from the last geometry but from an interpolated point with a corresponding interpolated gradient based on the previously calculated points on the surface. 

In iteration n the A matrix has dimension (n -1- 1) x (n -t 1), where n usually is less than 20. The coefficients c can be obtained by directly inverting the A matrix and multiplying it onto the b vector, i.e. in the subspace of the iterations the linear equations are solved by direct inversion , thus the name DIIS. Having obtained the coefficients that minimize the error function at iteration n, the same set of coefficients is used for generating an extrapolated Fock matrix (F ) at iteration n, which is used in place of F for generating the new density matrix. 

In order to accelerate the SCF convergence or to overcome convergence problems, SlNDOl includes three different weighting procedures, a level shift procedure, a direct energy minimization procedure, and Pulay s direct inversion in iterative subspace (DIIS) procedure. ... 

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