Inversion of the Iterative Subspace

Having generated a sequence of optimization steps c , the Direct Inversion of the Iterative Subspace (DIIS) method (Csdszar and Pulay 1984 Hutter et al. 1994 Pulay 1980,1982) is designed to accelerate convergence by finding the best linear combination of stored c, vectors. 

Ideally, of course, c +i is equal to the optimum vector Copt. Defining the error vector e, for each iteration as 

Instead of the ideal case O Eq. 7.85, in practice one minimizes the quantity 

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DIIS (direct inversion of the iterative subspace) algorithm used to improve SCF convergence... 

Included in the methods discussed below are Newton-based methods (Section 10.3.1), the geometry optimization by direct inversion of the iterative subspace, or GDIIS, method (Section 10.3.2), QM/MM optimization techniques (Section 10.3.3), and algorithms designed to find surface intersections and points of closest approach (Section... 

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. 

A simple (though not very efficient) solution is to use simple convergence control schemes which only admix the new solutions to the starting orbital set via a flexible parameter. Much more efficient schemes have been developed such as the direct inversion of the iterative subspace (DllS) by Pulay et al. [380-382], in which the new guess spinors for the next iteration step are expanded into a couple of previous solutions. This expansion can even very conveniently be done with the (approximate) Fock operators. 

BFGS = Broyden - Fletcher - Goldfarb - Shanno DFP = Davidson-Fletcher-Powell EF = eigenvector following GDIIS = geometry optimization by direct inversion of the iterative subspace LST = linear synchronous transit QST = quadratic synchronous transit RFO = rational function optimization. 

The GDIIS method (geometry optimization by direct inversion of the iterative subspace) is an alternative approach for predicting the change in the geometry that is comparable in efficiency to the quasi-Newton methods. A linear combination of the current and previous points is chosen so that the Newton step is a minimum ... 

Hamilton T P and Pulay P 1986 Direct Inversion In the Iterative subspace (DNS) optimization of open-shell, excited-state and small multiconfiguratlonal SCF wavefunctlons J. Chem. Phys. 84 5728... 

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... 

The SS-LMBW equation (33) was discretized in the interval—409.6 < z < 409.6 A on a linear grid with a resolution of 0.05 A. To position the interface at z = 0, the parameter in Equation (31) setting the average density of the coexisting phases was chosen as rj — 0.5.TheSS-LMBWequation (33)with the boundary conditions(26)wereconverged by using the modified direct inversion in the iterative subspace (MDBS) method [54]. 

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... 

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... 

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. 

The 3D-RISM-MCSCF approach has been applied to carbon monoxide (CO) solute in ambient water [33]. Since it is known that the Hartree-Fock method predicts the electronic structure of CO in wrong character [167], the CASSCF method (2 core, 8 active orbitals, 10 electrons) in the basis sets of double zeta plus polarization (9s5pld/4s2pld) augmented with diffuse functions (s- and p-orbitals) was used. Water was described by the SPC/F model [127] and the site-centered local pseudopotential elaborated by Price and Halley for CP simulation [40]. The 3D-RISM/KH integral equations for the water distributions specified on a grid of 64 points in a cubic supercell of size 20 A were solved at each step of the SCF loop by using the method of modified direct inversion in the iterative subspace (MDIIS) [27, 29] (see Appendix). 

The modified direct inversion in the iterative subspace (MDIIS) method combines the simplicity and relatively small memory usage of an iterational approach with the efficiency of a direct method. It comprises two stages minimization of the residual linearly approximated with last successive iterative vectors used as a current basis, and then update of the basis with the minimized approximate residual by a properly scaled parameter. 

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