Methods for optimizing large molecules. Part III. An improved algorithm for geometry optimization using direct inversion in the iterative subspace (GDIIS)

Abstract

The geometry optimization using direct inversion in the iterative subspace (GDIIS) has been implemented in a number of computer programs and is found to be quite efficient in the quadratic vicinity of a minimum. However, far from a minimum, the original method may fail in three typical ways: (a) convergence to a nearby critical point of higher order (e.g. transition structure), (b) oscillation around an inflection point on the potential energy surface, (c) numerical instability problems in determining the GDIIS coefficients. An improved algorithm is presented that overcomes these difficulties. The modifications include: (a) a series of tests to control the construction of an acceptable GDIIS step, (b) use of a full Hessian update rather than a fixed Hessian, (c) a more stable method for calculating the DIIS coefficients. For a set of small molecules used to test geometry optimization algorithms, the controlled GDIIS method overcomes all of the problems of the original GDIIS method, and performs as well as a quasi-Newton RFO (rational function optimization) method. For larger molecules and very tight convergence, the controlled GDIIS method shows some improvement over an RFO method. With a properly chosen Hessian update method, the present algorithm can also be used in the same form to optimize higher order critical points.

On the equivalence of LIST and DIIS methods for convergence acceleration

“…It is then possible to find a linear combination of consecutive parameter vectors p =  i c i p i so that the corresponding error vector  i c i e i approximates the zero vector in the leastsquares sense." The flexibility on the choice of e j has allowed the application of DIIS to accelerate the convergence of methods other than Hartree-Fock and Kohn-Sham DFT such as coupled cluster, 3 multireference wavefunctions, 14 planewave-based density functionals, 15 and geometry optimization algorithms as in, e.g., the popular gradient-DIIS [16][17][18] (GDIIS). We also note that Rohwedder and Schneider 5 have analyzed the DIIS method assuming a general, approximate, residuallike term for the error function.…”

Six-dimensional quantum dynamics of H2 dissociative adsorption on the Pt(211) stepped surface

“…7 In quantum chemistry, linear and nonlinear equations are often solved by using the direct inversion of iterative subspace ͑DIIS͒ method. 15,16 The DIIS method was originally developed to improve the local convergence of selfconsistant field calculations, but it has proven useful for the solution of many other problems in electronic structure theory including geometry optimization 17,18 and the solution of the coupled-cluster equations. 19 A short review on the properties and use of the DIIS method has recently been published.…”

Numerical Methods for Electronic Structure Calculations of Materials

“…According to [42], this version of the GDIIS method is quite ef£cient in the quadratic vicinity of a minimum. However, farther away from the minimum, the method is not as reliable and can fail in three major ways: convergence to a nearby critical point of higher order, oscillation around an in¤ection point on the potential energy surface, and numerical instability problems in determining the GDIIS coef£cients.…”

“…However, farther away from the minimum, the method is not as reliable and can fail in three major ways: convergence to a nearby critical point of higher order, oscillation around an in¤ection point on the potential energy surface, and numerical instability problems in determining the GDIIS coef£cients. In [42], Farkas and Schlegel give an improved GDIIS method which overcomes these issues and performs as well as a quasi-Newton RFO method on a test set of small molecules. On a system with a large number of atoms, their improved GDIIS algorithm outperformed the quasi-Newton RFO method.…”

“…We now give the derivation for the GDIIS method from [42]. The development of the GDIIS method is based on a linear interpolation (and extrapolation) of previous molecular geometries, r i , that minimizes the length of an error vector.…”