EF, BFGS, RF, and DIIS optimizations

Although 'Opt' is a robust scheme, the convergence speed is very slow in general. Much faster schemes based on a quasi Newton method are available for the geometry optimization. They are the eigenvector following (EF) method [36], the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [38], the rational function (RF) method [37], and a direct inversion iterative sub-space (DIIS) method [35], implemented in the Cartesian coordinate. In the EF and RF methods, the approximate Hessian is updated by the BFGS method. Thus, five geometry optimizers, Opt, EF, BFGS, RF and DIIS, are available in OpenMX Ver. 3.5, which can be specified by 'MD.Type'. The relevant keywords are listed below:

   MD.Type                     EF         # Opt|DIIS|BFGS|RF|EF
   MD.Opt.DIIS.History          3         # default=3
   MD.Opt.StartDIIS             5         # default=5
   MD.Opt.EveryDIIS            200        # default=200
   MD.maxIter                  100        # default=1
   MD.Opt.criterion          1.0e-4       # default=0.0003 (Hartree/bohr)

Then, you can control these schemes by two keywords:

   MD.Opt.DIIS.History      7       # default=7
   MD.Opt.StartDIIS         5       # default=5

The keyword 'MD.Opt.DIIS.History' specifies the number of the previous steps to update an optimum Hessian matrix. The default value is 7. Also, the geometry optimization step at which 'EF', 'BFGS', 'RF', or 'DIIS', starts is specified by the keyword 'MD.Opt.StartDIIS'. The geometry optimization steps before starting the these methods is performed by the steepest decent method as in 'Opt'. The default value is 5.

The initial step in the optimization is automatically tuned by monitoring the maximum force in the initial structure, while it was specified by the keyword "MD.Initial.MaxStep" in the version 3.2 (the keyword 'MD.Initial.MaxStep' is not available in OpenMX Ver. 3.5). As shown in the Fig. 9 which shows the number of geometry steps to achieve the maximum force of below 0.0001 hartree/bohr in molecules and bulks, in most cases the EF method seems to be the most robust and efficient scheme.

Figure: The number of optimization steps to achieve $10^{-4}$ hartree/bohr for (a) molecular systems and (b) bulk systems using four kinds of optimization methods.