Quasi-Newton methods BFGS optimization

Some of the most important variations are the so-called Quasi-Newton Methods, which update the Hessian progressively and therefore economize compute requirements considerably. The most successful scheme for that purpose is the so-called BFGS update. For a detailed overview of the mathematical concepts, see [78, 79] an excellent account of optimization methods in chemistry can be found in [80]. 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

By utilizing forces on FES, we can identify the SS and TS stmctures in solution with full optimization with respect to all coordinates of the solute molecules. For example, if we adopt the quasi-Newton method with the following Broyden-Fletcher-Goldfarb-Shanno (BFGS) procedure [26-29] for stractural optimization scheme in the FEG method, the i -F l)-th reactant structure is taken as,... 

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