BFGS procedure

TABLE XIII A COMPARISION OF THE MURTAGH-SARGENT (MS) PROCEDURE WITH A BROYDE-FANNO-GOLDFARB-SHANNO (BFGS) PROCEDURE. THE NOTATION IS THE SAME AS THAT OF TABLE VI. 

We have also examined here the use of approximate solutions of the coupled perturbed Hartree-Fock equations for estimating the Hessian matrix. This Hessian appears to be more accurate than any updated Hessian we have been able to generate during the normal course of an optimization (usually the structure has optimized to within the specified tolerance before the Hessian is very accurate). For semi-empirical methods the use of this approximation in a Newton-like algorithm for minima appears optimal as demonstrated in Table 17. In ab-initio methods searching for minima, the BFGS procedure we describe is the best compromise. 

There are three Hessian matrix elements to be solved for, but only two equations to be satisfied, so there is not a unique solution for the and several recipes have been proposed to find improved that satisfy (15.73). One commonly used recipe is the Broyden, Fletcher, Goldfarb, Shanno (BFGS) procedure Leach, p. 229). 

T can be minimized in exactly the same fashion in which E itself was minimized, e.g., by using the Murtagh-Sargent or the BFGS procedure already described. A generalized Newton-Raphson method may also be employed, assuming... 

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,... 

The BFGS (Broyden [42], Fletcher [124], Goldfarb [145], Shanno [379]) algorithm is an update procedure for the Hessian matrix that is widely used in iterative optimization [125]. The simpler Rm update takes the form... 

Although line searches are typically easier to program, trust region methods may be effective when the procedure for determining the search direction p is not necessarily one of descent. This may be the case for methods that use finite-difference approximations to the Hessian in the procedure for specifying p (discussed in later sections). As we shall see later, in BFGS quasi-Newton or truncated Newton methods line searches may be preferable because descent directions are guaranteed. 

We notice that even at optimization, none of these procedures yield particularly accurate Hessian matrices. For most of the systems we have examined, however, the BFGS method is most accurate, followed by the rank one method. Least accurate is the Greenstadt method. The modes least accurately represented are the softer modes. If this system had been given an initial geometry that had C3v symmetry, then only the 2a modes 3 and 4 of the tables, would be updated. 

In this paper we have reviewed and studied several Hessian update procedures for optimizing the geometries of polyatomic molecules. Of the several methods we have studied, many of which we have reported on here in detail, the most reliable and efficient is the BFGS. The weak line search outlined in this work is sufficient to ensure successful optimization even when starting geometries are generated through molecular mechanics [45] or ball and stick models [46]. 

The current best correction matrix Q is denoted by the acronym BFGS, coming from the four authors who independently proposed it Broyden, Fletcher, Goldfarb, and Shanno. It is the complementary formulation of the DFP. The procedure for getting it is reported here below. 

Practical tests confirmed that the BFGS updating procedure is the best of the rank-2 alternatives when one-dimensional searches are limited. Also for this reason, this formula is currently the most recommended of all the available ahematives. 

D. K. Agrafiotis and H. S. Rzepa, /. Chem. Res. (S), 100 (1988). Evaluation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) Variable Metric Method in Geometry Optimisation Using Semi-Empirical SCF-MO Procedures. 

The MS- and the BFGS-method are the basis for the optimization procedures incorporated into the program packages GAUSSIAN 88 and CADPAC . 

Oxo-7a-methyl-l7/ -benzoyloxy-zl5(10),9(11). 19-norandrostadiene treated with m-chloroperoxybenzoic acid in methylene chloride with ice-methanol cooling, the crude intermediate chromatographed on alumina (activity II) with toluene-ethyl acetate, and treated with BFg-etherate in methylene chloride-ether 3-oxo-7a-methyl-17y5-benzoloxy-z]4,9,ll.i9-norandrostatriene. Y 60%. - This mild procedure is suitable for acid-sensitive compounds. F. e. s. P. Wieland and G. Anner, Helv. 50, 1453 (1967). 

---