Quasi-Newton Algorithms

Although it was originally developed for locating transition states, the EF algoritlnn is also efficient for minimization and usually perfonns as well as or better than the standard quasi-Newton algorithm. In this case, a single shift parameter is used, and the method is essentially identical to the augmented Hessian method. 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

Boston, J. F. Fournier, R. L., "A Quasi-Newton Algorithm for Solving Multiphase Equilibrium Flash Problems", Paper presented at Miami AIChE Meeting, November 1978. 

Dixon, L.C.W., "Quasi-Newton Algorithms Generate Identical Points. Math. Programming, 1972, 2 (3), 383-387. 

J. C. Gilbert and C. Lemarechal, Math. Prog., 45,407 (1989). Some Numerical Experiments with Variable-Storage Quasi-Newton Algorithms. 

The SCF wavefunctions from which the electron density is fitted was calculated by means of the GAUSSIAN-90 system of programs [18]. The program QMOLSIM [3] used in the computation of the MQSM allows optimization of the mutual orientation of the two systems studied in order to maximize their similarity by the common steepest-descent, Newton and quasi-Newton algorithms [19]. The DIIS procedure [20] has been also implemented for the steepest-descent optimizations in order to improve the performance of this method. The MQSM used in the optimization procedure are obtained from fitted densities. This speeds the process. The exact MQSM were obtained from the molecular orientation obtained in this optimization procedure. 

Since a point of the MEP is uniquely defined by the value of the radius of the hypersphere, the step size appears not to be critical in our approach and one can carry out a fast uphill walk with a large step size. Every minimization step of the E function is an efficient procedure due to the application of a quasi-Newton algorithm and to the initial geometry guess, estimated as extrapolation from previous MEP points. 

Flg. Steps in finding the minimum on a quadratic surface using a quasi-Newton algorithm with... 

P. Culot, G. Dive, V. H. Nguyen, and J. M. Ghuysen, A quasi-newton algorithm for first-mder saddle-point location. Theor. Chim. Acta. 82(3-4), 189-205 (1992). 

Figure 2 Flowchart for quasi-Newton algorithms for geometry optimization (Reproduced from Ref. 2a, with permission from World Scientific Publishing)...

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