Experience with the Use of Optimization Schemes

Pattern search has also found some use in quantum chemistry. An early use was by Miller and Ruedenberg41 in their geminal calculations on the M. J. Box, Comput. /., 1966, 9, 67. 

The situation with respect to Powell s method is therefore far from clear and, in the absence of any systematic work to determine its effectiveness in a quantum chemical context, it seems to the Reporters that the method is always worth a preliminary try in an optimization problem particularly as it is so easy to program and so compact. 

The cost of extending integral programs is usually far from small, if one wishes them to perform efficiently, and if the program does not perform efficiently then the time taken in the optimization program to evaluate the gradient elements can become prohibitive. 

It is widely believed that, generally speaking, methods such as the Davidon-Fletcher-Powell method are superior to the Fletcher-Reeves method and, indeed, Fletcher suggests (see p. 82 of ref. 8) that typically the Fletcher-Reeves method will take about twice as many iterations as the Davidon-Fletcher-Powell method. 

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable 

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