Grid search method

The most intuitive approach to tackle the crystal structure solution by DST is to define a grid over the parameter space of interest and systematically explore every grid point. This method is simple and easily implemented in a computer code, and guarantees that the global minimum may be found, provided a suitably fine grid is used. However, it can be applied only when the parameter space is relatively small, namely when the number of DOFs necessary to describe the model is limited. 

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Conformational isomers represent minima on an energy surface, and all structures and the corresponding strain energies can be obtained by a careful analysis. This can be performed manually (such as in Sections 17.3 and 17.4) or automatically. An automatic procedure may involve a systematic search (grid search methods), a stochastic search (e.g., torsional Monte Carlo or cartesian stochastic, i.e., the random kick method) or molecular dynamics (see Chapter 5 and Section 16.5). Implemented in MOMEC is a random kick stochastic search module, and this has been shown to lead to excellent results, not only for conformational equilibria, but also for distributions of configurational isomers[37]. 

To approximately locate the global minimum i.e., to obtain a good quality crystal structure solution) in a reasonable amount of time, grid search methods should be replaced by the stochastic ones, based on a random sampling of the parameter space. This technique, called Monte Carlo MC), has been widely used in other scientific fields to simulate the behavior of complex systems. Its application to crystal structure determination from powder diffraction data has been developed by many authors the main strategies are outlined below. 

Fig. 5. The maximum figure of merit found with different protocols on systems with different number of composition (x) and noncomposition (z) variables (from Falcioni and Deem, 2000). The results are scaled to the maximum found by the grid searching method. Each value is averaged over scaled results on 10 instances of the random phase volume model with different random phases. The Monte Carlo methods are especially effective on systems with a larger number of variables, where the maximal figures of merit are more difficult to locate.

A five-dimensional grid search method is used to find the optimum values of the five design optimization variables. The following steps are used in the design procedure ... 

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