Using pattern search methods for surface structure determination of nanomaterials

Atomic-scale surface structure plays an important role in describing many properties of materials, especially in the case of nanomaterials. One of the most effective techniques for the determination of surface structure is low-energy electron diffraction (LEED), which can be used in conjunction with optimization to fit simulated LEED intensities to experimental data. This optimization problem has a number of characteristics that make it challenging: it has many local minima, the optimization variables can be either continuous or categorical, the objective function can be discontinuous, there are no exact analytical derivatives (and no derivatives at all for categorical variables) and function evaluations are expensive. In this study we show how to apply a particular class of optimization methods known as pattern search methods to address these challenges. These methods do not explicitly use derivatives, and are particularly appropriate when categorical variables are present, an important feature that has not been addressed in previous LEED studies. We have found that pattern search methods can produce excellent results compared to previously used methods, both in terms of performance and in locating optimal results.