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Top1. Introduction
A chemical cluster is an aggregate of between a few and millions of atoms or molecules. A better understanding of the properties of clusters is relevant for many areas, from protein structure prediction to the field of nanotechnology. The overall organization of a cluster leads to distinct physical traits from those of a single particle or bulk matter and a multidimensional function is used to describe the interactions occurring in the aggregate. This mathematical function, known as the Potential Energy Surface (PES), contains all the relevant information about the chemical system and models all reciprocal actions between particles (Doye, 2006).
The goal of cluster geometry optimization is to determine the optimal structural organization for the set of particles that compose a given aggregate, i.e., to position all atoms/molecules in the 3D space in such a way that the structure corresponds to the lowest potential energy. Due to efficiency reasons, model PES are usually adopted when studying large clusters. Pair-wise PES are simplified potentials that only consider the distance between every pair of particles composing the cluster to calculate the energy of the aggregate. The most common model potentials are Lennard-Jones (1931) and Morse (1929) functions. The latter is particularly relevant, since it provides accurate approximations of real materials (e.g., C60 molecules or alkali metal clusters) (Braier et al., 1990; Smirnov et al., 1999).
A PES usually generates a highly roughed energy landscape, with multiple funnel topography (Stillinger, 1999). Moreover, it has been proved that the global minimization of most PES is NP-hard, since the number of local minima increases exponentially as the clusters grow in size (Wille & Vennik, 1985). It follows that stochastic global optimization methods are the most effective approaches for identifying low energy configurations for different cluster compositions. In particular, Morse clusters define challenging benchmarks and are regularly adopted to access the effectiveness of new optimization methods (Doye & Wales, 1997). Examples of current state-of-the-art approaches for Morse cluster optimization are hybrid evolutionary algorithms (Johnston 2003; Pereira & Marques, 2009), dynamic lattice searching (Cheng & Yang, 2007) or population basin-hopping (Grosso et al., 2007).
In this paper we propose a Particle Swarm Optimization (PSO) algorithm for cluster geometry optimization. PSO algorithms were proposed by Kennedy and Eberhart (1995) and are inspired by the dynamics of social interactions. They maintain a set of particles (potential solutions) that travel across the search landscape defined by the problem being solved. The movement of each particle takes into consideration its own previous history and also relevant information gathered from neighbor solutions. PSO are particularly well suited to continuous optimization scenarios and they have been successfully applied to a large number of problems (Poli, 2008). Nevertheless, there are just a few simple proposals described in the literature reporting the application of PSO algorithms to cluster geometry optimization problems (Call et al., 2007; Hodgson, 2002).