## Introduction

The structural characterization of clusters or nanoparticles, consisting of a few to thousands of atoms or molecules,[1-3] is essential to rationalize their size- and composition-dependent properties and to link their unique properties to their atomistic structures. Experiments alone often provide an incomplete picture of the cluster structure and only by combining them with independent theoretical investigations can a complete description of the geometric arrangement and the corresponding properties be established.[4-6] However, to develop a realistic theoretical model for the cluster structure without using experimental information is far from trivial.

There are two basic issues that need to be considered when developing candidate structures for a certain nanoparticle. The first concerns the complexity of the overall optimization problem and the second is how to model the interatomic bonding adequately in a computationally efficient way. First, let us assume that a certain description of the interatomic interactions is used. In this case, the general problem is to explore the -dimensional potential energy surface (PES) of the *N*-atom cluster and find several minima with the ultimate goal of locating the overall lowest lying stationary point in energy, defined as the global minimum (GM).1 Hence, the task is to perform a global optimization (GO) of the potential energy2 as a function of all atom coordinates. How can this goal be achieved? Of course, the simplest way is to use heuristic information about the bonding in the system under study, that is, to perform a biased structure search. For example, bonding motifs found in the solid state or in molecules could be used as possible starting structures. Unfortunately, the novel properties of nanomaterials are closely related to their unique bonding and, consequently, assuming structures based on known compounds could result in erroneous structure predictions. The best known example that highlights this problem is C_{60}.[11] Until its discovery, carbon was known to exist in the sp^{2} and sp^{3} hybridized graphite and diamond modifications, consisting of hexagons and tetrahedra, respectively.[12] Based on these binding motifs, the wrong structure for the C_{60} cluster would be predicted. Why not simply generate every possible isomer and be sure that the GM is one of these candidate structures? The larger the cluster gets the greater is the number of (meta-)stable isomers, and a rough estimate indicates that the number of stable structures rises exponentially with the number of atoms.[10, 13] Although this kind of approach can be used for smaller clusters,[14-16] it becomes computationally infeasible for larger systems. The situation becomes even more complicated for so-called “nanoalloys,” which are clusters in which two or more metals are combined and whose properties can be tuned by varying their composition and chemical ordering, as well as their size and morphology.[17] For these nanoalloy clusters, locating the GM is complicated by the existence of a large number of homotops (inequivalent permutational isomers) for each geometric isomer.[18] Hence, the best approach is to perform an unbiased systematic exploration of the PES with a GO algorithm.

Besides the need for an efficient GO method, the other prerequisite to make reasonable predictions of cluster structures is to describe the PES realistically. Hence, the level of theory that is used for the actual GO has to accurately reflect the nature of the chemical bonding in the nanosized aggregate. For large nanoparticles, it is expected that the bonding is closely related to that of the bulk. Consequently, for a structural exploration of these large systems, model or empirical potentials (EPs) can be used, which approximately mimic the bonding in the solid state. Common examples are the Lennard–Jones (LJ),[19, 20] Born–Mayer,[21] Gupta,[22] Sutton–Chen,[23] and Murrell–Mottram[24] potentials for describing the bonding in van-der-Waals, ionic, metallic, and covalently bound clusters. The smaller the nanoparticle is, however, the more important a full quantum chemical description of the interatomic bonding becomes. Even for larger clusters, a quantum description of the bonding may be necessary, for example, when electronic shell closure leads to enhanced stability for a particular cluster size.[25] Unfortunately, the cluster size at which quantum chemical modeling becomes indispensable is not known *a priori* and varies for different systems. Hence, the most general or “first principles” approach to theoretically predicting the structure of clusters and nanoparticles is to perform a GO using a Born–Oppenheimer[26] (BO) PES created by using a commonly available density functional theory (DFT)[27-31]3 or *ab initio* electronic structure method.[32]4 However, the computational cost required to perform these “first principles” GOs is many times higher than using an EP. This is the reason why, until recently, most GO investigations used EPs, sometimes followed by refinement of candidate low energy isomers using electronic structure methods.[33] We shall define this methodology as the “classic” GO (or EP-GO) approach. Lately, improved search algorithms, more efficient quantum chemistry codes and highly parallelized programs using present day high-performance computer facilities, have enabled the application of the outlined GO scheme [what we will call the “new” or “first principles” GO approach (FP-GO)] for chemically relevant systems.

The review presented here will cover these recent developments and the applications of FP-GO codes to predict the geometric and electronic structures of clusters and nanoparticles. To give the reader an overview of available GO algorithms, the first section of this review will introduce the most common approaches for locating the GM on the PES. To be able to compare the performance of GOs using electronic structure methods and EPs, the operating principles of the various algorithms are presented using some EP-GO examples. In the following section, an overview is presented of recent FP-GO investigations, highlighting some of the novel structural motifs and bonding situations that were identified using this approach. Furthermore, the benefits, drawbacks, and the characteristic computational performance of the EP-GO and FP-GO approaches are compared and some concluding remarks are made about possible future developments in this research area.