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Keywords:

  • global optimization;
  • first principles methods;
  • genetic algorithms;
  • basin;
  • hopping;
  • clusters and nanoparticles

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies

Over the past decade, there has been a significant growth in the development and application of methods for performing global optimization (GO) of cluster and nanoparticle structures using first-principles electronic structure methods coupled to sophisticated search algorithms. This has in part been driven by the desire to avoid the use of empirical potentials (EPs), especially in cases where no reliable potentials exist to guide the search toward reasonable regions of configuration space. This has been facilitated by improvements in the reliability of the search algorithms, increased efficiency of the electronic structure methods, and the development of faster, multiprocessor high-performance computing architectures. In this review, we give a brief overview of GO algorithms, though concentrating mainly on genetic algorithm and basin hopping techniques, first in combination with EPs. The major part of the review then deals with details of the implementation and application of these search methods to allow exploration for global minimum cluster structures directly using electronic structure methods and, in particular, density functional theory. Example applications are presented, ranging from isolated monometallic and bimetallic clusters to molecular clusters and ligated and surface supported metal clusters. Finally, some possible future developments are highlighted. © 2013 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies

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 inline image-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 C60.[11] Until its discovery, carbon was known to exist in the sp2 and sp3 hybridized graphite and diamond modifications, consisting of hexagons and tetrahedra, respectively.[12] Based on these binding motifs, the wrong structure for the C60 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.

Basic Principles of the GO of Clusters and EP-GO Applications

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies

The aim of this section is to introduce the main characteristics of the most widely used GO algorithms using some illustrative EP-GO examples. However, it is not intended to cover all developments in the active research field of GO algorithms or to discuss other systems than clusters and nanoparticles. For this purpose, the reader is referred to more detailed monographs and reviews on GO algorithms.[8-10, 34-48]

The overall task of a GO is to find the GM on the PES of the nanoparticle system under study. Of course, it is far too time consuming to explore the complete PES with all its local minima and all connecting transition states (TS). This can only be done for restricted regions of the PES, very small model systems,[49] or simplified EPs[10] but these so-called disconnectivity graphs,[50-53] which are a powerful way of representing the PES, are not needed to search for low-lying minima. Hence, the computational method needs to effectively locate new minima and ideally improve the proposed cluster structures successively with respect to the potential energy. To fulfill this task, several highly efficient GO algorithms have been developed, which can be grouped according to their basic working principles.

First, we can distinguish biased and unbiased GO schemes, as briefly explained in the Introduction. For biased GO, some knowledge of the system's chemical bonding or favored structural motifs is used to speed up the global exploration of the PES. In special cases, this approach is highly beneficial and the required computational resources can be reduced dramatically.[54, 55] The problems connected with these methods were outlined in the Introduction, and we will only concern ourselves here with truly unbiased GO methods.

These can be divided into thermodynamically motivated Monte Carlo (MC), molecular dynamics (MD), and all other algorithms.5 The first mentioned method adopts ideas from thermodynamics to locate possible minima and ideally to obtain the GM. Hence, the temperature T is introduced as a parameter and a defined number of Markovian steps are repeated, either at fixed T or with a varying T-schedule. In general, in the limit of an infinite number of steps, the ensemble created by MC methods would resemble a thermodynamic distribution if a simple Metropolis criterion[56] is used.6 For sufficiently low T, this distribution will mainly consist of the geometric configuration of the cluster with lowest energy, that is, the GM. Similar in nature are the MD codes which solve the Newtonian, Langevin, or Brownian equations of motion to obtain information about the dynamics of the system and thereby move from one minimum to another.7 To locate the GM with MC or MD methods, it is important to increase and then decrease the temperature.8 Hence, these algorithms are equivalent to crystallizing the cluster structure slowly from a melt. For this so-called simulated annealing (SA) approach, MC[57] and MD[58] variants exist, which can be applied to the GO of cluster structures. The problem of long simulation times at low temperatures and the possibility of trapping in metastable minima has resulted in the development of the parallel tempering/replica exchange MC[59]/MD[60, 61] algorithms, which are commonly used in present day computations. However, because these GO algorithms are truncated after a certain number of steps, it is not known if the GM has definitely been located. In fact, even if the thermodynamic limit could be reached, it is not guaranteed that the GM would be found, that is, when starting with an unfavorable initial structure, it is not assured that the region on the PES where the GM is located can be found. This is only possible if all minima encountered during the GO are locally ergodic. An in-depth discussion of this topic and requirements for local ergodicity can be found in Refs. [ [9, 38, 39]. Alternative strategies exist that apply evolutionary principles or combine and extend the aforementioned methods to overcome some of their intrinsic limitations to find low-energy cluster structures effectively.

The last classification concerns the local energy optimization step in the global exploration of the energy landscape. No matter how the trial structures are created, there are two choices of what to do with these starting geometries. Either only the energy of the given nanoparticle geometry is calculated at the given level of theory, or a full local energy optimization of the cluster structure is performed. An extremum is always encountered using the second method, and analysis of the Hessian is performed to check for imaginary frequencies, indicating TS and higher-order saddle points. Thereby, the identification of a nearby minimum of the potential energy can be assured, though this procedure is computationally very time-consuming. Hence, many GO algorithms exist in two modifications, one only calculating the cluster energy at a certain geometry and the second relaxing this geometry using gradient-driven methods, to a nearby minimum.

In general, a vast number of different GO methods exist that can all be combined with electronic structure calculations. After having outlined the basic classifications, we cannot extensively describe all these algorithms here. Hence, the review is focused on GO methods that are most frequently used in combination with electronic structure methods. The genetic algorithm (GA) and basin hopping (BH) methods will be introduced and some details given in the next two sections. Some other EP-GO methods are briefly described in the section Other Methods. These include, for example, MD-based methods, particle swarm optimization (PSO), and the tabu search scheme, as these have only recently found application in combination with electronic structure methods.

Genetic algorithm

In a GA, evolutionary principles are adopted to identify the optimal cluster structure.[36] The geometry of the nanoparticle is represented by a certain set of coordinates or variables which are analogs of genes. The task is to globally optimize the values (alleles) for the complete set of genes, forming a chromosome. Because in nature this task is not achieved with a single individual but with a population consisting of many individuals, the same procedure is applied in the GA. Many cluster structures are generated randomly,9 forming the initial population with a predefined population size inline image. What to do next with the generated nanoparticles? This depends on whether a “Darwinian” or a “Lamarckian” optimization strategy is applied.[36] In the first scenario, the energy of the cluster structure is calculated exclusively, whereas the second method combines all GA steps with a subsequent local minimization. These “Lamarckian” algorithms have been found to improve the efficiency of the GA dramatically compared to “Darwinian” codes.[42] Hence, in practice, GO of clusters is generally carried out using “Lamarckian” GAs.

A schematic representation of the working principles of such a “Lamarckian”-GA, in this case the Birmingham cluster GA (BCGA),[36] is shown in Figure 1. Because every GA step is combined with a local minimization, rather than the actual PES (black solid line), the transformed (black dashed line) PES is investigated, as shown schematically for the generalized coordinates inline image in Figure 1. Some of the possible isomers are represented by colored circles and ellipses. The general idea of the GA code is to use pseudogenetic operators to sequentially improve the cluster geometry with respect to the potential energy. As a first step, all clusters are energetically (locally) minimized and ordered with respect to their fitness value fi, which is a function of the relative energy scale

  • display math(1)
image

Figure 1. Schematic representation of a “Lamarckian”-GA. The PES is shown (solid black line) as a function of the generalized coordinates inline image, whereas colored circles and ellipses are used as simple representations of possible cluster isomers. Due to the local minimization combined with the GA, the PES is transformed into a stepped energy surface (dashed black line). Randomly generated structures are used after geometry optimization to form the initial population of the GA. A fitness value is assigned to every nanoparticle, some of these are selected for subsequent mating and mutation operations and the offspring is again energetically optimized. The clusters with highest fitness value are selected for the next generation and this process is repeated until the GA has converged, generating a set of low-energy candidate structures. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Here, the indices correspond to the energy E of the ith individual and the clusters with the lowest (Emin) and highest energy (Emax) values in the current population. A more complete overview of the available fitness functions and their characteristics is given in Ref. [ [36]. In Figure 1, this process is described as “Assign Fitness.” After a fitness value has been assigned to each cluster in the population, a subset of nanoparticles is selected. Different selection strategies exist.[36] A frequently used selection scheme is the roulette wheel approach. One of the individuals is chosen randomly and its fitness value is compared to a random number between 0 and 1. If the fitness value is greater than this random number, the cluster is accepted for a genetic operation; otherwise, it is rejected and the processes are repeated until the required number of individuals have been found. Hence, the analogy between this algorithm and a roulette wheel with slot widths corresponding to the fitness values of the cluster is quite obvious. For most selection methods, a higher fitness value improves the probability of being selected. The other commonly used selection method is tournament selection, where two or more clusters are picked at random and the one with highest fitness is chosen to take part in crossover.

After the initial generation has been energetically minimized, the genetic operators are applied to the selected clusters. The two most important operators are mating and mutation. The number of offspring clusters and mutants is set to inline image, while the probability of applying a mutation is pmut. Hence, on average, the mating operator (crossover or recombination) is used for (1 − pmut)ntot and the mutation operator for pmutntot selected clusters.

The mating operator produces an offspring structure from two parent isomers by the “cut-and-paste” approach introduced by Deaven and Ho.[62] In the simplest form (one-point crossover), the two “parents” are cut at random positions and orientations and complementary fragments are pasted together (Fig. 1, “Crossover”), with retention of the total number of atoms and composition. For other crossover schemes, the reader is again referred to Ref. [ [36]. For the mutation operation, different methods exist, which all use the relevant operator on a single isomer. In general, a random change of the selected cluster is performed (Fig. 1, “Random Change”), i.e. atoms or fragments are moved randomly, a completely new random structure is introduced and, especially important for multimetallic/multicomponent systems, single atom or fragment identities are permutated without changing the coordinates.[36] For some nanoparticle families, the combination of different mutation schemes is highly beneficial as outlined below. After producing offspring and mutants, the new structures are energetically minimized, and all cluster isomers are ordered again with respect to their fitness values. Now the total number of clusters is larger than inline image and, consequently, only the fittest inline image clusters are selected for the next step (Fig. 1, “Natural Selection”). This process is equivalent to the elitist natural selection process in nature and ensures that the lowest energy structure cannot get worse from one cycle of the algorithm to the next.

In addition to the aforementioned energy-based selection strategy, sometimes geometrical criteria are used to maintain a high structural diversity in the population.[37] This means that, to avoid population stagnation, the isomers in the population are characterized by some geometrical descriptors such as moment of inertia, radial distribution function, and average and variance of atomic distances. Combined with the energy criterion, only the fittest structures with differing geometries are allowed to enter the next generation. The processes “Assign Fitness,” “Crossover,” “Random Change,” and “Natural Selection” act to form the next generation. Hence, the sequential improvement of trial cluster structures is possible by repeating all these processes for several generations until either the energy of the lowest lying isomer does not change for a predefined number of generation steps ( inline image) or the maximum number of generations is reached (mmax). The resulting low energy isomers are then extracted as candidates for further computational investigations (Fig. 1, “Lowest Lying Isomers”).

The description presented here is based on a generation-based GA approach. Another possibility is to perform a population- or pool-based GA. In this approach, the initial generation is smaller than inline image, and the pool of clusters is successively filled by generating offspring and mutants until a “natural selection” step is applied when the maximum pool-size is reached. Furthermore, the evolution of the algorithm is not measured by generations but how many times genetic operations are applied.[63] These two methods are in general very similar; however, the pool approach is easier to implement in parallel, because it does not have to wait for all population members in a particular generation to be relaxed locally.[43]

Typical “classic” applications of GAs are investigations of medium-sized and large clusters using the Born–Mayer,[64] van-der-Waals,[65] TIP4P,[66] or Gupta potentials.[67] To give the reader an idea of the typical computational parameters and performance, we briefly discuss the investigation of LJ[65] and bimetallic Cu-Au[67] (Gupta potential) clusters by GAs combined with EPs (GA-EP).

To test the efficiency of his “Lamarckian”-GA code using energy- and structure-based selection criteria, Hartke[65] investigated LJ clusters in the range 2–150 atoms, for which putative GM structures had been identified. For this task, inline image was set to 20 for all cluster sizes. All possible cluster pairs were used to generate offspring in a single generation, in addition to a 15% probability of random mutation. The global minima were found after 2–15 generations and consequently about inline image local optimizations had to be performed, resulting in a computational time of (102−105) s on a single 400-MHz Pentium-II computer. This number of local energy minimizations is very small when compared, for example, to approximately 18,000 minima and 21,000 TS for LJ75.[68] This is, therefore, a good example for highlighting the strengths of GOs. The number of potential minima rise dramatically as seen in the work of Darby et al.,[67] who explored the PES of pure and mixed Cu and Au clusters between 10 and 56 atoms using a Gupta potential. They used inline image, ntot = 24, and pmut = 0.1, with cluster replacement and exchange mutations, and ran the algorithm several times. Figure 2 shows a typical Evolutionary Progress Plot (EPP)[69] for Cu20 (Fig. 2a) and Cu10Au10 (Fig. 2b). Interesting information about the efficiency of the algorithm can be extracted from this EPP. The first observation is that, even for 20 atoms, it takes 65 generations to locate the putative GM for the pure cluster and more than 200 generations for the mixed cluster. Furthermore, the application of the exchange mutation operator (dashed line in Fig. 2b) speeds up the search dramatically (by a factor of nearly 10 with respect to generations), as this operation, by definition, explores some of the myriad, of homotops, whereas the other genetic operators cannot systematically account for this kind of isomerism. This is especially interesting with respect to combining this approach with “first principles” methods and demonstrates that, besides the need for an optimized GO code, only high-performance computer facilities can perform this many local minimization steps in a reasonable time period.

image

Figure 2. EPP (solid line) and putative GM for Cu20 (a) and Cu10Au10 (b). Here, V is used as variable for the potential energy and Cu and Au atoms are represented by copper and gold colored spheres. By applying the exchange operator in addition to the replacement mutation (dashed line), the GO for the mixed cluster becomes considerably more effective in finding the putative GM.[67] EPPs reprinted from Ref. [ [67], with permission from the American Institute of Physics. Putative GM structures provided by Chris Heard (Birmingham). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Basin hopping

A different approach is followed in the BH algorithm. This thermodynamically inspired (rather than population-based) GO code is a MC method, which uses random hopping moves combined with local minimizations to jump from one minimum of the PES to another. The method was first pioneered by Li and Scheraga[70] for the problem of protein folding and later more strictly formulated and applied by Wales and Doye[10, 71] to clusters. Its basic working principle is sketched in Figure 3 for the same PES as introduced for the GA method. Once again, the PES is illustrated as a function of the generalized coordinate inline image (black solid line) and the geometries corresponding to the minima are represented by colored circles and ellipses. In the BH approach, a random or otherwise generated geometry is used as the starting configuration. Because the BH method couples local optimization to MC moves, the structure is geometrically relaxed to the closest minimum. As for the “Lamarckian” GA method, again it is the transformed energy surface (Fig. 3, black solid line) defined by

  • display math(2)

rather than the original PES (Fig. 3, black dashed line), that is searched. In the next step, this local minimum structure is deformed randomly within a defined trust radius to avoid vaporization or fragmentation, that is, a Markovian process is introduced. The specific geometrical change of the nanoparticle by the Markovian operation depends on the nature of the investigated nanoparticle system. For example, the coordinates of all atoms could be changed randomly[71] or an exchange of atoms[72] or fragments could be performed. The resulting structure is again energetically minimized and, if the structural change was sufficient to overcome a nearby TS, the algorithm will move from one minimum to the next. Consequently, this algorithm allows hopping from one basin to the next on the PES (Fig. 3, “hopping”), thereby exploring the close vicinity of basins around the chosen starting structure. This is in contrast to the GA code, which jumps between regions of the PES via the crossover operation.

image

Figure 3. Schematic representation of the basic processes used in a BH algorithm. The colored circles and ellipses represent the isomers corresponding to individual basins. The energy is plotted as a function of the generalized coordinates inline image. During the search, the BH method hops (“hopping”) from one to another basin of the PES (solid black line). When the search algorithm becomes stuck in a basin, it jumps out (“jumping”) by a repetition of random atom displacements. Apart from the “jumping” moves, every “hopping” step is followed by a geometry optimization. Hence, it is the transformed energy surface (dashed black line) that is searched. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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To analyze the newly found minimum structure, the energy difference inline image between the initial and final isomers is compared using the Metropolis MC criterion.[56] The move is accepted if inline image or for inline image if exp [−ΔE/(kBT)] is larger than a randomly generated number between 0 and 1, where kB is the Boltzmann constant and T is the simulation temperature. The algorithm repeats these MC moves for a predefined number of steps and the lowest energy isomer found so far is the putative GM.

However, an intrinsic problem of the BH algorithm is that the code tries to move between minima by random changes of the geometry but, if a local basin10 is considerably lower in energy than all surrounding minima and is surrounded by high TS barriers, it becomes very unlikely that the search can escape from this “funnel” and, hence, it is trapped. Several methods have been developed to overcome this problem.[73, 74] One of these is shown in Figure 3 and is called “jumping.”[73] If the structure of the particle does not change for several MC steps, the temperature is raised to inline image, equivalent to always accepting a potential move, and the structure is changed several times without further geometry optimization. This allows the search to jump out of a deep basin. After this jumping move, the BH routine continues with normal hopping steps.

In addition to the classical BH and its extensions (e.g., the previously discussed “jumping” approach), other routines have recently been developed, which use descriptors to avoid trapping of the search algorithm in a funnel or basin. Two examples that have been used for clusters are the parallel excitable walkers (PEW) algorithm[75] and the population-based BH.[76, 77] The PEW algorithm simply executes several BH runs in parallel. After every BH step, the solutions of the different “walkers” are compared using various geometric descriptors. If the structures are not geometrically similar, the simple energy-based Metropolis MC criterion is used. However, if some solutions have similar descriptors, the energy difference is increased, effectively repelling these “walkers” and ensuring the sampling of different regions of the PES.[75] The population-based BH method introduces a population, in an analogous fashion to the GA, and also includes GA-like operations, such as mating and mutation, in addition to the standard BH steps. Other important differences to the PEW approach are the introduction of niches and a fixed population size. Hence, all newly generated cluster structures are compared to the old generation members. If the energy of a new structure is lower than that of an old individual and these clusters belong to the same niche, the cluster in the population is replaced by the newly found isomer.[76, 77]

Like the GA, the BH approaches have been applied to various nanoparticle families modeled by different kinds of EPs.[34, 71, 75-79] In the studies by Wales and co-workers on LJ clusters, the BH with inline image was used to explore the PES for the GM.[34, 71] Typically, 5000 MC steps were carried out for five runs, starting from randomly generated cluster structures. Two further GO attempts with only 200 MC steps were started from corresponding LJ clusters with N + 1 and N − 1 atoms, assuming that the GM for these sizes had already been found and removing or attaching an atom randomly. The Markovian process consisted of random displacements of all atoms, with a trust radius chosen so that 50% of MC moves are accepted. In this way, Wales and Scheraga were able to locate putative GM structures of all clusters in this size range but still needed inline image MC steps on average to locate these structures.[34] However, the number of steps required to locate the GM can only be known for these kind of model investigations. For an as yet unknown system, as many MC steps as possible have to be performed to increase the probability of finding the GM. For an EP, this is a “tractable” task11 but, combined with electronic structure methods, several hundred local optimization steps become immensely time consuming.

Other methods

In addition to the two approaches introduced above, several other methods, and variants exist, which we cannot discuss in detail. However, a brief introduction is presented here of some other methods which have been used recently for GO using electronic structure calculations.

One of the basic issues that nearly all of the GO algorithms described so far encounter, is to combine a more locally confined search of parts of the PES (containing previously found promising candidate structures) with a more stochastic exploration of as yet unvisited regions in a computationally effective way. Hence, re-examining a minimum only gives redundant information and wastes computational resources but confining the search to too small a neighborhood will not allow the algorithm to search for the GM in other funnels on the PES. This problem arises due to not storing information about previously identified structures, that is, minima which have been visited so far. Some algorithms try to overcome these problems by guiding the search using information about previously located isomers.

One example is the tabu search strategy which in principle can be combined with every GO method.[80, 81] The tabu search method (sometimes called “taboo”) uses historical information about the optimization problem to avoid generating the same trial solution more than once. In the case of nanoparticles, this means that the structures found during the search are characterized by several descriptors. This could be the energy at the minimum of the basin, the vibrational frequencies or some geometrical characteristics, for example those introduced for the GA. During the GO, this concept avoids the generation of structures with identical descriptors to those already identified. This strategy can be used, for example, to improve the performance of Langevin and Newtonian dynamics simulations by confining the MD trajectory to previously unvisited minima[82] or by only generating isomers with favorable geometric structures based on various descriptors.[83] This tabu search in descriptor space (TSDS) method not only uses the tabu search strategy to avoid generating identical structures but additionally interpolates the energy of the isomers based on the descriptors.[83] This implies a connection between the descriptors and the energy but can greatly reduce the number of computational steps for which an energy calculation is required.[83] The same strategy can also be used in combination with MD-based calculations. Goedecker,[84] for example, developed the minima hopping algorithm (MHA) which performs a MD simulation with a specially chosen kinetic energy12 to jump from one minimum to another, with a historic list of all previously visited minima, restricting the algorithm to explore unvisited regions of the PES. When a new minimum is located, the MHA performs a local geometry relaxation. In this way, Goedecker[84] was able to locate the GM of LJ38 very efficiently.

Another generation-based GO strategy, which is inspired by the social behavior of birds flocking or fish schooling and was designed to solve multidimensional optimization problems, is PSO.[85] It is based on the idea that the behavior of every individual (or agent) in the population is determined by swarm intelligence. This means that at every step each individual is influenced by the personal best result during the preceding search and the best result of the population. For the case of clusters, the best results are characterized by the energy and some geometrical or connectivity descriptors. Based on this shared information, the geometry of the individuals within the swarm is changed accordingly, and all individuals are attracted toward the best result encountered so far. If the cluster structures are generated randomly, this means that distinct regions of the PES are explored first but at a later stage the search is concentrated around the best known result. This nature-inspired GO code was successfully applied to LJ clusters using only single-point energy calculations[86] and local geometry optimization[87] and, at least for some of the LJ clusters, it proved to be more effective than BH and GA searches, based on the number of total energy calculations required. A related technique is the artificial bee colony (ABC)[88] algorithm, which has recently been applied for GO of LJ clusters.[89]

The FP-GO Approach for Clusters and Recent Examples

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies

So far, we have introduced the basic working principles for some selected GO algorithms. In this section, we will describe some recent investigations using these GOs, in combination with electronic structure methods, applied to clusters and nanoparticles. In general, the procedure is very similar to EP-GO methods, because an external quantum chemistry code (rather than an EP) is called by the GO routine to perform the energy calculation and (where appropriate) local optimization. In practice, however, several computational aspects need to be changed due to the much longer times required for these FP-GOs as compared to EP-GOs. Before highlighting some recent results from the literature, we will make some general comments about the FP-GO methods.

As described above, FP-GO is the most general method for searching the BO-PES for the GM and overcoming some intrinsic limitations of the EP-GO approach, in which one is not able to account for the quantum chemical nature of the bonding in clusters. A very nice illustration of the capabilities of FP methods for exploring the BO-PES is shown in Figure 4 for C60.[93] Kumeda and Wales explored possible rearrangements and constructed the disconnectivity graph (energy landscape) of C60 at two different DFT levels of theory. From this accurate predictions of the C60, PES and its TSs and funnels emerged. However, as discussed above, even for very small nanoparticles, several hundred energy calculations are required to reasonably explore possible minima on the PES. This limits the quantum chemical methods that can be used and increasing the cluster size further makes an investigation using advanced wavefunction-based methods with large basis sets infeasible. On the other hand, the unique bonding and structural motifs in some nanoparticle families, combined with the small energy differences13 between the different isomers, require sufficiently large basis sets and incorporation of exchange and correlation effects to make reliable predictions of the putative GM. For heavy elements, the situation becomes even more complicated, as scalar-relativistic and spin-orbit (SO) effects need to be considered.[96-98]

image

Figure 4. Disconnectivity graphs of C60 at the PW/LDA[90] (left) and PW/BLYP[91, 92] (right) level of theory, respectively.[93] Pentagonal and hexagonal faces are colored green and red, respectively. The energy scaling is indicated by the red vertical bar. Reprinted from Ref. [ [93], with permission from Elsevier. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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For these reasons, the overall optimization problem is often not solved in a single step by applying a high-level FP-GO algorithm, but rather a two step strategy is adopted. In the first step, a computationally “cheap” method and small but robust basis sets are required. This leaves only the Hartree–Fock (HF)[32] and DFT[27-31] methods, which can be used in practice.14 Although HF and DFT methods were used for the first attempts of FP-GOs,[100, 101] in the recent literature nearly all FP-GO investigations have been based on DFT methods. For light or heavy elements,15 all electron or basis sets optimized for effective core potentials (ECP)[102, 103] of typically double zeta quality and without or only a few diffuse and polarization functions are used, respectively. Although for light atoms typically all electron Pople-[104, 105] or Ahlrich[106, 107]-type basis sets can be used for the computations, the latter basis sets are used for heavy elements and combine an ECP describing the inner electrons with an explicit treatment of valence electrons to reduce the computational time. Commonly used basis sets for heavy elements are the Los Alamos,[108, 109] Stuttgart–Dresden,[110, 223] small Ahlrichs,[107] or plane-wave (PW)[102] basis sets which have the advantage that scalar-relativistic effects are incorporated within the ECP.[102, 103] 16 On the other hand, these “cheap” quantum chemistry methods require intense testing before the actual FP-GO stage to confirm the applicability of the method to the system under investigation. This does not only include the theoretical method and basis set but additionally all other computational settings such as force, energy, and electron density convergence criteria. Otherwise, the FP-GO will give wrong structures and structural motifs which can mislead the search for the GM in addition to wasting computational resources if the convergence criteria are chosen too tightly.

In the second step, energetically low lying isomers can be refined by consistently improving the basis set17 and tightening the convergence criteria. This should be followed by using a more accurate computational method or accounting for SO effects in the case of heavy elements. The number of energetically low lying isomers refined in the second step can introduce another source of uncertainty, as it is assumed that these isomers found by the low-level quantum chemistry computations will also be the best isomers at the higher level of theory. For DFT calculations, however, it is well-known that different functionals can alter the energetic ordering dramatically.[112] Therefore, care must be taken that the level of theory applied in the FP-GO and in the subsequent refinement stage are compatible.

In addition to these factors concerning the level of theory, the overall GO strategy must be adapted due to the computational resources required for these investigations. For EP-GOs, the search is repeated up to several hundred times and in each run up to several thousands of iterations are performed. From each of these runs, only the energetically lowest isomer is used for further refinement steps but still several isomers are available from the repetition of the GO.[33] Of course, this is not feasible for FP-GOs. Here, a decreased number of iterations are performed, for a small number (sometimes only one) of GO runs. From these calculations, several isomers are refined at the improved level of theory, but naturally the sampling of the PES cannot be as efficient as for EP-GO methods. Additionally, this entire procedure needs to be repeated for several spin multiplicities if spin restricted calculations are preformed.

Nevertheless, when the aforementioned factors and limitations are kept in mind, the combination of GO algorithms with electronic structure methods is a very powerful tool to identify novel structural motifs and electronic structures of clusters and nanoparticles, particularly for cluster systems for which no good guiding EPs exist. Here, we focus on recent developments in this field. Again, we emphasize that this review is not intended to be comprehensive but rather we aim to introduce the general methodology of FP-GOs and an overview of capabilities and methodological developments in the field.

Genetic algorithm

In the late 1990s, the first attempts were undertaken to overcome the intrinsic limitations of the GA-EP approach and improve structure predictions for small clusters by GA searches. First, this was done by modeling the intercluster bonding by DFT tight-binding computations[113] or on-the-fly reparameterization of EPs by comparison to a small number of ab initio calculations.[114, 115] These methods are still applied successfully to clusters.[116] Several years later, the first algorithms that directly linked GA codes with quantum chemistry packages allowed the investigation of small model systems with only a few light atoms.[117] In present day FP-GAs, it is possible to study nanoparticles with as many as 30–40 atoms,[37, 118] including heavy elemental clusters.[94, 119, 120]

A typical result of a FP-GA investigation is shown in Figure 5a for the cluster Ag8 using the BCGA and only using the energy criterion in the natural selection step.[94] Some structures that were found during the search are included in the same figure. For this small monometallic cluster, the following parameters were found to suffice: inline image, ntot = 0.8 × npop, pmut = 0.1 (random new structures), and mconv = 10, in combination with local optimization at the PW-ECP/PBE[121] level of theory. As shown in the EPP (Fig. 5a), the GA still needs five generations to find the structural motif of the putative GM and stops after 17 generations. Of course, the number of generations and local minimizations is much smaller than for the Cu20/Cu10Au10 example presented above (Fig. 2) but still 172 DFT local optimizations were performed on 24 processors18 requiring a total computational time of about inline image s per processor. This proof-of-principle investigation shows that the FP-GA reliably locates the GM, as this Td symmetry cluster (Fig. 5a) was proposed previously[123] as the putative GM.[94] However, the ultimate task of a GO algorithm is to identify the structures of so far unknown nanoparticles. Therefore, in addition to the monometallic Ag8 and Au8 clusters, all compositions of eight-atom mixed Ag-Au clusters were studied.[94] Even if the overall number of atoms does not change, the GO becomes considerably more difficult for nanoalloys due to permutational isomerism. This is illustrated by the EPP for Au4Ag4, shown in Figure 5b, in which some isomers found during the search are included. Using similar computational parameters for the bimetallic as for the monometallic cluster, as well as the application of the exchange mutation operator, the correct structure and homotop is not found until the 12th generation and 22 generations are required for the GA to converge, even though the structural motifs of Ag8 and Au4Ag4 are identical. This adds additional numbers of DFT optimizations and increases the overall computational time. However, with this systematic GA-DFT study of all compositions of small Au-Ag clusters, it was possible to identify a dopant-induced two-dimensional (2D)−3D transition on doping between 1 and 3 Ag atoms into Au8.[94] This would not have been possible with EPs, as these incorrectly favor 3D isomers for small gold clusters.[124] In other GA-DFT studies, Hong et al.[125] and Heard et al.[126] have extended this the concept of a dopant-induced geometry transition to all Au-Ag clusters with 5–12 atoms and eight-atom Cu-Au clusters, respectively.

image

Figure 5. EPP of Ag8 (top) and Au4Ag4 (bottom), with the energy E plotted relative to the energy of the GM isomer (E0) and only using the energy selection criterion (Ag—silver spheres, Au—gold spheres). Several low energy isomers of Ag8 (Au4Ag4) (corresponding generation numbers are indicated beneath the isomers) are included in the figure. The evolution of the lowest (min, black), average (mean, blue), and highest (max, red) energy in the population are depicted in the inset.[94] Reproduced from Ref. [ [94], with permission from The Royal Society of Chemistry. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Particularly in view of the subsequent structure refinement, the exclusive application of the energy criterion in the natural selection step is disadvantageous, as the final population only consists of the energetically lowest lying isomer. This is illustrated by the EPPs in Figure 5. The maximum, average, and minimum energies of the population coalesce for the final generation. Consequently, only a single structure is present in the final population. To avoid population stagnation and ensure structural diversity, geometric descriptors can be used.[37] As an example, the moments of inertia serve as descriptors for the cluster structure. This has been implemented in the BCGA and a typical EPP, and some isomers found for Na3Si11 are shown in Figure 6. 19 Now the maximum, average, and minimum population energy no longer converge to the same value, hence, various cluster isomers can be found in the final population. This is not only of great value in the later refinement step but Hartke[65] showed that a GA search with such evolutionary “niches” makes the algorithm more efficient overall. The structure located as the lowest energy isomer for Na3Si11 (Fig. 6) agrees with the previous work of Kumar and coworkers.[127] They investigated the neutral and anionic Na-Si system (with a maximum of three Na and 11 Si atoms), using with a population-based GA-DFT and applying energy and moment of inertia criteria in the natural selection step. However, in their investigation 1000–3000 genetic operations were applied and, consequently, the same number of DFT optimizations have been performed for every cluster size. In contrast, for the GA-DFT run shown in Figure 6, only about 370 DFT calculations were necessary until the algorithm converged to the same structure proposed in Ref. [ [127] as the putative GM. For FP-GOs in particular, this is a double-edged problem. On the one hand, the PES should be explored rigorously, performing as many GO steps as possible, but, on the other hand, significant computational resources can be “wasted” due to the immense computational expense of these methods. In this particular case, nearly a factor of 10 DFT calculations could be avoided, though this only becomes clear in retrospect. This example highlights an aspect common in FP-GOs: a fine balance, between the extensive exploration of many isomers versus the restrained and minimal use of computational resources, is required to make the FP-GO algorithm effective. Keeping these key aspects in mind, GA-DFT methods have identified a large number of unexpected structural motifs of clusters, very often in combination with state-of-the-art experimental methods.

image

Figure 6. EPP of Na3Si11 (Na—red spheres, Si—gray spheres) for the lowest (min, black), average (mean, blue), and highest (max, red) energy in the population, with the energy E plotted relative to the energy of the GM isomer (E0). Energy and moment of inertia selection criteria are used. In addition, several low energy isomers of Na3Si11 (corresponding generation numbers are indicated beneath the isomers) are shown. The located putative GM (isomer found in generation 13) agrees with predictions from Ref. [ [127]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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From the quantum chemical point of view, a small cluster of a single light main-group element in the gas phase is one of the simplest test systems possible. In this respect, cationic, neutral, or anionic boron clusters constitute an ideal model system which can be studied up to 30 atoms due to boron's small number of electrons.[128] In a benchmark study by Oger et al.,[118] the structures of B inline image clusters with inline image were investigated using GA-DFT calculations in conjunction with ion drift mobility measurements. For the GO, the def2-SVP/BP86[129] level of theory and population sizes up to inline image were used. However, still for some sizes, 100 generations with a 50% crossover rate did not succeed in finding an isomer which matches the measured experimental collision cross section. This is due to the fact that many low energy isomers consist of planar and bowl-like structural motifs, as shown in Figure 7a for several cluster sizes. Only a few isomers exhibit a wheel-like structure which, however, is the structural motif of the putative GM for inline image and inline image (Fig. 7a).[37, 132] These structures are unlikely to be generated from other isomers in the population by a crossover operation, as these exhibit completely different structural motifs. This makes the wheel-like structures very hard to find by a FP-GA.[132] Oger et al.[118] succeeded in finding the wheel-like structures by seeding the initial population with double or triple ribbons as well as quasiplanar and cylindrical structures with two to four rings. This resulted in improved convergence of the GA and, after further structure refinement, collision cross sections from theoretically predicted structures agreed with the experimental measurements to confirm the unexpected structural motifs of small boron cations. In a recent publication, Neiss and Schooss[132] showed how to avoid this structure seeding strategy by using experimental information to guide to search. Both of these approaches are examples of biased GO strategies.

image

Figure 7. Putative GM and low energy isomers found by the GA-DFT method for different systems. a) B inline image ( inline image),[118] b) Sn inline image ( inline image),[130] and c) Au8/F/MgO[141]. a) The GM (I) and low energy isomers (II;III) exhibit planar, bowl-like, or double-wheel like structures which cannot be reproduced by EPs. Reprinted from Ref. [ [118], with permission from Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. b) The GM (1) and low energy isomers (2;3) consist of fragments identified for small Sn cluster anions (green: pentagonal bipyramid of Sn7; red: tricapped trigonal prism of Sn9; blue: bicapped square antiprism of Sn10). Reprinted from Ref. [ [130], with permission from American Institute of Physics. c) The golden Au atoms for the various cluster isomers are bond to the F-center. Only the upper MgO layer (red: oxygen; gray: magnesium) is shown in color. Reprinted figure with permission from Ref. [131]. Copyright 2012 by the American Physical Society. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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More difficult systems for FP-GO approaches are clusters of heavy main group elements. This is due to the increasing importance of relativistic effects,[96-98] though, compared to transition metal clusters, generally only low spin multiplicities need to be considered.20 For example, cationic, neutral, and anionic tin clusters have been studied extensively by combining GA-DFT with various experimental methods.[130, 134-136] In an interesting case study, Schooss and coworkers[130, 136] investigated the structures of medium sized Sn cluster anions, combining GA-DFT calculations with trapped ion electron diffraction and collision-induced dissociation. Population sizes of 40 individuals and up to 32 “offspring” were required to locate several low energy isomers within 15–90 generations at the def-TVZP/TPSS[137] level of theory, to explain the experimental results. Nevertheless, the initial generations of larger clusters were seeded with structure fragments found for small tin anions, which helped the convergence of the GA. This procedure is justified by the resulting nanoparticle geometries, with the low energy isomers of Sn inline image, Sn inline image, Sn inline image, and Sn inline image shown in Figure 7b, highlighting the construction of larger clusters by small tin subunits. An unbiased GA-DFT search of neutral Sn clusters reinforces this partially biased GO strategy, because similar growth behaviour was identified.[135]

Using very similar strategies, the investigation of other main group and coinage metal clusters is possible. A selection of GA-DFT studies for these systems is presented in Table 1. We emphasize again that all these studies include scalar-relativistic effects (at most) at the GA stage. Only recently, the first studies that include SO effects or perform relativistic two-component calculations at the refinement stage were published for Pb[141] and Bi[143] cations.

The next stage of complexity is the GO of multielement clusters. This can include nanoalloys consisting of several metals, oxide species, or any other combination of two or more elements. Due to the permutational isomerism, the optimization problem becomes even more complex (see Figs. 5 and 6), and the investigated systems are typically smaller than for pure elemental clusters. Apart from the results presented above for coinage metal and Na-Si clusters, only the Sn-Bi system has been studied in great detail. In conjunction with electric beam deflection measurements, isomers found using the BCGA-DFT approach revealed a close connection of the neutral bimetallic cluster structures with inorganic Zintl ions Sn inline image (for inline image).[146, 147] FP-GA studies for multielement systems are quite new and, consequently, it is not surprising that only a few examples exist so far (Table 1). Apart from the nanoalloys, to date mostly metal oxide species have been investigated by GA-DFT in combination with infrared (IR) spectroscopy, because they serve as model systems for oxide surfaces or catalysts.[203, 204] Table 1 also includes some representative examples, but a more complete overview of GA-DFT for oxide clusters and surfaces can be found in Ref. [ [37].

So far, only FP-GA searches for gas-phase nanoparticles have been presented, as the presence of ligands or a surface prohibited the use of FP-GO methods until recently. To our knowledge, to date, the only GA-DFT-based GO of a ligand-protected cluster was published in 2010 by Xiang et al.,[161] who treated the ligands and the metal core explicitly in their GO approach. In the course of this study, the authors predicted the putative GM for [Ag7(SR)4] inline image and [Ag7(DMSA)4] inline image (R represents an organic residue and DMSA is meso−2,3-dimercaptosuccinicacid) clusters and rationalized the structures with an in depth discussion of the bonding.

Very similar to the problem of describing ligands is the treatment of surface effects in the GO of clusters. This is especially important for rationalizing observations in catalysis by mass-selected clusters deposited of well-defined surfaces.[203, 204] Recently, the first GA-DFT study of Au clusters on MgO was published to explain the increased reactivity toward CO oxidation of this system.[131] First, Vilhelmsen and Hammer studied the structures of Au8 located on F-centers of the MgO(100) surface (For convenience, we use the shortened notation Au8/F/MgO in the following). They used a combination of orbital-based and grid-based computation methods to speed up the GO and performed two independent GA-DFT searches, using the PBE and M06-L[205] functionals, respectively. The crossover and mutation operations were only applied to the clusters and not to the MgO surface, which was modeled by a 64-atom three-layer slab. Every newly generated cluster structure was not only characterized by its energy but also by geometrical descriptors to assure the investigation of atom configurations not located before.[131] In Figure 7c, the most stable structures of Au8 on F/MgO, encountered by the authors during their GA-DFT run, are presented. The planar A structure is the most stable using the PBE functional, whereas the M06-L functional favors the 3D C structure (Fig. 7c). In combination with experimental information and the trend that M06-L binds Au8 more strongly to the F-center, the authors inferred the most likely GM structure of Au8/F/MgO. In a second step, an O2 molecule was adsorbed onto the putative GM Au8/F/MgO structure and only the Au atoms not in contact with oxygen were geometrically optimized by using the GA-DFT routine. Using this procedure, the authors were able to identify putative GM structures of surface bound clusters21 and predict their fluxional behavior on O2 adsorption, thereby enabling a qualitative explanation of the observed reactivity.[131] Another study by one of the authors on the absorption behavior of Au8, Pd8, and Au4Pd4 in a metal-organic framework[162] is included in Table 1 but the number of GA-DFT investigations treating ligand-protected or surface bound clusters is very small at present.

Table 1. A selection of published FP-GO studies for clusters and nanoparticles
MethodSystem typeClusters
  1. a

    See Ref. [ [37] for a more detailed overview of oxide clusters and surfaces.

  2. b

    See Ref. [ [201] and citations therein for a recent review on Au and doped Au cluster anions.

  3. c

    PSO.

  4. d

    TSDS see http://www.yorku.ca/renef/ for further examples.

  5. e

    MHA.

  6. f

    CK see Ref. [ [202] for further examples. The table includes only some of the most recent investigations.

  7. g

    SA-MD.

  8. h

    GSAM.

  9. i

    SA-MC.

  10. Only the elemental composition of the clusters is given, while the investigated size range and charge state are avoided for the sake of clarity. Abbreviations: Surf = Surface, Lig = Ligand, MGE = Main-group element, TM = Transition metal.

Genetic algorithmMain-group elements and coinage metalsLi,[117] Cs,[119] B,[118] Al,[138] Ga,[139] Si,[140]
Sn,[130, 134-136] Pb,[141] P,[142] As,[142] Bi,[143]
Cu,[126, 144] Ag,[94, 125, 126] Au[94, 125, 126, 145]
NanoalloysNa-Si,[127] Sn-Bi,[146, 147] Ag-Cu,[126] Au-Cu,[126] Au-Ag[94, 125]
Oxidea and other systemsLi-O,[148] Be-O,[149] Mg-O,[150, 151] B-O,[152]
Al-O,[153] Ce-O,[154] W-O,[155] Li-F,[156]
Al-H,[157] B-C,[158] Li-Al-B,[159] H2O-H[160]
Complex systemsAg-Lig,[161] Au-Surf,[131, 162] Au-Pd-Surf[162]
Basin hoppingHomoatomicB,[163] Na,[164] Si,[165-168] Ge,[168] Sn,[168]
Co,[169] Rh,[170, 171] Cu,[167] Au[172, 173]
HeteroatomicC-Ti,[174] Co-Ar,[169] Au-S,[175]
Au-MGE,[176] Au-TM[177]b
Complex systemsH2O-MeOH,[178] Au-Lig,[179, 180]
Coinage metals-Surf[181-186]
Other methodsHomoatomicLi,[187]c,[87]dMg,[188]e B,[189]fSi[190]e, g Se,[100]g
Te,[191]g 13-atom clusters[192-194]c
HeteroatomicB-TM,[195]f Si-Al,[196]h Zn-Cu,[197]c
Cu-Sn,[198]c Sn-Pb,[199]c Li-F[200]i

Basin hopping

Early examples of BH searches using electronic structure methods, involved coupling to approximate quantum chemical methods such as the Hückel/tight binding-model or orbital-free DFT (like Thomas–Fermi DFT) calculations.[206] However, it was not until 2004–2005 that the first BH-DFT GO studies of silicon[165, 166] and boron[163] were reported and the first study of a heavy element cluster, bare Au20, was reported in 2006.[173] As for the GA, the BH algorithm has subsequently been coupled to many quantum chemistry codes and has been applied to numerous problems in cluster science.

The typical approach of BH-DFT searches is nicely illustrated in a recent case study by Jiang and Walter who explored the PES of Au38 and Au40.[172] Figure 8 shows a plot of all visited minima (equivalent to part of the transformed PES, in Fig. 3) and some local minimum structures of Au40 encountered during the GO. The BH starting structure (Fig. 8a) was generated from the putative GM of Au38 by manually adding two Au atoms.[172] Over 1000 BH steps were performed at the PAW/PBE level of theory, each consisting of a random change of every Cartesian coordinate by a step in the range [−70;70] pm, followed by local DFT energy minimization. This tour-de-force GO required 256 parallel cores on a Cray XE6 supercomputer, to complete approximately 1000 BH steps in 67 h. The two most stable structures (GM: Fig. 8b; higher energy isomer: Fig. 8c), which are separated by only 0.17 eV, evolved out of this investigation, both exhibit twisted, chiral pyramidal structures.[172] An in-depth analysis of the electronic structure for these isomers revealed that an electronic shell closing, leading to enhanced stability, very similar to the situation found for the famous Au20 tetrahedron,[173, 207] is responsible for this unexpected cluster shape.[172] This example highlights that the determination of the geometric and electronic structure of clusters, even up to these sizes, often require the interatomic bonding to be modeled by electronic structure methods.

image

Figure 8. Reprinted figure from Ref. [ [172], with permission from American Physical Society. Part of the transformed PES as a function of the number of BH steps, obtained from the BH-DFT investigation of Au40.[172] The initial cluster structure (a), the putative GM (b), and a low energy isomer (c) are included into the figure. The two last mentioned clusters exhibit a twisted pyramidal structure. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Similar to the GA applications presented above, the BH-DFT approach has been used to search for the GM of many cluster systems, ranging from homoatomic clusters of light and heavy elements, through heteroatomic systems (including nanoalloys) and clusters bound to ligands and surfaces, to mixed molecular clusters (Table 1). As discussed for the GA, the complexity of the overall optimization problem can vary due to the number of explicitly treated electrons or inequivalent atoms, in addition to the level of theory required to yield reasonable cluster structures. However, for simple homoatomic clusters, only a few BH-DFT investigations exist, with the exception of Si[165-168] and Au[201] which have been studied extensively (Table 1). All these studies proceeded in a very similar way, as described for Au40, and face the same problems as described in the FP-GA section. However, in contrast to the available GA-DFT studies at least, two investigations of pure transition metal clusters exist. In a first study, Reuter and coworkers have searched the PES of Co inline image and Ar-Co inline image clusters ( inline image), not only to optimize the geometry but also the lowest-lying spin configuration.[169] In combination with IR-multiphoton depletion (IR-MPD) experiments, the authors tried to identify the geometries of these small cationic Co clusters but satisfying agreement between theory and experiment is only obtained for a few sizes. A similar study, combining IR-MPD with BH-DFT calculations, was presented by Walsh and coworkers for Rh inline image ( inline image).[170, 171] They observed a significant dependence of the most stable putative GM and its spin multiplicity on the functional, making the identification of the geometric and electronic GM very challenging.[170] These case studies illustrate the problems FP-GOs face when investigating transition metal clusters using DFT methods. The multiple close lying spin states require a proper investigation using a multideterminant wavefunction-based method,[32] but this is presently infeasible for all but very small clusters.

In addition, several heteroatomic clusters have been studied using BH-DFT. Generally, FP-BH studies have been used to investigate the influence of a single main-group element,[176] transition metal,[177, 208]22 or a few sulfur[175] atoms on the geometric and electronic structure of Au clusters. A little different and, from a chemical point of view, more instructive, is the case study by Gao et al.[174] on [CTi7] inline image and [CTi7][BH4]2.The authors identify [CTi7] inline image as a stable cluster with a heptacoordinated carbon atom at 0 K in the gas phase using BH-DFT. In a next step, two [BH4] inline image counter ions are added and the BH-DFT is repeated, revealing the chemical stability of the [CTi7] inline image moiety. These BH calculations are required to make reasonable cluster structure predictions for future synthetic attempts in the solid state.

Beside the aforementioned case studies, most FP-GOs have been applied to atomic clusters. Hence, the coordinates of all atoms are treated as variables. However, in principle bond lengths, angles, and (especially) torsion angles could be varied—that is, using internal rather than Cartesian coordinates. This is of particular relevance for molecular clusters, avoiding the generation of high energy isomers, thereby making the GO algorithm more effective. However, it must be kept in mind that molecular clusters are mainly held together by weak van-der-Waals forces or hydrogen bonds. This prevented BH-DFT studies (as well as all other GO-DFT approaches) of molecular clusters for many years, since standard DFT methods fail to describe van-der-Waals interactions properly.[209, 210] Due to the development of a new generation of functionals,[209, 210] accounting for dispersion interactions in a semiempirical way, the first BH-DFT investigations of neutral and protonated H2O, CH3OH, and H2O-CH3OH clusters have been published.[178] Do and Beseley, for example, studied the structures of CH3OH clusters using BH-DFT. The putative GM identified for (CH3OH)6 and (CH3OH)7 at the 6–31+G*/B3LYP[211] and 6–31+G*/B3LYP+D[212] level of theory are shown in Figure 9. Although the number of hydrogen bonds does not change due to the incorporation of the dispersion interactions, the relative orientations of the CH3 groups differ considerably. In the absence of dispersion, the methyl groups tend to avoid each other (Fig. 9, upper part). Including dispersion, interactions result in the aggregation of the CH3 groups in the center of the ring-like structures formed by the CH3OH molecules (Fig. 9, lower part). Hence, this work clearly demonstrates that FP-GO studies of molecular clusters are possible, which is of special interest for the treatment of molecular clusters (including mixtures) where no parametrized potential exist so far.

image

Figure 9. Comparison of the putative GM for (CH3OH)6 and (CH3OH)7 obtained from two independent BH-DFT runs, using the B3LYP (upper row) and the B3LYP+D (lower row) functionals.[178] Carbon, oxygen, and hydrogen are represented by gray, red, and white spheres, respectively. The orientations of the methyl-groups change considerably when the DFT calculations include dispersion interactions. Reprinted from Ref. [ [178], with permission from American Institute of Physics. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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In addition to the investigation of molecular clusters, the FP-BH approach has recently been used to study the structures of several ligand-protected clusters or cluster-molecule complexes. Examples include BH-DFT studies of O2[180] and CO[179] decorated Au nanoparticles. However, as a final example, we want to review the BH-DFT exploration of the PES for surface bound clusters. In this respect, the pioneering work of Fortunelli and coworkers must be mentioned. They implemented a cluster approach for studying surface-bound metal clusters at an F-center on the MgO(100) surface. The surface is modeled by a finite MgO cluster, representing a fragment of the MgO crystal surface with the desired orientation and with a single F-center defect (the geometry of the MgO cluster is not changed during the GO). Then, the metal cluster is placed on this F-center, followed by GO of the metal cluster by BH-DFT. In this way, the structures of pure[181-183] and mixed[184, 185] coinage metal clusters, and their interaction with the F-center, have been studied.

Fortunelli and coworkers[186] have subsequently extended this approach to study the reaction of clusters on F/MgO by a method they termed Reactive GO (RGO). The main idea of the RGO approach is to follow the reaction path by BH-DFT and it has been demonstrated for the reaction of propene with O2 on Ag3/F/MgO. First, the cluster is placed on the surface and its structure is globally optimized. In the next step, an oxygen molecule is added to the cluster and in a subsequent BH-DFT run the position of O2 is optimized. To locate reaction intermediates or TS, Negreiros et al.[186] implemented the internal reaction coordinate algorithm[213] which follows a reaction path based on available eigenvectors. Hence, this BH does not change the structure randomly but follows a certain eigenvector until the TS barrier is overcome and a subsequent geometry optimization is performed. In this way, the authors observed a dissociation of O2 on Ag3/F/MgO what is in contrast to the gas-phase Ag3O2 cluster.[186] In the next step, propene is adsorbed on the Ag3O2/F/MgO system and by using RGO several reaction paths are identified. An example of a possible reaction path and several configuration encountered during the RGO are shown in Figure 10. Starting from the initial configuration (defined as the zero of energy), an energy barrier of 0.2 eV is overcome, leading to insertion of an oxygen atom into the C[DOUBLE BOND]C double bond (Fig. 10). Subsequently, propyleneoxide desorbs, leaving behind Ag3O/F/MgO. In this proof-of-principle study, Fortunelli and coworkers[186] demonstrated that, in addition to reliable structure predictions, FP-GO methods can be used to study chemically relevant reactions.

image

Figure 10. A possible reaction path and some stable structures encountered during the RGO of propene and O2 on Ag3/F/MgO.[186] Carbon, oxygen, and hydrogen atoms in the molecular fragments are represented by small yellow, red, and cyan spheres, respectively. The silver, magnesium, and (substrate) oxygen atoms are shown as large silver, blue, and gray spheres, respectively. In this example, a previously dissociated oxygen reacts with propene to form propyleneoxide and Ag3O/F/MgO. Reproduced from Ref. [ [186], with permission from The Royal Society of Chemistry. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Other methods

Beside the results presented above, based on the very popular GA and BH approaches, a number of other methods (for a selection see Table 1) have been combined with FP calculations and applied successfully for the GO of clusters.

One of the first methods to be used in this respect was SA in combination with Car–Parrinello simulations.[7] This method performs an ab initio MD simulation (e.g., in the NVT ensemble using the Nosé–Hoover thermostat[214]) for smoothly decreasing T, resulting in putative GM structures. In the classic study by Hohl et al.,[100] the PES of small Se clusters was explored. This method is still used for GO investigations of clusters,[191] but it is more frequently used to study the dynamics and finite temperature properties of nanoparticles[215] or amorphous solids.[191] Very similar in nature are SA-MC runs coupled to HF or DFT calculations. For instance, Doll et al.[200] used this SA Metropolis MC method combined with HF calculations followed by a random quench (rapidly decreasing T) to perform an extensive search for the GM of Li-F clusters. As only single-point calculations, with no local geometry optimizations are performed, up to 75,000 MC steps were required to carry out a reasonable sampling.

As briefly described above, the MHA method[84] combines MD with subsequent local optimization and a historical list of all previously found structures. In a first attempt, Goedecker and coworkers studied medium sized Si clusters, applying a combined EP- and DFT-MHA search, to decrease the computational resources required for the GO. MHA has also been successfully extended to perform GO of small and medium sized Mg clusters at the DFT level.[188]

Methods that perform predominantly single-point energy calculations (rather than local optimizations) found early application in conjunction with FP methods due to their reduced computational cost. In this respect, a very effective algorithm, as it uses geometric descriptors to interpolate the energy of candidate structures, is the TSDS algorithm introduced above.[83] Its first application in conjunction with DFT calculations was for GO of Li clusters.[187] This was followed by a large number of TSDS searches for various homoatomic and heteroatomic clusters.[192-194, 197, 198] Particularly interesting is a systematic search of 13-atom clusters of various main-group and transition elements.[192-194] Many EPs predict an icosahedron for this cluster size. In contrast to these predictions, Fournier and coworkers were able to show, using their TSDS-DFT approach, that there is a rich structural diversity of GM structures for 13-atom clusters of different elements.[192-194] However, it must be remembered that structure predictions of transition element clusters are very difficult, not necessarily due to the inability of the GO algorithm to locate the GM (on the DFT-PES) but also due to the reliability of the DFT calculations.

A problem encountered by many GO methods, at least in the first cycles of the algorithm, is that randomly generated structures often correspond to unphysical atom configurations. Therefore, the very first calculations often fail to converge or require more computational time compared to the subsequent FP computations. To overcome this problem, Boldyrev and coworkers developed a very simple but robust GO method called coalescence kick (CK),[202] similar to the multiple quench[9] or ab initio random search[47] algorithm used successfully in solid state physics.23 A large number of random structures are generated first. Every randomly placed atom is either connected or not (in the sense of being close enough to form a chemical bond) to its nearest neighbors. However, to form a reasonable starting structure, all the atoms should be part of a continuously connected network. Hence, the main idea of the CK algorithm is to identify the connected atoms (forming individual fragments) and randomly push all fragments to the center of mass until the cluster is no longer fragmented. This gives a large number of reliable candidate structures which are subsequently locally minimized using FP methods. The CK-DFT method has been used to predict the putative GM minimum for many cluster systems[202]: most recently, it was applied in conjunction with high-resolution photoelectron spectroscopy to predict the GM of pure and transition metal-doped boron cluster anions.[189, 195]

Besides the GA, there are many more nature inspired optimization algorithms but, to our knowledge, only PSO has been used in structure prediction of clusters, using electronic structure methods.[86, 87] For example, Lv et al.[87] extended their original PSO algorithm to explore the PES of clusters more thoroughly by performing several local PSOs at the DFT level of theory. In a local PSO, the information is diffused into small parts of the swarm enabling simultaneous but independent searching in different funnels of the PES. Combined with the seeding of randomly generated high energy isomers into the swarm, this “new” PSO proved to be very efficient for small and medium-sized Li clusters.[87]

Conclusion and Outlook

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies

In this review, we did not set out to compare all different GO schemes, but rather we wanted to illustrate the capabilities of present day GO algorithms, when coupled to electronic structure methods. We have shown that the number of FP-GO studies has increased considerably in recent years, especially for systems for which no reliable EPs exist. The applications range from light or heavy elemental clusters of up to 40 atoms, through heteroatomic systems to nanoparticles passivated by a ligand shell or deposited on surfaces. Even a systematic investigation of reaction pathways with FP-GO methods has become possible. Particularly, when coupled with state-of-the-art experiments, FP-GO methods can show their full capabilities. Due to the fact that the PES exploration cannot be as thorough as for EP-GOs and because unusual structural motifs are often encountered for small nanoparticles, nobody can expect that a GO, even for relatively small sizes, will be able to locate every GM structure from FP. In this respect, a close symbiosis between experiment and FP-GO investigation can be beneficial to both. On the other hand, FP-GOs have also been successfully used to explain experimental trends[131] or predict putative GM[94, 125] which still await experimental confirmation.

Despite all the work described above, a large number of problems as well as opportunities remain. At some stage, the entire range of cluster sizes, types, and compositions should be treatable with GO methods using the required level of theory. As we have shown, at present, FP-GOs can be used for nanoparticles with up to 30–40 atoms. Although these studies have given detailed insight into the electronic and geometric structures of these nanoparticles, and can be considered as benchmark investigations, the initial structures were generated by educated guesses or from the fusion of smaller cluster fragments. In many other FP-GOs starting from random structures, the maximum cluster size is about inline image atoms and typically inline image atoms for heteroatomic systems. On the other hand, EP-GOs for similar systems can treat nanoparticles with up to hundreds or thousands of atoms.[217-219] Structural refinement of many candidate (EP-)isomers of this size is possible at the DFT level of theory.[220] The drawback of this approach, as has been made clear throughout the article, is that for small clusters EPs may introduce errors due to the simplified description of the bonding. It is far from realistic, though, to expect that FP-GO will replace EP approaches completely. Therefore, a more desirable future task is bridging the size gap between FP- and EP-GO approaches. This would include performing FP-GO up to sizes and compositions at which low energy isomers agree with predictions from EP-GOs.24

This aim will probably be achieved with the help of developments of the GO algorithms. This could include using reasonable initial structures, for example using the CK method,[202] making use of symmetry to speed up the calculations[222] or using available information of previously visited minima (e.g., structural descriptors, energies, frequencies,…)[65, 83, 87] to assure the exploration of as many funnels of the PES as possible. Most importantly, the GO algorithm itself, not only the electronic structure calculation, must be parallelized to make optimal use of modern computer architectures. Combined with further improvements in the speed as well as performance of electronic structure methods, and with the increasing availability of high performance computer facilities, the future will definitely see the more frequent use of FP-GO investigations.

From what we have presented here, bridging the size gap of nanoparticles currently studied by FP-GO and EP-GO algorithms seems to be within reach. This would be of great use for basic research in cluster physics/chemistry and also for structure prediction from “first principles” of chemically or materials science relevant systems.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies

The authors are grateful to Detlef Schooss (Karlsruhe), Patrick Weis (Karlsruhe), Lasse Vilhelmsen (Aarhus), Michael Walter (Freiburg), Nick Besley (Nottingham), Chris Heard (Birmingham), Alessandro Fortunelli and coworkers (Pisa) for supplying original figures. S.H. is grateful to Rolf Schäfer (Darmstadt) for excellent mentoring and support during his PhD studies. R.L.J. acknowledges members of his research group and collaborators, past and present, for their contributions to his research, particularly in the areas of global optimization and the use of empirical potentials and DFT calculations to study metal clusters and nanoalloys. In terms of the development and testing of the BCGA-DFT code, special thanks are extended to Dr Andrew Logsdail (now at University College, London), Chris Heard (Birmingham), and Ali Shayeghi (Darmstadt).

  1. 1

    By a coordinate transformation, the rotational and translational degrees of freedom can be separated, reducing the dimensionality of the PES to (3N–5).

  2. 2

    If the temperature T is not zero, the free energy inline image must be considered, with E and S being the energy and entropy, respectively. We will only use the term “energy” throughout the manuscript. However, incorporating temperature effects are conceptually quite straight forward but computationally much more demanding.[7-10]

  3. 3

    In our definition, this does not include orbital-free DFT methods.

  4. 4

    Again, it can be argued that DFT is not a “first principles” or ab initio electronic structure method. For convenience, we will include DFT and ab initio approaches under the heading “first principles.”

  5. 5

    This also includes combinations of the two previously mentioned algorithm types.

  6. 6

    A short explanation of this criterion is given in the basin hopping section.

  7. 7

    If local minimization is not performed, then at finite T, the system will not be at a minimum but higher up in the potential well or funnel.

  8. 8

    Any other type of simulated temperature ramp is possible but the notation for these methods can be different and must be distinguished from tempering algorithms.

  9. 9

    The most common schemes are to generate random coordinates within a sphere, cubic box, or some other volume.

  10. 10

    Though not containing the GM.

  11. 11

    For LJ110 about 70 min are required on average on a single processor.[34]

  12. 12

    MHA assigns the kinetic energy of the atoms in the cluster randomly. However, if no new minima are found, the energy is increased until new stable isomers are located.

  13. 13

    The energy differences in small and medium clusters are of the order of several hundred meV or less. See for example the Refs. [[94] and [95].

  14. 14

    In general, every level of theory [Møller–Plesset perturbation theory (MPPT), coupled cluster (CC) calculations][32] and other semiempirical methods (e.g., tight binding, AM1,[99] inline image) could be used in conjunction with GO studies. Here, we restrict ourselves to HF and DFT methods, as MPPT/CC methods have not be used so far and in this review semiempirical methods are not considered as “first principles” approaches. Hence, FP-GO will be the synonym for GO investigations, searching a PES generated from HF or DFT calculations.

  15. 15

    We roughly distinguish between first and second row elements and all others. The former are described here as light elements and can be described by all-electron basis sets in FP-GO studies. The latter will be called heavy elements.

  16. 16

    Sometimes the projector augmented-wave (PAW) method is used which follows a different approach as described in Ref. [ [111]].

  17. 17

    Not necessarily changing the type of basis set but also possibly improving the quality (increasing BS size, adding polarization, or diffuse functions).

  18. 18

    The University of Birmingham's BlueBEAR high-performance computer.[122]

  19. 19

    The PW-ECP/LDA level of theory with a 35 Ry energy cutoff was used. The GA settings were: inline image, ntot = 0.8 × npop, and pmut = 0.2 consisting of 80% exchange mutations and 20% new structures.

  20. 20

    See for example, Ref. [ [133]] for the importance of relativistic effects on the polarizability of group 14 elements.

  21. 21

    Similar calculations were performed for Au6, Au10, and Au12 on F/MgO in Ref. [ [131]].

  22. 22

    See Ref. [ [201]] and references therein.

  23. 23

    However, for Morse clusters, the advantage of GA over an albeit simple random search has been demonstrated.[215]

  24. 24

    When no simple EPs exist, tight-binding models may replace the costly electronic structure methods. Another option would be the reparametrization of EPs using putative GM located by FP-GO investigations or employing the Gaussian approximation potentials (GAP)[221] method.

Biographies

  1. Top of page
  2. Abstract
  3. Introduction
  4. Basic Principles of the GO of Clusters and EP-GO Applications
  5. The FP-GO Approach for Clusters and Recent Examples
  6. Conclusion and Outlook
  7. Acknowledgments
  8. Biographies
  • Image of creator

    Sven Heiles obtained his Ph.D. in 2012 from the TU Darmstadt, Germany, where he worked in the group of Rolf Schäfer on structure discrimination of inorganic clusters by experimental beam deflection measurements in conjunction with GA-DFT calculations. During this time, he spent 6 months in the group of Roy L. Johnston at the University of Birmingham, where he assisted in the development of the GA-DFT code and carried out calculations on a number of cluster systems. His research focuses on experimental and theoretical characterization of gas-phase species, a topic he will continue to study with Evan R. Williams in his upcoming postdoctoral stay at the University of California, Berkeley. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

  • Image of creator

    Roy Johnston is Professor of Computational Chemistry in the School of Chemistry at the University of Birmingham, UK. His research spans the fields of computational nanoscience and nature-inspired computation. Examples include: the study of elemental and bimetallic clusters; the application of genetic and other nature-inspired algorithms to optimization problems in chemistry (including GA-DFT optimization); simulating the optical spectra and electron microscopy images of metal nanoparticles; DFT studies of the geometries, electronic structures, and reactivity of metal clusters and nanoalloys; and developing techniques for visualizing and analyzing the complexity of energy landscapes. He is the author of one book and editor of three others and has published over 170 journal articles and reviews. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]