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. 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]
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) 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, or plane-wave (PW) 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 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. 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.
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 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. 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. 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. 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: , ntot = 0.8 × npop, pmut = 0.1 (random new structures), and mconv = 10, in combination with local optimization at the PW-ECP/PBE 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 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 as the putative GM. 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. 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. This would not have been possible with EPs, as these incorrectly favor 3D isomers for small gold clusters. In other GA-DFT studies, Hong et al. and Heard et al. 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.
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. Reproduced from Ref. [ , 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. 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 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. 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. [  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.
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. [ . [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. In a benchmark study by Oger et al., the structures of B clusters with were investigated using GA-DFT calculations in conjunction with ion drift mobility measurements. For the GO, the def2-SVP/BP86 level of theory and population sizes up to 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 and (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. Oger et al. 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 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.
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 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 , Sn , Sn , and Sn 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.
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 and Bi 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 (for ).[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. [ .
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., 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] and [Ag7(DMSA)4] (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. 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 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. 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. Another study by one of the authors on the absorption behavior of Au8, Pd8, and Au4Pd4 in a metal-organic framework 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
|Genetic algorithm||Main-group elements and coinage metals||Li, Cs, B, Al, Ga, Si,|
|Sn,[130, 134-136] Pb, P, As, Bi,|
|Cu,[126, 144] Ag,[94, 125, 126] Au[94, 125, 126, 145]|
|Nanoalloys||Na-Si, Sn-Bi,[146, 147] Ag-Cu, Au-Cu, Au-Ag[94, 125]|
|Oxidea and other systems||Li-O, Be-O, Mg-O,[150, 151] B-O,|
|Al-O, Ce-O, W-O, Li-F,|
|Al-H, B-C, Li-Al-B, H2O-H|
|Complex systems||Ag-Lig, Au-Surf,[131, 162] Au-Pd-Surf|
|Basin hopping||Homoatomic||B, Na, Si,[165-168] Ge, Sn,|
|Co, Rh,[170, 171] Cu, Au[172, 173]|
|Heteroatomic||C-Ti, Co-Ar, Au-S,|
|Complex systems||H2O-MeOH, Au-Lig,[179, 180]|
|Other methods||Homoatomic||Li,c,dMg,e B,fSie, g Se,g|
|Te,g 13-atom clusters[192-194]c|
|Heteroatomic||B-TM,f Si-Al,h Zn-Cu,c|
|Cu-Sn,c Sn-Pb,c Li-Fi|
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. However, it was not until 2004–2005 that the first BH-DFT GO studies of silicon[165, 166] and boron were reported and the first study of a heavy element cluster, bare Au20, was reported in 2006. 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. 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. 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. 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. 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.
Figure 8. Reprinted figure from Ref. [ , 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. 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 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 and Ar-Co clusters ( ), not only to optimize the geometry but also the lowest-lying spin configuration. 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 ( ).[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. 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, 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, transition metal,[177, 208]22 or a few sulfur 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. on [CTi7] and [CTi7][BH4]2.The authors identify [CTi7] 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] counter ions are added and the BH-DFT is repeated, revealing the chemical stability of the [CTi7] 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. 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 and 6–31+G*/B3LYP+D 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.
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. 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. [ , 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 and CO 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 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. implemented the internal reaction coordinate algorithm 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. 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 CC double bond (Fig. 10). Subsequently, propyleneoxide desorbs, leaving behind Ag3O/F/MgO. In this proof-of-principle study, Fortunelli and coworkers demonstrated that, in addition to reliable structure predictions, FP-GO methods can be used to study chemically relevant reactions.
Figure 10. A possible reaction path and some stable structures encountered during the RGO of propene and O2 on Ag3/F/MgO. 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. [ , 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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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. This method performs an ab initio MD simulation (e.g., in the NVT ensemble using the Nosé–Hoover thermostat) for smoothly decreasing T, resulting in putative GM structures. In the classic study by Hohl et al., the PES of small Se clusters was explored. This method is still used for GO investigations of clusters, but it is more frequently used to study the dynamics and finite temperature properties of nanoparticles or amorphous solids. Very similar in nature are SA-MC runs coupled to HF or DFT calculations. For instance, Doll et al. 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 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.
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. Its first application in conjunction with DFT calculations was for GO of Li clusters. 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), similar to the multiple quench or ab initio random search 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: 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. 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.