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Application of a parallel genetic algorithm to the global optimization of medium-sized Au{Pd sub-nanometre clusters
  Eur. Phys. J. B (2018) 91: 34https://doi.org/10.1140/epjb/e2017-80314-2  T HE  E UROPEAN P HYSICAL  J OURNAL  B Regular Article Application of a parallel genetic algorithm to the globaloptimization of medium-sized Au–Pd sub-nanometre clusters  ,  Heider A. Hussein 1,2 , Ilker Demiroglu 1 , and Roy L. Johnston 1,a 1 School of Chemistry, University of Birmingham, Birmingham B15 2TT, UK 2 Department of Chemistry, College of Science, University of Kufa, Najaf, IraqReceived 31 May 2017 / Received in final form 30 June 2017Published online 12 February 2018c   The Author(s) 2018. This article is published with open access at Springerlink.com Abstract.  To contribute to the discussion of the high activity and reactivity of Au–Pd system, we haveadopted the BPGA-DFT approach to study the structural and energetic properties of medium-sized Au–Pd sub-nanometre clusters with 11–18 atoms. We have examined the structural behaviour and stabilityas a function of cluster size and composition. The study suggests 2D–3D crossover points for pure Auclusters at 14 and 16 atoms, whereas pure Pd clusters are all found to be 3D. For Au–Pd nanoalloys, therole of cluster size and the influence of doping were found to be extensive and non-monotonic in alteringcluster structures. Various stability criteria (e.g. binding energies, second differences in energy, and mixingenergies) are used to evaluate the energetics, structures, and tendency of segregation in sub-nanometreAu–Pd clusters. HOMO–LUMO gaps were calculated to give additional information on cluster stabilityand a systematic homotop search was used to evaluate the energies of the generated global minima of mono-substituted clusters and the preferred doping sites, as well as confirming the validity of the BPGA-DFTapproach. 1 Introduction Due to the environmental and energy challenges facingthe world, research in catalysis is particularly impor-tant. Nanometallic catalysts, in particular, often showsuperior performance compared to their bulk counter-parts [1,2]. These catalysts, which are widely used in the chemical industry, have motivated experiments on newmaterials on the nanoscale with high catalytic activityand/or selectivity [3–5] and have inspired the development of computational methods for predicting new catalystcandidates and optimizing their efficiency [6,7]. Although extensive research indicates that platinumand platinum-based nanostructures exhibit exceptionalelectrocatalytic activity, for example in direct alcohol fuelcells [8,9] their applications are somewhat limited due to the rarity of Pt in the earth’s crust and its consequenthigh cost. This has motivated researchers to look for alter-native metals (or alloy systems) to replace Pt, ideallykeeping the high performance but at lower cost [10–12]. The much higher natural abundance of Pd relative to Pt  Contribution to the Topical Issue “Shaping Nanocata-lysts”, edited by Francesca Baletto, Roy L. Johnston, JochenBlumberger and Alex Shluger.  Supplementary material in the form of one PDF file avail-able from the Journal web page at https://doi.org/10.1140/epjb/e2017-80314-2. a e-mail:  r.l.johnston@bham.ac.uk has led to researchers fabricating nanostructures based onPd [7,11]. Pd-based bimetallic nanostructures have also been used extensively, not only in the application to thedirect alcohol fuel cells (as the cathodic catalyst) [11] but also in many high-tech fields, due to their interesting mag-netic and electronic properties, as well as their catalyticproperties [2,8,13,14]. Changing the chemical order and composition of thebimetallic nanostructures can enhance and enable tuningof their catalytic properties [15,16]. The chemical order effect can lead to control of the catalytic properties (e.g.modifying relevant activation energy barriers) by tuningthe energy and spatial distribution of electrons at the sur-face of the clusters [2]. There are also economic reasons for mixing two metals, such as adding low cost metals(e.g. Co, Cu, and Ni) to balance the high cost of noblemetals. The enhancement of catalytic performance andpossible discovery of unique properties is still the mainforce driving the designing of novel nano-catalysts andsub-nanometre cluster catalysts. The Pd–Au bimetallicsystem, for example, exhibits high durability and catalyticactivity for many interesting chemical reactions, such asthe electro-oxidation of ethanol [17,18], the Suzuki cou- pling reaction [19], and the oxygen reduction reaction [20]. The catalytic activity and selectivity of sub-nanometre-and nano-clusters are also strongly affected by the sizeof the cluster and its electronic distribution. The srcinof size effects are still ambiguous [21], however, they are  Page 2 of  12 Eur. Phys. J. B (2018) 91: 34 usually attributed to the changing surface-area-to-volumeratio and the number and nature of exposed facets/surfacesites [22–24]. Gas-phase sub-nanometre clusters are simplified modelsthat can test a system’s suitability for specific applica-tions, at a reasonably high level of theory, before dealingwith more complex systems. However, even the optimisa-tion of gas-phase structures is not a trivial task. There isusually little energy separation between many competitiveisomers of metallic clusters [25], which may explain the challenges that face experimentalists in determining thepreferred isomer [26,27]. In addition, charge-neutral clus- ters are more difficult to investigate experimentally com-pared with anions and cations, as most characterizationsrely on mass spectrometry measurements, which creates aclear difficulty in separating and probing different size of neutral clusters [28,29]. In spite of the early investigations of neutral Au 2  [30]and Au 3  [31]. It is well known that experimental investiga-tions of medium sized neutral Au clusters are limited forthe reasons given above. Hence, a combination of experi-mental techniques with theoretical calculations has beenused effectively: for example, Fielicke and co-workers [28] reported the gas-phase structures of neutral Au 7 , Au 19 ,and Au 20  clusters tagged with Kr atoms. The case of Pd is similar as, unlike the smaller Pd clusters, stud-ies of medium sized clusters are relatively few and arethe object of some controversy in terms of identifyingthe structural characteristics [32]. Turning our atten- tion to medium-sized Au–Pd nanoalloys, a comprehensivestudy of structural motifs for all compositions have, toour knowledge, not been investigated before. However,Au-doped Pd clusters [33], (1:1) compositions [34], and nuclearities lower than 14 atoms [35] have previously been studied theoretically.In the Au–Pd system, doping an atom of one metalinto a pure cluster of the other metal often yields clus-ters with non-identical structures and properties [35–37]. This can increase the difficulty in finding global min-ima, in addition to the permutational isomers (homotops)effect [38]. There is additional complexity introduced by multi-directional bonding, unrestrained bond orders, andfluxional behaviour as a result of electron delocalization.In addition, alloying Pd with Au modifies the lattice dis-tance between host atoms and the low dimensionality of Au could influence the spatial arrangement of Pd atoms[39,40]. The DFT-based Birmingham Parallel Genetic Algo-rithm (BPGA-DFT) computational approach has beensuccessfully applied to search for low-lying isomers for var-ious sub-nanometre cluster systems. Applications include:pure clusters Ir 10 –Ir 20  [41], Rh 4 –Rh 6  [42], Au 4 –Au 10 ,Pd 4 –Pd 10  [37], Ru 3 –Ru 12  and Pt 3 –Pt 10  [43]; and bimetal- lic clusters (AuRh) 4 – 6  [42]. (AuPd) 4 – 10  [37], (AuIr) 4 – 6 [44], and (RuPt) 3 – 8  [43] as well as surface-supported [37,44] clusters. In this context, we present here a computational studyof the structural properties of binary sub-nanometre Au–Pd clusters, including a comparison to the pure clusters.Using the BPGA-DFT approach, the lowest energy struc-tures in the size range 11–18 atoms were calculated for allcompositions. This work also sheds some light on the ener-getics of these clusters and the underlying mechanisms of mixing in binary metallic systems on the sub-nanometreand nanoscale. Our findings for the Au–Pd system shouldprovide valuable information for Au–Pd catalysts and forfurther theoretical and experimental investigations. 2 Methodology The Birmingham Parallel Genetic Algorithm (BPGA-DFT) approach [41,45] was applied to investigate (at the DFT level) the lowest energy structures of Au–Pdsub-nanometre clusters with total number of atoms  N   =11–18, as well as pure Au N   and Pd N   clusters. BPGA-DFT is an open-source genetic algorithm [45], which is a parallel extension of the Birmingham Cluster GeneticAlgorithm (BCGA), a genetic algorithm for locating theglobal minima of small metal clusters directly at the DFTlevel [46]. Instead of generations, BPGA-DFT employs a poolmethodology to evaluate structures in parallel. In eachrun, multiple BPGA instances are implemented, and ineach instance, a set of processes are run in parallel andindependently [47,48]. Initially, the pool population is formed by generating a number of random isomers [41]. The ten generated structures forming the initial pool aregeometrically relaxed by local DFT energy minimization[47]. Once minimized structures are generated, the genetic algorithm crossover and mutation operations are appliedto members of the population.The clusters are selected for either crossover or muta-tion. The crossover operation involves selecting a pair of clusters from the pool, using the tournament selectionmethod, based on a fitness criterion, where the fittestisomers (those with the lowest DFT energies) are morelikely to be selected for crossover. Offspring clusters arethen generated using the cut-and-splice method intro-duced by Deaven and Ho [49]. There are two mutation operations, in which a single cluster is randomly selectedand either randomly chosen atoms are displaced or (forbimetallic clusters) the positions of a randomly chosenpair of non-identical atoms are swapped. After crossoverand mutation, the structures are locally energy-minimizedat the DFT level. The newly generated structures arethen compared energetically with existing structures inthe pool and the highest energy isomers are replaced byany lower energy isomers among the set of offspring andmutants.All the local energy minimizations mentioned abovewere conducted with gamma-point DFT calculationsemploying the Vienna ab initio Simulation Package(VASP) code [50]. Projected-augmented wave (PAW) pseudopotentials were used, with the (GGA) Perdew–Burke–Ernzerhof (PBE) exchange correlation functional[51,52]. A plane-wave basis set was implemented includ- ing spin polarization. The plane wave cut-off energy wastruncated at 400 eV. Methfessel–Paxton smearing, witha sigma value of 0.01 eV, was implemented to improveconvergence [53].  Eur. Phys. J. B (2018) 91: 34 Page 3 of  12 For pure Au and Pd and mixed Au-Pd clusters, thestability of each cluster, relative to neighbouring sizes, isindicated by the second difference in energy (∆ 2 E  ) whichis given by:∆ 2 E   =  E  ( A N  +1 ) + E  ( A N  − 1 ) − 2 E  ( A N  ) (1)where  E  ( A N  ) corresponds to the energy of the  N  -atomcluster and  E  ( A N  +1 ) and  E  ( A N  − 1 ) are the neighbour-ing clusters, with one atom more and one atom less,respectively.The effect of mixing Au with Pd atoms in nanoal-loys can be evaluated by calculating the mixing or excessenergy (∆) which is given by:∆ =  E  (Au m Pd n ) − mE  (Au N  ) N   − nE  (Pd N  ) N   (2)where  m  and  n  are the numbers of Au and Pd atoms,respectively,  E  (Au m Pd n )  is the total energy of the nanoal-loy Au m Pd n  whereas  E  (Au N  )  and  E  (Pd N  )  are the energyof pure metal clusters of Au and Pd, respectively, of thesame size ( N   =  m + n ).The average binding energy per atom ( E  b ) is given by: E  b  = − 1 N  [ E  ( Au m Pd n ) − mE  ( Au ) − nE  ( Pd ) ] (3)where E  ( Au )  and E  ( Pd )  are the electronic energies of singleAu and Pd atoms, respectively.The homotops (inequivalent permutational isomers)[38] are evaluated using: ∆ E   =  E  hom − E  GM   (4)where ∆ E   is the relative energy of the proposed homo-top and  E  hom  and  E  GM   are the electronic energies of aparticular homotop and the lowest energy isomer (globalminimum) of the cluster, respectively.Ignoring the symmetry, the number of homotops isgiven by: N  P  Au,Pd  =  N  ! m ! n ! =  N  ! m !( N   − m )! (5)where  N   is the total number of atoms,  m  is the numberof Au atoms, and  n  is the number of Pd atoms. 3 Results and discussions 3.1 Structures 3.1.1 Au clusters The putative global minima for pure Au clusters 11  ≤ N   ≤  18, are shown in Figure 1 and their energies, coor-dinates, and point groups are listed in Table S1 (see theSupporting Information).The lowest energy structures obtained for  N   = 11–13 and 15 Au clusters have planar (2D) configurations.The clusters deviate from planar to 3D structures at N   = 14 and for 16–18 atoms. The exact 2D–3D transi-tion point for neutral Au clusters is disputed theoreticallyand experimental evidence is scarce. Theoretical predic-tions of the 2D–3D crossover point have previously rangedfrom  N   = 7–14 atoms [54–57]. This range is consistence with the evolution of structure-symmetry for Au clustersreported here. The smallest 3D ground-state structure waspredicted previously to be Au 10  by David and co-workers[58], using the second-order Møller–Plesset perturbation theory (MP2) method. Employing hybrid DFT, Zanti andPeeters [35] arrived at the same conclusion showing a 3D structure for  N   = 10. This, however, disagreed with the2D Au 10  structure recently obtained by us [37] and previ- ously by different research groups using semi-local densityfunctional theory (DFT) [59] and coupled cluster singles doubles and perturbative triples [CCSD(T)] calculations[60,61]. Recently, Johansson et al., using a genetic algo- rithm and meta-GGA DFT, assigned a 3D structure asthe GM structure at N   = 12. However, they also suggestedtwo isoenergetic structures (2D and 3D) for N   = 11 atoms[62]. Three generic structure types can be identified for Auclusters (see Fig. 2): (i) a 2D close packed  planar  layer(analogous to the (111) face of fcc bulk gold) for  N   = 11–13 and 15; (ii) a condensed  flattened cage  structure for N   = 14 and 16 clusters; (iii) a pseudo-spherical  hollowcage  structure for  N   = 17–18. To explain the differencebetween the flattened and hollow cage structures, we cancompare the predicted structures of Au 16  and Au 17 , whichare shown in Figure 3. For Au 16  (flattened cage), thedimensions of the shortest two internal axes are 0.3 nmand 0.7 nm. This cage could accommodate a small atom(e.g. H, He, Ne, O or F). However, for Au 17  (hollowcage), the dimensions are both 0.6 nm and the cage couldaccommodate larger atoms, even an extra Au atom.The planar structures obtained for the global minima of Au N   ( N   = 11–13 and 15) clusters agree with the findingsof Fa et al. [63] concerning Au 11  and Au 12 . However, adifference is observed for Au 13  and Au 15 , which are pre-dicted to be 3D by Fa. The nearest low-lying isomer toour 2D Au 13  global minimum is predicted to be 3D, withan energy 0.5 eV higher than the GM. The lowest-energystructures of Au 14  shows flattened cage structures, as pre-viously reported [63,64]. The competitive isomer for the 3D GM of Au 14  (C 2 v) is also 3D (C 2 v), with a relativeenergy of only 9 meV. The lowest-energy structure of Au 15 is a 2D close packed layer, with C 2 v symmetry. The sec-ond most stable isomer is 3D (C 2 v), with relative energy0.19 eV. Having a flattened cage structure and C 2 v sym-metry, Au 16  is similar to the case of Au 14 , as previouslyreported [64]. In contrast, anionic Au 16  has been reportedto adopt a hollow cage structure [65]. The structural transition from flattened cage to hol-low cage configurations occurs at  N   = 17. The lowestenergy structure we have obtained for Au 17  is similar towhat has been reported for neutral [64] and anionic [65] Au 17 , showing a pseudo-spherical hollow cage structurewith C 2 v symmetry. The nearest competitive isomer tothe GM is also another hollow-cage structure, with a rela-tive energy of 0.08 eV and C 1  symmetry. The hollow cagestructure (D 4 d) observed for Au 18  is different from the  Page 4 of  12 Eur. Phys. J. B (2018) 91: 34 Au 11  Au 12  Au 13  Au 14 Au 15  Au 17 Au 16  Au 18 Fig. 1.  Putative global minimum structures for Au N   clusters, N   = 11–18. hollow cage structure reported by Bulusu and Zeng [64], though the latter agrees with the energetically compet-itive isomer that we have found, which is only 0.08 eVhigher in energy than the GM with Cs symmetry. 3.1.2 Pd clusters Figure 4 shows the putative global minima for Pd clusters:their energies, coordinates and point groups are listed inTable S1 (see the Supporting Information).Similar to small Pd clusters [37], the lowest energy structures obtained here for medium-sized Pd clusters areall 3D. As for pure Au clusters, the structural motifsadopted by Pd clusters are size-dependent; for medium-sized Pd clusters, a structural transition occurs at  N   = 15atoms from  bilayer  structures to  filled cage  structures.Global minima for the gas phase Pd 11 , Pd 13  and Pd 14 clusters are found to be distorted hexagonal bilayer (HBL)structures with C 2  symmetry whereas Pd 13  is found to beicosahedral fragment (C 2 ). These resemble the ground-state structures of the same sizes investigated previously[14,66]. The predicted Pd 11  structure is energetically pre-ferred over the next lowest-lying structure by only 2 meV.The competitive isomer for Pd 14  also has a distortedhexagonal bilayer (HBL) structure, but with lower sym-metry (C 1 ) and a relative energy of 0.09 eV. The globalminimum of Pd 13  reported here is found to be differentfrom the compact icosahedron structure predicted as theGM in references [67–69]. The GM for Pd 12  has a (C 1 ) buckled mono-planar(BMP) structure, which is similar to that observedrecently by Xing and co-workers [32]. The GM of Pd 15 is a buckled biplanar (BBP) structure. In the case of Pd N  clusters,  N   = 16–18 atom, the global minima are found tobe pseudo-spherical filled cage structures and agree withthe fcc-like growth pathway observed previously for 16–20atoms [14,32,70]. 3.1.3 Au–Pd clusters The global minima for all compositions of Au m Pd n  clus-ters, 11 ≤ m + n ≤ 18, are shown in Figures 5–8. Tables S2–S5 list the energies, coordinates and point groups (seethe Supporting Information). Fig. 2.  Evolution of structural motifs for Au clusters. Thestructural properties for sizes  N   = 4–10 are taken fromreference [37]. Fig. 3.  The shortest-axis lengths of the predicted hollow cagestructure of Au 17  and flattened cage structure of Au 16 . Pd 11  Pd 12  Pd 13  Pd 14 Pd 15 Pd 16  Pd 17  Pd 18 Fig. 4.  Putative global minimum structures for Pd N   clusters, N   = 11–18. As shown in Figure 5, all the predicted structures of  mono-gold-doped Pd clusters (Au 1 Pd n ,  n  =  N   − 1 = 10–17) are 3D. For  N   = 11, 12 and 18, replacement of a singlePd atom in Pd N   by an Au atom yields geometries whichare significantly distorted from the pure clusters. For sizes N   = 13, 15 and 16, the Au-doped Pd clusters are similarto their pure Pd species whereas for N   = 14 and 17 atoms,the Au-doped structures are quite different from the purePd clusters.Au 1 Pd n  n  = 10–13 show icosahedral (Ih) derivatives; asfor Au 1 Pd 10 , Au 1 Pd 11 , and Au 1 Pd 12  clusters, the moststable structure is an Ih fragment, while Au 1 Pd 13  is an Ihfragment which can be considered as an incomplete M 19
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