This journal is©The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2014 New J. Chem.

Cite this:DOI: 10.1039/c4nj00984c

Theoretical studies of structures and energiesof Pd, Au–Pd, and Au–Pd–Pt clusters

Xia Wu* and Yanjie Dong

Stacking fault (sf) and twin defects in Pd, Au–Pd, and Au–Pd–Pt clusters are theoretically studied, which

has affected the mechanical properties of face centered cubic (fcc) metal and alloy clusters. The stable

structures are located with Gupta potential by the density functional theory (DFT)-fitted and averaged

parameters. In Pd30–200 clusters, besides morphologies, such as Mackay icosahedra (Ih) and Marks deca-

hedra (Dh), fcc, sf-fcc, and twin-fcc motifs are identified. Furthermore, the growth pattern of twin-fcc is

based on the 50- and 79-atom motifs with high D3h symmetry, and the construction method for all

possible D3h twin-fcc is proposed starting from centered and uncentered truncated octahedra. Moreover,

a strong competition among fcc, sf-fcc, twin-fcc, Ih, and Dh is found in AumPdn (m + n = 50 and 61) and

55-atom Au–Pd–Pt clusters. The segregation phenomena of the Au, Pd, and Pt atoms in the Au–Pd–Pt

clusters are studied and further compared with Au–Pd, Au–Pt, and Pd–Pt clusters.

1. Introduction

Bi- and trimetallic (e.g., Au–Pd and Au–Pd–Pt) clusters haveraised considerable interest because their unique properties aredifferent from those of pure clusters (e.g., monometallic gold,palladium, and platinum) especially in the domain of catalysisand advanced functional nanomaterials.1–5 Nanoscale gold,palladium, and platinum clusters containing tens to thousandsof atoms have received great attention for their possible appli-cations in electronic, catalytic, magnetic, and optical fields.6–9

Their unique properties have been widely studied by experi-mental and theoretical approaches.10–13 For instance, Turneret al.10 reported that gold entities (B1.4 nm) derived from55-atom gold clusters and supported on inert materials areefficient and robust catalysts for the selective oxidation ofstyrene by dioxygen. Adsorption of an H2 molecule on PdN

clusters (N = 2–4, 7, 13, 19, and 55) was investigated usingdensity functional theory (DFT) with the hybrid PBE0 func-tional.11 Duchesne et al.12 presented the results of employingX-ray absorption spectroscopy to probe the local structure andelectronic properties of platinum nanoclusters reduced andstabilized using N,N-dimethylformamide.

Simulation and experimental studies revealed that stableordered structures of bimetallic Au–Pd clusters are Pdcore/Aushell.

14

For the studies of 40-atom Au–Pt clusters, the PtcoreAushell

configuration with two Au atoms capping the (100) facets wasmost energetically favored, and the layered (phase segregated)

configuration also had lower energy compared with theAucorePtshell and mixed configurations.15 In Pd–Pt clusters, amixed decahedral/close-packed (Dh-cp) structure was formed,and the segregation effects of Pd atoms to the surface were foundand corroborated by density functional theory (DFT) calculations.4

Fang et al.5 synthesized catalytic Au@Pd@Pt nanoparticles anddemonstrated that the high activity for electrochemical oxidationof formic acid was critically dependent upon the Pd-shell thicknessand the Pt-cluster coverage.

In face-centered cubic (fcc) metal and alloy clusters, stackingfault (sf) and twin formation have been reported to significantlyaffect their mechanical properties.16–18 In a Monte Carlo study,one or more stacking faults were found for the minimumenergy geometries of clusters with a tight binding model.19

For Au–Pd clusters with 52 atoms, fcc close packed structureswith a hexagonal close-packed (hcp) stacking fault werelocated.20 On the other hand, twins have been shown to beable to simultaneously increase the strength and ductility ofnanometals, which is attributed to the dislocation interactionwith and accumulation at twin boundaries.21 Using X-ray diffrac-tion and high resolution electron microscopy (HREM), for Auclusters, the structural transition from decahedron (Dh) to atwin-fcc may occur probably at no greater than B1.8 nm.22 Thefcc lattice is characterized by ABCABC. . . packing of {111} layersand the hcp lattice by ABAB. . . packing. The strain-free close-packed (scp) lattice by Cheng et al.23 contained fcc and hcp {111}layer sequences at each direction, which is also recognized astwin-fcc.

Optimization algorithms, such as genetic algorithm (GA),24

basin hopping (BH) method and its variants,25–27 fast annealingevolutionary algorithm (FAEA),28 random tunneling algorithm

School of Chemistry and Chemical Engineering, Anqing Normal University, Anqing,

246011, P. R. China. E-mail: xiawu@aqtc.edu.cn; Fax: +86-556-550-0090;

Tel: +86-556-550-0090

Received (in Montpellier, France)13th June 2014,Accepted 24th July 2014

DOI: 10.1039/c4nj00984c

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(RTA),29 dynamic lattice searching (DLS) methods,30 and heuristicalgorithm combined with the surface and interior operators(HA-SIO),31 were originally developed for the optimization ofpure elemental (monoatomic) clusters, e.g., Lennard-Jones (LJ)clusters,28–30 silver,32 and aluminum clusters.33 However, in binaryor ternary clusters, the ‘‘homotopic’’ problem, i.e., the samegeometrical configurations but different atomic arrangement bytwo or three type atoms,34 makes the optimization much morecomplicated. Therefore, the modifications of GA,34 BH algo-rithm,35,36 adaptive immune optimization algorithm (AIOA),37–39

and an evolutionary algorithm (EA)40 were developed for optimiz-ing binary LJ clusters,41 bimetallic Pt–Pd clusters,42 ternary LJclusters,43 Ar–Kr–Xe clusters,44 and Cu–Ag–Au clusters.45

In the present work, the Gupta potential is adopted for describ-ing the interactions between Au, Pd, and Pt clusters in Au–Pd andAu–Pd–Pt clusters. The stable structures of Pd30–200 clusters, AumPdn

(m + n = 50 and 61) clusters, and Au18(PdPt)32 clusters are investi-gated with DLS, DLS with constructed cores (DLSc), and AIOAmethods. The structural characteristics and the atomic distributionin Au–Pd and Au–Pd–Pt clusters are studied. Furthermore, theconfiguration of twin-fcc with D3h symmetry is discussed.

2. Methodology2.1. Gupta potential for Pd clusters, mixed Au–Pd, andAu–Pd–Pt clusters

Gupta potential is applied to consider the interatomic inter-actions of pure Pd clusters, bimetallic Au–Pd, and trimetallicAu–Pd–Pt clusters, which is based on the second momentapproximation to tight binding theory. This potential has anattractive many-body Vm(i) term formulated in the secondmoment approximation of the electron density of states withinthe tight binding scheme, and a Born–Mayer V r(i) term, whichdescribes the repulsive pair interactions. Gupta potential VN

with N atoms can be depicted in the following form:

VN ¼1

2

XNi¼1

VrðiÞ � VmðiÞf g (1)

VrðiÞ ¼XN

j¼1ð jaiÞAij exp �pij

rij

rð0Þij

� 1

! !(2)

VmðiÞ ¼XN

j¼1ð jaiÞxij

2 exp �2qijrij

rð0Þij

� 1

! !1=2

(3)

where rij is the distance between atom i and j. Parameters of Aij,r(0)

ij , pij, qij, and xij are usually fitted to the experimental values ofcohesive energy, lattice constant, and elastic constants for thefcc crystal structure at 0 K. Here, the homo-(Pd clusters) andheteronuclear (Au–Pd clusters) parameters fitted to DFT calcu-lations at solid phases were obtained from ref. 46 as listed inTable 1. The parameters for the trimetallic Au–Pd–Pt clustersare the average parameters introduced by Logsdail et al.,20

which are also listed in Table 1.

2.2. DLS and DLSc method for Pd clusters

The DLS method was developed by absorbing the basic ideasof static modeling and stochastic optimization algorithms.It starts from a randomly generated and locally minimizedstructure of a cluster. Next, a ‘‘lattice construction’’ operation isperformed to construct vacant dynamic lattice (DL) sites around thestarting structure. Then, a ‘‘lattice searching’’ operation attempts tofind a solution with lower energy by iteratively moving the atomswith higher energy to the vacant DL sites with lower energies. Therepetition of lattice construction and lattice searching operations isexecuted to find structures with lower energies. The DLS methodhas been successfully applied for the optimization of LJ clusters upto 500 atoms,30 MorseN (N r 240) clusters,47 (C60)N (N r 150)clusters,48 Ag clusters,32 and Al clusters.33 Therefore, it is employedto find the global minima of Pd30–100 clusters.

Furthermore, a variant method of DLS (i.e., DLSc) is appliedto find the stable structures of Pd100–200 clusters. DLSc is achievedthrough constructing the inner core of the starting structure toreduce the searching space. Note that the starting structure is theonly difference between DLS and DLSc. DLSc has been success-fully used to optimize the structures of aluminum clusters upto 800 atoms.49 For Pd clusters, there are mainly decahedron,icosahedron, and fcc motifs. The inner cores with the sametypes of configuration are thus adopted.

2.3. AIOA for Au–Pd and Au–Pd–Pt clusters

The putative global minimum structures of bimetallic Au–Pdand trimetallic Au–Pd–Pt clusters are optimized using heuristicAIOA,37–39 which is based on the evolutionary ideas of clonalselection principles. It takes the basic frame of a genetic algo-rithm, and its basic steps include an immune clone, a mutationoperation, and an updating operation. In the mutation operationof a generation (a repetition), 50% of the individuals are selectedwith energy-based mutation, and the other 50% are performedwith an atomic exchange operation, i.e., two types of atoms arerandomly selected and their locations are exchanged. Energy-based mutation is mainly utilized for geometrical isomers, andthe strategy of atomic exchange is for the homotopic problem inbinary clusters. In the updating operation, individuals with lesssimilarity and lower energy are kept in the updated gene library,which is performed by cluster similarity checking with a con-nectivity table (CT).38 Repetition of the selection, mutation, and

Table 1 DFT-fitted Gupta potential parameters for Pd clusters and bimetallicAu–Pd clusters and average parameters for Au–Pd–Pt clusters20,46

Compositions Aij (eV) xij (eV) pij qij r(0)ij (Å)

DFT-fit Au–Au 0.2019 1.8097 10.2437 4.0445 2.8840Pd–Pd 0.1653 1.6805 10.8535 3.7516 2.7485Au–Pd 0.1843 1.7867 10.5420 3.8826 2.8160

Average Au–Au 0.2016 1.79 10.229 4.036 2.8840Pd–Pd 0.1746 1.718 10.867 3.742 2.7485Pt–Pt 0.2975 2.695 10.612 4.004 2.7747Au–Pd 0.19 1.75 10.54 3.89 2.816Au–Pt 0.250 2.20 10.42 4.02 2.830Pd–Pt 0.23 2.2 10.74 3.87 2.76

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updating operations is performed to find the stable structure ofAu–Pd and Au–Pd–Pt clusters.

3. Results and discussion3.1. Structures and energies of Pd clusters up to 200 atoms

Fig. 1 shows the optimized structures of palladium clusters fromPd30 to Pd100. The structural distribution of Pd30–100 clusters canbe categorized into 2 amorphous structures, 4 Mackay icosahedral(Ih) structures, 48 Dh structures, 5 fcc structures, 4 sf-fcc struc-tures, and 8 twin-fcc structures.

From Fig. 1, amorphous structures are obtained at sizes ofN = 30 and 32. It can be seen that Pd30 and Pd32 have C3v and D2d

point group (PG) symmetry. Structures with Ih motifs are obtainedat N = 44 and 53–55. Obviously, the motif of Pd44 is a partial Ih Pd55

with an Ih PG. The structures of Pd53 and Pd54 are constructed byremoving one and two atoms from the vertexes of the outer-shell ofPd55. Including the truncated octahedral (TO) motif, 5 structureswith fcc motif are found at 38–40, 87, and 88. The structure of Pd38

is a complete TO, and one and two extra atoms of Pd39 and Pd40

are added on one and two (100) faces of Pd38, respectively. Thegrowth of Pd87 and Pd88 clusters is based on a complete TO with 79atoms. On the other hand, 4 sf-fcc structures are found at 61, 62,85, and 94. Among these structures, one stacking fault layer islocated on the surface. From the figure, twin-fcc structures arefound at 50–52, 79, 80, 91, 93, and 98. It should be noted that Pd50

and Pd79 have a D3h PG, which is considered as one of the highestsymmetries for a twin-fcc motif. It was found that the growth ofPd51 and Pd52 is based on Pd50, and one extra atom of Pd80 isadded on Pd79. Pd91, based on Pd79, has a C2v PG. Therefore, Pd50

and Pd79 both with D3h symmetry may be the complete configu-ration in the growth sequence of twin-fcc.

Moreover, in the size range of Pd30–100, Dh is the dominantmotif. In the small size range, clusters with (2,1,2) m-Dhmotifs50 are found at 31, 33–37, and 41–43. With the increaseof sizes, the extra atoms are gradually added to form a complete(2,2,2) m-Dh at Pd75. In the size range of 76–92, most structuresare based on Pd75. After Pd95, i.e., Pd95–97, Pd99, and Pd100, thestable motif changes to be (2,3,2) m-Dh.

Furthermore, the stable structures of Pd110–200 clusters withseveral tens of atoms are optimized due to the difficulties for

Fig. 1 Structures of putative global potential energy minima for Pd30–100 clusters.

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the optimization, by DLSc method with different inner cores asplotted in Fig. 2. From Fig. 2, it can be seen that Dh is still adominant motif for larger sized Pd clusters. On the other hand,Pd110, Pd120, and Pd200 clusters have twin-fcc motifs. Amongthese twin-fcc structures, the Pd200 cluster has C3v PG.

The finite difference of energy DE and the second finitedifference of energy D2E of the palladium clusters from Pd30 toPd100 are plotted in Fig. 3a and b, respectively. The DE and D2Ehave definitions as follows:

DE(N) = E(N) � EJ(N) (4)

D2E(N) = E(N+1) + E(N+1) � 2E(N) (5)

where EJ(N) = a + bN1/3 + cN2/3 + dN is a four-parameter fit of theenergy of global minimum. The value of DE shows the variationof the cluster stability with cluster size, and D2E measures thestability of an N-atom cluster structure with respect to itsneighboring cluster size. The negative peaks (or valleys) inFig. 3a and the positive peaks in Fig. 3b indicate particularlystable structures compared to their neighbors.

In Fig. 3a, eight apparent valleys at Pd38, Pd55, Pd64, Pd71,Pd75, Pd79, Pd91, and Pd95 can be clearly found, which corre-spond to the complete structural motifs or the magic numberclusters. In Fig. 3b, positive peaks corresponding to the nega-tive peak in Fig. 3a can be clearly found. However, there aresome other peaks at Pd48 (C2v PG) and Pd50 (D3h PG), which aremore symmetrical than their neighboring clusters.

This result is consistent with the results of the experimentaland theoretical simulation. For instance, using high resolutionelectron microscopy, fcc close-packed, icosahedral, decahedral,single-twin fcc, and amorphous of Pd nanoparticles were foundin the range of 1–5 nm.22 Johnston et al.42 located the globalminimal structures of Pd13–55 clusters, which is calculated basedon Gupta potential with parameters fitted to experimental values.By the comparison of Gupta potential with experimental-fittedparameters and DFT-fitted parameters in this work, only Pdclusters at N = 31, 45, 47, and 52 are different. For Gupta potentialwith experimental-fitted parameters, Pd31 and Pd45 have sf-fccmotifs, but for the case of DFT-fitted parameters they are deca-hedra. Pd47 clusters with experimental-fitted and DFT-fittedparameters are Dh and Ih, respectively. Corresponding motifsof Pd52 are sf-fcc and Dh with anti-layer, respectively.

3.2. Twin formation with a D3h PG in fcc metal clusters

To our knowledge, a D3h PG may be of high symmetry fora twin-fcc configuration. It is also recognized as one kind ofstacking fault in fcc metals. As discussed above, the motifs ofPd50 and Pd79 are twin-fcc with D3h PG. The 50-atom D3h motifsas in Pd50 are also found in other fcc metal clusters, e.g., Guptapotential for silver clusters,32 platinum clusters,42 Sutton–Chen(SC) potential for Ag, Ni and Au.51 In addition, a D3h PG is alsofound in 50-, 79-, and 201-atom Morse clusters.23

To better understand the structural characteristics of twin-fcc with a D3h PG, a simple construction method is proposedstarting from a complete truncated octahedron (TO). At first,according to Wulff’s construction, TO configurations can beconstructed from regular octahedra.49 A parameter n is used todefine a regular octahedron. For the octahedral structures, theatom number N = (2 � (n + 1)3 + (n + 1)/3) is 6, 19, . . ., 14 644when the number of shells (n) is 1, 2, . . ., 27. By truncatingm subshells atoms from each vertex of a regular n-shell octa-hedron, a complete TO structure is formed with the remainingatoms, and the clusters with the same m values belong to a‘‘family.’’ It is emphasized that octahedral and the corre-sponding TO structures are classified into centered, i.e., thereis an atom at the center, regular octahedron and uncenteredregular octahedron. Therefore, two kinds of twin-fcc with D3h

PG can be obtained from complete centered and uncentered TO.

Fig. 2 Structures of putative global potential energy minima for theselected clusters from Pd110 to Pd200.

Fig. 3 (a) The finite difference (DE) of the energy of palladium clusters fromPd30 to Pd100 for EJ(N) = 0.1638 + 1.3590N1/3 + 0.9229N2/3 � 3.8461N,(b) the second finite difference (D2E) of the energy of Pd30–100 clusters.

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Fig. 4 and 5 illustrate the structural transformation fromcomplete uncentered and centered TO to D3h twin-fcc, respec-tively. A 38-atom uncentered TO, as shown in Fig. 4a is consid-ered as an example to generate an uncentered twin-fcc. Fig. 4bshows the structure after a 901 rotation, and Fig. 4c shows thatstructure after cutting the layer 4 in Fig. 4b. Then, taking the layer3 in Fig. 4c as a symmetric plane, symmetric layers of layers 1 and2 are constructed, and a 50-atom uncentered twin-fcc with a D3h

PG is formed as in Fig. 4d. Fig. 5 shows the formation of centeredtwin fcc with a D3h PG. Fig. 5a plots the 79-atom centered TO, andFig. 5b shows the structure after a 901 rotation. After rotating 1801

for layers 4 and 5 around the axis drawn in Fig. 5b, a 79-atomcentered twin-fcc with D3h is finally constructed as illustrated inFig. 5c. On the other hand, in our previous work,49 it wasconcluded that if n was odd, the complete TO was uncentered;otherwise, the center was occupied by an atom, but it hadnothing to do with the value of m. All possible complete TOswithin 10 000 atoms were thus constructed by truncating dif-ferent m subshells atoms. Therefore, according to the aboveconstruction method, all twin-fcc structures with a D3h PGincluding centered and uncentered could be located from thesecomplete TOs.

3.3. Geometrical competition in Au–Pd clusters

The putative stable structures of AumPdn (m + n = 50) andAumPdn (m + n = 61) clusters are obtained, and Fig. 6a and b plotthe variation of the potential energies and the structural motifswith respect to the cluster size, respectively. Different symbolsare used to represent the motifs of the clusters in the Figure. InAumPdn (m + n = 50) clusters shown in Fig. 6a, there exist 11twin-fcc structures, 30 decahedral structures with close packinganti-layers (also recognized as Dh-cp by Johnston et al.51), and8 Ih structures. Furthermore, the Au50 cluster is also optimized,and twin-fcc with a D3h PG motif is found as in the Pd50 cluster.

From Fig. 6b it can be found that the structures of AumPdn

(m + n = 61) clusters can be categorized into four classes, i.e.,

Fig. 4 The formation of a 50-atom uncentered twin-fcc with D3h PGfrom a 38-atom complete TO.

Fig. 5 The formation of a 79-atom centered twin-fcc with D3h PG from a79-atom complete TO.

Fig. 6 Variation of potential energies and the structural distribution ofAumPdn (m + n = 50) clusters (a) and AumPdn (m + n = 61) clusters (b) withm. Different symbols are used for the different motifs. Au and Pd atoms arerepresented by gold and blue spheres, respectively.

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2 fcc for m = 3 and 4, 9 sf-fcc at m = 1, 2, 5, 8, 10, 11, 36, 54, and60, 39 Mackay icosahedra for m = 6, 9, 12–14, 16, 17, 19–21,23–35, and 37–52, and 10 decahedra at m = 7, 15, 18, 22, 53, and55–59. On the other hand, the motif of Au61 and Pd61 clusters isidentified as sf-fcc. The results indicate the structural variationfrom a sf-fcc of Pd61 to another sf-fcc of Au61. At Au1Pd60 andAu2Pd59, the extra Au atoms are located on the surface of thePd61 cluster. However, at Au3Pd58 and Au4Pd57, regular fccmotifs are formed. With the increase of m values from 6 to 52,the dominant motif is Ih, and the competitory configurationsinclude Dh and sf-fcc. The Dh motif is recognized as an anti-Marks decahedron (M-Dh), which has been reported for a localminimal LJ75 cluster.6 For the icosahedral configuration, e.g.,Au9Pd52, the growth is based on a 55-atom complete Ih.

Furthermore, the character of the stable structures in Au–Pdclusters is investigated by the order parameter (RA), which isadopted to analyze the distribution of different types of atomsin the bimetallic clusters.45 RA can be calculated by the averagedistance of a type of atoms from the center of a cluster, i.e.,

RA ¼1

nA

XnAi¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixi2 þ yi2 þ zi2

p(6)

where nA is the atomic number of type A in the bimetallic A–Bcluster, and xi, yi, and zi represent the coordinates of the atoms.Clearly, a small or large value of the order parameter indicates thatthe A atoms are at the center or surface of the cluster with asegregated pattern, and a medium value means a well-mixed cluster.

Fig. 7a and b show the variation of the order parameteralong with m for the studied AumPdn (m + n = 50) and AumPdn

(m + n = 61) clusters, respectively. From Fig. 7a, it can be found

that for each bimetallic cluster, RAu is larger than RPd. Thisindicates that Au atoms tend to occupy the surface sites. Thesegregation of Au atoms to the surface can be explained in termsof the lower surface energy and cohesive energy of Au, which isconsistent with the conclusion of studying 98-atom Au–Pdclusters.52 Simulation and experimental results also revealed thatstable ordered structures of Au–Pd clusters are Pdcore/Aushell.

12

Moreover, in Pd-rich Au–Pd clusters (i.e., m = 1–30), mediumvalues of RPd tell us that Au and Pd atoms are partially mixed inthe size range. By further investigation, it was found that a mixedAu–Pd surface is generally formed. The conclusion is confirmedby a study of Au–Pd clusters up to 50 atoms with experimental-fitted and DFT-fitted parameters.46 In addition, the same variationof order parameter can also be seen in Fig. 7b.

3.4. Structural characteristic in Au–Pd–Pt clusters

The putative stable structures of 50-atom trimetallic Au18PdnPt32�n

(n = 1–31) clusters are investigated, and the variation of thepotential energies and the typical motifs are shown in Fig. 8a.These 31 trimetallic motifs can be categorized into four classes,

Fig. 7 Variation of the order parameter RAu and RPd with m in AumPdn

(m + n = 50) clusters (a) and (m + n = 61) clusters (b).

Fig. 8 Variation of potential energies and the structural distribution (a), andvariation of the order parameter (b) of Au18PdnPt32�n (n = 1–31) clusters with n.Different symbols are used for the different motifs. Au, Pd, and Pt atoms arerepresented by gold, blue, and purple spheres, respectively.

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i.e., 14 decahedral structures with close packing anti-layers forn = 1–3, 7, 9–11, 14–16, 19, and 29–31, 6 twin-fcc structures forn = 4–6, 8, 12 and 13, 9 Ih structures at n = 17, and 21–28, and2 twin-fcc with anti-layers.

Fig. 8b shows the variation of the order parameterRA (A = Au, Pd, and Pt) along with the nPd for the Au18PdnPt32�n

(n = 1–31) clusters. In Fig. 8b, RAu and RPd are apparently largerthan RPt for all compositions. Small RPt values represent that Ptatoms tend to be in the inner cores of the clusters. For mostcompositions, RAu is bigger than RPd. It appears that Au atomsprefer to reside on the surface as in Au–Pd clusters. Further-more, medium values of RPd mean that Pd atoms are mostlylocated in the middle-shell, but also tend to dissolve into theAu and Pt regions.

Bimetallic Au–Pd, Au–Pt, and Pd–Pt clusters are con-sidered as a reference for comparison with trimetallic Au–Pd–Ptclusters.13,14,46,52 By analysis of mixing and segregation effects,surface segregation effects of Pd atoms in medium-sized Pd–Ptclusters are supported at the density functional theory (DFT)level.53 Analysis of the Leary tetrahedron (LT) structures withthe lowest excess energies reveal that they possess segregatedPdcoreAushell chemical ordering.52 A study of global optimiza-tion for Gupta potential and DFT reoptimization has shownthat the PtcoreAushell configuration is confirmed to be energeti-cally favored, compared to the AucorePtshell and mixed con-figurations.13 It is clear that the segregation phenomena of Au,Pd, and Pt in bimetallic clusters can be observed in Au–Pd–Ptclusters, mainly due to their different surface energies. To mini-mize the cluster surface energy, Au and Pd atoms prefer to resideon the surface and on the subsurface, respectively, because of thelower surface energy of Au (96.8 meV Å�2), compared with Pd(125–131 meV Å�2) and Pt (155–159 meV Å�2).14,46

4. Conclusions

The structural characteristics of Pd30–200 clusters and theatomic distribution in AumPdn (m + n = 50 and 61) clustersand Au18(PdPt)32 clusters are studied. Gupta potential is adoptedfor describing the interactions between Au, Pd, and Pt atoms inAu–Pd and Au–Pd–Pt clusters. The stable structures of Pd30–200

clusters are located by dynamic lattice searching and dynamiclattice searching with constructed cores methods, and they arecategorized into 2 amorphous structures, 4 Mackay icosahedralstructures, 48 decahedral structures, 5 face centered cubicstructures, 4 stacking fault face centered cubic structures, and8 twin face centered cubic structures. As one kind of stackingfaults in face centered cubic metals, twin face centered cubicstructures are found to be based on Pd50 and Pd79, which haveD3h symmetry. To better understand the structural characteris-tics of twin face centered cubic with D3h symmetry, a simpleconstruction method is proposed starting from a completetruncated octahedron. Furthermore, the structural competitionamong stacking fault face centered cubic, twin face centeredcubic, decahedron, and icosahedron was discussed in AumPdn

(m + n = 50 and 61) clusters, and Au18(PdPt)32 clusters, which are

investigated with an adaptive immune optimization algorithm.Moreover, the structural characteristics and the atomic distribu-tion in Au–Pd and Au–Pd–Pt clusters are studied.

Acknowledgements

This study is supported by National Natural Science Foundationof China (NNSFC) (Grant No. 21203002 and 21171008) and AnhuiProvincial Natural Science Foundation (No. 1308085QB29).

Notes and references

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