First principles study of small palladium cluster growth and isomerization

First Principles Study of Small PalladiumCluster Growth and Isomerization

CHEN LUO,1 CHENGGANG ZHOU,1,2 JINPING WU,1

T. J. DHILIP KUMAR,2 NADUVALATH BALAKRISHNAN,2

ROBERT C. FORREY,3 HANSONG CHENG4

1Institute of Theoretical Chemistry and Computational Materials Science, China University ofGeosciences, Wuhan, People’s Republic of China 4300742Department of Chemistry, University of Nevada at Las Vegas, Las Vegas, NE 891543Department of Physics, Penn State University, Berks-Lehigh Valley College,Reading, PA 19610-60094Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, PA 18195-1501

Received 12 December 2006; accepted 10 January 2007Published online 20 February 2007 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21315

ABSTRACT: Structures and physical properties of small palladium clusters Pdn up ton � 15 and several selected larger clusters were studied using density functional theoryunder the generalized gradient approximation. It was found that small Pdn clusters begin togrow 3-dimensionally at n � 4 and evolve into symmetric geometric configurations, such asicosahedral and fcc-like, near n � 15. Several isomers with nearly degenerate averagebinding energies were found to coexist and the physical properties of these clusters werecalculated. For several selected isomers, relatively moderate energy barriers for structuralinterchange for a given cluster size were found, implying that isomerization could readilyoccur under ambient conditions. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 107:1632–1641, 2007

Key words: Pd clusters; cluster evolution; cluster isomerization

Introduction

Transition metal (TM) clusters have been a subjectof active theoretical and experimental studies in the

recent decades [1–9]. At the most fundamentallevel, evolution of structure and physicochemicalproperties from small TM clusters to bulk metalshas been one of the central issues in condensedmatter physics research. Technologically, TM clus-ters serve as catalysts in many industrially impor-tant chemical processes. They are also used exten-sively in semiconductor and sensing devices,magnetic storage materials, and constitute impor-tant components of nanomaterials for various ap-plications [10–18]. TM clusters exhibit a wide vari-ety of geometries, isomers, and sizes. Many studies

Correspondence to: H. Cheng; e-mail: [email protected] grant sponsor: Research Foundation for Outstand-

ing Young Teachers, China.Contract grant sponsor: University of Geosciences, Wuhan.Contract grant number: CUGQNL0519.Contract grant sponsor: US Department of Energy.Contract grant number: DE-FG36–05GO85028.

International Journal of Quantum Chemistry, Vol 107, 1632–1641 (2007)© 2007 Wiley Periodicals, Inc.

have shown that the physicochemical properties ofthese clusters are critically dependent on thesestructural arrangements. Understanding of thestructural and property evolution of TM clusters isessential for design and development of novel ma-terials and processes for a broad area of applica-tions. Of particular interest are the palladium clus-ters, since they serve as excellent catalysts in manyheterogeneous catalytic systems. These clusters ex-hibit some unique physicochemical properties[12–15, 19], which have been the subject of in-tense theoretical and experimental studies [3, 4,13, 20 –29]. They show an unique magnetic prop-erty and promise for potential application onsupported metal catalysts. The properties of theseclusters are determined by their electronic struc-tures and, ultimately, by the cluster structuralarrangements.

Like other TM clusters, palladium clusters ex-hibit numerous isomers for a given size and thestructures of these clusters have been extensivelyreported in literature. Several theoretical calcula-tions have shown that many of these isomers pos-sess similar binding energies, suggesting that theymay coexist under certain experimental conditions.One interesting question concerning the relativestability of these isomers thus arises: under whatconditions would one isomer evolve into another?One of the objectives of the present study is toaddress this issue. Understanding of the isomerstructural exchange is important to gain insight intothe catalytic reactivity of palladium clusters. Aspalladium clusters grow, the transition of palla-dium clusters from atomic to bulk metallic behaviormay be monitored by photoemission spectroscopy[30]. In this study, we study Pdn cluster structuresand growth using density functional theory (DFT).Our main focus is to explore the cluster growthpathways, understand the transition from smallclusters to the bulk structure, quantify the ener-getics of isomeric structural exchange and, ulti-mately, gain insight into the chemical propertiesof palladium catalysts. We discovered severalPdn cluster structures that have not been previ-ously reported. These new cluster structures areenergetically favorable with higher or compara-ble binding energies than the previously reportedcluster structures of the same size. These findingsallow us to gain detailed insight into the clusterstructural evolution and the energetics for iso-meric structural exchange.

Computational Method

The present study utilized DFT under the gen-eralized gradient approximation (GGA) withPedrew–Wang’s exchange-correlation functional(PW91) as implemented in DMol package [31–35].Spin-polarization scheme was employed to dealwith the electronically open-shell system. The spinstates and the magnetic moments were then deter-mined based on the calculated energies of spin-upand spin-down orbitals. To describe the valenceelectrons, we employed a double numerical basisset augmented with polarization functions, and thecore electrons are described with an effective corepotential (ECP), which also accounts for the relativ-istic effect expected to be significant for heavy ele-ments in the periodic table. Electronic structurecalculations for systems containing TMs are knownto be notoriously difficult to converge. This is par-ticularly the case for TM clusters due to the largenumber of d-electrons involved. To enhance theSCF convergence, we initially used a small value ofthermal smearing (�0.001Ha) to obtain structuresclose to the equilibria and then sequentially re-moved the value until it reaches to zero. This guar-antees that the number of unpaired electrons is aninteger. All cluster geometries were fully optimizedwithout imposing symmetry constraints, except forthe icosahedral structures for which point groupsymmetries were utilized to simplify the calcula-tions. For a given cluster size, a variety of initialstructures were selected followed by energy mini-mization, yielding numerous stable isomers. Thecluster structural search scheme used in our calcu-lations is to add a new atom on the precedingcluster with the highest binding energy at variouspossible binding sites and then to identify the en-ergetically most stable site via geometry optimiza-tion. To avoid missing structures with more favor-able binding energies, additional caution was takenby placing the new addition to the preceding clus-ters with energies close to the highest binding en-ergy cluster followed by energy minimization.

The average cluster binding energy was calcu-lated using the following expression:

�En � �n E(Pd)�E(Pdn)]/n, �n � 2, 3, . . . �, (1)

where E(Pd) represents the atomic energy of Pdand E(Pdn) is the energy of Pdn cluster. Subse-quently, we evaluated the cluster ionization poten-

SMALL PALLADIUM CLUSTER GROWTH AND ISOMERIZATION

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tial (IP), electron affinity (EA) and magnetic mo-ment (�) defined by

IPn � E�Pdn�� E�Pdn�, (2a)

EAn � E�Pdn�� E�Pdn�, (2b)

� � �mu � md)/2, (2c)

where mu and md are the numbers of spin-up andspin-down electrons with mu � md, respectively.

For comparison purpose, we also performed cal-culations to evaluate the cohesive energy, which isidentical to the average bulk binding energy in thepresent case. This was done by imposing a periodicboundary condition in the DFT calculation usingthe primitive unit cell of palladium. The Brilliounzone integration was performed with 8 8 8k-points using the Monkhorst and Pack scheme.Both atomic coordinates and cell parameters werefully relaxed upon structural optimization, yieldingthe cell parameters within 0.01 Å of the experi-mental crystal structure.

Results and Discussions

For a given size of cluster, full structure optimi-zation usually yields numerous isomers. Figure 1displays the main isomeric structures obtained forPdn (n � 2–15). The corresponding average bindingenergies are shown underneath. A cluster with thehighest binding energy at a given size is placed atthe front. The highest binding energy structures forn � 2–10 have been reported previously [2, 3, 10,11], and our optimized geometries are in goodagreement with other DFT calculations. To the bestof our knowledge, the highest average binding en-ergy structures for n � 11–15 are reported for thefirst time in this work.

The calculated average binding energy of palla-dium dimer is 0.692 eV, while the experimentalvalue varies from 0.37 to 0.57 eV [36, 37], respec-tively, slightly lower than the calculated one. Thecalculated bond length of 2.555 Å is also slightlylonger than the experimental value of 2.48 Å [38].Other reported theoretical calculations gave thedimer bond distance of 2.52 Å [3] in close agree-ment with what is reported here. Pd3 is an isoscelestriangle, slightly distorted from the equilateral tri-angle due to the Jahn–Teller effect. For the samereason, Pd4 with the highest binding energy is a

slightly distorted tetrahedron. From Pd4 to Pd7, theclusters grow with bipyramid configurations withincreasing average bond distances and binding en-ergies. A previous study using the DFT-based tight-binding model [39] predicted that the planar Pd7-cis the most stable structure. However, our calcula-tion suggests that the planar geometry is consider-ably less stable than the bipyramid configuration.From Pd8, deviation from the symmetric bipyramidstructure occurs and the cluster growth appears tobe quite “irregular”. Nevertheless, the difference ofthe calculated average binding energies among theisomers is rather small, which suggests that theseisomers may coexist thermodynamically. For Pd9,Karabacak et al. [3, 4] predicted that Pd9-e structureis the most stable configuration using moleculardynamics and slow-quenching technique with anembedded-atom potential. Our DFT calculationpredicts that this is the least stable structure and theconfiguration with distorted double octahedronsfused with one facet is the most stable one. It isworth noting that Pd9-d can be regarded as a halficosahedron configuration with an average bindingenergy 0.037 eV lower than that of Pd9-a.

From Pd10, certain characteristics of icosahedralgeometry appear in most of our calculations. ForPd10, the highest average binding energy occurs fora structure that is similar to a half icosahedral con-figuration. Calculations by Futschek et al. [2] usingPAW coupled with plane-wave basis set as imple-mented in the VASP package suggests that Pd10-d isthe most stable cluster. Our study shows that theaverage binding energy of Pd10-d is only slightlylower than that of Pd10-a.

For Pd11, the highest binding energy structurePd11-a is constructed by three octahedrons, each ofwhich fuses one facet with the other two. The av-erage bond length of Pd11-a is less than that of theother isomers, indicating that this structure is themost close-packed. Pd11-e was previously predictedas the most stable structure [2]. Our calculationssuggest that the average binding energies of the n �11 isomers are close to each other but the close-packed structure Pd11-a is energetically most favor-able.

For the Pd12, the highest binding energy struc-ture Pd12-a is composed of two octahedrons and apentagonal bipyramid and has the shortest averagebond length, indicating an essentially close-packedstructure. Pd12-b is an icosahedral structure with avacant center. This structure was predicted to beunstable and will relax to a centered icosahedronwith a defect in the outer shell [2]. We found that it

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1634 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 107, NO. 7

FIGURE 1. The main optimized isomeric structures of Pdn (n � 2–15). The average binding energy per Pd atom isshown underneath.



could still stabilize without significant deformationbut its average binding energy is considerablylower than that of Pd12-a by 0.108 eV. Separately,Pd12-c [3] and Pd12-e [2] were also predicted as themost stable structure. Their average binding ener-gies are slightly lower than that of Pd12-a. We notethat Pd12-c can be viewed as a distorted icosahe-dron.

For Pd13, the highest binding energy structurePd13-a is built upon Pd10-a by adding three cappedatoms on the triangular facets. Pd13-b is a typicalcentered icosahedral structure with an averagebinding energy of 2.337 eV, nearly degeneratedwith that of Pd13-a. The icosahedral geometry waspredicted to be the most stable structure in previ-ous calculations [3, 10, 11, 14, 40]. Pd13-c is a cen-tered cube with four capped atoms on its side fac-ets. Pd13-d was also predicted to be the most stablestructure [2]. Our calculation shows that the differ-ence of average binding energies between Pd13-aand Pd13-d is only 0.003 eV.

For Pd14, the highest average binding energystructure Pd14-a takes a pentagon prismatic config-uration with three capped atoms. The close-packedPd14-b is formed by adding three atoms on threetriangular facets of Pd11-d. Pd14-c is a typical icosa-hedron by adding one atom on a trigonal facet.Pd14-d is close to a fcc-like cluster with slight distor-tion. The geometric configuration of Pd14-e is some-what similar to the icosahedral structure of Pd19(see Fig. 1) with the main difference being that therings are formed by five atoms in Pd19-ic instead offour atoms here. The difference in the calculatedaverage binding energies among the isomers is verysmall. We note in particular that all three geometricconfigurations, close-packed, fcc-like, and icosahe-dron, coexist for n � 14.

For n � 15, the highest average binding energycluster, Pd15-a, is a typical fcc-like structure with theatoms in the faces slightly out of plane. The averagebond length of Pd15-a is also much shorter than theother isomers, indicating a close-packed structure.The fact that Pd15-a is energetically most stable sug-gests that fcc-like growth may start dominating thecluster evolution at a small size. Adding one atomon two different square side facets of Pd14-b respec-tively yields the structures, Pd15-b and Pd15-c withnearly identical average binding energies. Thestructure of Pd15-d is somewhat similar to that ofPd13-b except the five-membered rings in Pd13-b arereplaced by six-membered rings in Pd15-d.

Figure 2 displays the calculated bondlength dis-tributions of the highest average binding energy

clusters. The majority of the bonds in the clustersfall into the range of 2.6–2.8 Å. The sharp increasefrom Pd3 to Pd4 results from the first structuraltransition from two dimensions to three dimen-sions. The bondlength increases slowly as the clus-ter size increases until n � 8 at which cluster growsin a rather “irregular” fashion. However, the gen-eral trend is that clusters preferably adopt geomet-ric configurations that are most close packed. As aconsequence, the bondlengths for n � 8 remainmore or less constant.

The energetic and topological similarity of theclose-packed, icosahedral, and fcc-like structuresobserved in small palladium clusters suggests thatstructural transitions from the close-packed config-urations to icosahedral and, ultimately, to the bulk-like fcc configurations could occur at a relativelysmall cluster size. To gain insight into the evolution,we performed calculations on four selected icosa-hedral structures for n � 13, 19, 25 and 55 and fourselected fcc-like structures for n � 14, 23, 32 and 38.The optimized structures as well as the calculatedaverage binding energies are shown in Figure 3.The choice of fcc-like structures is somewhat arbi-trary but it helps us to gain at least a qualitativeunderstanding of the cluster growth pathway.

Figure 4 displays the average binding energy ofthe highest average binding energy clusters to-gether with the selected icosahedral and fcc-likeclusters verses the inverse of cluster size. For1/n30, the average binding energy approaches tothe bulk cohesive energy. Several interesting fea-tures can be readily observed:

1. The average binding energy of small palla-dium clusters increases with the cluster size;

2. At around n � 13, the average binding ener-gies of both icosahedral and fcc-like structuresare comparable to that of the close packed“irregular” clusters, indicating that the transi-tion from the “irregular” structure to the ico-sahedral and fcc-like structures could occur ata small cluster size;

3. In the approximate range of 20 � n � 50, theicosahedral and fcc-like structures coexist butafter that the icosahedral structures appear todominate the cluster growth. Unfortunately,due to the computational intensity, we areunable to optimize the energy for large clus-ters using the DFT method. Therefore, it isdifficult to find out at what size of a cluster

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FIGURE 2. The calculated bondlength distribution of the highest average binding energy clusters.



transition from icosahedral geometry to thebulk structure occurs.

We note that Karabacak et al. [3] obtained ahighest average binding energy structure for Pd19.

Upon energy minimization of their structure, weobtained a structure of Pd19 that is energetically andtopologically similar to the icosahedral configura-tion. It is also worth noting that the structural tran-sition of small palladium cluster differs signifi-cantly from that of platinum [9] and copper [6–8].For platinum clusters, a structural transition fromtriangular clusters to icosahedral clusters occurs atthe cluster size approximately n � 19 and a lessenergetically favorable transition from triangularclusters to fcc-like clusters takes place around n �38. For copper clusters, it was found that structuraltransitions from the triangular growth clusters tothe icosahedral and fcc-like clusters occur at ap-proximately n � 16 and n � 32, respectively.

Results presented in Figure 1 suggest many iso-mers of palladium clusters exhibit nearly degener-ate average binding energies and thus these iso-mers could coexist from thermodynamic point ofview. An interesting question is how easy theseisomers could transform from one structure to an-other or, in other words, whether the structuralexchange of the isomers is kinetically feasible. It is

FIGURE 3. Selected icosahedral structures for n � 13, 19, 25, and 55 and fcc-like structures for n � 14, 23, 32,and 38. All structures were fully optimized with the average binding energies underneath.

FIGURE 4. The average binding energy of the highestaverage binding energy clusters and the selected ico-sahedral and fcc-like clusters verses the inverse ofcluster size.

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computationally very difficult to calculate the struc-tural exchange pathways, which could be numer-ous, and rigorous studies on the kinetics requirecalculations on the transition states along thesepaths. To gain qualitative or semi-quantitative in-sight into the feasibility of isomer structural ex-change, we selected a few isomers of various clustersizes to study the transformation between isomersalong the pathway defined by linear synchronoustransit (LST) using the structures of the isomersusing the following equation:

ri � � ri�1� � �1 � ��ri

�2�, �0 � � � 1�, (3)

where ri, ri(1), and ri

(2) are the coordinates of the ithatom along the LST pathway and of isomers (1) and(2), respectively, and i � 1, . . . , N (N is the numberof atoms of the isomers). Single point energy calcu-lations were then performed along the LST path-way. The structures and energetics so calculated ofcourse do not necessarily represent the minimumenergy path. However, they do represent a possibletransformation path with an energy barrier higherthan that of the minimum energy pathway. There-fore, the calculated activation energy can serve asthe upper bound of the true energy barrier of theminimum energy pathway.

We selected five pairs of isomers from Figure 1,Pd6-a/Pd6-b, Pd9-b/Pd9-c, Pd13-b/Pd13-c, Pd15-a/Pd15-b, and Pd15-b/Pd15-d, and calculated their LSTpathways. The results are shown in Figure 5. Thecoordinate correlations between isomers were care-fully considered so that the atom pairs would re-side in similar positions upon transformation. Nev-ertheless, it can be readily seen from Figure 5 thatstructures of some of the isomer pairs selected inthe present study are radically different. However,the calculated energy barriers for the selected LSTpathways are considerably moderate in most cases,suggesting that structural isomerization couldreadily take place at or near ambient temperature.Figure 5 also indicates that the isomerization bar-rier increases with cluster size as expected. It can beenvisaged that for large clusters, isomerization willbecome increasingly difficult. However, we specu-late that localized structural rearrangements couldstill occur without too much energy penalty.

Figure 6 displays the calculated total magneticmoments of the highest average binding energyclusters for n � 2–15. Earlier experiments suggestedthat small palladium clusters are nonmagnetic [41–43], while photoelectron spectra suggested that Pdn

(n � 7) clusters have Ni-like spin distribution [44] andpredicted that n � 8 is the critical size for magnetic–nonmagnetic transition [44, 45]. The present studysuggests that the cluster magnetism varies withcluster size and shape. Isomers with nearly degen-erate binding energies may differ significantly intheir magnetic moments. For example, for the close-packed structure, Pd13-a, our calculation yields no

FIGURE 5. Cluster isomerization energy of the se-lected pairs of isomers along the LST pathway.



magnetism, while for the icosahedral structure,Pd13-b, which is nearly degenerate with Pd13-a interms of average binding energy, the calculated totalmagnetic moment is 4�B/atom. This result agreeswell with the studies by Reddy et al., [46, 47] who alsofound that 13-atom clusters of 3d and 4d TMs adopt-ing an icosahedral symmetry have nonzero magneticmoments. Therefore, depending on specific experi-mental conditions, under which specific structures ofcluster isomers are produced, the measured magneticmoments could be very different. Our calculated clus-ter magnetism is essentially consistent with whatwere reported by T. Futschek et al., [2], Kumar et al.[48] and Barreteau et al. [39] for similar cluster struc-tures. The calculated IP and EA are shown in Figure7(a) and (b), respectively. The results show that thecalculated IPs decrease rapidly with cluster size fromn � 2 to n � 5 and then become steady as the size ofclusters increases. The calculated EAs generally in-crease with cluster size. We note that the EA values ofPd3 and Pd7 obtained from experiments [49–51] aresubstantially higher than our results. The underesti-mation of cluster EAs largely stems from fact that thebasis set used in the present computational methoddoes not include diffuse functions, which are knownto be essential for accurate electronic structure calcu-lations of anionic species.

Conclusions

We have performed extensive theoretical studieson small palladium clusters using DFT. Computa-tional search for energy minima on cluster potentialenergy surfaces yielded numerous isomers at agiven cluster size, which is an important feature

shared by many TM clusters. In particular, wefound many of these isomers are nearly degeneratein energy despite the fact that their topologicalstructures differ considerably, indicating that theseisomers could coexist thermodynamically. Kineti-cally, the estimated energy barriers of transformingstructures from one isomer to another were rathermoderate, suggesting that these isomers mayreadily exchange their structures at ambient condi-tions. The barrier increases with cluster size andthus the larger is the cluster the more stable theisomer structure will be. Nevertheless, we speculatethat localized structural arrangement in a large pal-ladium cluster could occur in view of the relativelysmall energy barriers for small isomer structuralexchange.

In the present study, we focused mostly on threestructural patterns of palladium clusters: the close-packed “irregular” structures, the icosahedralstructures and fcc-like structures. It was found evenat a small size of the clusters (n � 13), the isomersof these structural patterns are nearly degenerate.Unfortunately, the wide structural variety of largeclusters and the associated heavy computationalcost prohibit us to perform DFT calculations tostudy structural transitions at a large cluster size.Nevertheless, we expect that the icosahedral struc-tural pattern would dominate the cluster growth asthe size of clusters increases.

FIGURE 7. The calculated cluster ionization potential(IP) (a) and electron affinity (EA) (b).

FIGURE 6. The calculated total magnetic moments ofthe highest average binding energy clusters for n �2–15.

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One of the driving forces to study palladiumclusters is to understand their catalytic properties.Understanding the cluster structures and ener-getics is the first step toward understanding thestructure–activity relationships in many catalyticprocesses. We are currently studying the role ofsmall palladium clusters in catalyzing dissociativechemisorption of molecular hydrogen and the re-sults will be reported in due course.

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First principles study of small palladium cluster growth and isomerization

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