Theoretical Studies of Palladiumâ Gold Nanoclusters: Pdâ Au Clusters

J. Phys. Chem. C 2009, 113, 9141–9152

9141

Theoretical Studies of Palladium-Gold Nanoclusters: Pd-Au Clusters with up to 50 Atoms Faye Pittaway,† Lauro Oliver Paz-Borboń,† Roy L. Johnston,*,† Haydar Arslan,‡ Riccardo Ferrando,§ Christine Mottet,| Giovanni Barcaro,⊥ and Alessandro Fortunelli⊥ School of Chemistry, UniVersity of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom, Department of Physics, Zonguldak Karaelmas UniVersity, Zonguldak 67100, Turkey, Dipartimento di Fisica, UniVersita’ di GenoVa, and CNISM, Via Dodecaneso 33, 16146 GenoVa, Italy, CINaM-CNRS, Campus de Luminy, case 913, 13288 Cedex 09, Marseille, France, and Molecular Modeling Laboratory, Istituto per i Processi Chimico-Fisici del Consiglio Nazionale delle Ricerche (IPCF-CNR), Via G. Moruzzi 1, I-56124 Pisa, Italy ReceiVed: January 20, 2009; ReVised Manuscript ReceiVed: April 6, 2009

Global optimization of Pd-Au bimetallic clusters in the size range N ) 2-50 has been performed using a genetic algorithm, coupled with the Gupta many-body empirical potential (EP) to model interatomic interactions. Three sets of EP parameters have been examined in this work: (a) an average of pure Pd and Au parameters, (b) experimental Pd-Au-fitted parameters, and (c) DFT-fitted parameters. Stability criteria, such as binding energy and second difference in energy, have been used to determine the lowest energy structures, that is, the global minima (GM). DFT local relaxations have been performed on all the “putative” GM structures for 1:1 compositions of (Pd-Au)N/2 up to N ) 50 for the three sets of EP parameters. It is found that the average parameter set a leads to a PdcoreAushell segregation, whereas the fitted parameter sets b and c lead to more Pd-Au mixing. DFT reoptimization of the structures produced by potentials a, b, and c shows small differences in binding energies. In addition, 34- and 38-atom Pd-Au clusters were studied using these three Gupta potential parametrizations as a function of composition and analyzed in terms of their mixing energies and chemical order parameters. DFT relaxations were performed on the lowest mixing energy compositions, allowing us to have a clearer description of the energy landscape for all three EP parameter sets at these cluster sizes. For the compositions, Pd17Au17 and Pd19Au19, DFT calculations confirm that some degree of Au surface segregation is energetically preferred, though it is not necessarily complete PdcoreAushell segregation, as predicted by the average potential a. TABLE 1: Some Elemental Properties of Pd and Au

1. Introduction The desire to fabricate materials with well-defined, controllable properties and structures on the nanometer scale has generated considerable interest in bimetallic clusters or “nanoalloys”.1 Much research on nanoalloys has been driven by their actual and potential applications in a number of catalytic processes. Pd-Au nanoparticles, in particular, have been found to be efficient catalysts for a wide variety of chemical reactions, such as acetylene cyclotrimerization, the hydrogenation of hex2-yne to cis-hex-2-ene, hydrodechlorination of trichloroethene (TCE) in water, the low-temperature synthesis of hydrogen peroxide, and the reduction of CO and alcohols.2-6 Previous experimental studies of Pd-Au nanoalloys (e.g., see refs 3, 7-13) have shown that PdcoreAushell, AucorePdshell, 3-shell, and solid solution configurations can be generated, whereas theoretical studies (e.g., see refs 14 and 15) have indicated that the PdcoreAushell configuration is favored. Surface segregation in nanoalloys may be simply predicted on the basis of simple elemental properties, such as cohesive energy (Ecoh), surface energy (Esurf), atomic radius (ra), and electronegativity (χ).1 These quantities are listed for Pd and Au in Table 1. * Corresponding author. [email protected]. † University of Birmingham. ‡ Zonguldak Karaelmas University. § Universita’ di Genova, and CNISM. | CINaM-CNRS, Campus de Luminy. ⊥ Istituto per i Processi Chimico-Fisici del Consiglio Nazionale delle Ricerche (IPCF-CNR).

Pd Au

Ecoh/eV atom-1

Esurface/meV Å-2

ra/Å

χ

3.89 3.81

131 96.8

1.38 1.44

2.2 2.4

Preferential segregation of Au atoms to the surface of Pd-Au nanoalloys can be rationalized in terms of the marginally larger cohesive energy of Pd (favoring a Pd core, maximizing Pd-Pd bonds) and the smaller surface energy of Au (atoms with lower surface energy forming a surface shell lower in cluster surface energy). The smaller atomic radius of Pd also leads to preferential core location of Pd atoms as it helps to minimize bulk elastic strain. Finally, the slightly higher electronegativity of Au compared with Pd will lead to a certain degree of Pd to Au electron transfer, as found in density functional theory (DFT) calculations by Yuan et al.16 This ionic contribution may be expected to favor Pd-Au mixing, although we have previously shown for Ag-Au clusters that M-Au charge transfer (M ) metal) can also favor surface enrichment by the more negatively charged Au atoms.15,17 Of course, the arguments presented above are rather simple, and the fine details of cluster structure, chemical ordering (segregation or mixing), and surface site preferences depend critically on electronic structure. For example, recent DFT calculations on fcc-type cuboctahedral Pd-Au nanoparticles have indicated that surface Pd atoms occupy (111) rather than (100) facets, thereby maximizing the number of relatively strong surface Pd-Au bonds.18 In fact, the ionic contribution to Pd-Au

10.1021/jp9006075 CCC: $40.75  2009 American Chemical Society Published on Web 05/01/2009

9142

J. Phys. Chem. C, Vol. 113, No. 21, 2009

Pittaway et al.

TABLE 2: Gupta Potential Parameters Used in This Study (a) average parameters (average)

(b) experimental-fitted parameters (exp-fit)

(c) DFT-fitted parameters (DFT-fit)

parameter

Pd-Pd

Au-Au

Pd-Au

Pd-Pd

Au-Au

Pd-Au

Pd-Pd

Au-Au

Pd-Au

A/eV ζ/eV p q ro/Å

0.1746 1.7180 10.867 3.7420 2.7485

0.2061 1.7900 10.229 4.0360 2.8840

0.1900 1.7500 10.540 3.8900 2.8160

0.1715 1.7019 11.000 3.7940 2.7485

0.2096 1.8153 10.139 4.0330 2.8840

0.2764 2.0820 10.569 3.9130 2.8160

0.1653 1.6805 10.8535 3.7516 2.7485

0.2091 1.8097 10.2437 4.0445 2.8840

0.1843 1.7867 10.5420 3.8826 2.8160

bonding has been used to predict that Pd-Au bonds are stronger than both Pd-Pd and Au-Au bonds.16 These findings are consistent with earlier studies by Goodman and co-workers9 and by Jose´-Yacamań and colleagues,10 who have also combined TEM and HAADF-STEM measurements with image simulation and classical molecular dynamics simulations to study shell structure and defects in Pd-Au nanoalloys.11-13 In the present study, global optimizations are performed for Pd-Au bimetallic clusters in the size range of N ) 2-50 with the interatomic interactions modeled by the Gupta manybody empirical potential (with three different potential parametrizations investigated). Furthermore, DFT local relaxations are performed on all the “putative” GM structures for 1:1 Pd/Au compositions up to 50 atoms for all sets of EP parameters. For specific nuclearities (N ) 34 and 38), the structures and chemical ordering are investigated as a function of composition at both the Gupta and DFT levels. 2. Methodology 2.1. Genetic Algorithm. Global optimizations were performed using a genetic algorithm (GA), as encoded in the Birmingham Cluster Genetic Algorithm (BCGA) program,19 with interatomic interactions modeled by the Gupta many-body empirical potential.20 The GA parameters used in the BCGA were population size ) 40; crossover rate ) 0.8 (i.e., 32 offspring are produced per generation); crossover type ) 1-point weighted (the splice position is calculated on the basis of the fitness values of the parents); selection ) roulette wheel; mutation rate ) 0.1; mutation type ) mutate_move; and number of generations ) 400. The number of GA runs for each composition was 100. The high number of GA runs is made necessary by the relatively large size of the clusters and the presence of homotops. 2.2. Homotops. The term “homotop” was introduced by Jellinek et al. to describe AaBb nanoalloy isomers with a fixed number of atoms (N ) a + b) and composition (a/b ratio) and the same geometrical structure, but which differ in the way the different atom types are arranged.21 The number of homotops rises combinatorially with cluster size, which makes global optimization an extremely demanding task. A single geometrical isomer of an N-atom AB binary cluster gives rise to NPA,B homotops

PA,B )

N

N! N! ) NA!NB ! NA!(N - NA)!

(1)

where N is the total number of atoms and NA and NB are the number of atoms of type A and B, respectively. 2.3. Gupta Many-Body Potential. Empirical potentials have been developed to allow relatively rapid searching of large areas of configuration space. We use the Gupta potential, formulated by Cleri and Rosato,20 to model the interatomic interactions in nanoalloys. The Gupta potential is based on the second moment approximation to tight-binding theory and comprises an attrac-

tive many-body (V m) term and a repulsive pair (V r) term, obtained by summing over all N atoms N

Vclus )

∑ [V r(i) - V m(i)]

(2)

i

where V r(i) and V m(i) are defined as

[

N

Vr(i) )

j*i

and

[∑ N

Vm(i) )

(

r

)]

∑ A(a, b)exp -p(a, b) r0(a,ij b) - 1

j*i

(

(

ζ2(a, b)exp -2q(a, b)

(3)

))]

rij -1 r0(a, b)

1/2

(4)

where a and b refer to the element type. In eqs 3 and 4, rij represents the distance between atoms i and j in the cluster. The parameters A, r0, ζ, p, and q are fitted to experimental values of the cohesive energy, lattice parameters, and independent elastic constants for the reference bulk crystal structure at absolute zero. The parameter ζ is the many-body energy scaling parametersalso known as the hopping integral. 2.4. Pd-Au Parameters. Three sets of potential parameters have been adopted in this work (see Table 2): (a) one in which the heteronuclear Pd-Au parameters were obtained as averages of the pure Pd-Pd and Au-Au parameters, hereafter referred to as “average”;15 (b) one in which Pd-Pd, Pd-Au, and Au-Au parameters were simultaneously fitted to experimental properties of bulk Pd, Au, and Pd-Au alloys, hereafter referred to as “expfit”; and (c) one in which the parameters for Pd-Au heteronuclear interactions were fitted to DFT calculations, hereafter referred to as “DFT-fit”. The choice of set a, in which the Pd-Au parameters have been rounded to the precision where the arithmetic and geometric means are the same, was based on our earlier work on Pd-Pt nanoalloys, where averaged parameters were found to qualitatively reproduce the experimental Pd surface segregation.22 Recently, we have shown that the average Pd-Pt Gupta potential also gives results that are in reasonable agreement with DFT calculations.23,24 In parameter set b, the mixed parameters A and ζ have been fitted to the dissolution energies of one impurity Au atom substituted into bulk Pd and vice versa. These values are derived from the enthalpy curves in the Pd-Au phase diagram.25 The dissolution energies are taken as the slope of the mixing enthalpy curve on each side of the phase diagramsthat is, the Pd-rich phase for the dissolution energy of Au and the Au-rich phase for the dissolution energy of Pd. As the difference in size between the two elements is not negligible, we took into account possible relaxations around the impurity in our fitting procedure. The Pd-Au bulk phase diagram presents a complete series of solid solutions with some possible ordered L12 AuCu3-type phases near the composition, Au60Pd40. This means that there is a strong tendency to mix the two elements in the bulk. It is

Theoretical Studies of Palladium-Gold Nanoclusters

J. Phys. Chem. C, Vol. 113, No. 21, 2009 9143

interesting to note that the exp-fit parameter set b has a pair (repulsive) energy scaling parameter (A) that is larger for Pd-Au than for either Pd-Pd or Au-Au: in our previous work, a high heteronuclear A value was found to lead to layerlike segregation.22 However, set b also has a many-body (attractive) energy scaling parameter (ζ) that is greatest for Pd-Au: in our previous work, this was found to favor ordered heteronuclear mixing.22 As will be shown below, for set b, the many-body term wins out so that, overall, the fitted potential favors Pd-Au mixing, as expected from the fitting procedure detailed above. The pair and many-body range exponents (p and q) for Pd-Au interactions (and the atomic radii, ra) were taken as arithmetic means of the pure metal values. In parameter set c, the parameters have been fitted to the results of first-principles DFT calculations. This consists of taking the DFT cohesive energy curves of the pure systems and rescaling these to fit the experimental ones (to obtain the exact values of the cohesive energy, lattice parameter, and stickiness). The obtained rescaling factors are used to rescale the cohesive energy curves of the systems given by ordered intermetallic phases, which are then used to fit the parameters of the heteronuclear Pd-Au interaction. For the metals A and B (i.e., Pd and Au), the chosen ordered alloys are of the type AnBm, with m + n ) 4: in the case where m ) n ) 2, we chose the L10 ordering of the alloy, whereas in the case where m ) 1 and n ) 3 and vice versa, we chose the L12 ordering. 2.5. DFT Calculations. DFT calculations on finite Pd-Au clusters were performed using the NWChem 5.1 quantum chemistry package26 and the Perdew-Wang (PW91) exchangecorrelation functional.27 Spherical Gaussian-type orbital basis sets of double-ζ quality28,29 were used for Pd (7s6p5d)/[5s3p2d] and Au (7s5p5d)/[6s3p2d], combined with effective core potentials (ECP).30 Charge density fitting basis sets were used to speed the evaluation of Coulombic contributions:31 Pd (8s7p6d5f4g)/[8s6p6d3f2g] and Au (9s4p4d3f4g)/[8s4p3d3f2g]. All the calculations were performed spin-unrestricted, using a Gaussian smearing technique (with a smearing parameter of 1.36 eV) for the fractional occupation of the one-electron energy levels. For the fitting of the DFT-fit parameters, DFT calculations on the solid phases were performed using the PWscf (plane-wave self-consistent field) computational code32 employing ultrasoft pseudopotentials.33 A total of 10 and 11 electrons is explicitly considered for Pd and Au, respectively. The PBE exchange-correlation functional, which is a gradient-corrected functional, was used. Values of 40 Ry on the energy cutoff of the wave function and of 160 Ry on the energy cutoff of the electronic density have been shown to provide accurate results (1 Ry ) 13.606 eV) and have, thus, been employed. All the calculations have been performed by applying a smearing procedure of the energy levels with a Gaussian broadening of 0.002 Ry. The Brillouin zone has been described by choosing a (10,10,10) grid. 2.6. Energetic Analysis. The average binding energy of an N-atom cluster at the Gupta level is defined as

EGupta ) b

-Vclus N

(5)

where Vclus is the total cluster potential energy (eq 2). At the DFT level, the average binding energy per atom is obtained as the difference in total energy between the N-atom cluster and its constituent atoms, divided by N. When studying fixed-size bimetallic clusters as a function of composition, the mixing energy ∆NGupta is a useful quantity34

∆NGupta

)

Gupta Etotal (PdnAum)

Gupta Gupta Etotal (PdN) (AuN) Etotal -n -m N N (6)

where EGupta total (PdnAum) is the Gupta total energy (Vclus) for a given Gupta cluster and ( EGupta total (PdN))/(N) and ( Etotal (AuN))/(N) are the Gupta energies per atom calculated for the pure PdN and AuN clusters, respectively. Negative values of the mixing energy indicate favorable mixing and, thus, more stable clusters. ∆NDFT can be similarly defined at the DFT level.34 A final indicator of the relative stabilities of clusters is the second difference in energy, ∆2E, which is defined as follows at the Gupta level

∆2EGupta ) EGupta (Pd/Au)N+1 + EGupta (Pd/Au)N-1 b b b (Pd/Au)N 2EGupta b

(7)

for clusters of increasing size and fixed composition (e.g., for a 1:1 composition). Intense peaks in ∆2EbGupta can be related to relatively stable clusters. To analyze trends in chemical order as a function of size and composition, it is convenient to define an order parameter with the following characteristics: positive when phase separation (segregation) takes place, close to zero when disordered mixing occurs, and negative when mixing and layerlike structures coexist. The chemical order parameter, σ, is defined as

σ)

NPd-Pd + NAu-Au - NPd-Au NPd-Pd + NAu-Au + NPd-Au

(8)

where Nij (with i, j ) Pd, Au) is the number of nearest-neighbor i-j bonds. An order parameter of this type has proven to be useful in the description of short-range order in binary bulk alloys and surfaces.35 3. Results and Discussion 3.1. Gupta Potential Calculations on (Pd-Au)N/2 Clusters (N ) 2-50). Some of the putative GM for (Pd-Au)N/2 clusters (with N ) 2-50) are compared in Figure 1 for the average, exp-fit, and DFT-fit parameters. The symmetries and total energies of all the GM at the Gupta level (in this size range) are listed in Tables 1-3 in the Supporting Information (for the average, exp-fit, and DFT-fit parameters, respectively). The geometrical structures found for the three sets of parameters are often the same but generally exhibit different chemical ordering. From the structures shown in Figure 1, it is clear that the average parameters favor PdcoreAushell segregation (as found in our previous study15), whereas the exp-fit parameters favor more mixed configurations, with more Pd-Au bonds. The DFTfit parameters predict a degree of Au/Pd mixing between the two other potentials. A wide range of structural motifs are found after performing GA global optimizations for the three sets of parameters. Structural families include icosahedra (e.g., N ) 54), Marks’ decahedra (e.g., N ) 98), and fcc-type structures (e.g., the truncated octahedron, N ) 38).1,15,36-40 Figure 2a shows the calculated Gupta potential binding energies (EbGupta) for 1:1 Pd-Au nanoalloys with 2-50 atoms for the three sets of parameters. In each case, the binding energy is reported for the putative GM found by the BCGA. The binding energies (EbGupta) of the GM obtained for the exp-fit parameters are higher than those obtained for the average parameters but slightly higher than for the DFT-fit parameters, indicating that Pd-Au mixing is preferred (at the empirical potential level) in this size regime. Figure 2b shows plots of the corresponding second differences in binding energy (∆2EbGupta). Large positive peaks in ∆2EbGupta correspond to

9144


Pittaway et al.

Figure 1. Comparison of selected GM found for (Pd-Au)N/2 clusters with N ) 2-50 for the average, exp-fit, and DFT-fit parameters. Au and Pd atoms are denoted by yellow and gray colors, respectively.

Figure 2. Plots of (a) binding energies (EGupta ) and (b) second differences in binding energy (∆2EGupta ) for (Pd-Au)N/2 clusters with N ) 2-50 at b b the Gupta level, calculated using the average (black line), exp-fit (red line), and DFT-fit (blue line) parameters.

structures that have high stability relative to their neighbors. However, for Pd-Au nanoalloys, the ∆2EbGupta values are quite small (for the three sets of parameters) because of the small difference in the cohesive energies of Pd and Au (see Table 1). The ∆2EbGupta plots are similar for the three sets of potentials (which tend to find the same structure as GM, with different chemical order), where the discrepancies are due to the different structures and homotops stabilized by these potentials. 3.2. DFT Calculations. DFT local relaxations have been performed on all the putative GM structures for 1:1 Pd-Au clusters with up to 50 atoms for the three sets of parameters. Figure 3a shows a plot of the DFT binding energies (EbDFT) versus N. Here, one can see that there is not much difference at the DFT level between the GM isomers found for the average, exp-fit, and DFT-fit parameters. Figure 3b shows the differences in DFT binding energies between the fitted and average parameters (i.e., ∆EbDFT ) EbDFT(average) - EbDFT(fitted)), highlighting that, for some specific sizes, one or both fitted potentials may give rise to more stable structures (indicated by negative ∆EbDFT values) than the average parameters.

Structural Analysis. In Figure 4, the total number of bonds (nearest-neighbor contacts) of each type and the chemical ordering parameter (σ, see eq 8) for the three parameter sets are plotted as a function of cluster size (N) for 1:1 Pd-Au clusters. For the average potential, one can see that the numbers of Pd-Pd and Pd-Au bonds increase monotonically (quasilinearly) with N and are nearly equal (see Figure 4a). The number of Au-Au bonds also increases with N, but less steeply than for NPd-Pd and NPd-Au. For both fitted parameters (Figure 4b,c), the rate of increase in the number of bonds is in the order Pd-Au . Pd-Pd > Au-Au, indicating the greater Pd-Au mixing favored by the fitted potentials. The corresponding σ values for all three sets of parameters are plotted in Figure 4d. We find that, for the average parameters (Figure 4d), σ is positive for cluster sizes N > 10, corresponding to PdcoreAushell segregation. In contrast, for both fitted parameters, the σ values are negative for all cluster sizes, indicating that the clusters have significant Pd-Au mixing (NPd-Au > NPd-Pd + NAu-Au). 3.3. 34-Atom Pd-Au Clusters. Figure 5 shows the mixing Gupta ) calculated at the Gupta level (for the three sets energy (∆34



Figure 3. (a) DFT binding energies (EDFT b ) after local re-minimization of the putative GM for the average (black line), exp-fit (red line), and DFT-fit (blue line) Gupta parameters for (Pd-Au)N/2 clusters with N ) 2-50. (b) Differences in DFT binding energies (∆EDFT b ): exp-fit and DFT-fit vs average parameters, where negative values indicate more stable structures than those found using the average parameters.

Figure 4. (a-c) Numbers of Pd-Pd (black), Au-Au (red), and Pd-Au (green) nearest-neighbor bonds as a function of nuclearity (N ) 2-50) for the 1:1 composition of Pd-Au clusters for GM obtained using the average, exp-fit, and DFT-fit parameters. (d) Comparison of the chemical order parameter (σ) for the three sets of parameters: average (red), exp-fit (black), and DFT-fit (green).

of potentials) for all compositions of 34-atom Pd-Au clusters plotted against the number of Au atoms (m). We selected 34atom clusters to allow comparisons to be made with the results of our previous in-depth study of 34-atom Pd-Pt clusters.23,24 The tendency of the exp-fit and DFT-fit parameters to favor Pd-Au mixing is emphasized by the significantly more negative mixing energies obtained for these potentials. These large negative mixing energies (i.e., representing highly exothermic mixing) are reminiscent of the mixing energy curves obtained for 34-atom Pd-Pt clusters when weighting the heteronuclear Pd-Pt interactions toward the strongest (Pt-Pt) homonuclear interactions.24 The mixing energies (∆DFT 34 ) calculated at the DFT level (based on the GM found for the three sets of potentials) are shown in

Figure 6 for the composition range, Pd24Au10-Pd6Au29, which corresponds to the region of most negative mixing energies for these potentials at the Gupta potential level; see Figure 6 and Table 4 in the Supporting Information for a comparative analysis of GM structures for the three sets of potentials at the DFT Gupta curves, the level. Compared with the relatively smooth ∆34 plots are rather jagged, especially for both exp-fit and DFT∆DFT 34 fit parameter curves. It is not clear what the preference is at the DFT level for structures based on these potentials, though the exp-fit potential gives marginally lower mixing energy structures compared to the average and DFT-fit parameter GM. The jagged ∆DFT 34 plots are reminiscent of those obtained for 34-atom Pd-Pt clusters and may (as in the Pd-Pt case) indicate that there are differences between the energy ordering of structural motifs at

9146


Pittaway et al.

Figure 5. Bottom: Gupta potential mixing energies (∆Gupta 34 ) for 34-atom Pd-Au nanoalloys as a function of the number of Au atoms (m), over the entire composition range, for the average (black line), exp-fit (red line), and DFT-fit (blue line) parameters. Top: Dark blue squares indicate distorted icosahedral or poly-icosahedral structures; red squares, pIh-6 (“pancake-type”) structures; white squares denote incomplete decahedral structures.

Figure 6. DFT mixing energies (∆DFT 34 ) after local minimization of the putative GM for the average (black line), exp-fit (red line), and DFT-fit (blue line) Gupta parameters for 34-atom Pd-Au nanoalloys in the range of m ) 10-29 (m ) number of Au atoms).

the Gupta and DFT levels.23 The reason is that some of the motifs that are energetically close and crossing at the Gupta DFT curve obtained by level are disfavored by DFT so that the ∆34 reoptimizing the Gupta GM at the DFT level becomes irregular whenever such motifs are predicted as the Gupta GM. Further research in this area will include DFT re-minimization of other low-lying structural isomers, not just the Gupta global minima. The structures of the average, exp-fit, and DFT-fit GM (after DFT re-minimization) are shown in Figure 7 for the composition range, Pd17Au17-Pd10Au24. This composition range includes the most negative ∆DFT 34 values, as shown in Figure 6. In this region, all of the putative GM for the average parameters have incomplete decahedral structures (but no Dh-cp(DT) structures as found previously for 34-atom Pd-Pt clusters). This probably explains the rather smooth curve for the average parameters

DFT minimum at the (black line) in Figure 6, with the ∆34 composition, Pd10Au24. On the other hand, the structures derived from both fitted parameters tend to maximize the number of Pd-Au bonds; hence, they tend to form incomplete icosahedral or poly-icosahedral (here we classify as poly-icosahedral those clusters containing at least two elementary 13-atom icosahedra) arrangements in this composition range. It is interesting to note that the lowest ∆DFT 34 compositions correspond to two incomplete icosahedral structures obtained using exp-fit parameters (i.e., compositions of Pd13Au21 and Pd15Au19). When comparing these two structures with those found using the DFT-fit parameters, we notice that they have a distorted five-fold shape compared to the more asymmetric (amorphous) structures found using the DFT-fit parameters. It remains to be seen whether the Dh-cp (DT) motif (found as GM for 34-atom Pd-Pt clusters) will be



Figure 7. GM structures obtained after DFT optimization (for the three sets of Gupta parameters) for 34-atom Pd-Au clusters in the composition range, Pd17Au17-Pd10Au24.

Figure 8. Bottom: Gupta potential mixing energies (∆Gupta 38 ) for 38-atom Pd-Au nanoalloys as a function of the number of Au atoms (m) over the entire composition range for the average (black line), exp-fit (red line), and DFT-fit (blue line) parameters. Top: Light blue squares denote compositions at which TO structures were found, yellow denotes Oh-Ih (close-packed) structure, green denotes Ih-Mackay (pseudo-five-fold structures), and orange denotes pIh5 (five-fold “pancakes”), white denotes incomplete decahedral structures.

found as a GM for Pd-Au as this structure was not found during our GA global optimization procedure. 3.3. 38-Atom Pd-Au Clusters. Figure 8 shows the mixing Gupta ) calculated at the Gupta level (for the three sets energy (∆38 of parameters) for all compositions of 38-atom Pd-Au clusters plotted against the number of Au atoms, m. We selected 38atom clusters because size 38 is a “magic” number for the truncated octahedral (TO) structure, which has fcc packing (the bulk crystalline structure for both Pd and Au), and also for the poly-icosahedral, pIh-6 “six-fold pancake” structure. These structures, along with other structural motifs, have been studied in our previous work on 38-atom nanoalloy clusters.15,38,39 The tendency of the exp-fit parameters to favor Pd-Au mixing is again manifested in the much more negative mixing energies Gupta ) obtained for this potential, which is also found for the (∆38

values (see Figure 8). Less negative ∆Gupta values DFT-fit ∆Gupta 38 38 are found for the average parameters, for which the lowest ∆Gupta 38 value corresponds to the Au-rich composition, Pd14Au24. As shown in Figure 9 (which reproduces the GM structures found as the putative GM for the average, exp-fit, and DFT-fit parameters in the composition range, Pd19Au19-Pd13Au25 at the DFT level), this structure has regular Oh symmetry, consisting of an fcc-like truncated octahedron (TO) with an octahedral Pd6 core surrounded by a shell of 24 Au atoms, with 8 Pd atoms occupying the centers of the (111) facets. Gupta The corresponding minimum in ∆38 for the exp-fit and DFT-fit parameters is at Pd18Au20, which is consistent with these parameters favoring Pd-Au mixingswhich should be maximized around the 1:1 composition (although we still see a preference of Pd atoms to occupy core positions in the cluster

9148


Pittaway et al.

Figure 9. GM structures (for the three sets of Gupta parameters) for 38-atom Pd-Au clusters in the composition range, Pd18Au20-Pd13Au25.

for the three sets of potentials, i.e., PdcoreAushell segregation). The preferred sites for occupation by Pd and Au atoms are Gupta curves. Specifically, for evident from the shapes of the ∆38 Gupta ∆38 (average), there are straight sections corresponding to the first 24 Au atoms occupying surface (100) TO sites (square faces), the next 8 atoms occupying the centers of the surface (111) facets (hexagonal faces), and the final 6 atoms occupying Gupta (exp-fit), however, the octahedral core of the cluster. For ∆38 the shape of the curve is different: the first 12 Au atoms occupy half of the (100) sites, the next 8 occupy the (111) sites, the next 12 occupy the remainder of the (100) sites, and the final 6 occupy the octahedral core. For the average parameters (see Figure 9), the GM structures obtained for all compositions are TO with surface Au enrichment (i.e., idealized PdcoreAushell configurations), in agreement with our previous studies.15 In agreement with the DFT calculations16 and as discussed above for Pd14Au24, surface Pd atoms preferentially occupy (111) facets. TO structures are also found as GM structures for the DFT-fit parameters. For the exp-fit parameters (see Figure 9), the GM for most compositions are also TO, though there is a five-fold symmetry structure found as GM at the composition of Pd13Au25 (an icosahedral Mackaytype structure, previously described in ref 15). Again, the expfit parameters show significantly more Pd-Au mixing, though they tend to favor surface Pd atoms occupying (100) rather than (111) sites, in contrast to the findings of Yuan et al.16 This is probably due to an incorrect interplay between binding and equilibrium distance. The mixing energies (∆DFT 38 ) calculated at the DFT level (based on the GM found for all three parameter sets) are shown in Figure 10 for the composition range, Pd18Au20-Pd13Au25 (the DFT energies are point-group symmetries, structure types, and ∆38 listed in Table 5 in the Supporting Information). In contrast to DFT DFT shown in Figure 6, the ∆38 plots are quite the plots of ∆34 smooth, and it is also clear that, in this composition range, the isomers predicted to be the GM by the average parameters have consistently more negative mixing energies after DFT reminimization than those from the exp-fit and DFT-fit parameters. The smoothness of the plots and lack of fluctuations are probably because all of the structures (with the above-mentioned exception of Pd13Au25 (exp-fit)) have the same geometric structure (TO), as shown in Figure 9.

Figure 10. DFT mixing energies (∆DFT 38 ) after local re-minimization of the putative GM for the average (black line), exp-fit (red line), and DFT-fit (blue line) Gupta potentials for 38-atom Pd-Au nanoalloys, in the range of m ) 19-25 (m ) number of Au atoms). DFT plots for It should be noted that the minimum in the ∆38 the average and exp-fit parameters occurs at the composition, Pd15Au23-that is, one less Au atom than the minimum of ∆Gupta 38 (average) and three more Au atoms than the minimum of ∆Gupta 38 (exp-fit). Thus, for 38-atom Pd-Au clusters in this composition range, we can say that the average parameters produce TO homotops that are closer to the true DFT GM (for this structure type) than those found using the exp-fit parameters. It is interesting to note that the DFT-fit parameters also predict TO structures as GM, with a higher mixing of Pd-Au bonds compared to that of the GM structures predicted by the average parameters. The DFT-fit parameter ∆DFT 38 curve is rather smooth, DFT curves, with a minimum at the similar to the other two ∆38 composition, Pd14Au24 (a highly symmetrical Oh structure), though it also has a maximum at Pd18Au20. The increased mixing on going from the average to the expfit parameters has been confirmed by a detailed analysis (for 38-atom Pd-Au clusters) of the number of homo- and heteronuclear bonds and other mixing parameters as a function of composition. Figures 11 and 12 show the average nearestneighbor distance (ANND); the number of Pd-Pd, Pd-Au, and


Figure 11. Variation of average nearest-neighbor distance (ANND) as a function of the number of Au atoms (m) for 38-atom Pd-Au clusters, for GM obtained using the (a) average, (b) exp-fit, and (c) DFT-fit parameters.

Au-Au bonds; and chemical order parameter (σ) as a function of the Au content (m) for the GM found for the average, expfit, and DFT-fit parameters, at the Gupta level of theory. The average nearest-neighbor distance (ANND) plots (Figure 11) show a steady increase with increasing Au content due to the larger atomic radius of Au (see Table 1). For the average parameters, there is an increase in the slope at m ) 24; as for higher Au content, the additional Au atoms start to occupy the (111) facets and the core of the cluster-causing a greater rate of cluster expansion per added Au atom. For the exp-fit parameters, however, there is a steeper initial slope (as there are more Pd-Au bonds, the effect of adding Au atoms is initially greater), but the plot flattens out at m ) 20. The points at which the slope changes-that is, Pd14Au24 (average) and Pd18Au20 (exp-fit)-correspond to the maxima in the Pd-Au bond plot (Figure 12). For the case of the average nearestneighbor distance (ANND) plot for the DFT-fit parameters, we see a rather jagged behavior as this set of parameters leads to

J. Phys. Chem. C, Vol. 113, No. 21, 2009 9149 a high degree of mixing (Pd-Au bonds), but not as much as for the exp-fit parameters. Figure 12 shows that, for the average potential, NPd-Pd decreases monotonically, whereas NAu-Au increases with increasing Au content, m. NPd-Au increases to a maximum at Gupta (average) Pd14Au24, corresponding to the minimum in ∆38 observed in Figure 8. For the exp-fit parameters, NPd-Pd decreases monotonically until m ) 20, after which there are some fluctuations. In contrast to the average parameters, there are no Au-Au bonds up to Pd26Au12 as the Au atoms are dispersed on the surface of the cluster. The number of Pd-Au bonds is maximized at the composition, Pd18Au20, corresponding Gupta (exp-fit) observed in Figure 8. A to the minimum in ∆38 substantially analogous (even though slightly more irregular) behavior is observed for the DFT-fit parameters, confirming the qualitative similarity between exp-fit and DFT-fit parameters. Finally, Figure 12 shows plots of the chemical order parameter (σ) for the three sets of potentials. For the average parameters, σ is positive for all compositions, falling to a minimum value of 0 for Pd15Au23 and Pd14Au24. Positive σ values correspond to core-shell segregation. The fact that the plot goes through a minimum is inevitable because, even for core-shell segregation, there are a high number of Pd-Au bonds (which lead to a reduction of σ, see eq 8) for comparable numbers of Pd and Au atoms.24 The minimum at around m ) 24 is because of the extra Pd-Au mixing due to the Pd atoms occupying the (111) facets (correlating with the maximum in NPd-Au shown in Figure 12). For the exp-fit parameters, σ is negative in the region of m ) 13-27 (being 0 for m ) 12), indicating a high degree of Pd-Au mixing. Similar behavior is found for the DFT-fit parameters. The minimum in σ for the exp-fit and DFT-fit parameters occurs at Pd18Au20, corresponding to the maximum in NPd-Au, as shown in Figure 12. It is interesting to note that the minima in the mixing energy plots for both the average and fitted parameters correspond to the maximum number of Pd-Au bonds (and minima in σ) for different homotopic configurations. Thus, Pd14Au24 (average) has the maximum number of Pd-Au bonds, consistent with the PdcoreAushell ordering favored (energetically) by the average potential; whereas Pd18Au20 (fitted) maximizes Pd-Au bonding according to the tendency of the fitted parameters to favor more mixed homotops. In principle, the maximum possible number of Pd-Au bonds should be realized for the 1:1 composition, that is, for Pd19Au19, but it is possible that to achieve this maximum Pd-Au bonding would require a structure that is not energetically competitive. 3.4. DFT Investigation of Segregation in 34- and 38-Atom Pd-Au Clusters. Due to the vast combinatorial problem of systematically evaluating all possible homotops for a given nanoalloy structure and composition (see eq 1) and the impossibility of carrying out a reasonably rigorous homotop search at the DFT level, we have chosen to carry out a DFT analysis of “normal” and “inverted” homotops for the 1:1 composition of 34- and 38-atom Pd-Au nanoalloys, Pd17Au17 and Pd19Au19 (this approach has previously been used to study a range of nanoalloy clusters15,23). For the three sets of Gupta parameters (i.e., average, exp-fit, and DFT-fit), the “normal” homotop is the putative GM found (using the BCGA) for Pd17Au17 or Pd19Au19. In each case, an “inverted” homotop is generated by exchanging the positions of all Pd and Au atoms. All of the normal and inverted homotops are then energy-minimized at the DFT level. The normal and inverted homotops for Pd17Au17 and Pd19Au19 are shown in Figure 13 (for the three sets of parameters), along

9150


Pittaway et al.

Figure 12. Variation of the number of bonds (black ) Pd-Pd, red ) Au-Au, green ) Pd-Au) and chemical order parameter (σ) as a function of composition (number of Au atoms ) m) for 38-atom Pd-Au clusters, for GM obtained using the (a) average, (b) exp-fit, and (c) DFT-fit parameters.

with the total DFT cluster energies (in electronvolts) relative to the lowest energy isomer found for each nuclearity. For Pd17Au17, the isomer derived from the exp-fit potential (a quite distorted fragment of a 55-atom icosahedron structure, with a high number of Pd-Au bonds) is lower in energy (by 0.29 eV) than the average structure (an incomplete decahedron with more defined PdcoreAushell segregation) after DFT re-minimization, which is consistent with its more negative ∆DFT 34 value (see Figure 6). The DFT-fit parameters predicted a GM poly-icosahedral structure; but this structure is not energetically favorable as it is found to be approximately 0.40 eV higher in energy. DFT optimizations on the three inverted homotops reveal that they are higher in energy (ranging from ∼1.85 eV for the DFT-fit inverted to ∼4.08 eV for the average inverted structures). For Pd19Au19, the “GM” structures for these potentials are truncated octahedra (TO). In this case, the lowest energy structure corresponds to that derived from the average parameters (welldefined PdcoreAushell), which is lower in energy than the exp-fit and DFT-fit parameter-predicted structures by 0.74 and 0.81 eV, respectively. This is consistent with the more negative ∆DFT 38

value (see Figure 7b) of the Pd19Au19 average potential structure. As the normal and inverted isomers all have TO structures and because homotop inversion forces Pd and Au atoms to adopt unfavorable sites, it is not surprising that the inverted homotops have higher energies. It should be noted that, because of the size of the homotop search space, we cannot be sure that we have found the lowest energy homotop at the Gupta potential level, let alone at the DFT level. A possible way forward would be to couple DFT calculations with low-temperature basin-hopping Monte Carlo searching41 to perform a partial homotop optimization by permuting unlike atoms, an approach that has previously been applied to Gupta potential calculations for 34-, 38-, and 98atom nanoalloys.15,23,36-40 4. Conclusions Three parametrizations of the many-body Gupta empirical potential have been compared with regard to the geometrical structures and homotops that they stabilize for Pd-Au



Figure 13. DFT investigation of segregation effects in 34- and 38-atom Pd-Au clusters for compositions 17-17 and 19-19, respectively. The “normal” homotop is the lowest energy homotop found using the average or fitted Gupta parameters. In each case, the “inverted” homotop is generated by swapping the positions of all the Pd and Au atoms. The energies quoted are the total DFT cluster energies relative to the lowest energy isomer found for each nuclearity.

nanoalloy clusters with up to 50 atoms. The “average” parameters (where the Pd-Au parameters are obtained by averaging those for Pd-Pd and Au-Au interactions) are found to favor PdcoreAushell segregationsas these maximize the number of the stronger Pd-Pd bonds and expose the most Au atoms (which have lower surface energies). The fitted experimental (exp-fit) and DFT (DFT-fit) parameterssobtained by fitting Pd/Au dissolution energies and the three ordered phases of Pd-Au, respectivelysfavor Pd-Au mixing, generally leading to mixed Pd-Au nanocluster surfaces. Analysis of the 1:1 Pd-Au clusters with up to 50 atoms shows that there is no clear preference for any one set of parameters over the whole size range. Hence, we decided to compare the three potentials for fixed size (i.e., 34- and 38-atom Pd-Au clusters) with variable composition. From our DFT analysis of 34-atomPd-Au clusters, mixing DFT ) curves for the fitted parameters showed a rather energy (∆34 jagged behavior compared with those of the average parameters. This can be interpreted in the sense that global optimizations (at the EP level) using the fitted parameters tend to find incomplete icosahedra or distorted poly-icosahedral structures, whereas when using average parameters, incomplete decahedral motifs are found as the lowest energy structures. It is not clear what the preference is at the DFT level for structures based on these potentials, though the exp-fit potential gives marginally lower mixing energy structures compared to the average and DFT-fit potentials. For 38-atom nanoalloys, whose structures and chemical ordering have been studied as a function of composition, the average parameters have been found to yield isomers with lower energies at the DFT level than those of both fitted parameters (i.e., a preference for more PdcoreAushell-type structures at this specific cluster size). A more extensive homotop search (incorporating a limited BHMC search and DFT re-minimization) would be helpful to establish whether the average homotops are, in fact, the GM or if there are lower energy homotops at the DFT level. This would involve the DFT reminimization of low-lying Gupta isomers as well as the Gupta GM, as previously performed for a number of 38-atom nanoalloys.15 This is currently being investigated, along with a detailed analysis of the effect of systematically varying the Gupta

parameters on the structures and chemical ordering of 34- and 38-atom Pd-Au nanoalloys. Acknowledgment. L.O.P.-B. and R.L.J. acknowledge the National Service for Computational Chemistry Software (http:// www.nsccs.ac.uk) for the award of a CPU grant. Calculations were also performed on the University of Birmingham’s e-Science cluster and the BlueBEAR 1500+ processor highperformance computer cluster (http://www.bear.bham.ac.uk/). H.A. acknowledges the HPC-EUROPA project for a travel grant and CPU time on HPCx at EPCC (Edinburgh). Computer resources on the HPCx service were provided to L.O.P.-B. and R.L.J. via their membership of the UK’s HPC Materials Chemistry consortium and funded by EPSRC (portfolio grant EP/D504872). F.P. is grateful to the EPSRC for financial support, and L.O.P.-B. is grateful to CONACYT (Me´xico) for the award of a Ph.D. scholarship. We acknowledge support from ESF for the Workshop on Computational Nanoalloys within the Simbioma Programme. Supporting Information Available: Tables 1-5 list the energies of (PdAu)N/2 clusters with a 1:1 composition (N ) 2-50) and 34- and 38-atom Pd-Au clusters with variable composition, for both Gupta potential and DFT calculations, as well as their corresponding point-group symmetries. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. ReV. 2008, 108, 845. (2) Lee, A. F.; Baddeley, C. J.; Hardacre, C.; Ormerod, R. M.; Lambert, R. M.; Schmid, G.; West, H. J. Phys. Chem. 1995, 99, 6096. (3) Schmid, G. In Metal Clusters in Chemistry; Braunstein, P., Oro, L. A., Raithby, P. R., Eds.; Wiley-VCH: Weinheim, Germany, 1999; Vol. 3, p 1325. (4) Nutt, M. O.; Hughes, J. B.; Wong, M. S. EnViron. Sci. Technol. 2005, 39, 1346. (5) Edwards, J. K.; Solsona, B. E.; Landon, P.; Carley, A. F.; Herzing, A.; Kiely, C. J.; Hutchings, G. J. J. Catal. 2005, 236, 69. (6) Enache, D. I.; Edwards, J. K.; Landon, P.; Solsona, B. E.; Carley, A. F.; Herzing, A.; Watanabe, M.; Kiely, C. J.; Knight, D. W.; Hutchings, G. J. Science 2006, 311, 362.

9152


(7) Bo¨nnemann, H.; Brijoux, W. In Metal Clusters in Chemistry; Braunstein, P., Oro, L. A., Raithby, P. R., Eds.; Wiley-VCH: Weinheim, Germany, 1999; Vol. 2, p 913. (8) Remita, H.; Mostafavi, M.; Delcourt, M. O. Radiat. Phys. Chem. 1996, 47, 275. (9) Luo, K.; Wei, T.; Yi, C. W.; Axnanda, S.; Goodman, D. W. J. Phys. Chem. B 2005, 109, 23517. (10) Mejia-Rosales, S. J.; Fernandez-Navarro, C.; Pe´rez-Tijerina, E.; Blom, D. A.; Allard, L. F.; Jose´-Yacamań, M. J. Phys. Chem. C 2007, 111, 1256. (11) Jose´-Yacamań, M.; Pe´rez-Tijerina, E.; Mejia-Rosales, S. J. J. Mater. Chem. 2007, 17, 1035. (12) Pe´rez-Tijerina, E.; Gracia-Pinilla, M.; Mejia-Rosales, S. J.; OrtizMeńdez, U.; Torres, A.; Jose´-Yacamań, M. Faraday Discuss. 2008, 138, 353. (13) Ferrer, D.; Blom, D. A.; Allard, L. F.; Mejia-Rosales, S. J.; Pe´rezTijerina, E.; Jose´-Yacamań, M. J. Mater. Chem. 2008, 18, 2442. (14) Ascencio, J. A.; Liu, H. B.; Pal, U.; Perez, R. J. Phys. Chem. B 2006, 110, 5191. (15) Paz-Borboń, L. O.; Johnston, R. L.; Barcaro, G.; Fortunelli, A. J. Chem. Phys. 2008, 128, 134517. (16) Yuan, D. W.; Gong, X. G.; Wu, R. Phys. ReV. B 2007, 75, 085428. (17) Chen, F. Y.; Johnston, R. L. Acta Mater. 2008, 56, 2374. (18) Yuan, D. W.; Gong, X. G.; Wu, R. Phys. ReV. B 2008, 78, 035441. (19) Johnston, R. L. Dalton Trans. 2003, 4193. (20) Cleri, F.; Rosato, V. Phys. ReV. B 1993, 48, 22. (21) Jellinek, J.; Krissinel, E. B. In Theory of Atomic and Molecular Clusters; Jellinek, J., Ed.; Springer: Berlin, 1999; p 277. (22) Massen, C.; Mortimer-Jones, T. V.; Johnston, R. L. J. Chem. Soc., Dalton Trans. 2002, 4375. (23) Paz-Borboń, L. O.; Johnston, R. L.; Barcaro, G.; Fortunelli, A. J. Phys. Chem. C 2007, 111, 2936. (24) Paz-Borboń, L. O.; Gupta, A.; Johnston, R. L. J. Mater. Chem. 2008, 18, 4154. (25) Hultgren, R.; Desai, P. D.; Hawkins, D. T.; Gleiser, M.; Kelley, K. K. Values of the Thermodynamic Properties of Binary Alloys; American Society for Metals, Jossey-Bass Publishers: Berkley, CA, 1981; p 306.

Pittaway et al. (26) Kendall, R. A.; Apra`, E.; Bernholdt, D. E.; Bylaska, E. J.; Dupuis, M.; Fann, G. I.; Harrison, R. J.; Ju, J.; Nichols, J. A.; Nieplocha, J.; Straatsma, T. P.; Windus, T. L.; Wong, A. T. Comput. Phys. Commun. 2000, 128, 260. (27) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhaus, C. Phys. ReV. B 1992, 46, 6671. (28) Scha¨fer, A.; Huber, C.; Ahlrichs, R. J. Chem. Phys. 1994, 100, 5829. (29) ftp://ftp.chemie.uni-karlsruhe.de/pub/basen/. (30) Andrae, D.; Haeussermann, U.; Dolg, M.; Stoll, H.; Preuss, H. Theor. Chim. Acta 1990, 77, 123. (31) Weigend, F.; Haser, M.; Patzel, H.; Ahlrichs, R. Chem. Phys. Lett. 1998, 294, 143. (32) Baroni, S.; Del Corso, A.; de Gironcoli, S.; Giannozzi, P.http:// www.pwscf.org. (33) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (34) Ferrando, R.; Fortunelli, A.; Rossi, G. Phys. ReV. B 2005, 72, 085449. (35) Aguilera-Granja, F.; Vega, A.; Rogan, J.; Andrade, X.; Garcıá, G. Phys. ReV. B 2006, 74, 224405. (36) Rapallo, A.; Rossi, G.; Ferrando, R.; Fortunelli, A.; Curley, B.; Lloyd, L. D.; Tarbuck, G. M.; Johnston, R. L. J. Chem. Phys. 2005, 122, 194308. (37) Rossi, G.; Ferrando, R.; Rapallo, A.; Fortunelli, A.; Curley, B.; Lloyd, L. D.; Johnston, R. L. J. Chem. Phys. 2005, 122, 194309. (38) Rossi, G.; Rapallo, A.; Mottet, C.; Fortunelli, A.; Baletto, F.; Ferrando, R. Phys. ReV. Lett. 2004, 93, 105503. (39) Curley, B.; Rossi, G.; Ferrando, R.; Johnston, R. L. Eur. Phys. J. D 2007, 43, 53. (40) Paz-Borboń, L. O.; Mortimer-Jones, T. V.; Johnston, R. L.; PosadaAmarillas, A.; Barcaro, G.; Fortunelli, A. Phys. Chem. Chem. Phys. 2007, 9, 5202. (41) Doye, J. P. K.; Wales, D. J. J. Phys. Chem. A 1997, 101, 5111.

JP9006075

Theoretical Studies of Palladiumâ Gold Nanoclusters: Pdâ Au Clusters

Recommend Documents

Theoretical Studies of Palladiumâ Gold Nanoclusters: Pdâ Au Clusters