Site- and Structure-Dependent Cohesive Energy in Several Ag Clusters

J. Phys. Chem. C 2009, 113, 10907–10912

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Site- and Structure-Dependent Cohesive Energy in Several Ag Clusters D. Liu, Y. F. Zhu, and Q. Jiang* Key Laboratory of Automobile Materials, Ministry of Education, and School of Material Science and Engineering, Jilin UniVersity, Changchun 130022, China ReceiVed: February 26, 2009; ReVised Manuscript ReceiVed: April 23, 2009

The cohesive energy of Ag clusters at any site x with the magic number N [Ecx(N)] is determined through calculating the vacancy formation energy Evx(N) using density functional theory and theoretical modeling. It is found that Ecx(N) is quite distinct at different surface and interior atomic sites within the same cluster. Ecx(N) is also structure dependent. Ecx(N) values of the core atoms with an icosahedron structure are much larger than the corresponding bulk value Ec(∞) due to the pressure-induced d-d shells repulsion and the sp-d hybridization weakening. 1. Introduction Clusters are defined as quantum dots with tens or hundreds of atoms. The dimensional sizes of the clusters exactly meet the borderline between that of nanocrystals and that of molecules, which lead distinct electronic, optical, and catalytic properties of them.1-3 For example, gold clusters with 55 atoms supported on inert substrates are efficient and robust catalysts for the selective oxidation of styrene by dioxygen, while gold particles with diameters of 2 nm or above are completely inactive.4 Besides, clusters can be switched from a metallic into an insulating state by ligating the outer atoms to the structurestabilizing triphenylphosphine ligand shell via chlorine.5 Clusters have noncrystalline or quasi-crystalline structures, and most atoms of them are at the surface.1 Thus, the cohesive energy of clusters Ec(N) (N is the number of atoms of a cluster) is quite different from that of the corresponding bulk Ec(∞), which determines their novel properties. Since Ec(N) is difficult to determine experimentally, computer simulations with the first-principles6-8 and the embedded-atom methods9-11 have been used. The results show that Ec(N) > Ec(∞). Moreover, several theoretical models have been developed to determine Ec(N) values through considering the surface/volume ratio of the clusters.12-14 The above considered Ec(N) values are in fact the mean values of all atoms of clusters, while the cohesive energy of atoms at a special x site, Ecx(N), is still unknown. Ecx(N) values in some cases however are more important than Ec(N) ones because many properties, such as adsorption, catalysis, and optical behaviors of clusters, are exhibited by atoms at special surface locations of clusters.15,16 Moreover, alloy clusters show superior performance compared with a single-element cluster in catalytic17,18 and optical properties.19 To determine the degree of segregation or mixing in an alloy cluster, besides the surface sites, Ecx(N) of the interior sites also needs to be understood. Ecx(N) values cannot be directly obtained by using simulation techniques. The vacancy formation energy of the x site Evx(N) in a cluster however could be determined by simulation,20 which can be related to Ecx(N). For bulk crystals, the vacancy formation energy Ev(∞) is approximately a fraction of Ec(∞), such as Ev(∞)/ Ec(∞) ≈ -0.3 for the transition metals.21 For nanocrystals, a * To whom correspondence should be addressed. Fax: +86 431 85095876. E-mail: [email protected].

thermodynamic model has been deduced as Ev(N)/Ec(N) ) Ev(∞)/Ec(∞), where Ev(N) is the average value of Evx(N).22 Since clusters usually have noncrystalline and quasi-crystalline structures, this linear relationship may be invalid and the Ecx(N)-Evx(N) relationship still needs to be clarified. In this contribution, Evx(N) values of Ag clusters are first calculated through density functional theory (DFT) simulation. After that, the corresponding Ecx(N) values are determined by considering the Ecx(N)-Evx(N) relationship in terms of simulation results and a theoretical consideration. In addition, the partial density of state (PDOS) and Mulliken charge analysis of atoms at distinct sites are also determined to understand the corresponding electronic distributions. 2. Simulation Details and Theoretical Models The first-principles DFT23,24 provided by DMOL code25,26 was used for the simulation, in which each electronic wave function was expanded in a localized atom-centered basis set. Each basis function was defined numerically on a dense radial grid, using a state-of-the-art delocalized internal coordinate optimization scheme. The quality of geometry optimization of Fine was used, where the convergence tolerance of energy was 1.0 × 10-5 Ha (1 Ha ) 27.2114 eV), maximum force was 0.002 Ha/Å, and maximum displacement was 0.005 Å. We used the local density approximation (LDA) as the exchange-correlation value where Perdew-Wang parametrization27 with Ceperly and Alder28 (PWC) functions were taken. DFT Semicore Pseudopots (DSPPs)29 developed in DMOL3 was included via a semilocal pseudopotential for all-electrons calculations, where the effect of core electrons was substituted by a simple potential including some degree of relativistic effects. In addition, double numerical plus polarization (DNP)23 was chosen as the basis set with an orbital cutoff of 4.4 Å. For Ag clusters, we used smearing techniques30 to achieve self-consistent field (SF) convergence with a smearing value of 0.005 Ha. In all cases, spin polarization was introduced. In the bulk calculation, Ec(∞) ) -3.51 eV/ atom and the nearest atomic distance h(∞) ) 2.85 Å through optimizing the cell. For PDOS calculation, the empty band was chosen as 12. The DFT static simulation was carried out at 0 K, which means the system is adiabatic and could represent the real system. To calculate Ev(∞) values of the bulk (111) and (100) facets, the 3 × 3 × 7 slab for them were separately established. A vacuum region being larger than 12 Å was set

10.1021/jp901797w CCC: $40.75  2009 American Chemical Society Published on Web 05/28/2009

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Liu et al. been reported that at Cu bulk surfaces, the binding energy of the Cu adatoms at the hollow sites of the (111) and (100) facets keep the ratios as Eb(111)(∞)/Ec(∞) ) -0.64 and Eb(100)(∞)/Ec(∞) ) -0.8.36 Since the CN of these sites are separately Zs(111)(∞) ) 3 and Zs(100)(∞) ) 4 while the structure of Ag is also FCC, these sites should be similar to the kink sites of Ag IH3 and DH75 clusters. Thus, Ebk(13)/Ec(∞) ≈ -0.64 and Ebk(75)/Ec(∞) ≈ -0.80 are approximately valid and the assumption of Ebk(N) ≈ -Ec(N) is reasonable. With the above consideration on the new reservoir, Evx(N) is defined as

Figure 1. Ag cluster structures with different magic numbers N. The different surface and interior sites are marked with numbers, and the corresponding site names are shown in Table 1.

to separate adjacent slabs. The K-point was set as 3 × 3 × 1, which is fine for the slabs. Figure 1 shows the structures of Ag clusters with different N atoms. For metallic clusters, although icosahedron (IH) at N ) 13, 55, and 147 are favorable structures due to their fewer surface broken bonds, they could not be perfectly packed at some N values, which leads to stabilization of other structures,31-33 such as the truncated octahedron (TO) at N ) 38, and the truncated decahedron (DH) at N ) 75 and 101. The N values referred to above are named as magic numbers,1 where the related clusters have close-packing surfaces. Ev(∞) denotes the lowest energy to remove an atom from a selected site, and usually the atom is brought to an assumed reservoir which determines the atomic chemical potential.34 For single-element crystals, this potential is Ec(∞). The physical meaning behind this is that the removed atom is brought to a kink site at the surface.34 Although the cohesive energy of the under-coordinated kink site Eck(∞) is larger than Ec(∞), it has been widely reported that the binding energy of the atoms at the kink site Ebk(∞) is equal to -Ec(∞).34-36 This is mainly because when an atom is bound at the kink site, besides its cohesive energy turning from 0 to Eck(∞), the cohesive energy of the coordinated atoms also decreases since these atoms all gain one extra bond. Thus, in DFT simulation, Ev(∞) has been deduced as34

Ev(∞) ) E(1) + E(∞,N - 1,1) - E(∞,N,0) + Ec(∞)

(1) where E(1) is the total energy of a single atom and E(∞,N,0) and E(∞,N - 1,1) are separately the total energy of the super cell and that after the atom at x site is removed. Since clusters are quite different from bulk crystals both in structure and in energy, a new reservoir should be assumed for calculating Evx(N). With the same consideration as bulk crystals, it is simply assumed that the new reservoir is determined by Ec(N), where Ebk(N) ) -Ec(N). To confirm the validity of the above relationship, more detailed discussion is given in the following. At bulk surfaces, the coordination number CN of the atoms at the close-packed (111) facet and at the kink site are separately Zs(111) ) 9 and Zsk ) 6.37 Hence, three neighboring bonds of an atom should be broken when making a kink at the plane. For clusters, the average CN of the atoms at the close-packed surface Zs(N) ) 6-8,38 which brings out the CN of corresponding kink sites Zsk(N) being 3-5. For instance, for Ag IH13 and DH75 clusters, Ec(13)/Ec(∞) ) 0.61 and Ec(75)/Ec(∞) ) 0.8 with Zs(13) ) 6 and Zs(75) ) 7.4.38 Thus, Zsk(13) ≈ 3 and Zsk(75) ≈ 4. It has

Evx(N) ) E(1) + Ex(N - 1,1) - E(N,0) + Ec(N)

(2) where E(N,0) and Ex(N - 1,1) are the total energy of the cluster with N atoms and that after the atom at x site is removed. For transition metals, it has been found that Ev(∞) - Ev(111)(∞) ≈ Ecb(∞) - Ec(111)(∞) ) γ(111)(∞), where γ is the surface energy,39,40 which says that the difference of Ev between the bulk surface and the interior is essentially induced by the corresponding discrepancy in Ec. Following the same consideration, the difference of Evx(N) in the same cluster is assumed to be mainly decided by the discrepancy in Ecx(N), that is

Evy(N) - Evx(N) ≈ Ecy(N) - Ecx(N)

(3)

where the subscript y denotes another site in the cluster, which differs from x. Moreover, Ecx(N) could also be deduced as

Ecx(N) ) [NEc(N) -

∑ NzEcz(N)]/Nx

(4)

z

where Nx and Nz are the numbers of atoms of the x and z sites. Inserting eq 3 into eq 4, it reads

∑Ny[Evy(N) - Evx(N)]}/N

Ecx(N) ) Ec(N){

(5)

y

In eq 5, Ec(N) values could be obtained by our simulation; Evy(N) and Evx(N) values are calculated in terms of eq 2 and our simulation results. Moreover, the γ value of the cluster with N atoms γ(N) is deduced as38

γ(N) ) Ecs(N) - Eci(N)

(6)

where Ecs(N) and Eci(N) are separately the average cohesive energy of the surface and interior atoms of the considered clusters. 3. Results and Discussion On the basis of eq 1, Ev(∞) ) 1.18 eV, which agrees with experimental and simulation results of 1.11-1.24 eV for bulk Ag crystals.21 The calculated Evx(N) values of Ag clusters in light of eq 2 are shown in Table 1. The vertex sites have comparatively small Evx(N) values as Ev-vertex(N) ) 0.3-0.84 eV, or the sites are easy to form vacancy compared with other surface sites. This result is not difficult to understand since Zvertex(N) ) 6 except that at the notch vertex sites of DH structures. In addition, Evx(N) first increases with increasing N until a maximum appears at Ev-vertex(55) ) 0.84 eV, and then it

Cohesive Energy in Several Ag Clusters

J. Phys. Chem. C, Vol. 113, No. 25, 2009 10909

TABLE 1: Ec(N) and Ecx(N) (in eV/atom), Evx(N) (in eV), Nx, Zx(N), and hx(N) of Ag Clusters Obtained from DFT Simulation and Eqs 2 and 5, where hx(N) Denotes the Average Bond Length of an Atom at the x Site -Ec(N)

atomic sites

Nx

Zx(N)

Evx(N)

-Ecx(N)

hx(N) (Å)

IH13

2.13

TO38

2.64

IH55

2.79

DH75

2.82

DH101

2.9

IH147

2.99

(1) vertex (2) core (1) (111) facet (2) vertex (3) sublayer (1) edge (2) vertex (3) sublayer (4) core (1) (111) facet (2) top edge (3) notch edge (4) top vertex (5) notch vertex (6) lateral vertex (7) subtop vertex (8) subtop edge (9) sub-(111) facet (10) core (1) (111) facet (2) top edge (3) notch edge (4) lateral edge (5) top vertex (6) notch vertex (7) lateral vertex (8) subnotch edge (9) subtop vertex (10) subtop edge (11) sub-(111) facet (12) core (1) (111) facet (2) edge (3) vertex (4) subedge (5) subvertex (6) second sublayer (7) core

12 1 8 24 6 30 12 12 1 10 10 5 2 10 20 2 10 5 1 10 10 10 10 2 10 20 10 10 2 5 2 20 60 12 30 12 12 1

6 12 9 6 12 8 6 12 12 9 8 10 6 7 6 12 12 12 12 9 8 10 7 6 7 6 12 12 12 12 12 9 8 6 12 12 12 12

0.44 0.79 1.21 0.5 1.16 1.05 0.84 1.33 1.15 1.15 0.9 1.35 0.4 0.84 0.5 1.18 1.36 1.04 1.32 1.04 0.82 1.24 0.61 0.32 0.65 0.37 1.43 1.06 1.16 1.16 1.31 1.14 0.81 0.3 1.25 0.99 1.28 0.42

2.10 2.44 3.10 2.39 3.05 2.77 2.56 3.05 2.87 3.05 2.80 3.25 2.3 2.74 2.4 3.08 3.24 2.94 3.22 3.08 2.86 3.28 2.65 2.36 2.69 2.41 3.47 3.10 3.20 3.20 3.35 3.18 2.85 2.34 3.29 3.03 3.32 2.46

2.83 2.72 2.83 2.78 2.83 2.86 2.81 2.80 2.71 2.84 2.82 2.81 2.79 2.8 2.79 2.82 2.84 2.84 2.81 2.84 2.83 2.84 2.81 2.8 2.81 2.79 2.85 2.81 2.84 2.83 2.82 2.88 2.86 2.82 2.83 2.81 2.79 2.70

cluster

decreases. Namely, vacancy is easier to form at the vertex sites with increasing N when N > 55. As mentioned at the beginning, clusters would transform to nanocrystals as N further increases. Although nanocrystals usually keep the TO structure thermodynamically,41 the IH42 and DH43 structures could also be metastable due to some kinetic factors. Recent molecular dynamic simulation results show that Au nanocrystals in a size range of 3-8 nm have the “Chui icosahedron” (c-IH) structure,44,45 which is truncated from the IH structure where all atoms at the vertex site are removed. A thermodynamic model suggests that there is a critical size beyond which the c-IH structure is energetically more favorable than the IH structure.46 This conclusion corresponds to our result that clusters or nanocrystals with larger size tend to form vacancies at the vertex sites. For the atoms at (111) facets [Z(111)(N) ) 9] and edge [Zedge(N) ) 8] sites, Ev(111)(N) ) 1.04-1.21 eV and Ev-edge(N) ) 0.81-1.05 eV, which are much larger than those of the bulk (111) and (100) facets of Ev-(111)(∞) ) 0.77 eV and Ev-(100)(∞) ) 0.53 eV. Hence, vacancies are more difficult to form at these sites for clusters than for the corresponding bulk. For the interior sites of Ag clusters, most Ev-interior(N) values are in the range of 1-1.43 eV, which is approximately equal to or even larger than Ev(∞) ) 1.18 eV. Accordingly, vacancies are more difficult to form at some interior sites of clusters than those at the interior of bulk crystals. The exceptions are the core sites of IH13 and IH147 with Ev-core(13) ) 0.79 eV and Ev-core(147) ) 0.42 eV. Therefore, the core site of the IH structure may be unoccupied.

This result is in agreement with the simulation results that Au clusters with the IH structure may be hollow, being similar to fullerenes.47 Note that several bulk quasi-crystalline alloys consist of stacking of 5-fold symmetric cells like the IH clusters,48 while the icosahedral cell usually has a vacant core.49 These phenomena support our simulation results. On the other side, the calculated Ecx(N) values in terms of eq 5 are also shown in Table 1. For the surface sites with Zsx(N) ) 6, Ec-vertex(N) values are in the range from -2.3 to -2.56 eV with the only exception of Ec-vertex(13) ) -2.10 eV. Ec-vertex(N) does not show obvious size dependence like Ev-vertex(N). Similarly, Ecx(N) keeps almost the same with the same Zsx(N) when Zsx(N) ) 7-10. For the bulk surface, Ecs(∞) ≈ [Zs(∞)/ Z(∞)]1/2Ec(∞) has been demonstrated using simulation techniques.50 As shown in Figure 2, the Ecx(N) ≈ Zx(N) relationship is similar to the Ecs(∞) ≈ Ec(∞) one. Accordingly, for most surface sites of clusters, Ecx(N) ≈ Ecs(∞) when Zsx(N) ) Zs(∞). As mentioned above, even if Ecx(N) ≈ Ecs(∞), surface atoms of clusters could not easily be separated to form vacancy. This is because for forming a vacancy at the cluster surface, the chemical potential of the assumed reservoir has been changed from Ec(∞) to Ec(N), which increases the difficulty of vacancy formation. For most interior atoms of clusters, Ecix(N) ) -2.87 to -3.35 eV with ZIx(N) ) 12. Although Ecix(N) > Ec(∞), some interior sites are harder to form vacancy than bulk due to the change of the chemical potential of the reservoir from Ec(∞) to Ec(N). The

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Liu et al.

Figure 2. Ecsx(N) as a function of Zx(N), separately for IH13 (9), TO38 (b), IH55(2), DH75(1), DH101(0), and IH147(O). The solid line is determined by Ecsx(N) ≈ [Zsx(N)/Z(∞)]1/2Ec(∞).

TABLE 2: Ecs(N), Eci(N), γ(N), and Zs(N) of Ag Clusters Obtained from DFT Simulation and Eqs 5 and 6 clusters

Zs(N)

γ(N) (eV/atom)

γ(N) (J/m2)

-Ecs(N)

-Eci(N)

IH13 TO38 IH55 DH75 DH101 IH147

6 6.75 7.43 7.4 7.53 7.96

0.34 0.48 0.33 0.43 0.51 0.36

0.46 0.85 0.6 0.86 1.06 0.75

2.1 2.57 2.71 2.71 2.76 2.86

2.44 3.05 3.04 3.14 3.27 3.22

exceptions are the core sites of IH13 and IH147 with Ec-core(13) ) 2.44 eV and Ec-core(147) ) 2.46 eV where there are smaller vacancy formation energies of Ev-core(13) ) 0.79 eV and Ev-core(147) ) 0.42 eV. Through averaging the Ecsx(N) and Ecix(N) values, Ecs(N), Eci(N), and γ(N) values in terms of eq 6 are calculated and shown in Table 2. Both Ecs(N) and Eci(N) increase as N decreases. γ(N) values are in the range of 0.46-1.06 J/m2, which differs from our previous theoretical results a little where γ(N) ) 0.84-1.25 J/m2.38 In our early model, γ(N) ) -Ec(N){1 [Zs(N)/Z(∞)]1/2} was taken through considering the limit case of Ec(∞) f Ec(N) and Zs(∞) f Zs(N).38 In this equation, the average CN of the interior atoms Zi(N) is simply considered as Z(∞) while Zi(N) is indeed equal to Z(∞) ) 12. However, the effective average CN of the interior atoms Zi-eff(N) should be smaller than Z(∞) since Eci(N) > Ec(∞), which leads to the above difference on γ(N). If Z(∞) in the above equation is replaced by Zi-eff(N), γ(N) values drop and will correspond to the results shown in this work. To understand the corresponding electronic background of the obtained Ecx(N) data, PDOS of Ag clusters at any site x are separately shown in Figures 3 (for surface atomic sites) and 4 (for interior atomic sites). It is generally found that valence bands (VB) of surface atoms VBsurface and that of the interior atoms VBinterior are quite distinct. Compared with bulk VB (VBbulk), all VBsurface are enhanced at the higher energy end and depleted at lower energy end, which confirms that Ecsx(N) > Ec(∞). VBsurface with the same CN is usually similar in shape, for instance, VBs of the top vertex sites for IH55, DH75, and DH101 or VBs of the (111) facet sites for DH75, DH101, and IH147. This validates our prediction that Ecx(N) values are similar when CN is the same. When N ) 13, VB of the vertex atom is almost annihilated at the lower energy end compared with those of other vertex atoms, which supports the obtained larger Ec-vertex(13) value of -2.10 eV/atom. In Figure 4, being different from VBsurface, VBinterior moves to the lower energy side and the widths of them increase compared

Figure 3. PDOS of surface atomic sites for Ag clusters with distinct N values.

Figure 4. PDOS of interior atomic sites for Ag clusters with different N values. PDOS of their bulk counterparts is given for comparison.

with that of VBbulk. In bulk crystals, simulation results suggest that the shape of DOS does not change much under pressure but VBbulk decreases uniformly with bandwidth increasing.51 According to the Laplace-Young equation, pressure on the order of GPa magnitude is present on interior atoms of clusters or nanocrystals due to the existence of the surface stress.52 Thus, a red shift of VBinterior arises. In the viewpoint of valence electron interaction, the pressure in the clusters would induce the repulsion among d shells of interior atoms. It is known that bonding of transition metals essentially is induced by the overlapping among d shells accompanied with the hybridization of sp-d electrons.53 Although this interaction gives net attraction among atoms at bulk size with a bond length h(∞), it becomes repulsive in clusters where the bond length h(N) is smaller than h(∞).54 The average bond length at a specific x site in Ag clusters hx(N) is shown in Table 1. Indeed, hix(N) < h(∞) for all interior atoms with Ecix(N) > Ec(∞) where the red shift of VBinterior and bands widening are present. In Figure 4, inside the same cluster, VBcore usually has more red shift than VBsublayer, which corresponds to the results that hcore(N) < hsublayer(N) for almost all cases. Similar with VBsurface, VBsublayer with increasing Ec-sublayer(N) is also enhanced at the

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TABLE 3: Counts (CT) of 4d, 5s, and 5p Electrons and Charge Transfer of the Specific Atoms in Ag Clusters Obtained from DFT Simulation cluster IH13

atomic sites

(1) vertex (2) core TO38 (1) (111) facet (2) vertex (3) sublayer IH55 (1) edge (2) vertex (3) sublayer (4) core DH75 (1) (111) facet (2) top edge (3) notch edge (4) top vertex (5) notch vertex (6) lateral vertex (7) subtop vertex (8) subtop edge (9) sub-(111) facet (10) core DH101 (1) (111) facet (2) top edge (3) notch edge (4) lateral edge (5) top vertex (6) notch vertex (7) lateral vertex (8) subnotch edge (9) subtop vertex (10) subtop edge (11) sub-(111) facet (12) core IH147 (1) (111) facet (2) edge (3) vertex (4) subedge (5) subvertex (6) second sublayer (7) core Bulk

Nx

4d

5s

5p

5s + 5p

CT

12 1 8 24 6 30 12 12 1 10 10 5 2 10 20 2 10 5

9.828 9.796 9.763 9.808 9.736 9.785 9.804 9.726 9.658 9.754 9.766 9.742 9.795 9.773 9.804 9.727 9.723 9.726

0.886 0.478 0.842 0.894 0.69 0.878 0.936 0.664 0.641 0.837 0.872 0.802 0.94 0.9 0.919 0.717 0.707 0.705

0.3 0.558 0.414 0.31 0.504 0.364 0.278 0.533 0.602 0.417 0.38 0.458 0.279 0.356 0.301 0.498 0.512 0.5

1.186 1.036 1.256 1.204 1.194 1.242 1.214 1.197 1.243 1.254 1.252 1.26 1.219 1.256 1.22 1.215 1.219 1.205

-0.014 0.168 -0.019 -0.011 0.07 -0.027 -0.018 0.077 0.097 -0.008 -0.018 -0.002 -0.014 -0.029 -0.024 0.058 0.058 0.069

1 10 10 10 10 2 10 20 10 10 2 5

9.706 9.754 9.767 9.737 9.789 9.796 9.775 9.797 9.702 9.725 9.722 9.721

0.735 0.846 0.875 0.818 0.876 0.952 0.892 0.926 0.742 0.741 0.721 0.708

0.497 0.41 0.378 0.458 0.348 0.271 0.356 0.302 0.525 0.492 0.512 0.511

1.232 1.256 1.253 1.276 1.224 1.223 1.248 1.228 1.267 1.233 1.233 1.219

0.062 -0.010 -0.020 -0.013 -0.013 -0.019 -0.023 -0.025 0.031 0.042 0.045 0.06

2 20 60 12 30 12 12

9.699 9.765 9.776 9.805 9.715 9.726 9.668

0.764 0.87 0.876 0.957 0.74 0.704 0.751

0.504 0.39 0.365 0.265 0.515 0.512 0.56

1.268 1.26 1.241 1.222 1.255 1.216 1.311

0.033 -0.025 -0.017 -0.027 0.030 0.058 0.019

1 9.675 0.731 0.572 9.704 0.804 0.492

1.303 1.296

0.022 0

higher energy end affected by surface broken bonds. For the DH clusters, VBcore has only a little more red shift than VBsublayer, but the former has no enhancement at the higher energy end and their Ecix(N) values in the range from -2.94 to -3.47 eV/ atom are similar. For the IH clusters, VBcore with intensively enhanced the lower energy end has more red shift than VBsublayer, accompanied with much smaller hIH-core(N) values of 2.70-2.72 Å and much larger Ec-IH-core(N) values from -2.44 to -2.87 eV/atom. Table 3 shows the counts of 4d, 5s, and 5p electrons and charge transfer of specific atoms in Ag clusters. The surface of all clusters is negatively charged, and the interior of the clusters is in reverse, in agreement with the previous simulation results for Au-Ag clusters.55 This is because s/p electrons are depleted at the metallic bulk surface,37 which may weaken the sp-d hybridization and leads to an increase of Ecx(N). The exceptions are s/p electrons of the second sublayer and core atoms of IH147 with slight enhancement of Ecx(N). However, their sp-d hybridization may still be weakened due to the VB shift. Figure 5 shows the VBs of 5s, 5p, and 4d of core atoms of IH13, IH55, and IH147 structures. There is hardly sp-d hybridization for the core atoms of IH13 and IH147 with larger

Figure 5. PDOS of the 4d, 5s, and 5p orbitals for the core atoms of Ag IH13, IH55, and IH147 clusters.

values of Ec-core(13) ) -2.44 eV/atom and Ec-core(147) ) -2.46 eV/atom, where 5s, 5p, and 4d VBs have different red shift extents. For the core atom of IH55, the VBs of 5s, 5p, and 4d electrons are partly hybridized, which leads to a smaller value of Ec-core(55) ) -2.87 eV/atom. These results confirm that the increase of Ec-core is induced by weakening of the hybridization. 4. Conclusions In summary, Ecx(N) values of several clusters with magic number N are determined through calculating Evx(N) values using DFT simulation and some theoretical considerations. It is found that Evx(N) may be larger than Ev(∞) since the chemical potential of the reservoir increases to Ec(N). Although clusters may be more difficult to form vacancy, Ecx(N) > Ec(∞) for all surface and interior atoms. Ecsx(N) ≈ [Zsx(N)/Z(∞)]1/2Ec(∞) when N > 13 and Ecix(N) > Ec(∞), while Ec-core(N) values of IH structures are the largest. This phenomenon is induced by the interior pressure produced by d-d shells repulsion and sp-d hybridization weakening. Acknowledgment. The authors acknowledge financial support from the National Key Basic Research and Development Program of China (grant no. 2004CB619301). References and Notes (1) Baletto, F.; Ferrando, R. ReV. Mod. Phys. 2005, 77, 371. (2) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (3) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. ReV. 2008, 108, 846. (4) Turner, M.; Golovko, V. B.; Vaughan, O. P. H.; Abdulkin, P.; Berenguer-Murcia, A.; Tikhov, M. S.; Johnson, B. F. G.; Lambert, R. M. Nature 2008, 454, 981. (5) Boyen, H. G.; Ka¨stle, G.; Weigl, F.; Ziemann, P.; Schmid, G. Phys. ReV. Lett. 2001, 87, 276401. (6) Ko¨hler, C.; Seifert, G.; Frauenheim, T. Chem. Phys. 2005, 309, 23. (7) Barnard, A. S.; Curtiss, L. A. ChemPhysChem 2006, 7, 1544. (8) Medasani, B.; Park, Y. H.; Vasiliev, I. Phys. ReV. B 2007, 75, 235436. (9) Negreiros, F. R.; Soares, E. A.; de Carvalho, V. E. Phys. ReV. B 2007, 76, 205429. (10) Grochola, G.; Snook, I. K.; Russo, S. P. J. Chem. Phys. 2007, 127, 224704. (11) Grigoryan, V. G.; Alamanova, D.; Springborg, M. Phys. ReV. B 2006, 73, 115415. (12) Jiang, Q.; Li, J. C.; Chi, B. Q. Chem. Phys. Lett. 2002, 366, 551. (13) Nanda, K. K.; Sahu, S. N.; Behera, S. N. Phys. ReV. A 2002, 66, 013208.

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