Equilibrium Geometries, Stabilities, and Electronic Properties of the

Dec 30, 2010 - The density functional method with relativistic effective core potential has been employed to investigate systematically the geometrica...
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Equilibrium Geometries, Stabilities, and Electronic Properties of the Bimetallic M2-doped Aun (M = Ag, Cu; n = 1-10) Clusters: Comparison with Pure Gold Clusters Ya-Ru Zhao,† Xiao-Yu Kuang,*,†,‡ Bao-Bing Zheng,§ Yan-Fang Li,† and Su-Juan Wang† †

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China International Centre for Materials Physics, Academia Sinica, Shenyang 110016, China § Department of Physics, Baoji University of Arts and Science, Baoji 721007, China ‡

bS Supporting Information ABSTRACT: The density functional method with relativistic effective core potential has been employed to investigate systematically the geometrical structures, relative stabilities, growth-pattern behaviors, and electronic properties of small bimetallic M2Aun (M = Ag, Cu; n = 1-10) and pure gold Aun (n e 12) clusters. The optimized geometries reveal that M2 substituted Aunþ2 clusters and one Au atom capped M2Aun-1 structures are dominant growth patterns of the stable alloyed M2Aun clusters. The calculated averaged atomic binding energies, fragmentation energies, and the second-order difference of energies as a function of the cluster size exhibit a pronounced even-odd alternation phenomenon. The analytic results exhibit that the planar structure Ag2Au4 and Cu2Au2 isomers are the most stable geometries of Ag2Aun and Cu2Aun clusters, respectively. In addition, the HOMO-LUMO gaps, charge transfers, chemical hardnesses and polarizabilities have been analyzed and compared further.

1. INTRODUCTION In recent years, the transition bimetallic clusters, particularly coinage metals (Au, Ag, and Cu), have received much attentions because of their particular physical and chemical properties and potential technological applications in solid-state chemistry, materials science, microelectron, nanotechnology, catalysis, biology, and medicine.1-17 Therefore, a great number of experimental and theoretical works about pure and alloyed coinage metal clusters have been performed. For instance, the gas phase,18 matrix electron spin resonance (ESR),19 Raman studies,20 dissociation energies,21-24 bond lengths,21,25-28 vibration frequencies,21,28 and ionization potential21,23,28,29 of small coinage metal clusters have been experimentally reported. The geometric structure, stabilities, and electronic properties of gold, silver, and bimetallic gold-silver and gold-copper clusters have been theoretically investigated on the basis of density functional theory.31-34 In comparison to pure gold clusters, the AgAu and CuAu dimers have been studied using resonant two-photon ionization spectroscopy.35 Negishi et al. investigated the photoelectron spectroscopy of AgmAun- (m þ n e 4) clusters.36 Weis et al. performed density functional theory (DFT) calculations for structures of AgmAunþ (m þ n < 6) clusters.37 The stabilities and charge transfers of neutral gold-silver mixed clusters up to five atoms have been studied by Bonacic-Koutecky et al.38 Photoelectron spectra of AunCu- (n = 2-7) clusters have been carried out at a photon energy of 4.661 eV.39 Wang et al. r 2010 American Chemical Society

investigated the structures of MAun (n = 8-11; M = Ag, Cu) and AgnAu16- and CunAu16- clusters by combined photoelectron spectroscopy (PES) and DFT methods.40,41 The structures and the electronic properties of Au19X clusters (X = Li, Na, K, Ru, Cs, Cu, and Ag) have been studied by Ghanty et al.31 In light of the previous works, one can find that geometric structures, electronic structures, and properties of the clusters vary with their size and differ, sometimes dramatically, from those of the bulk. As it is known, sliver and copper atoms possess similar valence electron character with gold atom, that is, a completely filled d shell and a singly occupied s shell. However, their physical and chemical properties are different from gold due to relativistic effects; the s-d hybridization of silver differs from gold, but copper is similar to gold. A question arises: Do their structure and properties differ from those of the bare gold clusters and mixed cluster doped Ag and Cu atoms, and if so, what kind of regular changes exist in serial alloy clusters? In the current work, we have systematically reported a density functional theory investigation on the small M2Aun (M = Ag, Cu; n = 1-10) clusters comparing with the equivalent size pure gold clusters. The geometrical structures, growth-pattern behaviors, relative stabilities, HOMO-LUMO gaps, charge transfers, Received: September 12, 2010 Revised: November 14, 2010 Published: December 30, 2010 569

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Table 1. Theoretic and Experimental Values of the Bond Lengths (r), Dissociation Energies (De), Vibration Frequencies (ωe), and Vertical Ionization Potentials (VIPs) for the Au2, Ag2, Cu2, AgAu, and CuAu Molecules r (Å) cluster Au2 Ag2 Cu2

calc 2.54 2.58 2.23

De (eV)

expt 2.47

21

2.53

25

2.22

26 27

AgAu

2.57

2.50

CuAu

2.38

2.33 28

calc

expt

ωe (cm-1) calc

2.29

21

1.66

25

2.08

23

278

2.32

2.10

24

184

2.78

2.39 24

242

2.30 1.92 2.51

170 190

expt

VIP (eV) calc

expt

191

21

9.83

9.20 29

192

21

8.18

7.66 21

265

21

8.29

7.90 23

248 28

9.06

9.02

Figure 1. Most stable isomers of gold clusters for each size. The corresponding point-group symmetries and electronic states are presented by PW91P86 level.

8.74 28

chemical hardnesses, and polarizabilities have been analyzed and compared. We find some interesting tendency for the geometrical structures of the M2Aun (M = Ag, Cu; n = 1-10) clusters. Furthermore, the bonding energies, fragmentation energies, second-order difference of energies, HOMO-LUMO gaps, and chemical hardnesses show the same even-odd alternation tendency with cluster size. It is hoped that our work would be useful for understanding the influence of material structure on its properties and could offer relevant information for further experimental and theoretical studies. This paper is organized as follows: section 2 gives a brief description of the theoretical approach. The geometrical structures, relative stabilities, and electronic properties of Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters are given in section 3, Conclusions are summarized in section 4.

Ag-Au bond length of 2.57 Å, which is close to the previous DFT study of 2.60 Å,47 is longer than Au-Au bonds. It may be caused by the reason much stronger relativistic effects in the Au2 cluster exist than that in AgAu molecular. On the basis of calculated results, it is interesting to note that the dissociation energies of the mixed AgAu and CuAu clusters are bigger than that of pure Au2, Ag2, and Cu2 molecules. This fact can reflect that the stability of the alloyed clusters will be stronger than that of the corresponding bare clusters.

3. RESULTS AND DISCUSSIONS 3.1. Geometries and Growth Pattern Behaviors . A. Bare Gold Clusters Aun (n = 3-12). To investigate the effects of

impurity atoms on gold clusters, we first performed some optimizations and discussions on pure gold clusters Aun (n = 3-12) by using identical method and basis set. Taking lots of possible initial structures into account, the most stable isomers for each size are only selected and shown in Figure 1. From the figure, one can see that the geometric structures and electronic states are in good agreement with the previous results.30 In addition, the averaged atomic binding energies, fragmentation energies, the second-order difference of energies, VIPs, and VEAs of gold clusters are studied and compared with the available experimental values in the following. B. Bimetallic Silver-Gold Clusters Ag2Aun (n = 1-10). For Ag2Aun (n = 1-10) clusters, a large number of initial configurations are optimized and many stable isomers have been obtained. Here, the lowest energy isomers and few low-lying structures for each size are displayed in Figure 2. According to the total energies from low to high, these isomers are designated by na, nb, nc, and nd. (“n” is the number of Au atoms in the Ag2Aun clusters) Meanwhile, the symmetries and energy differences compared to each of the lowest energy isomers are also indicated in the figure. The possible Ag2Au geometries such as C2v and D¥h isomers are optimized as the stable structures. According to the calculated results, it is found that the lowest energy isomer is an acute-angle triangular structure (1a) with C2v symmetry, a 57.2 angle, and a 2.77 Å of Ag-Au bonds. Another C2v symmetry isomer (1b) with an apex angle of 145.6 is 0.09 eV higher in energy than the lowest energy 1a structure. This isomer is similar to the groundstate Au3 cluster. Frequency analysis results indicate that the linear isomer of Ag2Au is a transition state. For Ag2Au2 clusters, 2a and 2c can be viewed as two derivatives of a substituted structure of the ground-state Au4 isomer. The Y-shaped structure

2. THEORETICAL METHODS AND COMPUTATIONAL DETAILS All optimizations of the Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters are performed by using the GAUSSIAN 03 program package42 with the PW91P86 functional of the generalized gradient approximation (GGA).43 In present calculations, full electron calculation for our investigated systems is rather timeconsuming, so it is better to introduce the effective core potential (ECP) LanL2DZ basis set44-46 to describe the outermost valence electrons 5s25p65d106s1 for Au atom, 4s24p64d105s1 for Ag atom, and 3s23p63d104s1 for Cu atom, respectively. In searching for the lowest energy structures of M2Aun (M = Ag, Cu; n = 1-10) clusters, lots of possible initial structures, which include one-, two- and three-dimensional configurations, are considered starting from the previous optimized Aun and AgmAun geometries, and all clusters are relaxed fully without any symmetry constraints. Furthermore, vibration frequency calculations were performed to guarantee the optimized structures corresponding to a local minimum and provide zeropoint energy (ZPE). It is worth pointing out all optimized geometries are found to prefer the lowest spin state when every initial structure was optimized at various possible spin multiplicities. To test the reliability of our calculations, the bond lengths, vibration frequencies, dissociation energies, and vertical ionization potentials (VIPs) of the Au2, Ag2, Cu2, AgAu, and CuAu molecules are calculated and listed in Table 1. Our theoretical values are in good agreement with the experimental results reported available works.21-29 The bond length of the Cu2 cluster and the dissociation energy of the Au2 cluster are very consistent with the experimental values.21,26 We found that the 570

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show the appearance of 3D geometries. The 3c structure of the Ag2Au3 cluster emerges, which is obtained by adding symmetrically two Ag atoms to the ground-state Au3 isomer. In addition, three trapezoid isomers are optimized after replacing two Au atoms on different sites by impurity Ag atoms for the groundstate Au5 cluster. The boatlike 3D structure (4c) with C2v symmetry is 0.63 eV less stable than the planar Ag2Au4 structures (4a) substituted from the Au6 isomer. It is interesting to find that the ring D2h symmetry 5c structure can be viewed as a combination of two trapezoid 3d isomers. The pentagonal bipyramid 5d structure is obtained when two Au atoms of the Au7 cluster with D5h symmetry are substituted by two Ag atoms. Unfortunately, its D5h symmetry is lowered to be the Cs symmetry due to the Jahn-Teller effects. In light of the previous works about goldsilver clusters,33,34 our calculation results show good agreement with available structures. It must be pointed out that the planar D2h structure of the ground-state Ag2Au6 clusters, which is reported by Zhao and Zeng,34 is a transition state with one imaginary frequency. Performing further calculation, the most stable Ag2Au6 isomer with D2 symmetry is obtained. For the 3D structures 6d and 7c, they can be described as two and three Au atoms being face capped on the Ag2Au4 cluster with D4h symmetry, respectively. According to Figures 1 and 2, it is found that all of the lowest energy isomers of Ag2Aun (n = 1-8) are the substituted structures of the corresponding ground-state Aun (n = 3-10) clusters. However, the lowest energy 3d isomers 9a and 10a are more stable than the planar substituted structures (9b and 10b) of Au11 and Au12 clusters by 0.12 and 0.50 eV, respectively. The reason may be that the SO coupling effects are decreased in Ag2Aun clusters than corresponding Aunþ2 clusters. In analogy to the growth patterns of 6d and 7c, the 9a isomer can be described as one Au atom being top capped on the 8b structure and 10a is viewed as one Au atom being capped on the generated 9a structure. C. Bimetallic Copper-Gold Clusters Cu2Aun (n = 3-10). The most stable isomers and low-lying structures for each size of Cu2Aun (n = 3-10) clusters are presented in Figure 3. It is worth pointing out that the geometric structures of Cu2Aun isomers are found to be very similar to the Ag2Aun isomers. This fact may be explained because the sliver and copper atoms have a similar electronic structure. Therefore, the differences between Cu2Aun and Ag2Aun (n = 3-10) clusters have been described in after the state. When n = 3, the 3D geometric structure (3b) of Cu2Au3 appears, but this isomer is different from the 3c isomer of the Ag2Au3 cluster. In comparison with the lowest energy structure of the Ag2Au6 isomer, the same results have been obtained. The planar D2h structure Cu2Au6 is also a transition state, and then the D2 symmetry structure 6b is obtained after further calculation. However, it is different that the 6b is not the lowest energy isomer of Cu2Au6 clusters because its total energy is higher than that of the 3D structure (6a) by 0.09 eV. For Cu2Au7 clusters, the lowest energy isomer changes a 3D structure with C2v symmetry, which is obtained by adding a Au atom on the 6a isomer. The substituted structure of the ground-state Au9 cluster (7b) is 0.07 eV less stable than the most stable 7a isomer. From the Figure 3, one can find that the low-lying isomers of Cu2Aun cluster for n = 8, 9, and 10 are consistent with the corresponding Ag2Aun clusters. Comparisons to the single copper atom doped species,48 the stable structure of Cu2Aun is generated after one gold atom of the CuAunþ1 cluster is substituted by one copper atom.

Figure 2. Lowest energy and low-lying structures of the Ag2Aun (n = 1-10) clusters. The corresponding point-group symmetries, electronic states, and relative energies are shown by PW91P86 level.

(2b) is energetically less favorable than the lowest energy D2h rhombus 2a structure. When one Au atom is capped on the 1b structure, the Cs symmetry isomer (2d) is obtained. Starting from n = 3, the geometric structures of the Ag2Aun (n = 1-10) cluster 571

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3.2. Averaged Bonding Energies and Stabilities. To predict and compare relative stabilities of the most stable Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters, it is significant to investigate the averaged atomic binding energy Eb(n), the fragmentation energy D(n,n-1), and the second-order difference of energy Δ2E(n). For Aun clusters, Eb(n), D(n,n-1), and Δ2E(n) are defined as the following formula:

nEðAuÞ - EðAun Þ n Dðn; n - 1Þ ¼ EðAun - 1 Þ þ EðAuÞ - EðAun Þ Eb ðnÞ ¼

Δ2 EðnÞ ¼ EðAun - 1 Þ þ EðAun þ 1 Þ - 2EðAun Þ

ð1Þ ð2Þ ð3Þ

where E(Aun-1), E(Au), E(Aun), and E(Aunþ1) denote the total energy of the Aun-1, Au, Aun, and Aunþ1 clusters, respectively. For M2Aun clusters, Eb(n), D(n,n-1), and Δ2E(n) are defined as the following formula: Eb ðnÞ ¼

2EðMÞ þ nEðAuÞ - EðM2 Aun Þ nþ2

Dðn; n - 1Þ ¼ EðM2 Aun - 1 Þ þ EðAuÞ - EðM2 Aun Þ

ð4Þ ð5Þ

Δ2 EðnÞ ¼ EðM2 Aun - 1 Þ þ EðM2 Aun þ 1 Þ - 2EðM2 Aun Þ ð6Þ where E(M2Aun-1), E(Au), E(M), E(M2Aun), and E(M2Aun-1) denote the total energy of the M2Aun-1, Au, M, M2Aun, and M2Aunþ1 clusters, respectively. The calculated Eb(n), D(n,n-1), and Δ2E(n) values of the most stable Aun and M2Aun clusters against the corresponding number of the Au atoms are plotted in Figure 4. As shown in Figure 4, the averaged atomic binding energy, the fragmentation energy, and the second-order difference of energy of the Aun and M2Aun clusters show an even-odd alternation phenomenon with the cluster size. It indicates that the clusters, with even number atoms, keep higher stability compared with their neighboring clusters. The results for bare Aun clusters agree with the previous works by Li et al.30 In addition, it is found that the averaged atomic binding energy reveals a growing tendency with the size increasing for the M2Aun (M = Ag, Cu; n = 1-10) clusters. There are two visible peaks in the curve at n = 2 and 4, which hints that the M2Au2 and M2Au4 isomers may be the most stable structures for the M2Aun clusters. In view of the fragmentation energy and the HOMO-LUMO gap of M2Aun clusters, the particularly most stable geometry of Ag2Aun clusters can be assigned to the Ag2Au4 isomer because it is slightly higher in fragmentation energy than the Ag2Au2. However, the most stable geometry of Cu2Aun clusters is the Cu2Au2 isomer. Lastly, one can see that the averaged atomic binding energies for each size of Aun and M2Aun clusters exhibit a sequence of Cu2Aun > Aunþ2 > Ag2Aun except for Au4 and Au12 isomers, which indicates that the stability of copper-gold clusters is higher than that of bare gold clusters and then the bare gold clusters is more stable than the sliver-gold isomers. It is worth noting that this sequence is in agreement with the relation of dissociation energies for the CuAu, Au2, and AgAu molecules. 3.3. HOMO-LUMO Gaps and Charge Transfer. The HOMOLUMO energy gap reflects the ability for electrons to jump from occupied orbital to unoccupied orbital. In addition, a large gap corresponds to a high strength required to perturb the electronic structure; namely, a bigger gap indicates a weaker chemical reactivity. Therefore, HOMO-LUMO gaps for each of the most stable Aunþ2 and M2Aun

Figure 3. Lowest energy and low-lying structures of the Cu2Aun (n = 1-10) clusters. The corresponding point-group symmetries, electronic states, and relative energies are shown by PW91P86 level.

In conclusion, the optimized geometries reveal that the M2 substituted Aunþ2 clusters and one Au atom capped M2Aun-1 structure for different sized M2Aun (M = Ag, Cu; n = 1-10) clusters are dominant growth patterns. 572

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Table 2. HOMO-LUMO Gap and Average Static Polarizability of the Most Stable Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) Clusters HOMO-LUMO gap (eV)

polarizability (au)

Aunþ2

Ag2Aun

Cu2Aun

Aunþ2

Ag2Aun

Cu2Aun

n=1

0.29

0.36

0.40

138.62

133.20

114.68

n=2

1.00

1.87

1.89

156.06

144.82

130.76

n=3

0.28

0.33

0.33

200.25

195.37

178.44

n=4

2.20

2.42

1.84

231.85

223.00

209.64

n=5 n=6

0.22 1.61

0.26 2.01

0.24 1.78

294.25 317.97

285.15 304.73

269.17 280.19

n=7

0.18

0.17

0.19

391.36

398.42

322.87

n=8

1.35

1.54

1.45

437.25

431.68

414.57

n=9

0.15

0.19

0.17

499.57

406.01

395.38

n = 10

0.93

1.14

1.28

545.74

443.75

433.52

isomers

Figure 5. Size dependence of the HOMO-LUMO gaps for the lowest energy structures of Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters.

the most stable M2Aun (M = Ag, Cu; n = 1-10) systems are investigated, and the results are listed in Table 3. As shown in Table 3, the MP values for the M atoms in the M2Aun (M = Ag, Cu; n = 1-10) clusters are positive, indicating that the charge in the corresponding cluster transfer from M atoms to the Aun frames due to larger electronegativity of the Au than that of the M atom. This feature coincides with the experimental results of Weis et al.37 Exclusively, the MP values of the Cu atoms for Cu2Au8 isomers have -0.15 e-charges, which means that the Au atoms act as electron donor in Cu2Au8 isomers. In addition, the natural electron configurations of the most stable structures for M2Aun (M = Ag, Cu; n = 1-10) clusters are listed in Table 4. As seen in the table, we find that 9.90-9.97 electrons and 9.829.91 electrons occupy the 4d subshell of Ag atoms in Ag2Aun clusters and 3d subshell of Cu atoms in Cu2Aun isomers, respectively. The values reveal that the d obitals of M atoms in M2Aun clusters are dominant core obitals. The order of their ability to attract electrons is Ag > Cu, the same as the order of their oxidizing character. 3.4. Vertical Ionization Potential, Electron Affinity, and Chemical Hardness. Chemical hardness has been established as an electronic quantity which may be applied in characterizing the relative stability of molecules and aggregate through the

Figure 4. Size dependence of averaged atomic binding energies, fragmentation energies, and second-order difference of energies for the lowest energy structures of Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters.

(M = Ag, Cu; n = 1-10) isomers are tabulated in Table 2, and the relationships of gaps as well as the corresponding cluster size are plotted as the curves in Figure 5. As shown in the figure, one can note that the HOMO-LUMO gaps present a similar oscillating behavior. Namely, the clusters with an even number of atoms have a weaker chemical activity due to their larger gaps compared with their neighbors. Besides, we find the clusters Ag2Au4 and Cu2Au2 for different M2Aun systems have the largest HOMO-LUMO gaps of 2.42 and 1.89 eV, respectively. It means that Ag2Au4 and Cu2Au2 isomers possess dramatically enhanced chemical stability. Therefore, the Ag2Au4 and Cu2Au2 clusters can be seen as the most stable building blocks and can be selected as candidates of novel nanomaterials. The net Mulliken populations (MPs) on the M2Aun (M = Ag, Cu; n = 1-10) clusters can provide reliable charge-transfer information. Here, the Mulliken populations of the two M atoms in 573

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Table 3. Mulliken Charge Populations of the Different M Atoms of M2Aun (M = Ag, Cu; n = 1-10) Clusters, Where M(1) and M(2) Correspond to the Top (or Left) M and bBottom (or Right) M Atoms in Figures 2 and 3

Ag2Aun

Aunþ2

Ag(1)

Ag(2)

isomers

Ag2Au

0.2362

0.2362

Cu2Au

0.2763

0.2763

Ag2Au2

0.4821

0.4821

Cu2Au2

0.5453

0.5453

n=1

4.90 1.83 6.73 5.51 1.66 6.81 5.14 3.73 8.78

7.50

0.8703

n=2

6.24 1.91 8.15 7.03 1.70 8.20 5.52 2.80 8.32

8.60

0.2974

n=3

4.50 3.01 7.51 4.66 2.97 7.63 4.49 3.39 7.88

8.00

0.5028

n=4

6.81 2.18 8.99 6.47 2.21 8.68 6.44 2.39 8.83

8.80

n=5 n=6

4.11 3.50 7.61 4.13 3.43 7.56 4.00 3.54 7.54 5.47 2.76 8.23 5.63 2.41 8.04 5.61 3.06 8.68

7.80 8.65

n=7

3.54 3.42 6.96 3.78 2.92 6.70 3.66 3.73 7.39

7.15

0.1425

n=8

4.89 3.00 7.89 4.88 2.95 7.82 4.70 3.19 7.89

8.20

0.9882

n=9

3.69 3.20 6.89 3.68 3.47 7.15 3.40 3.97 7.37

7.28

n = 10

4.99 3.21 8.20 4.92 2.53 7.45 4.09 3.59 7.68

8.15

0.3168

Ag2Au4

0.8421

0.3727

Ag2Au5

0.3727

0.4301

0.4929

Cu2Au3

Cu(2)

Cu2Aun

isomers

Ag2Au3

Cu(1)

Table 5. Vertical Ionization Potential, Vertical Electron Affinity, Chemical Hardness of the Most Stable Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) Clusters

0.4030

Cu2Au4

0.2974

Cu2Au5

0.3553

Ag2Au6

0.2984

0.2984

Cu2Au6

0.8672

0.8672

Ag2Au7 Ag2Au8

0.1837 0.0735

0.2023 0.0735

Cu2Au7 Cu2Au8

1.0911 -0.1462

1.0911 -0.1462

Ag2Au9

0.5942

Ag2Au10

0.5942

0.5164

0.5164

Cu2Au9

1.1674

Cu2Au10

0.9882

cluster size

η

VEA VIP

η

VEA VIP

η

VEA VIP VIP 29

Table 4. Natural Electronic Configurations of the M Atoms in the Most Stable M2Aun (M = Ag, Cu; n = 1-10) Systems, Where M(1) and M(2) Correspond to the Top (or Left) M and Bottom (or Right) M Atoms in Figures 2 and 3 Ag2Aun Ag (1) isomers 5s

4d

Cu2Aun Ag (2)

5p

5s

4d

Cu (1) 5p

4s

3d

4p

Cu (2) 4s

3d

4p

n=1

0.75 9.97 0.10 0.75 9.97 0.10 0.76 9.91 0.14 0.76 9.91 0.14

n=2

0.53 9.96 0.21 0.53 9.96 0.21 0.55 9.91 0.28 0.55 9.91 0.28

n=3

0.61 9.95 0.15 0.61 9.93 0.35 0.61 9.90 0.21 0.65 9.84 0.50

n=4

0.61 9.94 0.27 0.61 9.94 0.27 0.63 9.87 0.37 0.63 9.87 0.37

n=5

0.61 9.92 0.44 0.60 9.93 0.28 0.65 9.82 0.68 0.63 9.85 0.38

n=6 n=7

0.58 9.93 0.33 0.58 9.93 0.33 0.53 9.86 0.74 0.53 9.86 0.74 0.60 9.95 0.08 0.53 9.92 0.38 0.49 9.85 0.87 0.49 9.85 0.87

n=8

0.53 9.90 0.70 0.53 9.90 0.70 0.53 9.83 0.78 0.53 9.83 0.78

n=9

0.43 9.94 0.30 0.43 9.94 0.30 0.48 9.82 1.21 0.42 9.87 0.51

Figure 6. Size dependence of the polarizabilities for the lowest energy structure of Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters. The linear fitting of polarizability for Aun clusters is shown in inset.

n = 10 0.42 9.93 0.55 0.42 9.93 0.55 0.56 9.85 0.66 0.56 9.85 0.66

fragmentation energy, second-order difference of energy, and HOMO-LUMO energy gap. 3.5. Polarizability. It is known that the static polarizability is a measure of the distortion of the electronic density and the information about the response of the system under the effect of an external static electric field. So it is very sensitive to the delocalization of the valence electrons and the structures. The average static polarizability is defined as30

principle of maximum hardness (PMH) proposed by Pearson.49 On the basis of a finite-difference approximation and the Koopmans theorem,50 the chemical hardness (η) is expressed as η ¼ VIP - EA

ð7Þ

where VIP is the vertical ionization potential and EA is vertical electron affinity.51,52 Therefore, the vertical ionization potential, electron affinity, and chemical hardness of the most stable Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters are calculated and listed in Table 5. From the table, one can see that the values of VIP, VEA, and η for each cluster show an obvious oscillating behavior with the increasing cluster size. Through the MHP of chemical hardness, the behaviors indicate that the even-numbered isomers with higher hardness are more stable than their neighboring odd-numbered isomers. It is worth noting that the calculated VIP values of pure gold cluster agree well with experimental data as expected.29 In addition, the Ag2Au4 and Cu2Au2 isomers show the largest chemical hardness of 6.81 and 7.03 eV listed in Table 4. This trend and the maxima values of chemical hardness are also in accord with the above analysis based on the

Æaæ ¼ ðaxx þ ayy þ azz Þ=3

ð8Þ

In light of previous research,53-56 we can find that the polarizability dependences on the basis set. Considering the structure optimization and time-consumption, the results of polarizability for our systems are obtained using LanL2DZ basis set. For the Au, Ag, and Cu atoms, our calculated values are 36.3, 51.3, and 48.4 au, respectively. These are in agreement with the reported experimental estimates 39.1 ( 9.8, 57.5 ( 14.4, and 41.2 ( 10.3 au.57 It is seen that the value of the Au atom is smaller than those of Ag and Cu atoms, which is caused by enhanced screening of the s electron by the d electrons in the former. In the current work, the polarizabilities of small size 574

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The Journal of Physical Chemistry A gold, silver-gold, and copper-gold clusters are shown in Figure 6. As exhibited in Figure 6, one can find that the polarizabilities of the gold cluster increase as a function of size n, modulated by this linearity, up to n = 13. This trend of variation of polarizabilities with size for gold clusters is similar to the water clusters reported by Ghanty and Ghosh.53 Overall, the values of Æaæ for silver-gold and copper-gold clusters display the same trend as that of the gold cluster when n < 6. Whereas, the shape transitions of the silver-gold at n = 6, 9, and 10 and copper-gold clusters at n = 6, 7, 9, and 10 are reflected by a significant decrease in comparison with that of the gold clusters. This may be caused by the reason these M2Aun (M = Ag, Cu) isomers are threedimensional structures but the corresponding Au8, Au9, Au10, and Au12 isomers are planar structures. In addition, it is noted that the polarizabilities of small size gold, silver-gold, and copper-gold clusters exhibit a sequence of Cu2Aun < Ag2Aun < Aunþ2, which means that the copper-gold clusters are more stable than the sliver-gold and pure gold clusters. Furthermore, from Figure 6, one can see that the polarizabilities of small size gold, silver-gold, and copper-gold clusters show slight oscillating behaviors with the cluster size increasing. In comparison with the VIP and hardness, the inverse correlations of the polarizabilities vs the ionization potential and hardness of our investigated systems are found, which agree with the results of lithium and sodium metal clusters by the Ghanty group.55

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’ ASSOCIATED CONTENT

bS

Supporting Information. Table showing higher energy isomer structures. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel./fax: þ86 28 85405515. E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the Doctoral Education Fund of Education Ministry of China (Grant No. 20050610011) and the National Natural Science Foundation of China (Grant Nos. 10774103 and 10974138). ’ REFERENCES (1) Pal, R.; Wang, L. M.; Huang, W.; Zeng, X. C. J. Am. Chem. Soc. 2009, 131, 3396. (2) Eachus, R. S.; Marchetti, A. P.; Muenter, A. A. Annu. Rev. Phys. Chem. 1999, 50, 117. (3) Zhao, Y.; Li, Z. Y.; Yang, J. L. Phys. Chem. Chem. Phys. 2009, 11, 2329. (4) Hou, S. M.; Zhang, J. X.; Li, R.; Ning, J.; Han, R. S.; Shen, Z. Y.; Zhao, X. Y.; Xue, Z. Q.; Wu, Q. D. Nanotechnology 2005, 16, 239. (5) Scaffardi, L. B.; Pellegri, N.; de Sanctis, O.; Tocho, J. O. Nanotechnology 2005, 16, 158. (6) Fournier, R. J. Chem. Phys. 2001, 115, 2165. (7) Yuan, D. W.; Wang, Y.; Zeng, Z. J. Chem. Phys. 2005, 122, 114310. (8) Zhao, S.; Ren, Y. L.; Ren, Y. L.; Wang, J. J.; Yin, W. P. J. Phys. Chem. A 2010, 114, 4917. (9) Torres, M. B.; Fernandez, E. M.; Balbas, L. C. J. Phys. Chem. A 2008, 112, 6678. (10) Hashmi, A. S. K.; Loos, A.; Littmann, A.; Braun, I.; Knight, J.; Doherty, S.; Rominger, F. Angew. Chem. 2009, 351, 576. (11) Neumaier, M.; Weigend, F.; Hamper, O.; Kappes, M. M. J. Chem. Phys. 2006, 125, 104308. (12) Autschbach, J.; Hess, B. A.; Johansson, M. P.; Neugebauer, J.; Patzschke, M.; Pyykk€o, P.; Reiher, M.; Sundholm, D. Phys. Chem. Chem. Phys. 2004, 6, 11. (13) Ackerson, C. J.; Jadzinsky, P. D.; Jensen, G. J.; Kornberg, R. D. J. Am. Chem. Soc. 2006, 128, 2635. (14) Shaw, C. F., III. Chem. Rev. 1999, 99, 2589. (15) Valden, M.; Lai, X.; Goodman, D. W. Science 1998, 281, 1647. (16) Felix, C.; Sieber, C.; Harbich, W.; Buttet, J.; Rabin, I.; Schulze, W.; Ertl, G. Phys. Rev. Lett. 2001, 86, 2992. (17) Kim, S.-H.; Medeiros-Ribeiro, G.; Ohlberg, D. A. A.; Williams, R. S.; Heath, J. R. J. Phys. Chem. B 1999, 103, 10341. (18) Leopold, D. G.; Ho, J.; Lineberger, W. C. J. Chem. Phys. 1987, 86, 1715. (19) Howard, J. A.; Sutcliffe, R.; Mile, B. Chem. Phys. Lett. 1983, 94, 561. (20) Moskovits, M. Chem. Phys. Lett. 1985, 118, 111. (21) Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979. (22) Morse, M. D. Chem. Rev. 1986, 86, 1049. (23) Rohlfing, E. A.; Valentini, J. J. J. Chem. Phys. 1986, 84, 6560. (24) Ackerman, M.; Stafford, F. E.; Drowart, J. J. Chem. Phys. 1960, 33, 1784. (25) Beutel, V.; Kr€amer, H. G.; Bhale, G. L.; Kuhn, M.; Weyers, K.; Demtr€oder, W. J. Chem. Phys. 1993, 98, 2699. (26) Balbuena, P. B.; Derosa, P. A.; Seminario, J. M. J. Phys. Chem. B 1999, 103, 2830. (27) Fabbi, J. C.; Langenberg, J. D.; Costello, Q. D.; Morse, M. D.; Karlsson, L. J. Chem. Phys. 2001, 115, 7543. (28) Bishea, G. A.; Pinegar, J. C.; Morse, M. D. J. Chem. Phys. 1991, 95, 5630.

4. CONCLUSION The geometrical structures, growth behaviors, stabilities, HOMO-LOMO energy gaps, charge transfers, chemical hardnesses, and polarizabilities of the Aunþ2 and M2Aun (M = Ag, Cu; n = 1-10) clusters are investigated theoretically at the PW91P86 level. All of the calculated results are summarized as follows. (1) The optimized geometries reveal that the M2 substituted Aunþ2 clusters and one Au atom capped M2Aun-1 structure for M2Aun (M = Ag, Cu; n = 1-10) clusters are dominant growth patterns. (2) The averaged atomic binding energy, the fragmentation energy, and the second-order difference of energy of the most stable M2Aun clusters exhibit an oscillatory behavior as a function of cluster size. According to our analysis, it is concluded that the small Ag2Au4 and Cu2Au2 isomers are the most stable geometries for Ag2Aun and Cu2Aun clusters, respectively. (3) On the basis of the calculated Mulliken populations, it is noticed that the charges are transferred from M atoms to the gold frames. In addition, the HOMO-LUMO energy gap, vertical ionization potential, vertical electron affinity, and chemical hardness show an obvious oscillating behavior with the cluster size increasing. The results also reveal that Ag2Au4 and Cu2Au2 isomers are the most stable structures for Ag2Aun and Cu2Aun clusters. (4) The polarizabilities of the small size silver-gold and copper-gold clusters display the same linear tendency with the gold clusters except for the 3D structures at n = 6, 7, 9, and 10. The values of polarizabilities exhibit a sequence of Cu2Aun < Ag2Aun < Aunþ2 for our investigated systems. Otherwise, the inverse correlations of the polarizabilities vs the ionization potential and hardness of our investigated systems are found. 575

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