A Theoretical Study on Small Iridium Clusters: Structural Evolution

Nov 19, 2010 - Simple cube evolution pattern is revealed for Ir2−15 clusters, ... Citation data is made available by participants in CrossRef's Cite...
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J. Phys. Chem. A 2010, 114, 12825–12833

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A Theoretical Study on Small Iridium Clusters: Structural Evolution, Electronic and Magnetic Properties, and Reactivity Predictors Jiguang Du,† Xiyuan Sun,† Jun Chen,‡ and Gang Jiang*,† Institute of Atomic and Molecular Physics, Sichuang UniVersity, Chengdu, Sichuan 610065, China, and State Key Laboratory of Surface Physics and Chemistry, Mianyang, Sichuan 621907, China ReceiVed: August 5, 2010; ReVised Manuscript ReceiVed: NoVember 8, 2010

The structural, electronic, and magnetic properties of iridium clusters with sizes of n ) 2-15 are investigated by employing the generalized gradient approximation of density functional theory. Simple cube evolution pattern is revealed for Ir2-15 clusters, as predicted by previous reports. It is remarkable that for Ir10, Ir11 clusters, new generated isomers with higher stabilities relative to those reported in previous studies are obtained. The even-sized clusters are more stable than the odd-sized species. The Ir-Ir bonds in the cubic Ir8 and Ir12 clusters, which are considered as the basic units in the structural evolution present covalent character. Starting from n ) 8, the magnetic moments of Irn clusters decrease sharply. The moments of magnetic clusters show 5d characters. The reactive site selectivity of studied clusters with n ) 5-15 is analyzed with condensed Fukui function. The capped atoms in certain clusters (Ir9, Ir10, Ir11, and Ir13) generally show extraordinary activity for both nucleophilic and electrophilic attack. 1. Introduction Several 5d transition-metal (TM) clusters, such as Pt, Au, and W, are found to be effective catalysts and present interesting cluster size effects on catalytic reactivity.1-5 Iridium, possessing the 5d76s2 valence electron configuration, is also situated among the same 5d metals as Pt, Au, and W. However, the studies on iridium as catalyst were limited to few reactivity experiments,6-11 due to the high cost of Ir resource. The catalytic behaviors of tetrairidium clusters on the solid supports (MgO) were investigated by Xiao et al.,12 and they found that more O2 molecules were chemisorbed on solid supported Ir4 than on metallic iridium particles, whereas the uptake of H2 and CO molecules on small iridium clusters was less than that on metallic Ir particles. Using a low-temperature field ion microscope, the shape and stability of Irn clusters with n ) 18-39 on the close-packed Ir (111) plane were investigated by Wang et al.13 to characterize the energetics of surface clusters. They reported that the high coordination hexagonal clusters Ir19 and Ir37 are far more stable than the less compact ones. In theory, Feng et al.14 performed Hartree-Fock (HF) calculations on Irn clusters with n ) 4, 6, 8, 10 to study the catalytic activity of selected clusters. They concluded that the reactivity of Ir4 and Ir6 clusters is size-dependent; nevertheless, the Ir8 and Ir10 clusters may be of size-independent reactivity. The structures of 8-10 group clusters (Ru, Rh, Pd, Ir, and Pt) with sizes of 4, 8, 9, 12, 13 are investigated by Zhang et al. using Density Functional Theory (DFT) calculations.15 For the Ir clusters with given sizes, the simple cubic geometries are favored. Adsorption and activation properties of small Pt, Au, and Ir clusters toward to NO molecule are investigated by Endou et al.16 with DFT approach. They found that the order of the energetical stability of the adsorption states of NO is Ir > Pt > Au and dependent on the neither the geometries of pentamers nor the cluster sizes considered in the reference. The CO, O2, * Corresponding author. E-mail: [email protected]. † Sichuang University. ‡ State Key Laboratory of Surface Physics and Chemistry.

and NO molecules and the cubooctahedral model of Ir13 cluster were selected as the adsorbates and model cluster, respectively, to investigate the catalytic reactivity of Ir13 clusters.17 The charge transfers between the surface atoms of cubooctahedral Ir13 cluster and the adsorbates indicated the strong interaction between them. Bussai et al.18 investigated the structures and binding energies of various isomers of Ir4 cluster and its complexes with one H atom based on relativistic density functional theory. As mentioned above, the previous reports on iridium clusters are limited to either selected sizes or examining only simple geometries. There are very few studies19 devoted to the understanding of the structural and electronic properties of the isomers of small iridium clusters. The insights of geometrical and electronic structures of small Ir clusters are essential for further grasp of catalytic reactions on Ir clusters. Therefore, in this Article, we expect to determine the low-lying structures of Irn clusters (n ) 2-15) by employing DFT approach and to find the structural evolution and stability of small iridium clusters. We also perform the analyses on density of states (DOS) and HOMO-LUMO gaps to gain insights into the electronic structures for the lowest-energy geometries of Ir2-15 clusters. The magnetic moments of small Ir clusters are predicted, which can be utilized in further experimental comparison. Additionally, the reactivity sites of several groundstate structures are determined on the basis of condensed Fukui function. 2. Computational Details All of our spin-unrestricted calculations are carried out with the Dmol3 package.20 The DFT method BPW91 under generalized gradient approximation (GGA), in which the Becke’s exchange functional (B)21 married with the correlation approach of Perdew and Wang (PW91),22 was used to estimate the exchange-correlation energy. This BPW91 functional has extensively been employed to successfully describe other transition metal clusters.23,24 Iridium atom has as many as 77 electrons; the all-electron relativistic calculations are time-consuming.

10.1021/jp107366z  2010 American Chemical Society Published on Web 11/19/2010

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Therefore, the DFT-based semicore pseudopotential (DSPP)25 including scalar relativistic effects, which is generated by fitting all-electron relativistic DFT results, is employed in our calculations. With the double numerical basis set augmented with polarization p-function (DNP),20 which has a computational precision being comparable with split-valence basis set 6-31g**,26 an accurate description for the valence electrons of Ir (5s25p65d76s2) could be obtained. In the generation of the numerical basis sets, a large global orbital cutoff of 6.0 Å is used. The geometry optimizations are performed with the convergence criterion as follows: 10-5 au for the total energy, 10-6 for electron density, and 2 × 10-3 and 5 × 10-3 au for gradient of force and atomic displacement, respectively. The higher criteria (1 × 10-4 au for gradient of force and atomic displacement) are utilized for several selected isomers, like Ir8, Ir10, and Ir11clusters. We choose the octupole scheme for multipolar expansion of charge density. Moreover, the direct inversion in a subspace method (DIIS) developed by Pulay27 is utilized to speed up the self-consistent field convergence. A large number of configurations have been explored during the geometry optimization. Details of the structure generation used in this work can be found in our previous report.28 The different spin states for a given configuration are also optimized to investigate the magnetic moments. The spin contamination may be induced when performing the spin-unrestricted calculations. Yet previous literature29,30 has shown that the Kohn-Sham orbitals provide the wave function with a much smaller spin contamination than does the unrestricted Hartree-Fock scheme. The spin contamination problem is of minor importance in our calculation. The vibrational frequencies are calculated to guarantee that the optimized structures correspond to local minima on the potential energy surface. For several clusters, some calculations are also performed with meta-GGA TPSSTPSS functional31 and LANL2DZ32 as implemented in the Gaussian 03 program33 for comparison. The stabilities of Ir2-15 clusters are estimated with binding energies per atom (Eb/atom), second-order differences of total energies (∆2E), and formation energies (Ef), and these quantities are evaluated using the following expressions:

Eb /atom (n) ) (nE[Ir] - E[Irn])/n

(1)

∆2E (n) ) E[Irn+1] + E[Irn-1] - 2E[Irn]

(2)

Ef (n) ) Eb[Irn+1] - Eb[Irn]

(3)

where E is the total energy of corresponding system with zeropoint energy corrections and Eb is the total binding energy evaluated with Eb ) (nE[Ir] - E[Irn]). 3. Results and Discussion A. Geometrical Isomers. The relative energies ∆E (with zero-point energy corrections) and coordinates of all optimized isomers are shown in Table S1 in the Supporting Information. These isomers are all confirmed as local minima with none imaginary frequency. According to the stability from low to high, the isomers are designated by na, nb, nc, and so on, where n is the Ir atom number. The molecular properties of these isomers, point symmetry, coordination numbers (CN), average bond lengths (Rav), binding energies per atom (Eb/atom), and atomic magnetic moments (M/atom), can be found in Table 1. There are many isomers obtained in the geometry optimization,

TABLE 1: Point Group (PG), Coordination Numbers (CN), Average Bond Lengths (Rav), Bonding Energy (Eb), and Atomic Magnetic Moments (M/atom) of Different Isomers of Small Iridium Clusters Rav Eb M/atom Rav Eb M/atom isom PG CN (Å) (eV) (µb) isom PG CN (Å) (eV) (µb) 2 3a 3b 4a 4b 4c 5a 5b 5c 6a 6b 6c 6d 7a 7b 7c 7d 7e 8a 8b 8c 8d 9a 9b 9c 9d 9e 9f 9g 10a 10b 10c

D∞h D3h C2V D4h Td C2V C4V Cs D3h D3h C2 Cs C2V C2V C3V C2 C3V C2V Oh D2h Cs D3d Cs Cs D3h Cs D3h C2 C2 C2V C2V C2V

1.0 2.0 2.0 2.0 3.0 2.5 3.2 2.4 3.6 3.0 4.0 3.7 4.0 3.7 3.4 3.4 4.3 4.3 3.0 3.5 3.5 4.5 3.1 3.6 4.6 4.6 4.7 3.6 4.0 3.4 3.0 4.8

2.27 2.44 2.41 2.40 2.53 2.46 2.51 2.43 2.56 2.47 2.57 2.54 2.58 2.56 2.53 2.53 2.58 2.62 2.42 2.51 2.51 2.59 2.45 2.52 2.60 2.59 2.60 2.58 2.56 2.48 2.43 2.60

1.69 2.28 2.24 2.82 2.76 2.66 3.06 3.02 2.99 3.40 3.27 3.22 3.17 3.50 3.46 3.42 3.41 3.30 3.84 3.66 3.59 3.49 3.86 3.81 3.72 3.71 3.69 3.69 3.65 3.99 3.93 3.91

2.00 1.00 0.33 2.00 0.00 0.50 1.00 1.00 0.60 1.00 0.67 1.00 1.00 1.57 1.29 1.00 0.14 1.57 0.00 0.75 0.00 0.00 0.33 0.78 0.33 0.11 1.22 0.33 0.78 0.40 0.40 0.20

10d 10e 10f 10 g 11a 11b 11c 11d 11e 12a 12b 12c 12d 13a 13b 13c 13d 13e 13f 13g 14a 14b 14c 14d 14e 14f 15a 15b 15c 15d 15e 15f

C2V D5h C2V C2V Cs C2V D3h Cs Cs D4h C2V D3h D2h C1 Cs Cs Cs Cs C4V Cs Cs C2V Cs C1 C2V C1 Cs C1 Cs Cs Cs Cs

3.8 3.0 4.4 4.4 3.6 3.1 3.5 4.2 4.7 3.3 3.7 4.0 3.8 3.4 3.7 3.4 4.0 4.3 3.7 4.2 3.6 3.9 4.1 3.3 3.6 3.3 4.4 4.1 3.9 3.9 3.6 3.6

2.52 2.43 2.58 2.61 2.52 2.44 2.56 2.55 2.59 2.45 2.49 2.53 2.53 2.46 2.50 2.46 2.49 2.56 2.50 2.55 2.48 2.51 2.54 2.45 2.48 2.45 2.55 2.54 2.51 2.51 2.48 2.48

3.88 3.84 3.81 3.79 4.02 3.97 3.96 3.92 3.90 4.25 4.16 4.15 4.02 4.21 4.20 4.20 4.16 4.16 4.14 4.13 4.30 4.26 4.22 4.22 4.21 4.20 4.28 4.27 4.26 4.26 4.25 4.24

0.60 0.00 0.40 0.40 0.27 0.27 0.45 0.27 0.09 0.17 0.17 0.17 0.50 0.23 0.23 0.23 0.08 0.08 0.38 0.08 0.14 0.14 0.14 0.14 0.14 0.29 0.07 0.07 0.20 0.07 0.20 0.20

yet for the sake of simplicity, only selected isomers with small relative energy are discussed in the following. To help the readers find the atomic reactivity site in the analysis of section E, the lowest-energy geometries with number labels for n ) 5-15 are depicted in Figure 1. There is no direct spectroscopic information for the bond length and vibrational frequency of Ir2 dimer in experiments. However, Lombardi et al.34 indirectly obtained the internuclear distances and frequencies of transition-metal dimers from a fit of force constants with dissociation energies. They obtained that the bond length and frequency of Ir2 dimer were 2.23 Å (with Pauling’s rule) and 280 cm-1, respectively. Our calculations show that the ground quintet state of Ir2 dimer corresponds to the bond length of R ) 2.27 Å, frequency of ωe ) 278 cm-1, fitting well with the results of Lombardi et al. The dissociation energy of Ir2 dimer in the present work is 3.38 eV, smaller than the binding energy of 5.06 eV obtained by Pawluk et al.19 at the PW91 level with the VASP program, but in excellent agreement with the experimental values: 3.46 ( 0.12 eV35 and 3.7 ( 0.7 eV36 (recommended in Morse’s review). Our calculated bond length, frequency, and dissociation energy of Ir2 dimer are also in agreement with previous theoretical results37,38 based on systemic DFT calculations. For Ir3 trimer, the equilateral triangle (3a) with quartet spin configuration is 0.13 eV more stable than the isosceles one (3b) with doublet state. The Fe3 and Ni3 trimers39 (first-row TM atoms, just above iridium) also favor equilateral triangle. The calculated vibrational frequencies of Ir3 are 181, 181, and 263 cm-1, about 100 cm-1 lower than that of Fe3 trimer (251, 279, and 458 cm-1).39 A

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Figure 1. The lowest-energy geometries with number labels of Ir5-15 clusters. The typical Fukui indices are also depicted in the figure.

planar D4h structure (4a) with atomic magnetic moment of 2.0 µb/atom is obtained as the ground state of the Ir4 cluster. However, the distorted tetrahedron is preferred for first-row tetramers (Fe4, Co4, and Ni4).39 The frequencies of the planar isomer are in the range of 104-237 cm-1, also much smaller than that of Fe4 (187-413 cm-1). One regular tetrahedron (Td), corresponding to ∆E ) 0.21 eV, comes next in energy. The butterfly-like structure 4c (triplet state) with ∆E ) 0.64 eV is also found to be stable from frequency analysis. We obtained three isomers in geometry optimization of Ir5 cluster. The square pyramid one (5a, C4V, 1.0 µb/atom) is 0.23 eV more favored in energy relative to the 5b isomer with one Ir atom capped on the bridge site of square (Cs, 1.0 µb/atom). The trigonal bipyramid (TBP) structure existing in the quartet state, which is generally most favorable for first-row pentamers (Fe5, Co5, and Ni5) and other 5d TM pentamers (Hf5,40 W5,28 Ta541), is however found to have the highest energy among the three isomers of Ir5 cluster. The lowest-energy geometry of Ir5 also shows lower frequencies of 114-248 cm-1, in comparison with Fe5 pentamer39 (105-397 cm-1). Ir6 cluster prefers a triangular prism with D3h symmetry and a total magnetic moment of 6.0 µb as the ground state. One distorted octahedron (6b) with quintet state and ∆E ) 0.82 eV comes next in energy. Both capped square pyramid (6c) and TBP (6d) configurations exist in septet state and have less stability. By capping one Ir atom on the square unit of triangle prism (6a), we obtain the lowest-energy structure (7a) of Ir7 cluster with atomic moment of 1.57 µb. The triangle capped prism (7b) with higher symmetry (C3V) and lower atomic moment of 1.29 µb/atom is obtained as the first excited state. The energy gap between 7a and 7b isomers is just 0.26 eV. Other isomers, bicapped square pyramid (7c, octet state) and capped octahedron (7d, doublet state), are energetically 0.58 and 0.63 eV, respectively, above the square capped prism structure (7a). The compact pentagonal bipyramid (PBP) con-

figuration with atomic moment of 1.57 µb and C2V symmetry has the lowest stability among all optimized isomers, corresponding to the relative energy up to 1.37 eV. The 8-atom iridium cluster favors a perfect nonmagnetic cubic structure (Oh) with side length of 2.42 Å as the ground state. The optimized side distance (2.42 Å) in the present work is well comparable with previous calculated value of 2.37 Å.15 This cube has binding energy as high as 3.84 eV. Earlier studies on Rhn clusters also indicated that Rh8 corresponds to one cube with side of 2.40 Å.42 Other isomers, 8b, 8c, and 8d, are 1.46, 2.06, and 2.82 eV, respectively, less stable than the cube. Among the isomers of Ir8, only the 8b exists in septet state, showing magnetism. For the case of Ir9, the edge capped cube (9a) with quartet spin state is 0.50 eV more stable than the face capped one with octet state (9b). The remaining isomers obtained in the optimization of Ir9 cluster correspond to the relative energies in the range of 1.28-1.96 eV and are overall much less stable. The 9c and 9f isomers favor the quartet state, and 9d and 9e isomers have the magnetic moments of 1.00 µb and 11.00 µb, respectively. One new structure (10a) with Ir2 dimer capped on the face of cube is found as the ground state (quintet) of Ir10 cluster in the present work. Previously, Pawluk et al.19 reported one edge disturbed cube (10b) to have the lowest energy. In our calculation, this edge disturbed cubic structure (10b) with quintet state is found to be less stable by 0.64 eV than the 10a isomer. These two isomers are again optimized with the TPSS/ LANL2DZ scheme to guarantee the credibility of our calculation. Our results indicate that the face-capped cube (10a) is 0.63 eV more stable the edge disturbed one (10b) at TPSS/LANL2DZ level. The pyramidal structure (10c) in triplet state with relative energy less than 1 eV also possesses considerable stability. Other isomers 10d (septet), 10e (singlet), 10f (quintet), and 10 g (quintet) have relative energies of more than 1 eV. The lowestenergy structure of the 11-atom cluster (11a) with moment of

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Figure 3. The average bonding energies per atom (Eb/atom) of groundstate structures of Ir2-15 clusters as a function of cluster size. Figure 2. The comparison of IR spectrum for three degenerate isomers of Ir13 cluster.

0.27 µb/atom can be described as one additional Ir atom capped on square unit of 10a. Based on the cubic unit, the 11b (sextet) and 11c (quartet) isomers are obtained and have small relative energies of 0.53 and 0.66 eV, respectively. It is noteworthy that with the missing of 11a isomer, Pawluk et al. obtained 11b isomer as the lowest-energy structure of Ir11 cluster in the previous investigation.19 Our calculations at the meta-GGA level (TPSS/LANL2DZ) for the Ir11 cluster show that the 10a isomer is more stable than 11b by 1.01 eV. The 11d, 11e isomers constructed on the basis of rhombohedron favor quartet and doublet states, respectively, and have high relative energies of more than 1 eV. The lowest-energy structure of the 12-atom iridium cluster (12a) with a triplet state is a two-layer cubic framework, of which the binding energy is as large as 4.25 eV, even bigger than that of 13-atom cluster (4.21 eV). Other isomers of Ir12 cluster, 12b, 12c, and 12d, correspond to relative energies in the range of 1.04-2.70 eV, suggesting the outstanding stability of cubic structure of Ir12 cluster. The 12b and 12c isomers favor the same triplet state, and 12d corresponds to high septet state. Recently, Hu et al.43 and Zhang et al.44 paid their attention for 13-atom transition-metal clusters. Hu et al. obtained one square face-capped simple cube (13b) as the ground state for Ir13, while Zhang et al. predicted a lateral side-capped simple cube (13a) to be most stable. In addition, early, Pawluk et al.19 recommended one top side-capped simple cube (13c) as the lowestenergy state of Ir13 cluster. Our calculations show that the lateral side-capped simple cube (13a) is energetically favored, in agreement with Zhang et al.44 The square face-capped (13b) and top side-capped (13c) simple cubes are calculated to be almost degenerate in energy with respect to the lowest-energy isomer (13a), only higher in total energy by 0.08 and 0.17 eV, respectively. The three degenerate isomers have the same magnetic moment of 3 µb. Zhang et al. also indicated that the three isomers have a moment of 3 µb, and the energy gaps between them are within 0.3 eV. The stability is, however, predicted as 13a > 13c > 13b in the report of Zhang et al., little different from our calculation. In fact, the energy gaps between these isomers are so small that they are likely to coexist in experiment. As we know, it is meaningful to theoretically assign the ground-state structure by comparing the experimental IR spectrum.45 Therefore, in the present work, the IR spectra of these isomers being comparable in energy are theoretically simulated and shown in Figure 2. Both 13a and 13b isomers exhibit two clear peaks. The difference between them is that

the second peak of 13a isomer is presented in the higher frequency relative to 13b. The 13c isomer has very different spectra character with three distinct peaks exhibited in low frequency. The analyses of IR spectra are expected to be usefully for further experimental comparison. We also look forward to further experimental identification for ground-state structure of Ir13 clusters. Other optimized isomers (13d, 13e, 13f, and 13g) of Ir13 clusters have the relative energies ranging from 0.57 to 1.01 eV. Among these isomers, the 13f isomer corresponds to higher spin state (sextet) than do the 13d, 13e, 13 g isomers existing in doublet state. Adding Ir2 dimer on different sites of two-layer cubic structure (12a), we obtain the 14a, 14d, 14e, 14f isomers, among which the 14a isomer with binding energy of 4.30 eV is energetically favorable in triplet state. The 14b isomer with triplet state built upon 13d is less stable than 14a by 0.59 eV, which comes next in energy. The 14c, 14d, 14e, and 14f isomers possess the relative energies ranging from 1.06 to 1.30 eV. The low magnetic moment (2.0 µb) is presented for 14c, 14d, 14e isomers; however, the 14f favors a quintet state. As for the Ir15 cluster, the 15b, 15c, 15d, and 15e isomers are obtained by adding one Ir atom on different sites of the lowestenergy structure (14a) of the Ir14 cluster, 0.23-0.50 eV less stable relative to the ground-state structure 15a constructed on the basis of the 14c isomer. The 15b and 15d isomers correspond to doublet spin state, and for 15c and 15e isomers, the quartet state is favored. The lowest-energy isomer (15a) with binding energy of 4.28 eV favors doublet spin multiplicity. The 15f isomer existing in quartet state is also built upon 14c, and 0.55 eV above the lowest-energy structure. B. Stability. We plot the average binding energies per atom (Eb) of the lowest-energy structures of Ir2-15 clusters as a function of cluster size in Figure 3, from which one can see that the calculated Eb increases monotonically as cluster sizes evolve. Slight lifts corresponding to even-number clusters are found in the Eb curve of Ir2-15 clusters. The binding energies (Eb) versus 1/n are also plotted in the inset of Figure 3. As we know, the Eb will converge to the cohesive energy of bulk Ir when cluster size n grows to an infinity value (lim 1/nf0). By simply fitting our calculated binding energies of Ir2-15 clusters, one polynomial expression (5.1-11.7 × [1/n] + 9.8 × [1/n]2) is obtained. It is clear that the extreme Eb ) 5.1 eV is far from the experimental cohesive energy per atom of Ir bulk (6.94 eV),46 implying that simple cubic growth patters will be invalid and significantly structural changes will be present for larger iridium clusters. Our finding agrees well with the experimental result,13 which indicates that the high coordination hexagonal clusters Ir19 and Ir37 are far more stable than the less compact ones. The relative stabilities of the lowest-energy structures of

Theoretical Study on Small Iridium Clusters

Figure 4. The second-order difference of total energy (∆2E) (a) and formation energy (Ef) (b) of the lowest-energy structures of studied iridium clusters as a function of cluster size.

studied iridium clusters are investigated in terms of ∆2E of Ef depicted in Figure 4. Small iridium clusters show obvious odd-even oscillations in stability. The even-numbered clusters are generally more stable than odd-numbered ones, especially for Ir8 and Ir12 clusters with perfect cubic motifs. As we know, the electronic configuration of isolated Ir atom is 6s25d7, however, due to the electron transfers from 6s to 5d orbitals; the bonding Ir atoms in the clusters have 6s1(5d,6p)8-like electronic populations as shown in Figure 5 (the average electronic numbers per Ir atom). Therefore, the delocalized s electrons in even-number clusters favor the closed shell, being responsible for their enhanced stabilities. Previous investigation47 indicated that the Run clusters with 5s14d7 atomic configuration also show evident odd-even alternations in stability. Also, the geometrical structures of even-number clusters consisting of squares play an important role in reinforcing their stability. In addition, as shown in Figure 5, the average s and d electron numbers show clear inverse oscillation trends from Ir2 to Ir8, and incline to converge at one constant (7.6 for 5d, 1.05 for 6s) for large clusters (n > 8). This phenomenon indicates that s and d electronic structures dominate the stabilities of small iridium clusters (n < 8), whereas the geometrical structure may play a primary role in the determination of the stability for large clusters. Moreover, the 6p electrons ranging from 0.07 (Ir2) to 0.39 (Ir15) increase as the cluster sizes evolve, indicating that 6p electrons are gradually involved in the chemical bondings. Like the case of tungsten clusters,28 the s-d hybridizations are mainly formed in the bonding Ir atoms of small clusters, and the s, p-d hybridizations are presented in the larger sizes. C. Electronic Properties and Chemical Bondings. The energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is a useful parameter to study the nonmetallic-metallic transition for clusters.48 The HOMO-LUMO gaps for R (spin-up) and β (spin-down) electron states are plotted in Figure 6, from which one can see that energy gaps decrease in principle with the increase of cluster size, showing a trend of nonmetallic-metallic

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Figure 5. The average electron numbers per Ir atom of the lowestenergy structures of Ir2-15 clusters.

Figure 6. The energy gaps between HOMO and LUMO for different spin states of the lowest-energy structures for small iridium clusters.

transition. No clear odd-even oscillations are found for the HOMO-LUMO gaps of Ir2-15 clusters. Therefore, there is no direct correlation between gaps and stability. Starting from the Ir8 cluster, the gaps (below 0.4 eV for spin-up and 0.2 eV for spin-down states, respectively) sharply decline, suggesting the metallic bulk-like characteristic for larger clusters. This is expected as large clusters incline to bulk behavior. It is also noteworthy that the HOMO-LUMO gap cannot be accurately predicted within the DFT framework.49,50 Our reported results should be justified by further experiments. To further gain insights into the size-dependent electronic properties of iridium clusters, we analyze the atomic orbitalprojected partial density of states (PDOS) for representative clusters: Ir6, Ir8, and Ir12. Figure 7 shows the s, p, d-projected PDOS of these clusters. The total DOS near the Fermi level is mainly derived from the contributions of d electrons going with a small quantity of contributions from p electrons, indicating the d electrons are largely concerned in chemical relativity of

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Figure 8. The 3D (isovalue ) 0.03 au) and 2D presentations for deformation density of the most favorable cubic isomer of Ir8 and Ir12 clusters.

Figure 7. The atomic orbitals-projected partial density of states (PDOS) for selected clusters (Ir6, Ir8, and Ir12).

iridium clusters. The contributions from s electrons to total DOS are very small, even negligible near Fermi level. However, in the energy region below -6 eV, the complex s and p-d hybridization states are exhibited with the domination of s electrons. The s-domination peak of Ir12 clusters is present around -7.2 eV below those of Ir6 (-6.4 eV) and Ir8 (-6.6 eV) clusters, indicating that the energy levels gradually broaden with the increase of cluster size. The Ir8 cluster shows the more sharp peaks in DOS relative to Ir6 and Ir12 clusters. This phenomenon can be interpreted by the higher degeneracy degree of energy levels in Ir8 cluster stemming from the higher geometry symmetry. Such phenomenon suggests the structural sensitivity of electronic structures of Ir clusters. It is noteworthy that the DOS of Ir8 cluster presents the higher electron distributions near Fermi level than Ir6 and Ir12. Therefore, the 8-atom Ir cluster is expected to have high chemical reactivity in further experimental applications as catalyst. To analyze the bonding natures of iridium clusters, we calculated electron deformation density, which is denoted as the differences between the total electron density of cluster and the density of isolated atoms. For the sake of simplicity, the deformation density of typical Ir8 and Ir12 clusters with perfect simple cube is analyzed. Figure 8 displays the three-dimensional (3D) surface and 2D contour (with different plane defined) of electron deformation density for Ir8 and Ir12 clusters. It is clear from the 3D-representation that both Ir8 and Ir12 clusters present fairly uniform density with the significantly high density distributions being revealed between Ir atoms. The 2D contours with plane through square (left panel) and principal axis (right panel), respectively, also show high accumulated charge (red) between Ir atoms. These results indicate that the covalent bonding character is exhibited in the cubic structures of iridium clusters. It is noteworthy that the average bond lengths of evensized clusters ranging from 2.27 to 2.48 Å (shown in Table 1) are generally shorter than those of odd-sized clusters (2.44-2.56 Å), suggesting the stronger Ir-Ir interactions in even-sized clusters. In some odd-sized isomer with cube unit like 11a, there also exist many shorter bonds (2.41, 2.43, and 2.45 Å) relative

Figure 9. The spin gaps ∆1 and ∆2 of Ir2-15 clusters.

to the average value of 2.52 Å. This also indicates that some covalent bondings are formed in these clusters. D. Spin Stability and Magnetic Moments. Two spin gaps defined as ∆1 ) -[εHOMOmajority - εLUMOminority] and ∆2 ) -[εHOMOminority - εLUMOmajority] are calculated to estimate the magnetic stability of small iridium clusters. These quantities represent the energy required to move an infinitesimal amount of charge from the HOMO of one spin to the LUMO of the other.51 The values of both spin gaps have to be positive for the clusters to be magnetically stable. Figure 9 shows the spin gaps for the lowest-energy structures of Ir2-15 clusters, and one can see that all the studied iridium clusters are magnetic stable because ∆1 and ∆2 are both positive for these clusters. Both spin gaps show decrease trends from n ) 2 to 8. Moreover, the Ir4 and Ir7 clusters with high magnetic moments correspond to the higher spin gaps (∆1 and ∆2) in comparison with other sizes. There are no available experimental data for the magnetic moments of iridium clusters. In the present work, the magnetic moments per atom of Ir2-15 clusters shown in Figure 10 are also obtained. We expect the results can be used in further experiments for comparison. As shown in Figure 10, the atomic magnetic moments (µb/atom) of Ir2-15 clusters are closely connected with cluster sizes. Relatively high moments are found for small clusters with n < 8. There is a sharply decrease from Ir7 (1.57 µb/atom) to Ir8 (0 µb/atom) in the curve of magnetic moments. Starting from Ir10, the moments gradually decrease as the cluster sizes increase. The previous theoretical values19 are also plotted in Figure 10 for comparison. It is clear that similar trends (small-size clusters have high atomic moments) are in principle presented between Pawluk’s and our results. Figure 11 shows the atomic orbital-projected partial density of states (PDOS) of majority-spin and minority-spin for the most stable structures of typical magnetic clusters, Ir4 and Ir7. From

Theoretical Study on Small Iridium Clusters

J. Phys. Chem. A, Vol. 114, No. 49, 2010 12831 E. The Local Site Reactivity. In this section, the local reactivity of small Ir clusters is analyzed by means of Fukui indices.53 This method relates the reactivity of one chemical species with respect to nucleophilic/electrophilic attack to the charge density and is a successful way of measuring the reactivity of regions of clusters.54-56 Parr and Yang53 define the Fukui function as the partial derivative of the electron density with respect to the total number of electrons at a constant potential:

f(b) r )

Figure 10. The computed magnetic moments per atom as a function of cluster size; previous theoretical prediction values19 are also provided for the sake of comparison.

( ∂F(∂Nb)r )

υ(b) r

Because of the discontinuity of the derivative of above equation, two different Fukui functions can be defined by applying the finite difference approximation:

f+(b) r )

( ∂F(∂Nb)r )

+

f-(b) r )

( ∂F(∂Nb)r )

-

V(b) r

V(b) r

The f +(r) and f -(r) Fukui functions measure the reactivity toward a nucleophilic and electrophilic attacks, respectively. The condensed Fukui function proposed by Yang57 can be obtained with the following expression:

f+ k ) qk(N + 1) - qk(N) fk ) qk(N) - qk(N - 1)

Figure 11. The spin-resolved PDOS of typical clusters (Ir4 and Ir7) with high magnetic moments. The spin population of every atom is shown in the inset; the atoms with the same spin configuration are depicted with the same color.

this figure, one can see that the contributions from d-projected PDOS to magnetic moments of Ir4 and Ir7 clusters are dominating, whereas the s- and p-projected PDOS only contribute a small quantity of unpaired electrons. The s and p splitting, which reinforces the d moments, is primarily found in low energy regions. We also calculated the detailed atomic spin configurations based on Mulliken population,52 which are shown in the inset of Figure 11. In the Ir4 cluster, every Ir atom has the same spin configuration of 6s0.246p0.125d1.64, presenting evident d character with 82% contribution to total moments. As for the Ir7 cluster, one top four-coordinated Ir atom and other fourcoordinated Ir atom in the prism, corresponding to spin populations of 6s0.196p0.125d1.47 and 6s0.136p0.105d1.42, respectively, have the higher moments than do these three-coordinated ones (6s0.146p0.095d0.88). The 5d contribution to total moments is 81%, also substantial.

where qk is the electronic population of atom k in a certain chemical species. The values of qk can be calculated from a Mulliken population52 or from a numerical integration procedure such as Hirshfeld charge.58,59 The latter scheme is utilized in the present work. The calculated Fukui indices (FI) for Ir5-15 clusters are presented in Table 2. The numbers in this table correspond to the local sites of atoms in Figure 1. For Ir5 clusters, condensed FI indicates that the capped Ir atom (Ir1) favors nucleophilic attack such as CO molecule. The Ir4, Ir5 sites favor both nucleophilic and electrophilic attack, showing amphiphilic character. Six Ir atoms in Ir6 cluster do not show evident differences in selectivity. For the Ir7 cluster, Ir3 atom capped on the square face of prism is the most reactive site for electrophilic guest molecule (such as O2) absorption, and the Ir4 and Ir5 are good nucleophilic sites. All the atoms in Ir8 cluster except Ir6 favor amphiphilic attack. The Ir6 atom is slightly more favored for nucleophilic molecules binding. As noted from the FI of Ir9 cluster, the atom Ir9 capped on the cube is viable for both nucleophilic and electrophilic attack. Like the case of Ir9, two capped atoms (Ir1, Ir3) of Ir10 clusters show higher f +(r) and f -(r) indices, indicating the feasibility of nucleophilic and electrophilic attack. For the Ir11 cluster, the site Ir11 is the most favorable site in front of any chemical attack. Four atoms (Ir2, Ir6, Ir7, Ir8) located on the middle layer of Ir12 are inactive sites. Ir4 and Ir11 sites may be available for electrophilic molecule absorption. The two-coordinated atom (Ir13) in the Ir13 cluster is most preferable for any chemical attack. The Ir9, Ir10, Ir13, and Ir14 atoms in Ir14 cluster show the higher FI values relative to remaining atoms; therefore, the

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Du et al.

TABLE 2: Calculated Fukui Indices of the Lowest-Energy Geometries of Ir5-15 Clustersa Ir5 Ir6 Ir7 Ir8 Ir9 Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 a

+

f ff+ ff+ ff+ ff+ ff+ ff+ ff+ ff+ ff+ ff+ f-

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.211 0.175 0.167 0.160 0.131 0.132 0.125 0.125 0.111 0.110 0.125 0.129 0.066 0.065 0.101 0.103 0.061 0.054 0.091 0.091 0.096 0.093

0.211 0.215 0.167 0.160 0.131 0.132 0.121 0.124 0.111 0.110 0.073 0.076 0.111 0.113 0.045 0.043 0.020 0.017 0.039 0.042 0.028 0.022

0.184 0.197 0.167 0.170 0.155 0.169 0.121 0.124 0.069 0.072 0.125 0.129 0.050 0.045 0.103 0.099 0.098 0.087 0.091 0.091 0.047 0.045

0.211 0.215 0.167 0.170 0.160 0.151 0.121 0.124 0.107 0.109 0.073 0.076 0.050 0.045 0.103 0.115 0.095 0.095 0.064 0.059 0.047 0.045

0.184 0.197 0.167 0.170 0.160 0.151 0.121 0.124 0.117 0.118 0.073 0.076 0.066 0.065 0.103 0.099 0.091 0.089 0.064 0.059 0.096 0.093

0.167 0.170 0.131 0.132 0.149 0.129 0.117 0.118 0.114 0.109 0.111 0.103 0.044 0.043 0.042 0.044 0.021 0.022 0.030 0.036

0.131 0.132 0.121 0.124 0.069 0.072 0.114 0.109 0.103 0.105 0.044 0.043 0.042 0.044 0.039 0.042 0.096 0.097

0.121 0.124 0.107 0.109 0.073 0.076 0.103 0.105 0.044 0.040 0.042 0.044 0.021 0.022 0.028 0.022

0.193 0.181 0.114 0.109 0.110 0.112 0.101 0.103 0.088 0.086 0.098 0.097 0.096 0.097

0.114 0.109 0.110 0.112 0.103 0.099 0.096 0.107 0.098 0.097 0.030 0.036

0.119 0.125 0.103 0.115 0.097 0.095 0.085 0.094 0.057 0.064

0.103 0.099 0.093 0.103 0.085 0.094 0.057 0.064

0.134 0.135 0.103 0.096 0.095 0.092

0.103 0.096 0.098 0.097

0.098 0.097

The numbers in this table correspond to the local sites of Ir atoms in Figure 1.

guest molecule absorption is likely to occur at these sites. For the Ir15 cluster, the surface atoms numbered 1, 5, 7, 9, 13, 14, 15 in Figure 1 are more viable for chemical attack. The values of f +(r) and f -(r) presented by these surface atoms are very close, indicating the amphiphilic character. F. Comparison with Tungsten Clusters. In this section, we will compare the iridium clusters with tungsten clusters studied previously.28 That the small tungsten clusters favor the compact atomic arrangement is very different from small Ir clusters, which show a simple cubic evolution. The larger tungsten clusters (starting from n ) 14) are formed on the basis of the bulk body center cubic (bcc) unit. The binding energies of tungsten clusters increase monotonous without even-odd oscillation as the sizes fill out, by W15, the binding energy is 5.82 eV, 67% of bulk cohesive energy, while for Ir15 is it 62%. Therefore, the convergence to bulk is slower for Ir clusters. The stabilities of Ir clusters present evident even-odd oscillation, differing from W clusters, which show higher stabilities on W8, W12, and W15. For Ir clusters with n < 8, the s and d electrons show inverse oscillation population and are responsible for the stabilities, while for larger clusters with n > 8, the atomic motif effect mainly determines the stabilities. As for small W clusters, the electronic ordering effect along with atomic motif effect determines the stability. Both Ir and W clusters present the metallic behavior for the larger clusters with small HOMOLUMO gaps. Moreover, the 5d electrons play a dominate role in the chemical reactivity for both Ir and W clusters.

experimental identification. The stability of studied iridium clusters is investigated in terms of binding energy per atom, second difference energy, and formation energy. Our calculations show that even-sized clusters are more stable than the oddsized ones, because the s electrons in even-sized clusters favor the closed shell. The s and d electronic configurations dominate the stability of small clusters (n < 8), whereas the geometrical structure may play an important role in the determination of the stability for large clusters. Analysis on HOMO-LUMO gaps indicates that metallic behavior is presented for large clusters. Ir8 cluster is predicted to have strong chemical reactivity and selectivity from DOS. There exist many covalent bondings in the isomers with cube motif. The atomic magnetic moments of Ir2-15 clusters are calculated and found to be closely connected with cluster sizes. Small clusters possess the higher moments relative to large clusters. The 5d electrons play a dominating role in the determination of magnetism of small iridium clusters. The analysis on local site reactivity is performed using condensed Fukui function. The site selectivity of clusters with n ) 5-15 is obtained. Generally speaking, the capped atoms in certain clusters (Ir9, Ir10, Ir11, and Ir13) are extraordinary active for both nucleophilic and electrophilic attack. Supporting Information Available: Coordinates of all optimized isomers and corresponding relative energies ∆E (with zero-point energy corrections). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes

4. Conclusion The geometrical isomers and electronic and magnetic properties of small iridium clusters with sizes of n ) 2-15 are systemically investigated with the BPW91/DZP/DSPP scheme. The simple cube growth pattern is found for small iridium clusters and will disappear for large-size clusters. We obtain new isomers with the lower energy relative to the ones found as the lowest-energy structures in previous investigation19 for Ir10, Ir11 clusters. For three low-lying isomers of Ir13 with close energy, the IR spectra are simulated, looking forward to further

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