Lowest-energy structures of (WO3)n (2 ≤ n ≤ 12) clusters from first-principles global search

Chemical Physics Letters 544 (2012) 7–12

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Lowest-energy structures of (WO3)n (2  n  12) clusters from first-principles global search Linwei Sai a,b,c, Lingli Tang a,b,c, Xiaoming Huang a,b, Guibin Chen a,d, Jijun Zhao a,b,⇑, Jun Wang c a

College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, Dalian University of Technology, Ministry of Education, Dalian 116024, China c School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China d Jiangsu Key Laboratory for Chemistry of Low-Dimensional Materials and Department of Physics, Huaiyin Normal University, Huaian 223300, China b

a r t i c l e

i n f o

Article history: Received 3 January 2012 In final form 22 June 2012 Available online 11 July 2012

a b s t r a c t Using genetic algorithm combined with density functional theory calculations, we performed global search for the most stable structures of (WO3)n clusters for n = 2–12. Small (WO3)n clusters with n = 3 or 4 adopt ring-like configurations with W–O alternating arrangement. Starting from (WO3)8, the tungsten oxide clusters transform to symmetric spherical-like cages. The relative stability, HOMO–LUMO gap, electronic states of these (WO3)n clusters were discussed. Analysis of wavefunctions of frontier orbitals and electron density of states shows that the valence bands are dominated by the 2p electrons from oxygen and the conduction bands are mainly contributed by the 5d states from tungsten. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Tungsten oxides are important materials with diverse technological applications, such as electrochromic devices [1], chemical sensors [2], and especially acid–base and redox catalysts [3–6]. Tungsten oxide clusters supported on substrates of other oxide like ZrO2 [7–9], Al2O3 [10], SiO2 [11], TiO2 [12], and SnO2 [13] stand for an important class of catalysts owing to their high catalytic activity in partial oxidation of alcohols, selective reduction of nitric oxide, and metathesis of alkenes. However, it is a great challenge to identify the catalytic active sites and elucidate the detailed reaction mechanisms in the catalytic processes from both experimental and theoretical aspects. Experimental studies of gas-phase clusters, coupled with state-of-the-art ab initio calculations, provide fundamental insights into the nature of active species and the complex catalytic processes. Meanwhile, evolution from microscopic clusters to bulk solids may involve constantly structural reconstruction and significant size-dependent variations in the physical and chemical properties. It is thus fundamentally important to explore the ground-state structures for (WO3)n clusters of different sizes and to elucidate when and how the bulk-like structures are formed. Such valuable knowledge may also shed some light on the growth behavior of various tungsten oxide nanostructures. Previously, photoelectron spectroscopy (PES) technique combined with ab initio calculations were employed to investigate ⇑ Corresponding author at: College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China. E-mail address: [email protected] (J. Zhao). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.06.050

the stoichiometric (WO3)n (1  n  6) clusters [14–21]. A C3v structure with three identical W@O bonds was predicted as the global minimum for the WO3 monomer [14–17]. The reported lowestenergy structures of (WO3)n (2  n  4) clusters were planar ring structures with Dnh symmetry [14–16,19–22]. Every W atom is tetrahedral coordinated with two bridge W@O double bonds and two terminal W–O single bonds [14–16,19–22]. Interestingly, these clusters resemble the structural feature of bulk WO3 crystals and may be considered as an embryonic form of bulk tungsten oxide [14]. For the larger (WO3)n clusters (n = 5, 6), Li and Dixon [15] presumed a ring structure for (WO3)5 and a cage configuration with Oh symmetry for (WO3)6 (in which every W atom forms one W@O bond and four W–O bonds) and discussed the geometry parameters and bonding characteristics of these clusters in details. The structures of anionic ðWO3 Þ n clusters were also investigated theoretically and experimentally. The structure of ðWO3 Þ 2 was reported to be nearly the same as the neutral one, while the additional electron induces significant geometry changes for the (WO3)– and ðWO3 Þ 3 clusters [14,16,17,19,20,23]. The out-of-plane angle in the WO 3 cluster is smaller than the neutral one by about 10° and the O@W@O angle increases by about 8° [16,17]. For the ðWO3 Þ 3 cluster, the W–W bond length and W–O–W angle dramatically decreases by about 0.3 Å and 20°, respectively, while O–W–O angle increases by about 20° [20,23]. Similar to (WO3)4, ðWO3 Þ 4 also adopt a planar eight-membered ring skeleton but its symmetry degrades from D4h for the neutral cluster to C2v [14,21]. Among the small (WO3)n clusters, (WO3)3 exhibits exceptionally high stability and attracts especial attention. PES spectra show an appreciable HOMO–LUMO gap of 3.3 eV for (WO3)3, which is larger

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L. Sai et al. / Chemical Physics Letters 544 (2012) 7–12

than the other ones, i.e., 1.3 eV for WO3 monomer, 2.8 eV for (WO3)2 and 2.89 eV for (WO3)4 [21]. Furthermore, the reactions of C1–C4 aliphatic alcohols on the (WO3)3 clusters were studied using temperature-programmed desorption, infrared spectroscopy and density functional theory (DFT) calculations [24,25]. As an efficient catalyst for dehydration of alcohols, (WO3)3 effectively lower the reaction energy barriers [26,27]. Besides the gas-phase clusters, monodispersed (WO3)3 clusters from direct evaporation of WO3 solid were deposited on the rutile TiO2(1 1 0) substrates [26,27]. This system was reported to be thermally stable up to at least 750 K, and the cyclic structure of (WO3)3 cluster characterized by STM observation coincided with DFT calculations [24,25]. The successful synthesis of stable mono-sized (WO3)3 clusters supported on metal oxide substrates suggests new possibility of model catalysts. In addition to the stoichiometric (WO3)n clusters, nonstoichiometric WxOy clusters have also been studied using PES technique as well as ab initial calculations [17,19,21–23,28–34]. The structures and stability of the oxygen deficient clusters, e.g., WxO3x1ðWx O 3x1 Þ (2  x  4), are complicated due to the competition between W–O, W–W, and W@O bonds; and those isomer structures with rather different W–O topological connections may be nearly energetically degenerate [19,21,23,29]. For example, Wang et al. reported two W4 O 11 isomers with energy difference of only 0.09 eV, that is, one Cs structure with four bridge O atoms and no W–W bond, another C2v structure with three bridge O atoms and one W–W bond [21]. The structures of oxygen sufficient WxO3x+1 (2  x  4) clusters can be obtained via substituting one terminal O atom in WxO3x by an O2 unit [21–23]. But they are not the lowest-energy structures for the anionic Wx O 3xþ1 clusters; the structures with cleaved O–O bonds were reported to be more stable by 1 eV [21,23]. Generally speaking, as the number of O atoms increase, sequential oxidation of W atoms was observed [17,21,23,30]. The 5d electrons from W atoms gradually transfer to the O atoms, stabilizing the remaining d electrons and increasing their binding energies. PES spectra showed higher electron binding energy (>5.0 eV) for O-sufficient tungsten oxide clusters arising from strongly binding oxygen-2p electrons [17,21,23]. In spite of the extensive studies of small tungsten oxide clusters, little is known about the larger (WO3)n clusters with n  7. Knowledge of their geometrical structures and electronic properties is crucial for understanding the catalytic mechanism of WO3 materials. Moreover, the most stable structures considered in previous theoretical works were constructed by chemical intuition rather than unbiased global search; thus they are not guaranteed to be the truly global minimum structures. In this work, we combined genetic algorithm with first-principles DFT methods to globally explore the lowest-energy structures of (WO3)n clusters up to n = 12. The structural motifs, relative stability, and electronic properties of these tungsten oxide clusters are also discussed as functions of cluster size.

2. Computational methods Unbiased search of the most stable structures of (WO3)n clusters (n = 1–12) was carried out using genetic algorithm (GA) [35–38] incorporated with all-electron density functional theory, as implemented in the DMol3 [39] program. All cluster structures were fully relaxed with DFT without any symmetry constraint. The double numerical basis including d-polarization function (DND) and the generalized gradient approximation (GGA) with the Perdew– Burke–Enzerhof (PBE) [40] functional were adopted. Since tungsten is a heavy element, we include the relativistic effect in the all electron self-consistent field calculations. Vibrational analysis on the lowest-energy configurations and some metastable isomers

of these (WO3)n clusters was carried out to ensure that there are no imaginary frequencies corresponding to the saddle points on the potential energy surface and to compute the zero-point energies (ZPE), which was included in the energy difference between structural isomers. In the GA search, we created and maintained a population with 16 individuals for each sized (WO3)n cluster. The initial configurations in the GA population were generated randomly from scratch. In each GA step, two parents were selected randomly to generate the offspring cluster via a ‘cut and splice’ crossover operation [36]. The child cluster was then relaxed by DFT method described above. To preserve the diversity of GA population, we used two indices I1 and I2 to check if two individuals are isomorphic, which are defined as:

I1 ¼

X X r2Wi þ 2r 2Oj i

j

and

I2 ¼

X X 2r 4Wi þ r 4Oj ; i

j

here rWi (rOj) means the distance of i-th W (j-th O) atom from the original point (defined by the geometry center of cluster). Two isomers were regarded as isomorphic if their I1 and I2 indexes differ by less than 3%. To ensure a reliable global search, the GA search lasts for 2000 iterations with a mutation probability of 30%. Details about application of GA in cluster physics can be found in previous review articles [36–38]. 3. Results and discussion 3.1. Lowest-energy structures The lowest-energy structures of (WO3)n clusters (n = 2–12) determined from our GA-DFT global search are plotted in Figure 1, along with several selected metastable isomers. Table 1 summarizes the coordination numbers of W and O atoms for these most stable configurations of the (WO3)n clusters. From our DFT optimization at PBE/DND level of theory, the WO3 molecule adopts a NH3-like geometry (C3v) with W–O bond length of 1.714 Å and O–W–O bond angle of 107.1°, in line with previous ab initio calculations [14]. As shown in Figure 1, the most stable structure of (WO3)2 dimer owns D2h symmetry, in which the two W atoms are connected by two bridged O atoms, leaving four terminal O atoms on the two sides. The current equilibrium W–O bond length (1.913 Å) and W@O bond length (1.698 Å) compare well with previous bond lengths of 1.986 and 1.761 Å [14], respectively. In Ref. [14], Sun et al. also discussed the structural evolution of W2Ox (1  x  6) with oxygen content and found that the W–W bond becomes longer as the number of O atoms increases and it eventually break up to six O atoms. For reference, inclusion of relativistic effect (for the heavy tungsten element) has moderate influence on the geometry parameters of WO3 molecule and clusters. Using the non-relativistic all-electron calculations, the W–O bond length and O–W–O bond angle are 1.773 Å and 108.3°, respectively. As for WO3 dimer with D2h configuration, the equilibrium W–O and W@O bond lengths are 1.983 and 1.755 Å, respectively. Consistent with the previous finding [14,20], the ground state configuration of (WO3)3 trimer from our calculations is formed by a W3O3 triangle skeleton with W/O alternating arrangement and six terminal O atoms attached to three W atoms (see Figure 1). Such W–O–W triangle subunit is also found in the crystal structure of WO3 solid of hexagonal phase [41]. The W–O–W bond angle of 134.8° and O–W–O bond angle of 104.8° in the (WO3)3 cluster

L. Sai et al. / Chemical Physics Letters 544 (2012) 7–12

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Figure 1. Lowest-energy structures of (WO3)n clusters (2  n  12) and several metastable isomers (labeled as 5b, 6b, 10b). The Cartesian coordinates of all these clusters are provided in the Supplementary materials.

Table 1 Statistics of coordination numbers of W and O atoms in the lowest-energy structures of (WO3)n clusters (n = 1–12). n

2 3 4 5 6 7 8 9 10 11 12

W

O

CN4

CN5

CN6

CN1

CN2

CN3

CN5

2 3 4 2 2 2 – – – – –

– – – 3 2 – 8 9 10 11 12

– – – – 2 5 – – – – –

4 6 8 8 6 7 8 9 10 11 12

2 3 4 6 12 13 16 18 20 22 24

– – – 1 – – – – – – –

– – – – – 1 – – – – –

deviate from the bulk values of 150.5° and 89.5° of hexagonal phase by about 15°. The lengths of terminal W@O bonds and W– O bridge bonds are 1.695 and 1.888 Å, respectively, which are close to the previously reported values of 1.706 and 1.908 [20] or 1.735 and 1.927 Å [14]. For reference, the W–O bond lengths in bulk WO3 solids are 1.887 Å for hexagonal phase and 1.880 Å for monoclinic phase [41]. This means that the length of W–O bridge bonds even in small (WO3)3 cluster already approaches the bulk W–O bond lengths. Following the ring-like motif of (WO3)3, the most stable configuration of (WO3)4 has a quadrilateral framework with C4v symmetry, in which each of the four W atoms is bridged by two O atoms on the two sides and also attached with other two terminal O atoms. This structure is slightly different from previous studies [14,21], which predicted a D4h structure with higher symmetry. From our DFT calculation at PBE/DND level, the C4v structure is 0.05 eV lower in energy than the D4h isomer. While the energy difference becomes as small as 0.003 eV using B3LYP functional and basis sets of 6–31 G(d,p) for O and LANL2DZ for W, making these two isomers nearly isoenergetic. The eight W/O atoms in the quadrilateral skeleton do not locate at the same plane, but with W–O–W inflection angle of 144°. Similar quadrilateral structural units also exist in the bulk tungsten trioxide solids of most phases [41]. As shown in Figure 1, one can see that the triangle and quadrilateral WxOx (x = 3, 4) frameworks are the major building components of the larger (WO3)n clusters.

Starting from n = 5, (WO3)n clusters transforms from quasiplanar to three-dimensional (3D) configurations. The lowestenergy configuration of (WO3)5 (5a in Figure 1) has C3h symmetry, in which five W atoms form a trigonal bipyramid (three W atoms in the middle plane and two W atoms on the top and bottom). In the geometry center, there is an O atom bonded to three W atoms in the middle plane. This structure model is similar to that of solid WO3 [47,48]. Previously, Li and Dixon [15] compared ring-like and chain-like configurations of (WO3)5 and found that the ring structure (5b in Figure 1) is more stable. Indeed, the ring configuration with lower Cs symmetry is 0.06 eV more stable than the D5h one proposed by Li and Dixon [15]. According to our PBE/DND calculations, this ring isomer (Cs) is 0.67 eV less favorable than the present trigonal bipyramid (C3h) structure. However, it is noteworthy that the energy differences of various isomers might be functional dependent, basis set dependent and possibly even code dependent. Similar to (WO3)5, the lowest-energy structure of (WO3)6 from our global GA-DFT search is an O-centered C2v basket-like configuration. It prevails the previously reported cage-like structure with Oh symmetry [15] (shown in Figure 1 as 6b) by 0.36 eV. We further verified this finding by recomputing with other two exchange–correlation functions, i.e., BP [42,43] and BLYP [43,44]; both of which revealed that the Oh isomer is less stable than the C2v ground state by 0.31 and 0.23 eV, respectively. These undisclosed most stable structures clearly indicate that an unbiased global search is necessary for locating the lowest-energy structures of (WO3)n clusters even for relatively smaller sizes like n = 5 and 6. Beyond n = 7, there was no first-principles study of (WO3)n clusters within the best of our knowledge. For the ground state structure of (WO3)7, the major tungsten framework is a trigonal prism capped with one W on the quadrilateral face. It can be also obtained from the metastable (WO3)6 cage with Oh symmetry (6b in Figure 1) by adding one W atom as a new vertex; afterwards the cluster symmetry degrades to C2v. In the upper center of cluster, there is a five-coordinated oxygen atom, which bonds to the largest number of W atoms among all clusters explored. Roughly speaking, the medium-sized (WO3)n clusters with n = 5–7 can be considered as a kind of O-centered structural motif that is similar to WO3 solid. Interestingly, this motif does not continue at n = 8. Instead, the most stable structure of (WO3)8 is a hollow cage with D4d symmetry. The eight W atoms, along with eight bridge O atoms, form two stagger W4O4 squares; and these two squares are interconnected by eight O atoms. Each of the rest eight

L. Sai et al. / Chemical Physics Letters 544 (2012) 7–12

3.2. Electronic properties To explore the relative stability of (WO3)n (1 6 n 6 12) clusters, we calculated the binding energy, the normalized clustering energy (NCE) [32], and second-order energy difference and plotted them as functions of cluster size n in Figure 2(a) and (b), respectively. One can see that the binding energy increase with cluster size rapidly up to n = 6 and then becomes nearly saturated with only slight increment for n = 7–12. The binding energy of the largest (WO3)12 cluster is only 0.08 eV/atom less than the bulk value (see Table 2), implying that it is rather close to the bulk limit of 8.83 eV/atom (experimental value: 6.32 eV/atom [45]), which also can be seen by the NCE. Furthermore, on the energy second differences shown in Figure 2(b), there are three distinct peaks at n = 3, 6 and 9, suggesting that these clusters are relatively more stable than their neighboring sized ones. On the other hand, the local minima of D2E(n) at n = 5 and 8 can be related to the planar-to-3D structural transformation at (WO3)5 and the emergence of cage configurations at (WO3)8. Figure 2(c) depicts the HOMO–LUMO gaps of (WO3)n clusters (n = 1–12), compared with the band gap calculated for monoclinic tungsten trioxide solid. Except that the WO3 monomer possesses a relatively low gap of 1.48 eV, the HOMO–LUMO gaps for all other clusters are significantly larger, i.e., between 2.3 and 4.0 eV and are significantly higher than the theoretical value for WO3 solid (1.50 eV). This effect of enhanced band gaps can be attributed to the quantum size effect that is commonly seen in semiconductor nanoclusters. Among the explored (WO3)n clusters, the largest gaps of 3.5–4.0 eV are found for the small clusters with n = 3–5; while the insensitive gap variation in the size range of n = 8–12 can be attributed to the continuation of cage structural motif. As representatives, the wavefunctions for HOMO-2, HOMO-1, HOMO, LUMO, LUMO + 1, LUMO + 2 orbitals of (WO3)3 and (WO3)12 clusters are plotted in Figure 3. First, all these molecular orbitals are rather delocalized and distribute over the entire cluster in different symmetric manners. The top occupied orbitals (e.g., HOMO-2, HOMO-1, HOMO) mainly locate on the O sites and the bottom unoccupied orbitals (e.g., LUMO, LUMO + 1, LUMO + 2) distribute more on the W atoms. This observation is further supported by the analysis of the electron density of states (DOS), which shows that the electronic states around the Fermi level are mainly con-

Binding energy (eV)

8.8

a

8.6 8.4 8.2 8.0 7.8 1.0

Δ 2E(n) (eV)

O atoms attach to one W atom, forming a terminal W–O bond. Such hollow-cage motif dominates the ground state structures for all the rest (WO3)n clusters with n = 8–12 (see Figure 1). For example, (WO3)9 adopts a tricapped trigonal prism framework structure (D3h) as ground state configuration. The most stable structure of (WO3)10 with D4h symmetry can be viewed as a finite tube originated from the Oh cage of (WO3)6. The lowest-energy structure of (WO3)12 is a high symmetric cage (Oh), in which twelve W atoms constitute a cuboctahedron. Removing one WO3 vertex from this (WO3)12 cuboctahedron results in the skeleton of ground state of (WO3)11 cluster, whose symmetry is reduced to C2v. Further removing one WO3 vertex from (WO3)11 leads to a metastable structure of (WO3)10 (10b in Figure 1), which only lies 0.06 eV above the ground state. As summarized in Table 1 and Figure 1, in all these cage configurations for (WO3)n clusters with n = 8–12, every W atom are coordinated with five O atoms; whereas one third of O atoms are terminal atoms (CN = 1) and two thirds of O atoms are bridge ones (CN = 2). For comparison, in the crystalline WO3 solids of all phases, every W atom is six coordinated with O and every O atom bond with two W atoms [41]. In other words, the coordination of present cage structures already resemble the bulk solids except for the existence of surface terminal O atoms as a natural consequence of finite clusters.

b

0.5 0.0 -0.5 -1.0

HOMO-LUMO gap (eV)

10

-1.5 4.0 3.5

c

3.0 2.5 2.0 1.5 2

4

6 8 Cluster size n

10

12

Figure 2. (a) Binding energy per atom, (b) second differences of cluster energies, and (c) HOMO–LUMO gap (eV) of (WO3)n clusters (1  n  12). In graph (c), dash line labels the band gap (1.50 eV) of monoclinic WO3 solid from our DFT calculations with the same PBE/DND level. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2 Binding energy (Eb), normalized clustering energy (NCE), HOMO–LUMO gap (eV), average numbers of s, p, d electrons for W and s, p electrons for O calculated for (WO3)n clusters (n = 1–12) and bulk WO3 solid of monoclinic phase. n

1 2 3 4 5 6 7 8 9 10 11 12 Bulk

Eb (eV/ atom)

NCE (eV)

Gap (eV)

W Ns

Np

Nd

O Ns

Np

7.75 8.38 8.56 8.59 8.64 8.71 8.72 8.72 8.73 8.72 8.74 8.75 8.83

– 2.54 3.23 3.39 3.56 3.85 3.91 3.87 3.92 3.88 3.96 4.01 4.11

1.48 2.37 3.76 3.55 3.93 3.22 3.11 2.76 2.83 2.63 2.77 2.83 1.50

0.29 0.26 0.24 0.25 0.30 0.33 0.32 0.38 0.33 0.33 0.33 0.33 0.36

0.35 0.50 0.51 0.50 0.54 0.58 0.57 0.59 0.59 0.59 0.58 0.57 0.60

3.75 3.44 3.41 3.40 3.32 3.23 3.23 3.24 3.24 3.23 3.22 3.22 2.97

1.94 1.93 1.92 1.92 1.92 1.91 1.91 1.90 1.90 1.90 1.90 1.90 1.90

4.57 4.64 4.67 4.67 4.67 4.69 4.69 4.69 4.69 4.69 4.70 4.71 4.75

tributed by O-2p electrons and W-5d electrons. As shown in Figure 4 the valence bands are dominated by the O-2p states and the conduction bands are mainly contributed by the W-5d states, and there is certain hybridization between O-2p and W-5d electrons. Interestingly, the electronic states for the large (WO3)12 clusters rather resemble the DOS of monoclinic WO3 solid computed using the same theoretical scheme (see Figure 4). This also coincides

L. Sai et al. / Chemical Physics Letters 544 (2012) 7–12

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Figure 3. Isosurfaces for HOMO-2, HOMO-1, HOMO, LUMO, LUMO + 1, LUMO + 2 orbitals of (WO3)3 (upper and middle) and (WO3)12 (lower) clusters. Yellow and blue colors denote the wavefunction phases. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The W–O interaction can be further discussed via Mulliken population analysis [46]. As shown in Table 2, for (WO3)n clusters with n P 3, the on-site charge and orbital occupancy are rather insensitive to cluster size and already close to the bulk values. On average, each W atom loses about 1.8 electrons (mainly from the 5d orbital) and each O atom gains about 0.6 electrons (mainly on the 2p orbital) for (WO3)312 clusters, indicating the partially ionic feature of W–O bonding (corresponding to the W/O alternating structural patterns) and the dominance of p-d interaction between W and O atoms.

30

O-2p W-5d

(WO3)3

20

Partial Density of States (electrons/eV)

10

0 80

4. Conclusions

(WO3)12

60 40 20 0 20

WO3-bulk

15 10 5 0

-8

-6

-4

-2

0

2

4

6

Energy (eV) Figure 4. Electron density of States of (WO3)3 and (WO3)12 clusters compared with WO3-bulk solid of monoclinic phase.

with previous argument about the similarity of WO3 clusters and the bulk solids [14].

The lowest-energy structures of (WO3)n clusters (n = 2–12) have been determined by DFT calculations combined with genetic algorithm. The ground state structures for (WO3)5 and (WO3)6 are more stable than the previously reported ones. All these stable cluster structures adopt W–O alternating arrangements. The structural growth patterns can be divided into three stages: planar ring-like configurations for n = 2–4; O-centered 3D structures for n = 5–7; symmetric hollow cages for n = 8–12. In the latter case, the main structural building units (triangle and quadrilateral W–O frameworks) and coordination numbers of W/O atoms rather resembles the crystalline WO3 solids. From the computed binding energies, HOMO–LUMO gaps and energy second difference, (WO3)3 stands out as a magic cluster with high stability. The electronic gaps for small clusters with n = 3–5 are higher than the bulk value by 2– 2.5 eV, whereas the theoretical gap values are around 2.7 eV for larger clusters (n = 8–12). Analysis of electron density of states and wavefunctions revealed that the occupied states come from oxygen-2p electrons and the unoccupied states from tungsten-5d electrons. Charge transfer and p-d hybridization between W and O atoms are the major bonding mechanism in the (WO3)n clusters. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11134005, 11174101), the Fundamental Research Funds for the Central Universities of China (No.

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