Structural and Electronic Trends among Group 15 Elemental

Jun 19, 2009 - Systematic quantum chemical studies on the elemental nanostructures were performed to obtain periodic trends for their stabilities, str...
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Structural and Electronic Trends among Group 15 Elemental Nanotubes Antti J. Karttunen,*,† Jukka T. Tanskanen, Mikko Linnolahti, and Tapani A. Pakkanen Department of Chemistry, UniVersity of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland ReceiVed: March 17, 2009; ReVised Manuscript ReceiVed: May 22, 2009

We have investigated the structural and electronic characteristics of single-walled nanotubes composed of group 15 elements P, As, Sb, and Bi. Systematic quantum chemical studies on the elemental nanostructures were performed to obtain periodic trends for their stabilities, structural principles, and electronic properties. By studying P, As, Sb, and Bi nanotubes up to 5.4, 5.0, 5.7, and 6.0 nm in diameter, respectively, we found their structures and stabilities to converge smoothly toward their experimental bulk counterparts. A rhombohedral structural model was found to be favored over an orthorhombic one for all the elements, but in accordance with the experimental characteristics of the elements also the orthorhombic model was found to be important for phosphorus. All studied group 15 nanotubes were found to be semiconducting. Introduction The group 15 elements phosphorus, arsenic, antimony, and bismuth are all known to appear as layered materials.1 The layered structure of the bulk materials makes the elements attractive targets for controlled modification at the nanoscale, aiming toward preparation of elemental nanostructures. The thermodynamically most stable form of phosphorus is the orthorhombic black phosphorus, which is composed of sheets of puckered six-membered rings (Figure 1).2 Notably, a recently introduced kinetically controlled synthesis technique has made black phosphorus commercially available in gram quantities for the first time,3 greatly facilitating further modification of the bulk material into nanostructured modifications such as polyhedral cages.4 The orthorhombic modification of black phosphorus can be converted into a rhombohedral layered structure by applying high pressure (Figure 1).5 For the heavier group 15 elements, As,6 Sb,7 and Bi,8 the rhombohedral structure is the thermodynamically most stable form, although an orthorhombic structural modification is known for arsenic, as well.9 Furthermore, ultrathin bismuth nanofilms adopting the orthorhombic structural motif have been recently prepared in ultrahigh vacuum.10 The preference of the rhombohedral bulk structure for the heavier congeners of group 15 has been suggested to result from decreased s-p mixing in comparison to phosphorus.11 Another important periodic trend among the group 15 elements is the increasing metallic character of the elements when moving from P to Bi. While the orthorhombic black phosphorus is a semiconductor with a band gap of 0.33 eV,12 As, Sb, and Bi are semimetals.13 The layered bulk materials of group 15 elements are structurally related to graphite, and analogously to the structural relationship between the single layers of graphite and carbon nanotubes the individual sheets of the layered group 15 materials can be rolled into single-walled nanotubes. In the case of phosphorus, there are previous theoretical studies focusing on nanotubes derived from rhombohedral monolayer sheets.14 The properties of arsenic nanotubes are much less well-known, and there is not much theoretical or experimental data concerning * To whom correspondence should be addressed. E-mail: [email protected]. † Present address: Department of Chemistry, Technische Universita¨t Mu¨nchen, Lichtenbergstrasse 4, 85747 Garching, Germany.

Figure 1. Orthorhombic and rhombohedral crystal structures of group 15 elements (top) and a top view of the corresponding monolayer sheets (bottom).

them. Tubular nanomaterials composed of antimony15 and bismuth16 have actually been experimentally prepared by means of rational synthesis techniques, and rhombohedral bismuth nanotubes have also been studied theoretically.17 However, as the previous theoretical studies on phosphorus and bismuth nanotubes have focused only on the rhombohedral structural motif, and there is very little detailed theoretical knowledge on arsenic and antimony nanotubes, the periodic trends among the group 15 nanotubes are less well-known. Here we investigate the structural and electronic trends among single-walled nanotubes composed of group 15 elements phosphorus, arsenic, antimony, and bismuth. The elemental nanostructures are investigated by means of a systematic quantum chemical study, considering various tubular structural motifs for all four elements. Computational Details The studied one- and two-dimensionally periodic structures were fully optimized without any symmetry constraints by using the B3LYP hybrid density functional method.18 Karlsruhe splitvalence basis set with polarization functions (def2-SVP) was applied for all atoms.19 Generally, basis sets such as def2-SVP, which are originally developed for molecular calculations,

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contain diffuse basis to model the tails of wave functions. However, in periodic calculations, where the whole space is filled with basis functions, such diffuse functions are usually unnecessary and lead into numerical difficulties and/or several degradations of performance.20 In the case of phosphorus, the molecular basis set was modified for periodic calculations by reoptimizing the outermost s and p exponents variationally for a monolayer sheet of orthorhombic black phosphorus (Rs: 0.1050 f 0.1200; Rp: 0.1366 f 0.1516). The basis set optimization was performed with the help of the Billy program.21 For the heavier elements, modification of the basis set was not necessary. The 28-electron and 60-electron relativistic effective core potentials (ECP) were used to describe the core electrons of antimony and bismuth, respectively.22 Gaussian03 software was used for the geometry optimizations.23 The periodic nature of the studied systems was taken into account by applying the direct-space fast multipole methods for periodic calculations implemented in Gaussian03.20 For the k-point sampling of the reciprocal space, the default scheme for generating a dense and accurate k-point mesh was applied.20 For example, the applied scheme results in a mesh of 101 k-points in the reciprocal space for the rhombohedral phosphorus armchair tubes, where the dimension of the unit cell in the periodic direction is about 3.35 Å. If the number of k-points is quadrupled, the resulting changes in total energy are only around 10-3 kJ/mol/atom, showing a good convergence with respect to the k-point sampling. Cartesian coordinates of the unit cells used in the calculations are listed in the Supporting Information. Density of States (DOS) plots were prepared at the B3LYP/def2-SVP level of theory by using the CRYSTAL06 software package.24 Results and Discussion The studied group 15 nanotubes are composed of sheets of six-membered rings that have been rolled into a tubular shape. Hence, they are structurally related to carbon nanotubes (CNTs) rolled up from a two-dimensional graphene sheet, and it is convenient to describe them by using the naming convention of CNTs.25 The studied nanotubes are described by a chiral vector (n,m), which defines how the tube is rolled up from a two-dimensional lattice of six-membered rings. Here we focus on (n,0) and (n,n) tubes, also known as “zigzag” and “armchair” tubes, respectively. Despite the close structural relationship between the group 15 nanotubes and carbon nanotubes, there are also some important differences between them. While in the carbon nanotubes the six-membered rings adopt a planar configuration, in the group 15 elemental nanotubes the sixmembered rings pucker so that the lone pair repulsion is minimized. The two different kinds of puckered group 15 monolayer sheets, derived from orthorhombic and rhombohedral crystal structures (Figure 1), can both be rolled into nanotubes. Representative examples of the four structural motifs of group 15 nanotubes considered in this study are illustrated in Figure 2. We studied P, As, Sb, and Bi nanotubes up to 5.4, 5.0, 5.7, and 6.0 nm in diameter, respectively. In terms of chiral vector (n,m), the largest studied tubes were the rhombohedral and orthorhombic (48,0) tubes composed of phosphorus. The strain energies of the nanotubes were determined by comparing them with the corresponding infinite rhombohedral monolayer sheets, which can be considered as strain-free reference structures. The energetic trends of the studied nanotubes are shown in Figure 3 (for tabulated data, see the Supporting Information). The general stability trends of the different structural motifs are similar for all elements. For tube diameters larger than 2 nm, the rhombohedral zigzag and armchair tubes are practi-

Figure 2. Representative examples of the structural motifs of the studied rhombohedral and orthorhombic group 15 nanotubes. The (n,m) values are the chiral vectors defining how each tube has been rolled up from a two-dimensional lattice of six-membered rings.

cally equally stable, in accordance with the previous theoretical studies on similar rhombohedral phosphorus and bismuth nanotubes.14,17a In contrast to the rhombohedral nanotubes, the studied orthorhombic armchair tubes are clearly favored over the zigzag ones. The small orthorhombic zigzag tubes are strained, because rolling an orthorhombic monolayer sheet in the direction of (n,0) chiral vector results in unfavorably long bond distances between the atoms located on the outer shell of the nanotube. Overall, as the diameters of the nanotubes increase, all four studied structural families of group 15 nanotubes approach the respective monolayer sheet reference structures both energetically and structurally. For As, Sb, and Bi, the orthorhombic tubes are clearly less favorable than the rhombohedral tubes of similar size. However, in the case of phosphorus, the largest studied orthorhombic (30,30) tube is only about 0.8 kJ/mol/ atom less stable than the rhombohedral (24,24) tube with similar diameter. The orthorhombic monolayer sheet of phosphorus is

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Karttunen et al. TABLE 1: Bond Lengths and Angles for the Infinite Rhombohedral and Orthorhombic Monolayer Sheets of Group 15 Elements P bond length (Å) bond angle (°)

2.29 93.8

As

rhombohedral sheet 2.52 2.91 93.0 92.4

Sb

Bi 3.07 92.3

orthorhombic sheet bond lengths (Å) 2.25, 2.29 2.50, 2.51 2.90, 2.91 3.05, 3.09 bond angles (°) 96.6, 104.3 94.9, 102.6 94.2, 100.8 94.3, 98.9

Figure 3. Relative total energies of the studied group 15 elemental nanotubes. The energies in kJ/mol/atom are given relative to the corresponding rhombohedral monolayer sheets.

calculated to be 0.5 kJ/mol/atom less stable than the rhombohedral monolayer sheet, suggesting that the interlayer interactions present in the layered bulk material are responsible for

making the orthorhombic black phosphorus favored over the rhombohedral structural alternative. For comparison, in the case of As, Sb, and Bi, the energy differences between the rhombohedral and orthorhombic monolayer sheets are considerably larger, 3.8, 3.6, and 3.8 kJ/mol/atom in favor of the rhombohedral structure, respectively. The favored rhombohedral group 15 nanotubes become less strained when moving down for P to Bi. For example, the strain energies calculated for the rhombohedral (18,18) tube are 0.9, 0.7, 0.3, and 0.3 kJ/mol/atom for P, As, Sb, and Bi, respectively. The smaller strain of the nanotubes composed of the heavier congeners of group 15 is also in accordance with the experimental results, with antimony and bismuth nanotubes having already been synthesized.15,16 Overall, the strain energies of less than 1.0 kJ/mol/atom calculated for the large group 15 nanotubes are very small. For comparison, at the B3LYP level of theory used here, the strain energy of the experimentally known white phosphorus (P4) is 13.7 kJ/mol/atom, although taking entropy effects into account is expected to increase the relative stability of P4. Considering the relative diameters of the different types of tubes, the orthorhombic zigzag tubes have larger diameters than rhombohedral zigzag tubes with the same chiral vector, while in the case of armchair tubes the rhombohedral tubes are larger than the corresponding orthorhombic ones (for tabulated data, see the Supporting Information). The bond length and angle data shown in Table 1 provide further insights into the periodic structural trends of the studied elemental nanostructures of group 15 elements. The trends illustrated for the monolayer sheets in Table 1 apply for the respective nanotubes, as well. For each element, the bond lengths in the rhombohedral and orthorhombic sheets are fairly similar, but the bond angles show much more variation. The bond angles decrease when moving down from phosphorus to bismuth, especially for the orthorhombic structures, which are energetically less favorable in comparison to the rhombohedral structures. As expected, the metallic character of the group 15 nanotubes and monolayer sheets increases when moving down from P to Bi. The band gaps of the rhombohedral P, As, Sb, and Bi monolayer sheets are 3.3, 3.0, 2.4, and 1.3 eV, respectively, and for the orthorhombic monolayer sheets the corresponding values are 2.2, 2.1, 1.8, and 1.0 eV. Hence, the orthorhombic structures possess considerably smaller band gaps than the rhombohedral ones. In line with the energetical and structural behavior of the studied nanotubes, their band gaps also approach the gap values calculated for the corresponding monolayer sheets as the tube diameter increases (see the Supporting Information for tabulated data). Overall, the band structures calculated at the B3LYP level of theory suggest all studied group 15 singlewalled nanotubes to be semiconducting. The B3LYP functional has been shown to predict band gaps of semiconductor materials fairly well, although it usually overestimates the gaps slightly.26 Concerning the experimental results on group 15 nanotubes, Li

Trends among Group 15 Elemental Nanotubes et al. have found a transition from semimetal to semiconducting for bismuth nanotubes when the wall thickness is decreased from 100 to 15 nm.16d However, it should be noted that the experimental results have been obtained for multiwalled tubes while only single-walled tubes have been considered here. In addition to calculating the band gaps for all studied group 15 nanotubes and monolayer sheets, we investigated their electronic characteristics by the means of density of states (DOS) analysis. DOS plots for the rhombohedral and orthorhombic monolayer sheets of the studied elements are shown in Figure 4. The DOS plots illustrate some interesting periodic trends, such as the decreasing amount of s-p mixing when moving down from P to Bi.11 In the case of phosphorus and arsenic, the DOS plots show s-p mixing in the valence region, especially for the s-dominated region of the phosphorus DOS plots. However, for antimony, only very minor s-p mixing is observed for the orthorhombic structure, and in the case of bismuth the mixing is practically absent. As suggested previously,11 the s-p mixing is probably an important contributing factor to the fact that in ambient pressures black phosphorus exists as orthorhombic layered structure, while the heavier congeners adopt the rhombohedral structure. Rolling the monolayer sheets of phosphorus into tubular shape does not remove the s-p mixing illustrated by the monolayer DOS plots. Single-walled group 15 nanotubes similar to single-walled carbon nanotubes25 have not been synthesized to date, with the experimentally known antimony15 and bismuth16 nanotubes being multiwalled species structurally related to the corresponding layered bulk structures. In the case of phosphorus, the orthorhombic bulk structure suggests that the multilayered orthorhombic nanotubes might be favored over the rhombohedral ones. Hence, systematic computational studies on multiwalled group 15 nanotubes could provide especially interesting insights into the chemistry of group 15 nanotubes. Unfortunately, the dispersive van der Waals type interlayer interactions present in layered systems cannot be properly described with the commonly available density functional methods.27 The interlayer interactions could be accounted for by using correlated ab initio methods such as MP2, but program packages for performing periodic MP2 calculations are not yet generally available. Such calculations would also be computationally very expensive. A computationally more efficient approach would be the application of the dispersion-corrected DFT methods (DFT-D),28 which have recently been tested for periodic systems, as well.29 As a preliminary investigation of the structural principles of the multilayered tubes, we studied which of the single-walled phosphorus tubes considered here could fit together to form multilayered tubes with reasonable interlayer distance. Using a notation where (n1,m1)@(n2,m2) denotes a multilayered tube with a (n1,m1) tube enclosed within a (n2,m2) tube, the smallest feasible multilayered tubes were found to be rhombohedral zigzag (8,0)@(16,0), rhombohedral armchair (6,6)@(11,11), orthorhombic zigzag (12,0)@(22,0), and orthorhombic armchair (6,6)@(14,14). The tubes proposed here were constructed from fairly small single-walled tubes because the diameters of large nanotubes can be varied more easily by making small changes to the bond lengths and angles and therefore a proper geometry optimization would be necessary to make any reasonable estimates on large multilayered tubes. The experimentally known antimony and bismuth nanotubes have been synthesized via low-temperature routes such as hydrothermal reduction processes. The interest in the nanostructured modifications of antimony and bismuth arises from their unique electronic transport properties which suggest them

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Figure 4. DOS plots for the orthorhombic and rhombohedral monolayer sheets of group 15 elements. The band gap is denoted with a dotted line within each plot. The densities of states were expanded with 12 Legendre polynomials as implemented in the CRYSTAL06 software.

to possess promising thermoelectric applications.16d The nanostructures of phosphorus and arsenic have attractive applications, as well. For example, considering the recently demonstrated applicability of black phosphorus as an anode material for Liion batteries,30 new anode materials based on nanostructured modifications of black phosphorus could possess high application potential. Several experimental techniques have already been proposed for the preparation of phosphorus nanotubes.14a In addition, as black phosphorus is now readily available in

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gram quantities,3 novel synthetic methods based on the direct modification of the bulk material might be feasible. Arc discharge methods used in synthesis of graphitic carbon nanostructures31 and laser ablation techniques used to produce phosphorus clusters32 are some examples of such methods, although the high-temperature nature of these techniques should be taken into consideration. Conclusions We have studied the structural and electronic trends of singlewalled, allotropic group 15 nanotubes. The structures and stabilities of the investigated rhombohedral and orthorhombic nanotubes were found to converge smoothly toward their bulk counterparts. All studied nanotubes were found to be semiconducting. The nanotubes composed of heavier group 15 elements bismuth and antimony were found to be less strained with respect to their bulk counterparts than tubes composed of phosphorus and arsenic. The rhombohedral structural model was found to be favored over the orthorhombic one for all the elements, but in the case of phosphorus the orthorhombic structural motif cannot be neglected if multiwalled tubes are to be considered. The obtained structural characteristics and periodic trends are expected to help in the experimental characterization of novel one-dimensional group 15 nanostructures such as phosphorus and arsenic nanotubes. Acknowledgment. Funding from the Academy of Finland and the Finnish Funding Agency for Technology and Innovation (Tekes BNN-project) is gratefully acknowledged. Supporting Information Available: Data tables for the data illustrated in Figure 3 and Cartesian unit cell coordinates of the phosphorus nanotubes. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements; Butterworth-Heineman: Oxford, United Kingdom, 1997; pp. 479-482. (b) Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements; ButterworthHeineman: Oxford, United Kingdom, 1997; pp 550-551. (2) (a) Hultgren, R.; Gingrich, N. S.; Warren, B. E. J. Chem. Phys. 1935, 3, 351–355. (b) Brown, A.; Rundqvist, S. Acta Crystallogr. 1965, 19, 684–685. (3) (a) Lange, S.; Schmidt, P.; Nilges, T. Inorg. Chem. 2007, 46, 4028– 4035. (b) Nilges, T.; Kersting, M.; Pfeifer, T. J. Solid State Chem. 2008, 181, 1707–1711. (4) (a) Karttunen, A. J.; Linnolahti, M.; Pakkanen, T. A. Chem.sEur. J. 2007, 13, 5232–5237. (b) Karttunen, A. J.; Linnolahti, M.; Pakkanen, T. A. ChemPhysChem 2007, 8, 2373–2378. (c) Karttunen, A. J.; Linnolahti, M.; Pakkanen, T. A. ChemPhysChem 2008, 9, 2550–2558. (5) (a) Jamieson, J. C. Science 1963, 139, 1291–1292. (b) Kikegawa, T.; Iwasaki, H. Acta Crystallogr., Sect. B: Struct. Sci. 1983, 39, 158–164. (6) Schiferl, D.; Barrett, C. S. J. Appl. Crystallogr. 1969, 2, 30–36. (7) Barrett, C. S.; Cucka, P.; Haefner, K. Acta Crystallogr. 1963, 16, 451–453. (8) Cucka, P.; Barrett, C. S. Acta Crystallogr. 1962, 15, 865–872. (9) Smith, P. M.; Leadbetter, A. J.; Apling, A. Philos. Mag. 1975, 31, 57–64. (10) (a) Nagao, T.; Sadowski, J. T.; Saito, M.; Yaginuma, S.; Fujikawa, Y.; Kogure, T.; Ohno, T.; Hasegawa, Y.; Hasegawa, S.; Sakurai, T. Phys. ReV. Lett. 2004, 93, 105501. (b) Nagao, T.; Yaginuma, S.; Saito, M.; Kogure,

Karttunen et al. T.; Sadowski, J. T.; Ohno, T.; Hasegawa, S.; Sakurai, T. Surf. Sci. 2005, 590, L247–L252. (11) Seo, D.-K.; Hoffmann, R. J. Solid. State. Chem. 1999, 147, 26– 37. (12) (a) Keyes, R. W. Phys. ReV. 1953, 92, 580–584. (b) Goodman, N. B.; Ley, L.; Bullett, D. W. Phys. ReV. B. 1983, 27, 7740–7450. (13) Gonze, X.; Michenaud, J.-P.; Vigneron, J.-P. Phys. ReV. B 1990, 41, 11827–11835. (14) (a) Seifert, G.; Herna´ndez, E. Chem. Phys. Lett. 2000, 318, 355– 360. (b) Cabria, I.; Mintmire, J. W. Europhys. Lett. 2004, 65, 82–88. (15) (a) Wang, D.; Yu, D.; Peng, Y.; Meng, Z.; Zhang, S.; Qian, Y. Nanotechnology 2003, 14, 748–751. (b) Hu, H.; Mo, M.; Yang, B.; Shao, M.; Zhang, S.; Li, Q.; Qian, Y. New J. Chem. 2003, 27, 1161–1163. (16) (a) Li, Y.; Wang, J.; Deng, Z.; Wu, Y.; Sun, X.; Yu, D.; Yang, P. J. Am. Chem. Soc. 2001, 123, 9904–9905. (b) Liu, X.; Zeng, J.; Zhang, S.; Zheng, R.; Liu, X.; Qian, Y. Chem. Phys. Lett. 2003, 374, 348–352. (c) Yang, B.; Li, C.; Hu, H.; Yang, X.; Li, Q.; Qian, Y. Eur. J. Inorg. Chem. 2003, 3699–3702. (d) Li, L.; Yang, Y. W.; Huang, X. H.; Li, G. H.; Ang, R.; Zhang, L. D. Appl. Phys. Lett. 2006, 88, 103119. (17) (a) Su, C.; Liu, H.-T.; Li, J.-M. Nanotechnology 2002, 13, 746– 749. (b) Qi, J.; Shi, D.; Zhao, J.; Jiang, X. J. Phys. Chem. C 2008, 112, 10745–10753. (c) Qi, J.; Shi, D.; Jiang, X. Chem. Phys. Lett. 2008, 460, 266–271. (18) (a) Becke, A. D. J. Chem. Phys. 1993, 98, 5648–5652. (b) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623–11627. (19) (a) Schaefer, A.; Horn, H.; Ahlrichs, R. J. Chem. Phys. 1992, 97, 2571–2577. (b) Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297–3305. (20) Kudin, K.; Scuseria, G. E. Phys. ReV. B 2000, 61, 16440–16453. (21) Towler, M. Billy, version 4.0; University of Cambridge: Cambridge, 2003; http://www.tcm.phy.cam.ac.uk/∼mdt26/crystal.html. (22) Metz, B.; Stoll, H.; Dolg, M. J. Chem. Phys. 2000, 113, 2563– 2569. (23) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford CT, 2004. (24) Dovesi, R.; Saunders, V. R.; Roetti, C.; Orlando, R.; ZicovichWilson, C. M.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N. M.; Bush, I. J.; D’Arco, Ph.; Llunell, M. CRYSTAL06 User’s Manual; University of Torino: Torino, 2006. (25) Odom, T. W.; Huang, J.-L.; Kim, P.; Lieber, C. M. J. Phys. Chem. B 2000, 104, 2794–2809. (26) Muscat, J.; Wander, A.; Harrison, N. M. Chem. Phys. Lett. 2001, 342, 397–401. (27) Kohn, W.; Meir, Y.; Makarov, D. E. Phys. ReV. Lett. 1998, 80, 4153–4156. (28) Grimme, S. J. Comput. Chem. 2005, 25, 1463–1473. (29) Ugliengo, P.; Zicovich-Wilson, C. M.; Tosonic, S.; Civalleri, B. J. Mater. Chem. 2009, 19, 2564–2572. (30) Park, C.-M.; Sohn, H.-J. AdV. Mater. 2007, 19, 2465–2468. (31) Kra¨tschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature 1990, 347, 354–358. (32) Sˇedo, O.; Vora´e`, Z.; Alberti, M.; Havel, J. Polyhedron 2004, 23, 1199–1206.

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