Threadlike Tin Clusters with High Thermal Stability Based on

ARTICLE pubs.acs.org/JPCC

Threadlike Tin Clusters with High Thermal Stability Based on Fundamental Units Haisheng Li,† Hongbo Du,† Weiguang Chen,† Q. Q. Shan,† Q. Sun,† Z. X. Guo,‡ and Yu Jia*,† †

Center for Clean Energy and Quantum Structures, and School of Physics and Engineering, Zhengzhou University, Zhengzhou 450052, China ‡ Department of Chemistry, University College London, London WCIH 0AJ, U.K. ABSTRACT: First-principles calculations using the density functional theory (DFT) have been carried out to study the geometric and electronic structures of Snn (n = 10
50) clusters. Our findings show that tin clusters (mainly based on Sn10 and Sn15 units) usually favor threadlike growth mode, rather than that of multibranch germanium or amorphous-like lead. Besides, a biatomby-biatom oscillation with crossovers is shown, with the period between crossovers about ten atoms. Further studies on the electronic structures reveal that the charges of the highest occupied molecular orbital of Sn15 cluster accumulate at both ends, which can form strong covalent bonds with other units and give rise to a threadlike growth mode. Ab initio molecular dynamics simulations show that Snn (n = 10
50) clusters are usually stable even when the temperature is higher than the melting point of tin bulk. Our studies may provide some insight for an experiment to fabricate tin nanowires.

1. INTRODUCTION Nanoscale technology develops rapidly as more and more powerful experimental apparatus appears in which thermal stability is an important quality standard. Because of the large surfacevolume ratio, nanoparticles usually manifest lower melting temperature than their bulk counterparts.1
10 However, recent studies showed that Gan,11 Snn,12
18 and Pbn18 clusters obtained the opposite thermal character because of their covalent binding property. Therefore, these clusters are expected to have unique properties on binding forms, growth patterns, nonmetal
metal transitions, and so on. For group-IV elements, previous studies showed that silicon,19
25 germanium,25
32 and tin33
39 clusters adopted prolate growth mode when cluster sizes were lower than 27, 40, and 35, respectively, while near-spherical structures were predicted for all sizes of lead clusters.40
42 Furthermore, tin has three bulk phases (α, β, and brittle tin43
45) as temperature varies, which is very important in alloys. For the above reasons, the growth mode divergence between tin clusters and the other group-IV elements, together with their thermal stabilities, is worth studying. Experimentally, the photoionization mass spectroscopy analysis showed that Sn7 and Sn10 were magic clusters.33 Combining the trapped ion electron diffraction with density functional theory calculations, Lechtken39 et al. found an unexpected growth mode based on the extraordinary stability of the building blocks Sn9, Sn10, and Sn15. Cui46 et al. discovered that Sn122
is a highly stable icosahedral (Ih) cage cluster. Using multicollision-induced dissociation (MCID), Breaux17 et al. found that the dissociation energy of Sn20 was relatively low, due to the high stability of the Sn10 unit. Shvartsburg12 et al. demonstrated that tin cluster ions with 10
30 atoms were solid at 50 K above the melting point of bulk tin, which indicated that nanodevices manufactured from r 2011 American Chemical Society

such materials may retain structural integrity and functionality at much higher temperature than anticipated. By ion mobility measurements, Shvartsburg34 et al. concluded that cationic tin clusters rearranged toward near-spherical geometries from n = 35, through a series of more compact structural families and then all clusters fell into a number of distinct families with a new one starting roughly every ten atoms. By using photoelectron spectroscopy, a semiconductor-metal transition was observed at n = 42, beyond which no energy gaps were observed, and the PES spectra became featureless and continuous.37 From calorimetric experiments, the crossover between the spherical-like and elongated neutral tin structures was estimated to be located around n = 45
85.35 Theoretically, Oger38 et al. found that the optimum structures assigned for tin cluster anions up to 15 atoms resembled the structures predicted for neutral tin clusters, demonstrating the presence of clear size-dependent trends. Majumder et al. showed that the binding energies of Snn (10e n e 20) clusters are only about 11% less than the calculated bulk value36 and the dominant channel for the fragmentation of Snn+ (11< n < 20) clusters was the fission into two subclusters.47 An ab initio molecular dynamics study indicated that the melting temperature of Sn10 may be 1000 K or more above the bulk melting point, due to the stability of the covalent tetracapped trigonal prism (TTP) unit.13
15 Chuang16 et al. found that the melting temperatures of Snn (n = 6, 7, 10, and 13) clusters were all above the melting temperature (505 K) of bulk tin. Pushpa18 et al. also found that bonds in Sn

Received: August 23, 2011 Revised: December 11, 2011 Published: December 13, 2011 231

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and Pb clusters were shorter and stiffer than that in the bulk, and these clusters melted at temperatures higher than that in the bulk. However, the underlying essential physical mechanism of the above intriguing phenomena12
18,33
39 in tin clusters is still open: (1) what are the binding forms and growth modes of larger tin clusters, and (2) what is the reason for the abnormal thermal stability in tin clusters? To answer these questions, we performed DFT calculations and found that tin clusters preferred the threadlike growth mode up to 50 atoms. Furthermore, strong covalent bonds between Sn15 and the other units make the threadlike clusters stable even when the temperature is higher than the melting point of tin bulk.18

2. DETAILS OF CALCULATION METHODS Our spin-polarized calculations are carried out using the Vienna ab initio simulation package,48,49 with the exchangecorrelation energy corrections described by the Perdew
Burke
Ernzerh50 parametrization. The interactions between the valence electrons and the ionic cores are described by the projector augmented wave method,51,52 the geometric structures are optimized by the conjugated gradient (CG) method,53 and the wave function is expanded in a plane wave basis with the energy cutoff of 103 eV. The larger simple cubic unit cells are used for larger cluster calculations to make sure that there is at least 12 Å between clusters in the neighboring cells. The Brillouin zone is represented by the Γ point, the total energy is converged to 10
4 eV for the electronic structure relaxations, and the convergence criterion for the force on each ion is 0.02 eV/Å. In the process of determining a ground state structure, we have employed three complementary approaches. First, the reported structures (Si, Ge, Sn, Pb, and Ru19
42,54,55), as well as some new structures we can contrive are taken into consideration in which the multibranch Ge and thread-like Sn structures are found to be the low-lying isomers. Second, Snn+m is obtained by randomly adding Snm on the comparatively stable isomer of the Snn cluster optimized in the first step. Third, the ab initio molecular dynamics (MD) simulations (5 ps, with 1 fs/step) in the canonical ensemble are carried out on some low-lying isomers, which have been verified as a powerful method to find the ground states in our previous works.42,54 To test the thermal stabilities and find more configurations, the temperature is set to be as high as 2100 K until the optimized final structure is different from the input one. By MD simulations, the structures reported previously and some new structures can always be obtained, especially for smaller clusters. Actually, the last two steps are performed circularly until no more stable structure is found. As a test, the lattice constant and cohesive energy on bulk tin in the diamond structure at 0 K are calculated to be 6.65 Å and 3.19 eV/atom, in agreement with the experimental values 6.49 Å and 3.14 eV/atom, respectively.56

Figure 1. (a, b) The structures and binding energies of the tin clusters. Sn10a and Sn15 are in orange, Sn10b in green, and the others in pink.

agreement with ref 36. The structures and binding energies of the most stable isomers of Snn (n = 10
50) clusters are presented in Figure 1a,b, with the cutoff 4 Å for sticks connecting two atoms. As seen in Figure 1a,b, Snn (n = 10
50) clusters mainly adopt the threadlike growth mode (based on Sn10 and Sn15 units), rather than multibranch germanium25
32 or amorphous-like lead isomers,42 with the relatively large binding energy starting roughly every five atoms. To guide the eyes, we plotted Sn10a and Sn15 in orange, Sn10b in green and the others in pink. From Sn10 to Sn14 (other than Sn12), the clusters usually have a Sn10a or Sn10b unit. Sn15 (Sn16) can be regarded as the couple of two Sn10a (Sn10b) units, with five (four) sharing atoms, while Sn17 is the couple of Sn10a and Sn10b units, with three common atoms. With more neighboring atoms on average, Sn18 is a magic cluster. For Snn (n = 19
23), independent Sn10a or Sn10b appears. More coordinated

3. STRUCTURE AND ENERGETIC STABILITY 3.1. Structures and Binding Energy. To elaborate the stabilities, we calculated the average binding energy of a given cluster according to the following formula

Eb ¼
½EðSnn Þ
nEðSnatom Þ=n

ð1Þ

where E(Snatom) and E(Snn) represent the total energies of a single Sn atom and Snn cluster, respectively. The structures of the ground states of Snn (n = 2
9) in our calculations are in good 232

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Figure 2. Some most stable isomers of charged structures and their relative stability to the same sizes as those in Figure 1.

Figure 3. (a) Second-order difference of energies, Δ2E = En+1 + En
1
2En; (b) Energy gap (eV), which is the energy difference between the lowest unoccupied and the highest occupied molecular orbital.

Sn24 has much lower binding energy than Sn25 and Sn26, showing that the formation of less coordinated threadlike structures is not an accident. From n = 24, we found that the couples of the Sn15 unit with independent Sn15 or Sn10b units are relatively stable. Besides, there are a few other growth modes shown in the following. Sn26 (the ground state of Ge2630) is 0.10 eV more stable than Sn26b (coupled by a distorted Sn16 and Sn10b). As a component in Sn31, Sn16 will distort to Sn16b (coupled by a Sn10a and Sn10b), and integral Sn18 occasionally appears in larger clusters. Sn32 is mainly coupled by a regular and two distort Sn10b units in sequence, forming a closed loop. Also favored for Gen,25,32 the structure of Snn (n = 33 and 37) clusters are more stable than their neighbors. Sn44 is 0.12 eV more stable than the Sn44b (the ground state of Ge4431). With two atoms circled by three Sn15 units, a plate like Sn47 has lower binding energy than its neighbors, showing that compacter structures are not necessarily favored. We also constructed a Sn50b isomer, with a ten atom core from diamond bulk and four Sn10b units at almost tetrahedral orientations, which is 0.71 eV less favorable than Sn50 (in Figure 1b). It is worthwhile to note that the optimized ground states of Snn (n = 18, 21, 35) and Sn50b (not in previous papers) are also found to be the gound state of Gen clusters. The threadlike growth mode of Snn (n = 10
50) may change the common sense that metal clusters usually adopt compact growth patterns, and the Sn15 unit may also be expected to fabricate nanowires. To support the credibility of the threadlike growth mode, we compare them with the structures favored for Ge clusters,25
32 and find that the ground states of Sn clusters usually have larger average bond lengths, more nearest neighbor atoms, and smaller HOMO
LUMO gaps. These characters indicate that Sn clusters have stronger metallic characters than Ge clusters, though the threadlike structures seem to have a stronger covalent bonding character than multibranch Ge structures. Our results agree with ion mobility measurements, which showed that Snn (n = 35
65) clusters gradually rearrange toward near-spherical geometries from prolate structures, passing through several intermediate structural families.34 The calculated threadlike structures up to n = 50 are also in agreement with a calorimetric experiment on Snn clusters, which manifested that the crossover between the spherical-like and elongated neutral tin structures was estimated to be located around n = 45
85.35 3.2. Threadlike Charged Structures. In experiment, clusters usually appear as the charged states,37 making them meaningful

to study. Therefore, we calculated the charged states and found that, the ground states of the charged states usually have similar structures with their neutral counterparts, with the exceptions shown in Figure 2. The ground state of Sn10
(in Figure 2) is about 0.21 eV more stable than the structure of Sn10
(structure in Figure 1a). Sn12
is similar to neutral Sn12, with the center atom closer to surface. We also obtained a perfect icosahedra Sn122
(bond length 3.13 Å and diameter 5.95 Å), which is in accordance to the results of Cui et al.46 Sn14
has a C2v symmetry and Sn22
has a Sn10b unit. Our calculated Sn20
and Sn25
(structure in Figure 1a) are in good agreement with the results of A. Lechtken et al.,39 while the calculated Sn18
(structure in Figure 1a) and Sn23
are more favored than that in ref 39. Coupled by a Sn15 and Sn18, Sn33
(in Figure 2) is 0.10 eV more stable than the structure of Sn33
in Figure 1b. For the ground state of cations, Sn13+ has a pentagonal bipyramid, and Sn16+ is the couple of Sn10a and Sn10b. Sn26+ is the couple of Sn10b and Sn16+, which is also the ground state of Sn26
, and Sn27+ is the ground state of Sn20
connected by seven atoms. In conclusion, the ground states of charged Snn (n = 10
50) clusters still adopt threadlike growth mode. 3.3. Energetic Stability Oscillations. To exhibit the relative stabilities of Snn (n = 10
50) clusters, we display the secondorder difference of energies Δ2 E ¼ Enþ1 þ En
1
2En

ð2Þ

where En represents the total energy of the Snn cluster. The second-order difference of energies in Figure 3a shows a strong biatom-by-biatom oscillation, and the period between the crossovers (at n = 15, 25, 33, and 41) is about ten atoms, in close agreement with ion mobility measurements.34 Besides, the clusters with an atom number of the integral multiple of five usually exhibit peaks in the Δ2E curve, indicating the importance of Sn10 and Sn15 units. Figure 3b shows that the highest occupied molecular orbital
lowest unoccupied molecular orbital (HOMO
LUMO) gaps decrease (with some oscillations) as cluster size increases. The oscillation of the HOMO
LUMO gaps and the second-order difference of energies are similar, except that the gaps of clusters with the Sn15 unit are usually lower than those with the Sn10b unit. In addition, the small HOMO
LUMO gaps for Sn15, 233

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Figure 4. Changes of bond number (relative units) with bond length distribution. The two dotted lines denote bond lengths 2.8 and 4 Å, respectively.

Figure 6. (a) Averaged charge density differences, ΔF, along the axis of the bonds between two atoms, with the bond center at zero; (b) The atom distribution of Sn20.

in a narrower segment, making it a better candidate in building larger clusters.

4. ELECTRONIC STRUCTURES AND THERMAL STABILITY 4.1. HOMO Charge Distribution. The HOMO is usually the most active molecular orbital in a molecules reaction. To explain why tin clusters usually have some specific units and the existence of specific bonding orientations between different units, we plot the charge (10
3 e) distributions of the HOMO of tin clusters in Figure 5. As seen in Figure 5, the charge of Sn10a accumulates at the three bottom atoms, giving a good explanation for the formation of Sn20. Compared with that of the Sn10a unit in Sn20, the charge distributions of the Sn15 unit in Snn (n = 25, 30, 40 and 45) have more changes, indicating strong binding forms between Sn15 and the other units. Besides, the charges of clusters (n = 15, 25, 30, 40 and 45) containing Sn15 units usually accumulate at the open end of the Sn15 unit, leading to the formation of threadlike clusters. Furthermore, the charge distributions of Sn16, Sn16b, and Sn18 are less concentrated at the ends than that of Sn15, making them harder to be the stable units in larger clusters. Compared with Sn44b, the HOMO charge of the Sn50b distributes unevenly, showing its relatively asymmetrical configuration. In a word, with special configurations and HOMO distributions, Sn15based clusters may be more favorable to fabricate nanowires. 4.2. Binding Characters in Different Units. To elaborate the relationship between the geometrical structures and the binding characters of tin clusters, we calculate the difference of the electronic charge

Figure 5. Charge (10
3 e) distribution of the HOMO in tin clusters.

Sn30, and Sn45 show relatively high activity of Sn15-based clusters, which may be longer to form nanowires. 3.4. Bond Distribution. To reveal why tin clusters favor some specific units, we gave the bond distribution function (BDF) in Figure 4 in which the two dotted lines denote the borderlines 2.8 and 4 Å. BDF shows the changes of bond numbers with respect to bond length. By measuring the bond length distribution, we find that less coordinated atoms always relate to shorter bond length. Taking Sn10a, for example, the line at 2.9 Å represents the three bonds with the top atom, and the line at 3.34 Å stands for the three bonds in the upper triangle. Sn10b has a rhomb mismatched with a square, and the bond larger than 3.4 Å corresponds to the shorter diagonal of the rhomb. The relatively short bonds in Sn15 distribute in a narrow segment, leading small distortions when coupled with other units. With four bonds at 3.65 Å, Sn16a will distort to more compact Sn16b in larger clusters. Sn18 has more nearest neighbor atoms on average, and the bonds near 4 Å are the short diagonals relating to the top and bottom atoms. The four bonds larger than 3.4 Å in Sn20 are between the two Sn10a units, indicating the weaker binding forms between the two Sn10a units. Larger clusters begin to have longer bonds, and the bond length distribution of the multiunit coupled Snn (n = 35, 40 and 50) cluster seems more continuous than the monounit Sn15based Snn (n = 30 and 45) cluster. In a word, compared with other clusters, the bond length in Sn15 is shorter and distributes

ΔF ¼ FðSCÞ
FðSPÞ

ð3Þ

of Sn20 in Figure 6a. Here, ΔF is the averaged charge density difference along the axis of the bonds in which the bond center has been shifted to zero, F(SC) is the charge density obtained by a self-consistent calculation, and F(SP) is the superposition of the atomic charge for the same structure. As presented in Figure 6a, electrons accumulate at the center of the bonds, revealing a covalent binding character in the Sn20 cluster. Less coordinated atoms (atoms 1 and 3) usually form shorter bond lengths with other atoms, corresponding to stronger covalent bonds. Besides, the covalent character in the Sn10a unit is stronger than that between Sn10a units (atoms 4 and 6), making Sn20 easy to fragment into two Sn10a and explaining the low intensities of Sn20 in the mass spectra.28,33 The difference of the electronic charge and the atom distribution of Sn40 are shown in Figure 7a,b. On the one hand, the 234

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For Ts < 505 K (melting point of bulk tin), Ed is obtained at 505 K (triangles). Taking Sn10a, for example, when we selected the final structure of Sn10a after a 5 ps MD simulation (at Ts = 2000 K) to optimize (at 0 K), we obtained the initial Sn10a, and in the same way, we got Sn10b after MD simulation (Ts + 100 K = 2100 K). Correspondingly, we gave Ed = 0.46 eV in Figure 8b, which means that Sn10b is 0.46 eV less stable than Sn10a. As seen in Figure 8a, relatively integral Snn (n = 10
18, 24 and 33) clusters with compact structures have the highest thermal stability, while multiunit clusters containing Sn10a or Sn18 usually have the lowest thermal stability. By MD simulations, new structures can also be found. For example, at 1000 K, the symmetry is broken and a new Sn15 (the ground state for Ge15) is 0.03 eV unstable, whose HOMO
LUMO gap is 0.45 eV larger than the most stable Sn15 (in Figure 1a). At 800 K, the more compact Sn18 in Figure 1a is optimized from MD simulation, with the initial structure similar with that in ref 36, and a structure like Sn26b (in Figure 2) is optimized from Sn26 (in Figure 1a). During MD simulations, rotations are also found between units in some clusters, such as Sn20 and Sn35, adding more choices to find the most stable isomers. In the following, we will focus on the fragmentation behavior. Sn17 starts to break into a Sn10a and a bicapped pentagonal Sn7 at 1200 K, and Sn20 begins to split into two Sn10a units at 750 K, close with previous results.17 For MD simulations from 505 to 800 K, independent Sn10a is separated from Snn (n = 20, 22, 25, 37, 41, and 44), while other than some distortions, there is no obvious fragmentation behavior for other sizes. Though the thermal stability definition is somewhat arbitrary, it does show that Sn10b- and Sn15-based Snn (n = 10
50) clusters are usually stable when the temperatures are higher than the melting point of tin bulk. The calculated thermal stability is in agreement with the relative mobility experiment that tin cluster ions with 10
30 atoms remain solid at 50 K above the melting point of bulk tin.12 With relatively high thermal stability, Sn15 is expected to be a potential unit to fabricate nanowires. To elaborate the reason that some clusters are energetically favorable, we calculated the fragmentation energy

Figure 7. (a) Same as that in Figure 6a. (b) The atom distribution of Sn40.

Figure 8. (a) Thermal stability temperature Ts of Sn clusters, after MD at Ts + 100 K (Ts), the structures (do not) change; (b) Ed after MD at (Ts + 100 K), the energy difference between the optimized final state and the initial structure. For Ts < 505 K, Ed is found at 505 K (triangles). (c) Fragmentation energy.

Ef ðnÞ ¼ EðpÞ þ EðqÞ
EðnÞ

ð4Þ

in Figure 8c, of an n-atom cluster into p-atom and q-atom (n = p + q) fragments in which the fission is assumed to occur along the lowest energy pathways with no activation barrier.36 We find that our calculated dissociation paths are in accordance with the lowenergy collision experiment47 and the calculated results of Majumder et al.36 Besides, Snn (n = 10, 15, and 18) clusters with large Ef usually correspond to the mass abundance spectra, while the low fragmentation energies for Snn (n = 17, 19 and 20) corroborate the low intensities observed in the mass spectra.28,33 For larger clusters, Sn30 and Sn45 also have larger Ef, indicating that Sn15-based clusters have higher stability.

covalent bonds between Sn15 units and the other clusters (with the maximum between Sn15 units in purple) are the strongest, explaining why Sn15 can be a basic unit in building larger clusters. On the other hand, the bonds in the Sn15 unit are relatively weak. Compared with Pb clusters, the covalent binding character in Sn clusters is more significant.42 Figures 6 and 7 indicate that stronger binding forms between Sn15, and the other units make Sn15 a stable unit to form larger threadlike clusters. 4.3. Thermal Stability. Besides the energetic stability, thermal stability is another aspect to judge a cluster. In experiment, temperature is usually a key factor to produce needed materials. Huang et al. showed that energetically favored silicon monoxide may be more likely to dissociate under the growth temperature of the silicon monoxide nanowires.57 Li et al. showed that tin nanowires usually grew at temperatures higher than the melting point of tin bulk.58 To test the thermal stabilities and find more configurations, we carried out the ab initio molecular dynamics (MD) simulations (5 ps, with 1 fs/step) in the canonical ensemble on tin clusters. We defined a critical temperature (Ts) in Figure 8a to characterize the thermal stability. At Ts, the cluster can keep its configuration, while at (Ts +100 K), the cluster begins to change.

5. CONCLUSIONS In summary, first-principles calculations using density functional theory (DFT) have been carried out to study the geometric and electronic structures of tin clusters. Our results show that threadlike tin clusters (mainly based on Sn10 and Sn15 units) are usually significantly more favorable than the compact growth modes up to n = 50. Besides, the stabilities of Snn (n = 10
50) clusters manifest a biatom-by-biatom oscillation with some crossovers, and the magic cluster size appear at n = 10 N, in agreement with experimental results.34 Furthermore, the charges 235

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of the HOMO of the Sn15 cluster accumulate at both ends, which can form strong covalent bonds with other units and make Sn15based clusters stable. For the thermal stability, ab initio molecular dynamics simulations also reveal that Snn (n = 10
50) clusters are usually stable when the temperature is above the melting point of tin bulk, with Sn10a as the basic fragmentation at higher temperatures. Finally, the ground states of charged Snn (n = 10
50) clusters still adopt a threadlike growth mode. Our results may present a potential way to fabricate tin nanowires in experiment.

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’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank Dr. P. F. Yuan, Dr. F. Wang, and Dr. S. F. Li for helpful discussions. The work was partly support by the NSF of China (Grant No. 10974182) and partly by OYF of Henan, China. ’ REFERENCES (1) Schmidt, M.; Kusche, R.; Kronm€uller, W.; von Issendorff, B.; Haberland, H. Phys. Rev. Lett. 1997, 79, 99–102. (2) Schmidt, M.; Kusche, R.; von Issendorff, B.; Haberland, H. Nature 1998, 393, 238–240. (3) Schmidt, M.; Donges, J.; Hippler, Th.; Haberland, H. Phys. Rev. Lett. 2003, 90, 103401–103404. (4) Haberland, H.; Hippler, Th.; Donges, J.; Kostko, O.; Schmidt, M.; von Issendorff, B. Phys. Rev. Lett. 2005, 94, 035701–035704. (5) Zorriasatein, S.; Lee, M. S.; Kanhere, D. G. Phys. Rev. B 2007, 76, 165414–165421. (6) Hock, C.; Bartels, C.; Straßburg, S.; Schmidt, M.; Haberland, H.; von Issendorff, B.; Aguado, A. Phys. Rev. Lett. 2009, 102, 043401–043404. (7) Nu~nez, S.; Lopez, J. M.; Aguado, A. Phys. Rev. B 2009, 79, 165429–165432. (8) Starace, A. K.; Neal, C. M.; Cao, B. P.; Jarrold, M. F.; Aguado, A.; Lopez, J. M. J. Chem. Phys. 2009, 131, 044307–044317. (9) Starace, A. K.; Cao, B. P.; Judd, O. H.; Bhattacharyya, I.; Jarrold, M. F. J. Chem. Phys. 2010, 132, 034302–034310. (10) Zhang, Y.; Chen, H. S.; Liu, B. X.; Zhang, C. R.; Li, X. F.; Wang, Y. C. J. Chem. Phys. 2010, 132, 194304–194312. (11) Chacko, S.; Joshi, K.; Kanhere, D. G.; Blundell, S. A. Phys. Rev. Lett. 2004, 92, 135506–135509. (12) Shvartsburg, A. A.; Jarrold, M. F. Phys. Rev. Lett. 2000, 85, 2530–2533. (13) Joshi, K.; Kanhere, D. G.; Blundell, S. A. Phys. Rev. B 2002, 66, 155329–155333. (14) Joshi, K.; Kanhere, D. G.; Blundell, S. A. Phys. Rev. B 2003, 67, 235413–235420. (15) Krishnamurty, S.; Joshi, K.; Kanhere, D. G.; Blundell, S. A. Phys. Rev. B 2006, 73, 045419–045429. € g€ut, S.; Chelikowsky, J. R.; Ho, (16) Chuang, F. C.; Wang, C. Z.; O K. M. Phys. Rev. B 2004, 69, 165408–165419. (17) Breaux, G. A.; Neal, C. M.; Cao, B. P.; Jarrold, M. F. Phys. Rev. B 2005, 71, 073410–073413. (18) Pushpa, R.; Waghmare, U.; Narasimhan, S. Phys. Rev. B 2008, 77, 045427–045432. (19) Mitas, L.; Grossman, J. C.; Stich, I.; Tobik, J. Phys. Rev. Lett. 2000, 84, 1479–1482. (20) Jarrold, M. F.; Constant, V. A. Phys. Rev. Lett. 1991, 67, 2994–2997. (21) Li, S. F.; Gong, X. G. J. Chem. Phys. 2005, 122, 174311–174317. (22) Hellmann, W.; Hennig, R. G.; Goedecker, S.; Umrigar, C. J.; Delley, B.; Lenosky, T. Phys. Rev. B 2007, 75, 085411–085415. 236

dx.doi.org/10.1021/jp208121s |J. Phys. Chem. C 2012, 116, 231–236