3998
J. Phys. Chem. C 2008, 112, 3998-4004
Storage of Hydrogen Adsorbed on Alkali Metal Doped Single-Layer All-Carbon Materials Adolfo Ferre-Vilaplana* High-Performance Computing & Visualization Aided Scientific & Engineering Applications Group, Instituto Tecnolo´ gico de Informa´ tica, Ciudad Polite´ cnica de la InnoVacio´ n, Camino de Vera s/n, Edificio 8G, 46022 Valencia, Spain, and Departamento de Sistemas Informa´ ticos y Computacio´ n, Escuela Polite´ cnica Superior de Alcoy, UniVersidad Polite´ cnica de Valencia, Plaza Ferra´ ndiz y Carbonell s/n, 03801 Alcoy, Spain ReceiVed: August 28, 2007; In Final Form: December 4, 2007
Controversial experimental results have suggested that the presence of alkali metals in graphitic adsorbents would intensify hydrogen adsorption. Mainly by use of density functional theory methods, several authors have confirmed this intensification, which has been explained by a significant charge transfer from dopand, located on top of the center of a ring (the hollow configuration), to substrate. However, in this work, it was found that, for each distance of the hollow configuration of the Li-graphene interaction, depending on the initial guess, not only one but two qualitatively different self-consistent field solutions would be possible. One of them would concentrate charge in the region between lithium and graphene. The other one would be just the opposite. At fully correlated second-order Mo¨ller-Plesset level, for each distance, the latter was found to be significantly more stable than the former. So, it seems that the charge transfer from lithium to graphene described by several authors would not take place. The set of results reported here would provide evidence that solutions for the interaction of alkali metals with single-layer all-carbon materials would be extremely sensitive to initial guesses and correlation treatments. It is concluded that, in the case of alkali metal doping being capable of significantly intensifying physisorption of molecular hydrogen on single-layer all-carbon materials, the postulated charge transfer from dopand (located on top of the center of a ring) to substrate would not be the mechanism.
1. Introduction To use it as a clean fuel, the storage of hydrogen adsorbed on carbon has been extensively investigated.1-4 Chemisorption of atomic hydrogen on sp2 hybridized carbon nanostructures can be achieved.5,6 Though, due to the fact that lower desorption temperatures would be required, hydrogen physisorption, in molecular form, would have been considered a much more relevant mechanism for the application. However, physisorption of molecular hydrogen on single-layer all-carbon materials (of the order of 0.1 eV)7,8 would not be sufficient (0.3 eV would be required)9,10 as a mechanism around which to develop efficient hydrogen storage systems. Experimental results11 have suggested that the presence of alkali metals in carbonaceous adsorbents would intensify hydrogen adsorption. Despite the fact that it has been pointed out12,13 that the high hydrogen uptake measured by Chen et al.11 would be due to impurities, mainly using density functional theory (DFT) methods, several authors14-20 have confirmed the suggested hypothetical intensification. In fact, it has been predicted17 that on Li-doped graphene and single-walled carbon nanotubes (SWCNTs) hydrogen physisorption would be even two times larger than on pristine materials. A significant charge transfer from alkali metal (located on top of the center of a ring) to substrate, which, depending on the dopand specie and author, would be in the range of 0.3-1.0 e, would have been identified as the ultimate mechanism responsible for the mentioned intensification. It has even been argued14 that experimental evidence of the postulated charge-transfer would * To whom correspondence should be addressed. E-mail: aferre@ dsic.upv.es.
exist.21,22 However, the fully correlated second-order Mo¨llerPlesset23 (MP2) calculations reported here do not support the described view. Given that, for a number of interacting bodies greater than two, the accurate solution of a n-body molecular complex can represent a formidable challenge,24 in this research, the attention was centered on the dopand-substrate interaction. Because lithium has been the most investigated dopand14-20,25-27 and because planar graphene would be the most basic single-layer carbonaceous substrate, the Li-graphene interaction was reviewed. Finally, because it has been found14,17,25-27 to be the most favorable binding site and because, in order to perform computational intensive treatments, limiting the number of calculations would be desirable, only the configuration under which the lithium atom is located on top of the center of an aromatic ring (the hollow configuration) was taken into account. Mainly by use of DFT, the hollow configuration of the Ligraphene interaction has been described15,17,25-27 as a relatively intense interaction, presenting a relatively short equilibrium distance, explained by a significant charge transfer from lithium to graphene (Table 3). However, though recent DFT developments28 have tried to describe nonbond interactions, classical DFT functionals would not be capable of capturing long-range dispersion forces.10 Under the local density approximation (LDA) and under the generalized gradient approximation (GGA), DFT can find binding energy or repulsion, respectively. And, such as can be observed in Table 3, even using relatively similar LDA treatments (as parametrized by Perdew and Wang29 and as parametrized by Vosko et al.30), DFT can predict very different interaction energies (1.6 vs 0.88 eV).
10.1021/jp0768874 CCC: $40.75 © 2008 American Chemical Society Published on Web 02/21/2008
Storage of Adsorbed Hydrogen
Figure 1. Coronene-like (C24H12) planar sheet used as cluster-modeled graphene. Smallest balls represent hydrogen atoms, while largest balls represent carbon atoms. The purpose of the hydrogen atoms is to passivate the dangling bonds of the polycyclic sp2 hybridized carbon structure.
Alternatively, wavefunction based treatments can be considered. Zhu and Lu26 discarded their own MP2 results, corresponding to the interaction of alkali metals with different polycyclic aromatic hydrocarbons (PAHs), arguing they would be unusual. And, Cabria et al.17 borrowed conclusions arrived at by Okamoto and Miyamoto,31 from MP2 calculations, whose weaknesses have been discussed.7 So, a concluding MP2 treatment, to the Li-graphene interaction, has not yet been applied. Modeling graphene as a coronene-like (C24H12) planar sheet (Figure 1) and using correlation consistent basis sets, in this work, running spin-polarized fully correlated MP2 calculations, the hollow configuration of the Li-graphene interaction was reinvestigated. Under the rigid monomer approach, it was found that, for each distance of the considered configuration of the target interaction, depending on the initial guesses, not only one but two qualitatively different self-consistent field (SCF) solutions would be possible. One of them would concentrate charge in the region between lithium and graphene. The other one would be just the opposite. At the MP2 level, for each distance, the latter was found to be significantly more stable than the former. It was found that, under the hollow configuration, the Li-graphene interaction energy would be only of the order of 20% of what has been previously published. The set of results reported here would provide evidence that solutions for the interaction of alkali metals with single-layer all-carbon materials would be extremely sensitive to initial guesses and correlation treatments. It is concluded that, in the case of alkali metal doping being capable of significantly intensifying physisorption of molecular hydrogen on single-layer all-carbon materials, the postulated charge transfer from dopand (located on top of the center of a ring) to substrate would not be the mechanism. 2. Computational Methods Preliminary Li-benzene calculations revealed that, to get reasonably accurate quantitative charge transfer predictions,
J. Phys. Chem. C, Vol. 112, No. 10, 2008 3999 basis sets of at least triple-ζ quality would be required, that the augmentation of basis sets with diffusion functions, at least for the closest carbon atoms to the lithium atom, would be desirable, that the inclusion of diffusion functions in the lithium atom model would not give rise to a significantly better interaction description increasing significantly the basis set superposition error (BSSE), and that the consideration of the core correlation effects, at least for the lithium atom, could end up having a certain effect on the quantitative predictions quality. By assumption that the above summarized preliminary conclusions and modeling graphene as a coronene-like (C24H12) planar sheet (Figure 1), for which the coordinates of Jeolaica and Sidis32 were used (C-C and C-H bond lengths of 1.415 and 1.084 Å, respectively), mainly under the rigid monomer approach, for the superposition of atomic densities and for the assembled from fragments initial guesses, in this work, using NWChem 4.7,33 different Hartree Fock (HF) and spin-polarized fully correlated MP223 calculations were performed. Binding energies were estimated under the supermolecular approach. BSSEs were compensated by means of the counterpoise method.34 For the complex and the lithium atom monomer doublet wave functions were requested. For the coronene-like monomer spin-polarized singlet wave functions were obtained. All the MP2 calculations were performed under the C2v point group symmetry. Because MP2 density calculations were not available in NWChem 4.7, MP2 densities were evaluated using Gaussian 03.35 Given that the assembled from fragments initial guess was not available on Gaussian 03, to perform MP2 density calculations, the approach of using a projection of a solution based on relatively unsaturated basis sets on a much more saturated basis set as initial guess for a new self-consistent field (SCF) HF procedure, after which MP2 corrections were applied, was taken. It was verified that, under the above-described procedure, using Gaussian 03, virtually the same solution as that under the assembled from fragments initial guess, using NWChem 4.7, was reached. Quantitative charge transfers from lithium to graphene were estimated from Mulliken and natural orbital population analysis. Charge displacements were calculated as the difference between the density of the complex and that of the monomers. BSSEs, at density level, were compensated by means of using dimercentered basis sets for the calculation of the monomer densities. Charge displacements were visualized using Open DX. The correlation consistent family of spherical basis sets ccpVXZ36 (XdD and XdT) and the specifically designed for taking into account not only valence-valence but also valencecore and core-core correlation effects cc-pCVXZ37 (XdD and XdT), augmented with diffusion functions when required, were used for modeling the complex. Hydrogen atoms were modeled using cc-pVXZ basis sets. With the exception of the carbon atoms of the central aromatic ring, carbon atoms and lithium were modeled using cc-pCVXZ basis sets. The central aromatic ring was modeled using aug-cc-pCVXZ basis sets. Because ccpCVXZ and aug-cc-pCVXZ basis sets were not available in Gaussian 03, to perform MP2 density calculations, they were downloaded from the Extensible Computational Chemistry Environment Basis Set Database. Under the supermolecular approach the interaction energy of a dimer is given by
∆E ) E(D) - [E(M1) + E(M2)] where E(D) represent the energy of the dimmer and E(M1) and E(M2) that of the monomers. The counterpoise compensated
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Ferre-Vilaplana
Figure 2. Li-graphene total energy prospective results, for the lithium atom located on top of the center of an aromatic ring, estimated under the rigid monomer approach, running HF and spin-polarized fully correlated MP2 calculations and using double-ζ quality basis sets and two different initial guesses. (a) MP2 results. (b) HF results. Triangles, using the superposition of atomic densities initial guess. Circles, assembling initial guesses from calculations performed on fragments.
interaction energy and the counterpoise compensation energy are given by
∆EBSSE_corrected ) EDCBS(D) - [EDCBS(M1) + EDCBS(M2)] BSSE ) [EDCBS(M1) - EMCBS(M1)] + [EDCBS(M2) EMCBS(M2)] where EDCBS( ) and EMCBS( ) mean that the corresponding energy calculation has to be performed using dimmer-centered and monomer-centered basis sets, respectively.
3. Results and Discussion Using double-ζ quality basis sets, under the rigid monomer approach, for different distances of the hollow configuration of the Li-graphene interaction, the total MP2 energy prospective results displayed in Figure 2a (triangles) were obtained. Said curve suggests that a stationary point representing a relatively short equilibrium distance, in good agreement with previously published results (Table 3), would exist. However, because, for each point, in the SCF HF stage of the calculation, the highestoccupied and lowest-unoccupied molecular orbitals were both of them found degenerated, warning messages, pointing out that initial guesses obtained from the superposition of atomic
Storage of Adsorbed Hydrogen
J. Phys. Chem. C, Vol. 112, No. 10, 2008 4001
Figure 3. Li-graphene BSSE-corrected binding energies, for the lithium atom located on top of the center of an aromatic ring, estimated under the rigid monomer supermolecular approach, modeling graphene as a coronene-like sheet, using correlation consistent basis sets specifically designed for taking into account not only valence-valence but also core-valence and core-core correlation effects, assembling initial guesses from calculations performed on fragments and running spin-polarized fully correlated MP2 calculations. Triangles, using double-ζ quality basis sets. Circles, using triple-ζ quality basis sets.
densities, which is the default in NWChem 4.7,33 could end up giving rise to nonvariational energies, were generated. For that reason, a more educated initial guess, assembled from calculations performed on fragments, which is also available in NWChem 4.7, was tried, obtaining the results displayed in Figure 2a (circles). Moreover, starting from a far away distance, the strategy of using a previous solution as initial guess for a shorter distance, until reaching the shortest distance, was also tried, obtaining virtually the same results displayed in Figure 2a (circles). For comparison, also the HF results corresponding to the MP2 results presented in Figure 2a are displayed in Figure 2b. Figure 2b shows that, for each distance of the hollow configuration of the Li-graphene interaction, depending on the initial guess, not only one but two qualitatively different SCF HF solutions would be possible. One of them (triangles) would concentrate charge in the region between lithium and graphene. The other one (circles) would be just the opposite. In spite of the evident relevance of the mentioned fact, to our knowledge, this has not yet been described. Said circumstance, and not only the fact itself of using MP2, instead of DFT, would be the true origin of the surprising conclusions arrived at in this work. Starting from initial guesses, SCF procedures variationally minimize parametrized trial wavefunctions. Given that SCF procedures do not perform global optimizations, different initial guesses having different attractors give rise to different solutions. Universally optimal initial guesses do not exist. For that reason, reasonably competent software packages provide not only one but several ways of assembling initial guesses. Though, usually, only one of them would be statically designated as the default initial guess. Optimal initial guesses to describe relatively intense interactions (having the trend of locating charge in regions between atoms) could become inappropriate (giving rise to nonvariational energies) for describing some weak interactions (such as the one investigated in this research).
Under the hollow configuration, for the Li-graphene interaction, a dual solution can be obtained not only at HF level but also when using DFT. For instance, for the lithium atom located 2.2 Å on top of the center of the coronene-like cluster modeled graphene, under the B3LYP/6-31g* chemistry model, using the Harris functional (which is the default initial guess in Gaussian 0335), a 2-B2 initial guess, followed by a 2-B2 final solution, corresponding to -929.363635 hartrees of energy, giving rise to +0.59 electrons as Mulliken population for the lithium atom, were reached. However, for the same configuration and distance, using INDO (which was the default initial guess in Gaussian 98), a 2-A1 initial guess, followed by a 2-A1 final solution, corresponding to -929.372430 hartrees of energy, giving rise to -0.21 electrons as Mulliken population for the lithium atom, were obtained. Figure 2a shows that, for each distance of the hollow configuration of the Li-graphene interaction, the total MP2 energies obtained under the assembled from fragments initial guess are significantly more stable. However, because the mentioned results were obtained under the rigid monomer approach, additional calculations were performed. Starting from the initial distance of 1.8 Å, under the superposition of atomic densities initial guess, the lithium atom and the central aromatic ring of the cluster modeled graphene were completely relaxed at HF level. The total HF and fully correlated MP2 energies corresponding to the final configuration reached by the above-mentioned optimization process, estimated at double-ζ and triple-ζ quality basis sets levels and using two different initial guesses, are summarized in Table 1. It can be observed in Table 1 that, even for the most favorable configuration arrived at under the superposition of atomic densities initial guess, the total MP2 energy obtained under the assembled from fragments initial guess was found to be significantly more stable. So, according to Figure 2a (circles),
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Ferre-Vilaplana
TABLE 1: Li-Graphene Total Energy Prospective Results (hartrees), for the Final Configuration Reached after Relaxing the Lithium Atom and the Central Aromatic Ring under the Superposition of Atomic Densities Initial Guess, Estimated Running HF, and Spin-Polarized Fully Correlated MP2 Calculations at Double-ζ and Triple-ζ Quality Basis Sets Levels and Using Two Different Initial Guesses
double-ξ quality basis sets
superposition of atomic densities initial guess
assembled from fragments initial guess
-923.426637 -927.442251 -923.621513 -928.545355
-923.421707 -927.492401 -923.616459 -928.590610
HF MP2 HF MP2
triple-ξ quality basis sets
TABLE 2: Li-Graphene Equilibrium Distances, BSSE-Corrected Binding Energies, BSSE Corrections, and Percentage of the Correction with Respect to the Total Uncorrected Binding Energy, for the Lithium Atom Located on Top of the Center of a Ring, Estimated under the Rigid Monomer Supermolecular Approach, Modeling Graphene as a Coronene-Like Sheet, Using Correlation Consistent Basis Sets Specifically Designed for Taking into Account Not Only Valence-Valence but Also Core-Valence and Core-Core Correlation Effects, Assembling Initial Guesses from Calculations Performed on Fragments and Running Spin-Polarized Fully Correlated MP2 Calculations
double-ξ basis sets triple-ξ basis sets
equilibrium distance (Å)
BSSE-corrected binding energy (µhartree)
BSSE correction (µhartree)
BSSE correction/BSSE uncorrected binding energy (%)
3.2 3.2
4893 4918
2897 1352
37 22
TABLE 3: Li-Graphene Interaction Energy, Equilibrium Distance, and Lithium to Graphene Charge Transfer, for the Lithium Atom Located on Top of the Center of an Aromatic Ring, Estimated by Different Authors
Dubot and Cenedeseb Khanta et al.c Zhu and Lu.d Cabria et al.e Valencia et al.f this work
method
binding energy (eV)
equilibrium distance (Å)
charge transfera (e)
semiempirical periodic DFT molecular DFT periodic DFT periodic DFT molecular MP2
1.7 1.6 0.64-1.36 0.52-0.88 1.01-1.55 0.13
2.1 1.64 1.71-1.74 1.83-187 1.63-1.67 3.2
0.72 not quantifiedg 0.40-0.44 0.5 0.89-1.0 -0.01h
Positive values represent charge transfer from lithium to graphene. b On the basis of a sophisticated linear combination of atomic orbitals scheme.14 c On the basis of plane waves expansions under the LDA approximation as parametrized by Perdew and Wang.25 d Modeling graphene as different polycyclic aromatic hydrocarbons, using 6-31(d,p) basis sets under the B3LYP functional and neglecting BSSEs.26 e On the basis of plane wave expansions under the LDA approximation as parametrized by Vosko et al.17 f On the basis of plane wave expansions under the LSDA approximation and under the gradient-corrected PBE approximation to the exchange-correlation energy.27 g Charge displacements were visualized. A significant net charge loss on top of the lithium atom and a net charge gain in the region between lithium and graphene was described. h Estimated from a natural orbital population analysis of a solution having 22% of BSSE. a
stationary points having relatively short equilibrium distances would not exist. Moreover, parts a and b of Figure 2 and Table 1 suggest that, for the kind of interaction investigated here, solutions could be sensitive not only to initial guesses but also to correlation treatments. Compensating BSSEs by means of the counterpoise method, under the assembled from fragments initial guess and under the rigid monomer supermolecular approach, for different distances of the considered configuration of the Li-graphene interaction, BSSE-corrected binding energies were estimated at double-ζ and triple-ζ basis sets levels (Figure 3). Values corresponding to stationary points are summarized in Table 2. It was found that, under the hollow configuration, the Li-graphene interaction energy would be only of the order of 20% of what has been previously published (Table 3). Given that, as has been shown, solutions for the kind of interaction investigated here could be sensitive to correlation treatments, realistic charge-transfer estimations would have to be based on correlated density calculations. Moreover, because standard quantitative charge-transfer estimation methods would be based on solutions including BSSEs, in order to perform said estimations, solutions having relatively low BSSEs would be desirable. Running spin-polarized fully correlated MP2 density calculation, in Gaussian 03,35 at triple-ζ quality basis sets level, for the equilibrium point of the hollow configuration of the Li-
graphene interaction, Mulliken and natural orbital charge transfers from lithium to graphene of -0.06 and -0.01 e were found, respectively. Additionally, because, even at triple-ζ quality basis sets level, BSSE represents 22% of the total uncorrected binding energy (Table 2), the charge displacements provoked by the considered configuration of the Li-graphene interaction, calculated as the difference between the density of the complex and that of the monomers, compensating the BSSE at density level by means of using dimer-centered basis sets for the calculation of the monomers densities, were visualized (Figure 4). The most salient characteristic of the charge displacements visualized in Figure 4 would be the slight charge concentration on top of the lithium atom core region jointly to the slight charge depletion in the region between lithium and graphene. Excluding the core region of the lithium atom, said concentrations and depletions would be in the range of (1.0 × 10-3 e/au.3 From a qualitative point of view, the charge displacements found in this work would be just the opposite to the ones described by Khanta et al.,25 Cabria et al.,17 and Valencia et al.,27 who described a net charge loss on top of the lithium atom and a net charge gain in the region between lithium and graphene. From a quantitative point of view, the charge displacements reported here would be of an order of less significant magnitude. Given that a significant charge transfer from lithium to graphene seems not to take place, our results suggest that, in the case of alkali metal doping being capable of significantly
Storage of Adsorbed Hydrogen
J. Phys. Chem. C, Vol. 112, No. 10, 2008 4003 in the case of alkali metal doping being capable of significantly intensifying physisorption of molecular hydrogen on single-layer all-carbon graphitic surfaces, the postulated charge transfer from dopand (located on top of the center of a ring) to substrate would not be the mechanism.
Figure 4. Charge displacements provoked by the Li-graphene interaction, for the lithium atom located in the equilibrium point on top of the center of an aromatic ring, on a plane perpendicular to graphene containing the lithium atom, estimated from densities obtained running spin-polarized fully correlated MP2 calculations at triple-ζ quality basis sets level. The BSSE at density level was compensated by means of using dimer-centered basis sets for the calculation of the monomer densities. Excluding the core region of the lithium atom, concentrations and depletions are in the range of (1.0 × 10-3 e/au.3 Neutral regions are green colored, charge concentrations are colored from yellow to red, and charge depletions are blue. Isocurves correspond to the concentration and depletion values (1.0 × 10-4, (2.0 × 10-4, and (5.0 × 10-4 e/au.3
intensifying physisorption of molecular hydrogen on single-layer all-carbon materials, the postulated charge transfer from dopand (located on top of the center of a rig) to substrate would not be the mechanism. 4. Conclusions Under the hollow configuration, the Li-graphene interaction has been described as a relatively intense interaction explained by a significant charge transfer from lithium to graphene. However, in this work, it was found that, for each distance of the considered configuration of the target interaction, depending on the initial guess, not only one but two qualitatively different SCF solutions would be possible. One of them would concentrate charge in the region between lithium and graphene. The other one would be just the opposite. At the MP2 level, for each distance, the latter was found be significantly more stable than the former. So, it seems that the significant charge transfer from lithium to graphene described by several authors would not take place. It was found that, under the hollow configuration, the Li-graphene interaction energy would be only of the order of 20% of what has been previously published. Therefore, the considered configuration of the target interaction would be better described as a much more conventional weak interaction. Because the experimental results reported by Rao et al.21 and Lee et al.22 were obtained from atom intercalated bundles and ropes of carbon nanotubes, in which a single intercalated atom could interact simultaneously with two nanotubes, and because in this work only the hollow configuration of the Li-graphene interaction was taken into account (of three which were possible), our results would not necessarily be in conflict with experiments.21,22 The set of results reported here would provide evidence that solutions for the interaction of alkali metals with single-layer all-carbon materials would be extremely sensitive to initial guesses and correlation treatments. It is concluded that,
Acknowledgment. This work was supported in part by Consellerı´a de Empresa, Universidad y Ciencia de la Generalitat Valenciana, from Spain, under Grant GV05/262. The support of the Centro de Ca´lculo of the Universidad Polite´cnica de Valencia, providing access to the high-performance computing resources required in this research, is gratefully acknowledged. NWChem Version 4.7, as developed and distributed by Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352 USA, and funded by the U.S. Department of Energy, was used to obtain some of these results. Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 02/25/04, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory, which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, USA, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy under contract DE-AC0676RLO 1830. Contact Karen Schuchardt for further information. References and Notes (1) Dillon, A. C.; Jones, K. M.; Bekkedhal, T. A.; Kiang, C. H.; Bethune, D. S.; Heben, M. J. Nature 1997, 386, 377-379. (2) Meregalli, V.; Parrinello, M. Appl. Phys. A 2001, 72, 143-146. (3) Lamari Darkrim, F.; Malbrunot, P.; Tartaglia, G. P. Int. J. Hydrogen Energy 2002, 27, 193-202. (4) Liu, C.; Cheng, H. M. J. Phys. D: Appl. Phys. 2005, 38, R231R252. (5) Yildirim, T.; Gulseren, O.; Ciraci, S. Phys. ReV. B 2001, 64, 075404. (6) Nikitin, A.; Ogasawara, H.; Mann, D.; Denecke, R.; Zhang, Z.; Dai, H.; Cho, K.; Nilsson, A. Phys. ReV. Lett. 2005, 95, 225507. (7) Ferre-Vilaplana, A. J. Chem. Phys. 2005, 122, 104709. (8) Ferre-Vilaplana, A. J. Chem. Phys. 2005, 122, 214724. (9) Li, J.; Furuta, T.; Goto, H.; Ohashi, T.; Fujiwara, Y.; Yip, S. J. Chem. Phys. 2003, 119, 2376. (10) Lochan, R. C.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 1357-1370. (11) Chen, P.; Wu, X.; Lin, J.; Tan, K. L. Science 1999, 285, 91-93. (12) Yang, R. T. Carbon 2000, 38, 623-641. (13) Pinkerton, F. E.; Wicke, B. G.; Olk, C. H.; Tibbetts, G. G.; Meisner, G. P.; Meyer, M. S.; Herbst, J. F. J. Phys. Chem. B 2000, 104, 94609467. (14) Dubot, P.; Cenedese, P. Phys. ReV. B, 2001, 63, 241402. (15) Froudakis, G. E. Nano Lett. 2001, 1, 531-533. (16) Deng, W. Q.; Xu, X.; Goddard, W. A. Phys. ReV. Lett. 2004, 92, 166103. (17) Cabria, I.; Lo´pez, M. J.; Alonso, J. A. J. Chem. Phys. 2005, 123, 204721. (18) Sun, Q.; Jena, P.; Wong, Q.; Marquez, M. J. Am. Chem. Soc. 2006, 128, 9741-9745. (19) Zhang, Y.; Scanlon, L. G.; Rottmayer, M. A.; Balbuena, P. B. J. Phys. Chem. B 2006, 110, 22532-22541. (20) Mpourmpakis, G.; Tylianakis, E.; Froudakis, E. Nano Lett. 2007, 7, 1893-1897. (21) Rao, A. M.; Eklund, P. C.; Bandow, S.; Thess, A.; Smalley, R. E. Nature 1997, 388, 257-259. (22) Lee, R. S.; Kim, H. J.; Fischer, J. E.; Lefebvre, J.; Radosavljevic, M.; Hone, J.; Johnson, A. T. Phys. ReV. B 2000, 61, 4526. (23) Mo¨ller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618-622. (24) Valiron, P.; Mayer, I. Chem. Phys. Lett. 1997, 275, 46-55. (25) Khanta, M.; Cordero, N. A.; Molina, M. L.; Alonso, J. A.; Girifalco, L. A. Phys. ReV. B 2004, 70, 125422. (26) Zhu, Z. H.; Lu, G. Q. Langmuir 2004, 20, 10751-10755. (27) Valencia, F.; Romero, A. H.; Anciloto, F.; Silvestrelli, P. L. J. Phys. Chem. B 2006, 110, 14832-14841. (28) Xu, X.; Zhang, Q.; Muller, R. P.; Goddard, W. A., III. J. Chem. Phys. 2005, 122, 014105. (29) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244.
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