J. Phys. Chem. B 2001, 105, 11227-11232
11227
Theoretical Study of Oxygen Adsorption on Graphite and the (8,0) Single-walled Carbon Nanotube Dan C. Sorescu and Kenneth D. Jordan* Department of Chemistry and the Center for Molecular and Materials Simulations, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15260
Phaedon Avouris IBM Research Center, T. J. Watson Research Center, Yorktown Heights, New York 10598 ReceiVed: June 18, 2001; In Final Form: August 28, 2001
Spin-polarized density functional calculations are used to study the adsorption of O atoms and O2 molecules on graphite and on a (8,0) single-walled carbon nanotube. An O atom is found to bind to graphite and to the outside and inside surfaces of the nanotube to give stable epoxide-like structures. Of these, the most stable is the epoxide on the outer surface of the nanotube, with a calculated adsorption energy of 44 kcal/mol. In the case of O2, both physisorbed and chemisorbed species are identified. The O2 molecule is predicted to only weakly physisorb (adsorption energy ≈ 0.9 kcal/mol) to the graphite and the (8,0) nanotube surfaces. However, these adsorption energies are expected to be underestimated due to inadequate treatment of the dispersion interactions. The chemisorbed species are metastable in the sense that they lie energetically above the isolated X 3Σg+ O2 and graphite or nanotube systems. In the case of the outer wall of the nanotube, the chemisorbed species is predicted to lie only 10.2 kcal/mol above the reactants. The reaction of O2 with the nanotube surface to give two epoxide groups is predicted to be slightly exothermic.
1. Introduction The electronic properties of carbon single-wall nanotubes (SWNTs) can be appreciably altered by the presence of adsorbed molecules.1 This has important ramifications for device applications involving SWNTs, and it has led to considerable interest in the possible use of SWNT's as the basis of chemical sensors.1,2 Oxygen, in particular, has been found to influence electronic properties of SWNTs, with the electrical resistance, the thermoelectric power, and the local density of states all depending on oxygen exposure.1,3 Most strikingly, it has been reported that small-gap semiconducting nanotubes can be made metallic upon exposure to a small amount of oxygen.1 The increased electrical conductivity induced by oxygen adsorption has been attributed to an increase in the local density of states and a shrinking of the band gap of the nanotube. The sensitivity of carbon nanotubes to oxygen exposure has been also demonstrated by means of nuclear magnetic resonance (NMR) measurements.4 In particular, it has been found that SWNTs possess two spin-lattice relaxation times: a fast component, attributed to metallic tubes and a slow component, tentatively attributed to semiconducting tubes. Both relaxation times were found to increase significantly upon oxygen exposure. It was suggested that this could be the result of fluctuations in the local magnetic fields caused by the adsorption of the paramagnetic oxygen molecules on the surfaces of SWNT bundles.4 The spin-lattice relaxation times could also be influenced by changes in the electronic properties of the SWNTs brought about by oxygen adsorption. The interactions of O2 with carbon nanotubes have been investigated in several recent theoretical studies.5-7 Peng and Cho,5 using density functional (DFT) calculations within the local density approximation (LDA), found that an O2 molecule
molecule physisorbs on the outer surface of a (10,0) nanotube with a binding energy of 0.1 eV and with the adsorption process being accompanied by a transfer of 0.09 e- from nanotube to the O2 molecule. Although it was not stated explicitly, the adsorbed O2 presumably retained its spin triplet character. The effect of adsorbed oxygen on the electronic properties of semiconducting carbon nanotubes was analyzed in more detail by Jhi et al.,6 who also used the LDA method. These authors reported that O2 molecule in its triplet ground state is physisorbed to the outer wall of a (8,0) single-walled nanotube with an adsorption energy of about 0.25 eV. On the basis of the calculated density of states it was suggested that the adsorbed oxygen molecules can dope the semiconducting tubes with hole carriers and that conducting states are present near the band gap. Still more recently, Zhu et al.7 have used the self-consistent charge density-functional-based tight-binding (SCC-DFTB) method together with density functional calculations within both the LDA and generalized-gradient approximation (GGA) to study the interaction of O2 with the (5,5) armchair and (9,0) zigzag nanotubes. It was reported that there were several O2 adsorption sites with adsorption energies ranging from 0.5 to 0.6 eV. However, details about the geometrical structures were not reported. Nor is it clear which theoretical method was use to produce these binding energies. With the exception of the GGA calculations of Zhu et al.,7 the calculations described above used the LDA which tends to overestimate binding energies. Moreover, the GGA calculations of Zhu et al.7 employed localized orbital basis sets which could also lead to an over binding due to basis set superposition error. In the present study, density functional calculations using the GGA and plane-wave basis sets are used to study adsorption
10.1021/jp0122979 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/23/2001
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Sorescu et al.
of O atoms and O2 molecules on the surfaces of a single layer of graphite and a (8,0) SWNT. Both chemisorbed and physisorbed species are considered, and adsorption on both the inner and outer surface of the nanotube is examined. 2. Computational Methodology The electronic structure calculations were carried out spinpolarized density functional theory, employing the PW91 generalized gradient approximation (GGA) of Perdew et al.,9 plane-wave basis sets, and slab models with periodic boundary conditions. Only the valence electrons were represented explicitly in the calculations; the valence-core interactions being described by ultra-soft (Vanderbilt-type) pseudopotentials10 developed by Kresse and Hafner.11 An energy cutoff of 396 eV was used in the plane-wave expansion. This value was taken as a compromise to ensure both energy convergence and to reduce the computational time and memory requirements. Our tests for the binding energy of O2 on (8,0) nanotube have shown that a convergence within 0.3 kcal/mol can be achieved at this cutoff energy. The Brillouin zones for the nanotube and graphite supercells were sampled with 2 × 1 × 2 and 2 × 2 × 1 Monkhorst-Pack12 meshes, respectively. These sampling schemes led to two irreducible k-points for the nanotube and six irreducible k-points for graphite. The calculations were carried out using the Vienna ab initio simulation package (VASP).13-15 The planar graphite surface was simulated as a single sheet of C atoms in a hexagonal surface unit cell with dimensions 7.38 × 7.38 × 8.0 Å (see Figure 1a). For the nanotube calculations, the supercell contained four hexagonal rings along the tube axis (see Figure 1b) placed in an orthogonal box with dimensions of 12 × 15 × 8.52 Å. The same supercells were used for the systems with an adsorbed O atom or O2 molecule as for the isolated graphite and nanotube species. For the nanotube, the longest cell dimension was taken along the plane that contains the nanotube axis and the adsorbed O or O2. The equilibrium configurations were determined by relaxation of all atoms in the supercell. The adsorption energies were determined according to the expression
Eads ) E(X) + E(slab) - E(slab-X)
(1)
where E(X) is the energy of an isolated O atom or O2 molecule, the latter at its optimized geometry, E(slab) is the total energy of the relaxed slab in the absence of the adsorbed species, and E(slab-X) is the energy of the slab-adsorbate system. A positive Eads value corresponds to a stable structure. 3. Results and Discussion A1. Adsorption of O and O2 on the Graphite Surface. The calculated adsorption configurations of O and O2 on the graphite surface are depicted in Figure 2a and 2b. The key geometrical parameters together with the calculated adsorption energies are summarized in Table 1. An O atom interacts with the graphite surface to give rise to a highly stable spin singlet epoxide-like structure, consistent with experimental findings.16 The optimized structure has CO distances of 1.472 Å, and the CC bond in the three-membered ring is elongated to 1.500 Å compared to the 1.420 Å value in graphite itself. This species is calculated to be stable by 44 kcal/mol. We also optimized a spin-triplet epoxide structure, and find it to be less stable than the singlet by about 36 kcal/mol. The unpaired spin density of the triplet
Figure 1. Side view of the supercell models used for: (a) the graphite surface, and (b) the (8,0) nanotube. For visualization purposes several unit cells have been represented.
is localized mainly on the graphite rather than the epoxide functionality. For the O2/graphite system, the O2 molecule was constrained to lie parallel to the surface. Two different types of adsorbed O2 species were identified. One is a weakly adsorbed physisorbed species, with the O2 molecule located 3.36 Å from the surface, retaining its spin triplet character, and bound to the surface by only 0.89 kcal/mol. Chemisorbed singlet and triplet O2 species chemically bonded to an underlying C-C bond were also characterized. These species are metastable, in the sense that they are unstable with respect to X 3Σg- O2 and the bare graphite surface. The more stable, singlet chemisorbed species is predicted to lie energetically 50.4 kcal/mol above the reactants. It has four-membered ring system, with C-O bond lengths of 1.50 Å and associated O-O and C-C bond lengths typical of those of singly bonded species. The triplet chemisorbed species is less stable by 17.3 kcal/mol. It has a much longer O-O distance (2.03 Å), and can be viewed as consisting of two weakly interacting O atoms strongly bound to adjacent C atoms. Table 1 also reports charges, estimated by means of a Mulliken-like population analysis,17 on the chemisorbed O and O2 species. For the most stable singlet species the calculations predict a charge transfer of 0.48 and 0.36 e- between the graphite surface and the adsorbed O and O2, respectively.
Oxygen Adsorption on Graphite
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TABLE 1: Calculated Structural Parameters and Adsorption Energies of O and O2 Adsorbed on the Graphite and (8,0) Nanotube Surfaces system/configurationa
spin stateb
isolated O2 O/graphite O bridge O bridge O2/graphite O2 parallel* O2 parallel O2 parallel O/(8,0) nanotube O bridge, outside O bridge, outside O bridge, inside O Top, inside O2/(8,0) nanotube O2 parallel, outside* O2 hexag., outside* O2 bridge, outside* O2 tilted* O2 along axis, inside* O2 parallel, outside O2 parallel, outside O2 parallel, inside
T
a
R(C-C) (Å)
R(C-O) (Å)
R(O-O) (Å)
qc (e-)
Eads (kcal/mol)
1.237
S T
1.500 1.463
1.472 1.534
T S T
1.420 1.529 1.522
3.362 1.501 1.451
S T S S
1.471 1.466 1.481 1.481
1.468 1.477 1.414 1.440
T T T T T S T S
1.412
3.310 3.430d 3.148 3.146 3.186 1.483 1.511 1.555
1.412
1.528 1.524 1.493
1.238 1.510 2.033
-0.48 -0.49
43.97 7.88
-0.36 -0.27
0.89 -50.35 -67.68
-0.48 -0.49 -0.49
1.237 1.240 1.237 1.239 1.243 1.510 1.482 1.528
-0.01 -0.01 -0.28 -0.30
68.73 39.88 25.99 20.96 0.70 0.87 0.52 0.49 -2.97 -10.16 -36.03 -76.44
b
Physisorbed configurations are designated with a * symbol. The abbreviations S and T designate the singlet and triplet states, respectively. Charges on the O or O2 entities calculated using the Mulliken procedure are reported for a subset of the structures. d For the hexagonal physisorbed species, there is a small spread in C-O distances, and the average is reported.
c
Figure 2. Adsorption configurations of an O atom (a) and an O2 molecule (b) on the graphite surface.
A2. Adsorption of an O Atom on the (8,0) Nanotube. An O atom is found to bind to both the outer and interior surfaces of the (8,0) SWNT to give stable epoxide structures. A pictorial view of these configurations is given in Figure 3a and 3b. For the outer surface of the nanotube, as for the graphite surface, the singlet epoxide structure is calculated to be about 29 kcal/ mol more stable than the triplet. However, the epoxide species on the outer surface of the (8,0) SWNT are much more stable (by 25 and 32 kcal/mol for the singlet and triplet species, respectively) than their counterparts on the graphite surface. The unpaired spin density of the triplet species is delocalized over several carbon atoms in the vicinity of the epoxide group. For O atom adsorption on the interior of the tube, only the singlet species was considered. Both an epoxide-like structure and a structure with the O atom bound to a single C atom (Figure 3c) were located. Of these, the epoxide structure is more stable by 5.0 kcal/mol. The adsorption energy associated with the interior epoxide is only 26.0 kcal/mol, much less than that (68.7 kcal/mol) on the outer surface, and about 18.0 kcal/mol less stable than the epoxide structure on the graphite surface. These results show that the energy for binding an O atom to a
Figure 3. Pictorial view of the adsorption configurations of an O atom on the (8,0) nanotube: (a) epoxide bridge configuration on the outside surface; (b) epoxide bridge configuration on the inside surface; (c) the on-top configuration on the inside surface.
graphitic surface depends strongly on its curvature. The stability of the on-top structure to displacements of the O atom away from the on-top position was not examined. Thus, it is not known whether this on-top structure is a minimum or a transition state for migration of the O atom between adjacent epoxide sites. The Mulliken population analysis indicates a sizable (∼0.3 e-) electron transfer from the nanotube to the chemisorbed O atom. A3. Adsorption of an O2 Molecule on the (8,0) Nanotube. Our GGA calculations predicted that it is energetically unfavorable by nearly 3 kcal/mol to insert an O2 molecule in the interior of the (8,0) SWNT. (The minimum energy structure has the O2 molecule aligned along the tube axis). For O2 physisorbed on the exterior surface of the (8,0) nanotube, four orientations were
11230 J. Phys. Chem. B, Vol. 105, No. 45, 2001
Figure 4. Pictorial view of the adsorption configurations of an O2 molecule on the (8,0) nanotube: (a) chemisorbed configuration; (b) physisorbed “hexagonal” configuration with the O2 molecule sitting above the C1, C2, C3, and C4 atoms or a ring; (c) physisorbed vertical bridging configuration; and (d) physisorbed tilted configuration.
considered. These included a structure parallel to the surface similar to the chemisorbed species depicted in Figure 4a, but further displaced from the surface, and the three orientations shown in Figure 4b and 4d. The triplet O2 molecule is found to be weakly bound to the surface in all of these orientations, with the structure depicted in Figure 4b being most stable, with a calculated adsorption energy of 0.87 kcal/mol, essentially identical to that for O2 on the graphite surface. For these weakly bound species, the charge transfer between the nanotube and the O2 molecule is very small (∼0.01 e-). Our calculated adsorption energies for the physisorbed species are appreciably less than those reported in prior theoretical studies. We believe that the larger physisorption energies found in these prior studies resulted from the use of the LDA procedure or to limitations of the LCAO basis set. On the other hand, none of the DFT studies, including our own, properly treated dispersion interactions. We expect that a proper treatment of dispersion interactions would result in appreciably stronger adsorption energies for O2 physisorbed on both the graphite and (8,0) SWNT surfaces. Indeed, model potential calculations using a site-site LennardJones interaction between the C and O atoms give an adsorption energy of 2.7 kcal/mol for O2 on graphite and 2.1 and 6.5 kcal/ mol on the outer and inner walls of the (8,0) nanotube, respectively.18 Although, there are sizable uncertainties associated with the model potential results, they do underscore the possibility that it is energetically favorable for O2 molecules to adsorb in the interior of the (8,0) nanotube as well as on its outer surface. We have also optimized by means of DFT calculations chemisorbed forms of O2 on the outside (both singlet and triplet) and inside (singlet) surfaces of the nanotube. The singlet chemisorbed O2/nanotube species have geometrical structures similar to that of the singlet chemisorbed O2/graphite species. On the other hand, the triplet chemisorbed O2/nanotube species retains a short (∼1.48 Å) OO bond in contrast to the triplet chemisorbed O2/graphite species. The reaction to produce the singlet chemisorbed O2 species on the outer wall of the nanotube is calculated to be endothermic by only 10.2 kcal/mol. Thus, the outer wall of the (8,0) nanotube is much more reactive toward O2 than the graphite surface for which the chemisorbed
Sorescu et al. product is calculated to be unstable by 50.4 kcal/mol. In contrast, O2 chemisorbed on the inner wall of the nanotube is about 26 kcal/mol less stable than on the graphite surface. B. Minimum Energy Path for Approach of an O2 Molecule to the (8,0) Nanotube Surface, Parallel to the Nanotube Axis. The fact that we are able to optimize structures for the chemisorbed species indicates that there is a barrier to desorption of the chemisorbed forms of O2 from the graphite and nanotube surfaces. To obtain estimates of the barrier in the case of O2 chemisorbed on the outer nanotube surface, the energy was calculated for the lowest singlet and triplet states of the O2/nanotube system as a function of the distance of the O2 molecule from the nanotube. The calculations were performed over a grid of separations, with the O2 molecule being constrained to sit above a specific CC bond, parallel to it and to the nanotube axis, and with the other degrees of freedom being optimized. The optimizations were done for the singlet state, with the resulting geometries being used to carry out calculations on the triplet state. (At a subset of the O2/nanotube separations, geometry optimizations were also carried out on the triplet state to check that the resulting potential is indeed similar to that obtained using singlet optimized geometries.) At large separations of the O2 molecule from the surface, the singlet state is not well described by the Kohn-Sham DFT procedure. However, since our main interest is for O2-nanotube separations where there is significant interaction between the O2 molecule and the nanotube, this is not a major limitation. The resulting potential energy curves are shown in Figure 5a, and information of the variation in the optimized CO and OO distances is given in Figure 5b. At large separations from the surface, the energy difference between the singlet and triplet surfaces is calculated to be only 9.92 kcal/mol, smaller than the experimental 3Σg+ f 1∆g excitation energy (22.4 kcal/mol19) of O2. This is due to the above-mentioned inadequacy of the Kohn-Sham DFT method for describing the singlet state of the isolated O2 molecule. Both the singlet and the triplet states rise in energy as the O2 molecule is brought in from a large distance to a separation of about 1.85 Å from the surface. The energy then drops as the O2-nanotube separation is decreased to about 1.4 Å, and it then rises as the separation is further decreased. As can be seen from Figure 5, the potential energy minimum and the barrier occur at nearly the same O2-nanotube separations for the singlet and triplet potentials. Moreover, although the singlet chemisorbed structure is much more stable that the triplet, the barrier maxima on the two potential energy curves occur at nearly the same energy. The potential energy curves shown in Figure 5 correlate closely with the variation in the OO and underlying CC bond lengths as the CO distances are decreased. Specifically, the OO and underlying CC distances remain relatively constant as the O2 molecule is brought in from large O2-nanotube separation to about 2.15 Å. As the O2-nanotube separation is decreased from 2.15 Å to about 1.7 Å, the OO and underlying CC distances rapidly increase by about 0.25 and 0.11 Å, respectively. This corresponds to the growing barriers in the potential energy curves. These distances then plateau (i.e., stay roughly constant) as the O2-nanotube separation further decreases down to about 1.4 Å. This corresponds to the descent from the barrier down to the region of the potential energy minima of the chemisorbed species. As the O2-nanotube separation is further decreased, the OO and CC bond lengths rapidly increase, corresponding to the rapid rise in the energy.
Oxygen Adsorption on Graphite
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11231
Figure 5. Minimum energy path for formation of the chemisorbed O2 species on the (8,0) SWNT starting from isolated O2 and nanotube species. The energy paths for the singlet and triplet states are given in panels (a) and (b), respectively. The pathway is calculated keeping the O2 molecule above a specific CC bond and aligned parallel to the nanotube axis, optimizing all other degrees of freedom at each separation of the O2 molecule from the surface. The sum of energies of isolated nanotube and isolated triplet O2 molecule is taken as the reference. Variations of the O-O bond and underlying C-C bond lengths from calculations on the singlet state are given in panels (c) and (d), respectively.
slightly exothermic. However, there is evidence that the PW91 exchange-correlation functional overestimates the strength of CO bonds, so it is likely that the reaction is actually endothermic by a few kcal/mol.20 Although we have not calculated the reaction pathway for going from the chemisorbed O2 species to two epoxide groups, we expect it to be sizable. Moreover, the overall barrier in going from gas-phase O2 and the unreacted nanotube to these epoxide products is expected to be comparable or larger than for chemisorption of the O2. Thus, this process is expected to require high temperatures or the presence of defects to be important. 4. Conclusions
Figure 6. Relative energies of the oxygen/graphite and oxygen/(8,0) nanotube systems. The zeroes of energy are taken to correspond to the sum of the energies of an isolated triplet O2 molecule and a single layer of graphite or the (8,0) nanotube. All results are from PW91 DFT calculations.
On the basis of the potential energy curves in Figure 5, it seems possible that formation of the chemisorbed singlet O2 species on the (8,0) nanotube could proceed via intersystem crossing from the triplet species. However, due to the high barrier, formation of the chemisorbed species is unlikely at room temperature and in the absence of defects. A correlation diagram showing the relative energies of the various O2/graphite and O2/(8,0) nanotube/species is given in Figure 6. From this figure it is seen that our theoretical calculations predict that the reaction of O2 with the (8,0) nanotube to produce two epoxide groups on the surface is
Density functional theory in conjunction with slab models and periodic boundary conditions has been used to investigate the adsorption of O atoms and O2 molecule on the graphite and (8,0) nanotube surfaces. The interaction of an O atom with the graphite SWNT surfaces is found to be highly exothermic, with the adsorption energy calculated to be 44 kcal/mol on graphite and 69 and 26 kcal/mol on the outer and inner walls of the (8,0) SWNT, respectively. On the basis of these results, we conclude that the reaction of O2 with the outer wall of a (8,0) nanotube to give two spatially well-separated epoxide groups is exothermic, although such a process has a large barrier. The calculations also reveal that O2 can chemically bind to the graphite and SWNT surfaces, forming a ring structure with an intact O-O bond. This species is metastable and highly unstable on graphite and on the inner surface of the nanotube. However, on the outer SWNT surface, it is calculated to be unstable (endothermic) by only 10.2 kcal/mol. The adsorption
11232 J. Phys. Chem. B, Vol. 105, No. 45, 2001 of O2 to form these chemisorbed species is predicted to have large (∼50 kcal/mol) barriers. Our gradient-corrected DFT calculations predict only weak physisorption of triplet O2 on the graphite and outer SWNT surfaces, with the calculated adsorption energy being less than 1 kcal/mol. At this level of theory, it is predicted to be energetically unfavorable to insert an O2 molecule in the interior of the nanotube. However, a limitation of DFT calculations is the failure to describe long-range dispersion interactions. Indeed, force-field calculations using Lennard-Jones interactions give physisorption energies of 2.1 kcal/mol on the outer surface of the nanotube and 6.5 kcal/mol for an O2 inside the tube. The latter value is in fairly good agreement with the 9 kcal/mol experimental value for O2 bonding to SWNT bundles. Of course, in the case of nanotube bundles the physisorbed O2 molecules could be external to the tubes but interacting with more than one nanotube. Our GGA calculations give very small charge transfer (e0.01 e-) between the nanotube and the O2 molecule for the physisorbed species. However, it is possible that, with a proper treatment of dispersion, larger charge transfer would be found. A Mulliken population analysis reveals that significant charge transfer (0.3-0.5 e-) occurs between the (8,0) nanotube and the chemisorbed O atom or O2 molecule. Acknowledgment. The calculations were performed on the Cray T3E at Pittsburgh Supercomputer Center and on the IBM RS/6000 cluster in the Center for Molecular and Materials Simulations, University of Pittsburgh. The latter computers were funded by grants from IBM and NSF. We thank K. Karapetian for carrying out the force-field calculations.
Sorescu et al. References and Notes (1) Collins, P. G.; Bradley, K.; Ishigami, M.; Zettl, A. Science 2000, 287, 1801. (2) Kong, J.; Franklin, N. R.; Zhou, C.; Chapline, M. G.; Peng, S.; Cho, K.; Dai, H. Science 2000, 287, 622. (3) Bradley, K.; Jhi, S.-H.; Collins, P. G.; Hone, J.; Cohen, M. L.; Louie, S. G.; Zettl, A. Phys. ReV. Lett. 2000, 85, 4361. (4) Tang, X.-P.; Kleinhammes, A.; Shimoda, H.; Fleming, L.; Bennoune, K. Y.; Sinha, S.; Bower, C.; Zhou, O.; Wu, Y. Science 2000, 288, 492. (5) Peng, S.; Cho, K. Nanotechnology 2000, 11, 57. (6) Jhi, S.-H.; Louie, S. G.; Cohen, M. L. Phys. ReV. Lett. 2000, 85, 1710. (7) Zhu, Z. Y.; Lee, S. M.; Lee, Y. H.; Frauenheim, T. Phys. ReV. Lett. 2000, 85, 2757. (8) Cohen, M. L. Phys. Scri. 1982, T1, 5. (9) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Perdersen, M. R.; Singh, D. J.; Frolhais, C. Phys. ReV. 1992, B 46, 6671. (10) Vanderbilt, D. Phys. ReV. 1990, B 41, 7892. (11) Kresse, G.; Hafner, J. J. Phys. Cond. Matter 1994, 6, 8245. (12) Monkhorst, H. J.; Pack, J. D. Phys. ReV. 1976, B 13, 5188. (13) Kresse, G.; Hafner, J. Phys. ReV. 1993, B 48, 13 115. (14) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (15) Kresse, G.; Furthmu¨ller, J. Phys. ReV. 1996, B 54, 11 169. (16) He, H.; Klinowski, J.; Forster, M.; Lerf, A. Chem. Phys. Lett. 1998, 287, 53. (17) (a) Segall, M. D.; Pickard, G. J.; Shah, R.; Payne, M. C. Phys. ReV. 1996, B54, 16 317. (b) Segall, M. D.; Pickard, G. J.; Shah, R.; Payne, M. C. Mol. Phys. 1996, 89, 571. (18) Karapetian, K. unpublished results. These calculations employed the Lennard-Jones potentials from reference: Travis, K. P.; Gubbins, K. E. Langmuir 1999, 15, 6050. (19) Huber, K. P.; Herzberg, G. In Molecular Spectra and Molecular Structure: Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979. (20) Steckel, J. A.; Jordan, K D.; Avouris, Ph., submitted to J. Phys. Chem.
