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J. Phys. Chem. C 2008, 112, 1396-1400
Analytically Calculated Polarizability of Carbon Nanotubes: Single Wall, Coaxial, and Bundled Systems Edward N. Brothers, Artur F. Izmaylov, and Gustavo E. Scuseria* Department of Chemistry, Mail Stop 60, Rice UniVersity, Houston, Texas 77005-1892
Konstantin N. Kudin Department of Chemistry and Princeton Institute for the Science and Technology of Materials (PRISM), Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed: October 11, 2007
We present a comprehensive theoretical study of the polarizability of nanotubes and nanotube bundles modeled as periodic systems. Both static and dynamic fields are considered via our recent implementation of analytical methods for polarizability. The dynamic polarizability results in this article are nanotubes’ response to the time-dependent electric fields, in contrast to earlier studies that focused on only static fields. Results are obtained using both density functional and Hartree-Fock theories with a Gaussian basis set and periodic boundary conditions. In addition to elaborating on previous finite-field static polarizability calculations, we investigated how polarizability varies with the field frequency spanning a range of energies up to the band gap values, as well as how polarizabilities are affected by nanotube interactions. These findings have interesting implications for nanoelectronic applications and provide guidance for experimental studies of nanotubes’ polarizability.
E(t) ) [eiωt + e-iωt]E0
I. Introduction There have been a number of theoretical studies of the polarizability of carbon nanotubes,1-11 including a series of works by the authors of this paper.12-14 These studies have interesting implications for nanoelectronics, as theoretical calculations indicate how nanotubes respond to electric fields, which has been identified as important for a host of potential applications. For example, nanotube polarizability will be important in the emerging uses of nanotubes as electrochemical devices and field emission devices,15 and nanotube polarizability/ shielding in bundled arrangements will be critical for understanding electronic properties of flexible nanotube sheets.16 Nanotube polarizability is also critical for understanding the behavior of nanotubes as molecular sensors, as Girardet and co-workers have shown that explaining molecular adsorption on nanotubes requires including their polarization.17,18 In addition, theoretical studies also provide insight into new phenomena, such as highly efficient electric field shielding that we have reported for internal nanotubes in multiwall systems, even when the outer nanotube is a semiconductor; i.e., shielding arising from a phenomenon different from that of a Faraday cage.13 However, all previous nanotube work using quantum chemical methods has considered the response to a constant, nonoscillating field calculated through a finite difference method; i.e., the numerical static polarizability. Recent advances in our research group have allowed efficient calculation of the analytical polarizability of periodic systems using Gaussian basis sets with a procedure analogous to the methods used to calculate molecular polarizabilities,19 including the response of periodic systems to harmonically oscillating/dynamic/time-dependent fields. More formally, dynamic polarizability is the response of electronic structure to fields that change over time as
(1)
where E0 is field strength, ω is frequency, and t is time. Response to these oscillating fields is termed dynamic polarizability, and to our knowledge, this study is the first time ab initio dynamic polarizabilities have been calculated for nanotubes. Dynamic polarizabilities are expected to be closer to experimentally obtained results, since real fields are not static. Therefore, one of the initial reasons for undertaking this work was to examine the changes in electronic response of carbon nanotubes when dynamic electric fields are utilized in place of static electric fields. This will allow a new evaluation of published trends for polarizability of nanotubes. Another goal is to examine the polarizabilities of finite bundles of carbon nanotubes. Although three-dimensional (3D) bundles have been examined in the past with plane wave codes (typically in the context of subsequently using those 3D results with a Clausius-Mossotti correction20 to extrapolate to the results for a single nanotube), in this paper, we shall examine simple test systems such as two parallel nanotubes or a three nanotube bundle to understand the changes in polarizabilities (versus a single nanotube) that will be observed experimentally in bundles. This will provide a route to analyzing finite bundles’ polarizabilities. Finally, a third goal of this paper is one that has only recently become technically feasible: namely, calculating the change in the polarizability as the electric field frequency is increased. This property has previously not been explored for nanotubes, and the trend of this data provides a great deal of information about nanotube electronic structure and response properties. II. Method All calculations reported in this paper were carried out using the development version of the GAUSSIAN21 suite of programs
10.1021/jp709931r CCC: $40.75 © 2008 American Chemical Society Published on Web 01/11/2008
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TABLE 1: Static Polarizabilities of Nanotubes with Various Basis Setsa transverse
longitudinal
nanotube STO-3G 3-21G 6-31G(d) STO-3G 3-21G 6-31G(d) (5,0) (7,0) (8,0) (10,0)
12 18 23 34
15 23 28 40
16 24 30 42
280 371 477
375 419 604
376 444 615
a All polarizabilities are in Å3 and were calculated with the TPSS density functional. (5,0) nanotubes do not have longitudinal polarizabilities listed, as their band gaps are small enough to make this value numerically suspect.
modified locally to calculate the response to static and dynamic electric fields19,23 with periodic boundary conditions (PBC).22 Although it would be possible to perform nonperiodic calculations and extrapolate those results to include cap effects on polarizabilities, extremely long fragments would be necessary due to the slow convergence of polarizability;24 hence, the reliance on PBC. All nanotube geometries were generated by TubeGen25 and used without geometry optimization, because previous experience has shown polarizability trends are not crucially sensitive to small geometric changes when compared with sensitivity to functional choice.12 The zigzag nanotubes (10,0), (19,0), (20,0), (22,0), (23,0), (25,0), and (26,0) were investigated. (Metallic nanotubes26 were excluded because standard perturbation theory is inapplicable in the zero band gap case.) The aforementioned series of nanotubes was also used to construct coaxial systems. The (10,0) nanotube sits at almost van der Waals distance inside the (19,0) nanotubes,27 and thus, this was the tightest coaxial system considered. In the cases with larger outer nanotubes, e.g., (10,0) inside (23,0), the inner nanotube was perfectly centered. The nonempirical density functional28 of Tao, Perdew, Staroverov, and Scuseria (TPSS)29 was chosen for the density functional theory (DFT) calculations. One potential criticism of using DFT for polarizabilities is the well-known overestimation of this property in extended systems by most commonly used density functionals.30 To remedy this, we have included some Hartree-Fock31 calculations to ensure there is no change in the discovered trends, utilizing methods developed in our research group specifically to increase the speed of the (notoriously slow) Hartree-Fock (HF) PBC calculations.32-34 A Gaussian-type basis consisting of 2s1p atomic orbitals (STO-3G)35 was chosen as a compromise between speed and accuracy. This basis set is sufficient to demonstrate trends and allows a larger test set of nanotube systems of greater size to be studied than would be possible with large basis set. As can be seen from the results in Table 1, larger basis sets can increase the calculated results by up to 20% but do not signifigantly change the trends in the data. The aims of this paper required two separate sets of dynamic field calculations. A field frequency of 0.005 a.u. (equivalent to ∼0.13 eV, ∼32.9 THz, or ∼911 nm) was used to calculate dynamic polarizability for all nanotubes in this study. This frequency is approximately one-third of the band gap of the (26,0) nanotube, which had the smallest band gap of any nanotube in the study. If the electric field frequency chosen is larger than the band gap, polarizability is infinite, and in fact, if the electric field frequency is larger than one-half of the band gap, there can be significantly slower convergence.19 Thus, one set of calculations involved a range of nanotubes investigated at the same relatively small frequency. In the second set of calculations, a range of frequencies was used to investigate one
Figure 1. Transverse polarizabilities of single-wall nanotubes calculated with TPSS/STO-3G. The red points are static polarizabilities, and the blue points are dynamic polarizabilities. Note that the closeness of the static and dynamic polarizabilities results in overlap of the points.
Figure 2. Transverse polarizabilities of coaxial carbon nanotubes calculated with TPSS/STO-3G. The red points are static polarizabilities, and the blue points are dynamic polarizabilities.
nanotube. Frequencies of up to 90% of the band gap were used with a (10,0) nanotube, allowing us to observe the change in polarizability with change in the field frequency. III. Results The simplest trend in nanotube polarizability is the transverse polarizability of a single-wall nanotube, which has been shown in several works to be dependent on the square of the radius of the nanotube.1,8,12 Thus, this polarizability is completely independent of band gap or other electronic structure properties. Because of this, we expected that the transverse polarizability would remain unchanged in a dynamic electric field, and this expectation was shown to be correct by the results in Figure 1, where the static and dynamic transverse polarizability trends for bare carbon nanotubes are shown to be very close. We also compare static and dynamic transverse polarizability of coaxial nanotubes, since previous work by the authors13 and confirmed by others8 has indicated that coaxial or multiwall nanotubes should have static polarizabilities equal to the polarizability of the outermost nanotube by itself; i.e., the inner nanotube should be almost perfectly shielded by the outer nanotube. This effect must persist in oscillating fields to be useful or significant in practical application. Figure 2 demonstrates that the shielding continues with dynamic fields, and as in the case of the bare nanotube, and dynamic and static transverse polarizabilities are quite similar for coaxial nanotubes due to band gap independence of transverse polarizability.
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Figure 3. Longitudinal polarizabilities of single-wall nanotubes calculated with TPSS/STO-3G. The red points are static polarizabilities, and the blue points are dynamic polarizabilities.
Figure 4. Longitudinal polarizabilities of coaxial carbon nanotubes calculated with TPSS/STO-3G. The red points are static polarizabilities, and the blue points are dynamic polarizabilities.
Static longitudinal polarizabilities are heavily gap-dependent, and thus, the change from static to dynamic polarizabilities should result in significant alteration of the electronic response. Although there is still some difference of opinion in the literature over the governing trend for the static longitudinal nanotube polarizability, we have found the trend of static polarizability to be proportional to the inverse band gap when divided by the number of atoms in the unit cell. We believe that physically, this trend is intuitive, because polarizability can fundamentally be seen as electrons rearranging in a field, the number of atoms determines the number of electrons available for response, and the band gap defines the ease of rearrangement. When this trend is extended to dynamic polarizabilities, it still holds, albeit with a different slope, as seen in Figure 3. For the (10,0) nanotube, the points almost overlap due to the small size of the dynamic field versus the band gap of the nanotube. The difference between static and dynamic tubes becomes more pronounced as the ratio of the frequency of the external field versus the band gap becomes larger, which explains the larger slope of the dynamic polarizability trend. Please note that other possibilities for the trend of longitudinal polarizability were examined, such as that put forward by Louie et al.,1 and none provided even a qualitatively better fit than the trend used in Figure 3. For the longitudinal polarizability of coaxial nanotubes, the results are just the sum of the polarizabilities of the constituent nanotubes, as can be easily seen in Figure 4. This was previously noted for static fields,8 and this additive relation holds for dynamic polarizabilities, as well. Thus, we show conclusively that dynamic polarizabilities behave in roughly the same way as static polarizabilities, and thus, the trends from static polarizabilities are still valid and useful. To examine the variation of longitudinal polarizability with field frequency, a single (10,0) nanotube was used with various frequencies that ranged from 0 to 90% of the band gap, the results of which are presented in Figure 5. These results can be fit with the equation
Figure 5. Longitudinal dynamic polarizability of a (10,0) nanotube calculated with TPSS/STO-3G at various field frequencies.
R)
A +C B-ω
(2)
where ω is specified as the percent of the static band gap value, and the best fit parameters are A ) 20.2, B ) 102, and C ) 0.786. The parameter in the denominator can be considered as the effective band gap for a given field, and because the fit results in a value of approximately 102%, it can be argued that the effective band gap is slightly larger than simply the direct
band gap. In addition, there is a physical conclusion that can be drawn from the form of the trend. If all electron density on a semiconducting nanotube were equally polarizable, it would be expected that the trend would have the form of A/(B - X). However, eq 2 works very poorly if the additive parameter C is not included. This can be taken to be the portion of polarizability that arises from electronic density whose response is independent of the field frequency. This makes sense, because at low (compared to band gap) frequencies, the electrons that are being affected are only those in the valence or near-tovalence bands. This is related to the observation we made previously that most nanotube longitudinal polarizability arises from occupied-virtual pairings of the highest occupied crystalline orbital (HOCO) or HOCO - 1 with the lowest unoccupied crystalline orbital (LUCO) or LUCO + 1 elements of the dipole matrix.14 Finally, a brief observation on the trend of polarizability versus frequency should be included. The polarizability verses frequency curve has effectively two different regions, one of which is linear and the other, asymptotic. The change in slope occurs after a certain field frequency (as a percentage of the band gap) has been reached. Given a sample of semiconducting nanotubes, it should be possible to determine the average band gap of the sample by measuring the polarizability at various frequencies and noting the variation in response.
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Figure 6. The ratio of polarizabilities calculated with TPSS/STO-3G of parallel (10,0) carbon nanotube versus intertube spacing. The internanotube spacing is measured from the center of each nanotube in units of radius of the nanotubes (in this case, 3.93 Å). The red points are DFT static polarizabilities, and the green points are calculated with HF. Squares are longitudinal polarizabilities, circles are polarizability in the direction of intertube separation, and triangles are polarizabilities perpendicular to intertube spacing. A line with a value of 2 (the value if bundling the nanotubes would have no effect on polarizability) is included to guide the eye.
Figure 7. The ratio of polarizabilities of parallel (10,0) carbon nanotube versus intertube spacing in a 2D sheet calculated with TPSS/STO-3G. The internanotube spacing is measured from the center of each nanotube in units of radius of the nanotubes (in this case, 3.93 Å). The red points are static polarizabilities. Squares are longitudinal polarizabilities, circles are polarizability in the direction of intertube separation, and triangle are polarizabilities perpendicular to intertube spacing. A line with a value of unity (the value that would be obtained if the 2D sheet’s polarizability was equivalent to the bare nanotube) is added to guide the eye.
At this point, we have shown that the trends for static and dynamic fields are related. We now turn our attention to larger systems: specifically, to bundled nanotubes, which will be treated with static polarizabilities, because they have been shown in the first part of this paper to be sufficient for elucidating trends. The minimum nanotube bundle is parallel nanotubes, i.e., a two-nanotube bundle. We utilized a pair of (10,0) nanotubes at various intertube spacings starting at the graphite inter-sheet van der Waals distance.27 Rather than talk about raw polarizabilities, the results become more intuitive when considered as a ratio to the results of a single (10,0) nanotube, which we plot in Figure 6. For example, a ratio of exactly two indicates the parallel nanotubes are returning values equivalent to a single nanotube; i.e., there is no alteration in polarizability caused by nanotube interaction. As would be expected from multiwall studies, the longitudinal polarizability has a value almost exactly twice that of the bare nanotube. The transverse polarizability perpendicular to the intertube spacing is reduced versus the single nanotube, which can be explained physically by recalling that a field in that direction would cause charge to build up on one side of the nanotube, causing the polarization to compete with electronic repulsion between the adjacent nanotubes, thereby reducing polarizability. The transverse polarizability in the direction of the nanotube spacing deserves the most explanation. This polarizability is significantly enhanced, even at large internanotube separations. These large distances, coupled with the short range of the STO3G basis set, allow us to rule out a charge-transfer event, either physically motivated or induced by basis set superposition error. The polarizability enhancement also continues when HF theory is used, implying that this is not some artifact of the van der Waals-like attraction exhibited by TPSS.37 To explain this increase in one transverse polarizability, we consider an electric field as a pair of point charges, one positive and one negative. Now we place inside this field two nanotubes, whose electronic structure is in the zero-field arrangement. If one nanotube’s electronic structure is allowed to relax, a dipole will form across the nanotube, which will be in the direction
TABLE 2: Static Polarizabilities of 2D and 3D Arrays of (10,0) Nanotubesa level of theory HF DFT
array
Rzz
Ryy
Rxx
1D 2D 3D 1D 2D 3D
477 478 479 246 246 246
33 61 65 30 49 52
32 24 66 30 22 52
a Done with the STO-3G basis set. The TPSS functional was used for DFT calculations. All values are presented in Å3. The z direction is the longitudinal direction for nanotubes, y is the direction of intertube spacing in a 2D sheet, and x is the direction perpendicular to a sheet of nanotubes.
opposite the point charge’s dipole. If we now allow the second nanotube to relax, it will polarize to a greater degree than the first nanotube due to a beneficial interaction with the other nanotube’s induced dipole. This electric field increase from the polarized adjacent nanotube is the root cause of the enhanced nanotube polarizability. The changes we have seen in the parallel nanotubes are even more vivid in 2D and 3D arrays of nanotubes. If we examine the 2D sheets, the degree of modification versus a single nanotube is even larger than for the parallel nanotubes, as shown in Figure 7. Even at large internanotube spacings, we see significant enhancement of one transverse polarizability (in the direction of the sheet of nanotubes) and significant reduction in the other transverse polarizability (in the direction perpendicular to the sheet). Note that the 2D calculations with various spacings were all done with DFT to keep the computational cost reasonable. As a final demonstration of this behavior of bundles, calculations with 2D and 3D lattices at the van der Waals distance using both DFT and HF were conducted and are presented in Table 2. HF and DFT results follow exactly the same trends, with 2D sheets performing as discussed above. In the 3D infinite bundles, all transverse polarizabilities are equal and enhanced versus a single nanotube, because the 3D arrays are such that each nanotube has a hexagonal set of nearest neighbors, resulting in all transverse directions being equivalent.
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TABLE 3: Static Polarizabilities of Different Spatial Arrangements of (10,0) Nanotubes arrangement
Rzz
Ryy
Rxx
single sheet bundle
477 1433 (3.00) 1435 (3.01)
33 129 (3.85) 118 (3.53)
33 85 (2.54) 93(2.76)
a The arrangements correspond to the three nanotubes in a plane (sheet) or arranged in a triangle (bundle), along with the single nanotube for comparison. Raw polarizabilities are in Å3, and ratios of bundles versus the single nanotube is given in parenthesis. All calculations were performed with TPSS/STO-3G. See the caption of Table 2 for an explanation of directions x, y, and z.
Thus, in addition to providing simple rules for single and multiwall/coaxial nanotubes, we now have simple rules and trends for infinite bundles of carbon nanotubes. The final question is, then, what happens in larger finite bundles. Although it is true that calculating large finite bundles is currently computationally intractable, a model system consisting of three nanotubes can reveal a great deal of detail about the expected behavior and is a reasonable calculation with our implementation. There are two reasonable spatial arrangements for three carbon nanotubes: either as a sheet with all nanotubes in a plane or as a triangular bundle. Static polarizability calculations were conducted on both of these arrangements, and the results are presented in Table 3. In the sheet and in the triangular bundle, the same phenomenon was seen as in the two parallel nanotubes, i.e., enhancement in one transverse direction, reduction in another, and additive longitudinal polarizabilities. In the triangular bundle, the effects are less pronounced, resulting in smaller changes versus bare nanotubes, but the alterations are still present. Thus, our calculations demonstrate that nanotubes will exhibit enhanced polarizabilities in real physical systems, because nanotubes typically appear experimentally as finite bundles. IV. Conclusion This paper covers a wide range of polarizabilities for nanotubes, whether single, multiwall, or in a bundle, and greatly extends the range of studied nanotube polarizability, and also reports dynamic polarizabilities of nanotubes. We find that dynamic and static polarizabilities have similar trends, single and coaxial nanotube polarizabilities behave as previously described, and bundled nanotubes have simply explainable behaviors that typically result in polarizability having enhanced values versus single nanotubes. These trends now await experimental confirmation. Future work will address substituted carbon nanotubes,38,39 which may be more amenable to experimental observation. Acknowledgment. This work was supported by NSF-CHE0457030 and the Welch Foundation. References and Notes (1) Benedict, L. X.; Louie, S. G.; Cohen, M. L. Phys. ReV. B: Condens. Matter Mater. Phys. 1995, 52, 8541. (2) Guo, G. Y.; Chu, K. C.; Wang, D-s.; Duan, C-g. Comput. Mat. Sci. 2004, 30, 269. (3) Zhou, X.; Chen, H.; Zhong-Can, O.-Y. J. Phys.: Condens. Matter 2001, 13, L635. (4) O’Keeffe, J.; Wei, C.; Cho, K. App. Phys. Lett. 2002, 80, 676. (5) Li, Y.; Rotkin, S. V.; Ravaioli, U. Nano Lett. 2003, 3, 183. (6) van Faasssen, M.; Jensen, L.; Berger, J. A.; de Boeji, P. L. Chem. Phys. Lett. 2004, 395, 274. (7) Jensen, L.; Åstrand, P.-O.; Mikkelsen, K. V. Nano Lett. 2003, 3, 661. (8) Kozinsky, B.; Marzari, N. Phys. ReV. Lett. 2006, 96, 166801.
(9) Jensen, L.; Esbensen, A. L.; Åstrand, P.-O.; Mikkelsen, K. V. J. Comput. Methods Sci. Eng. 2006, 6, 353. (10) Mayer, A. Phys. ReV. B: Condens. Matter Mater. Phys. 2007, 75, 045407. (11) Sebastiani, D.; Kudin, K. N. ACS Nano, submitted. (12) Brothers, E. N.; Kudin, K. N.; Scuseria, G. E.; Bauschlicher, C. W., Jr. Phys. ReV. B: Condens. Matter Mater. Phys. 2005, 72, 33402. (13) Brothers, E. N.; Kudin, K. N.; Scuseria, G. E. J. Chem. Phys. 2006, 124, 041101. (14) Brothers, E. N.; Kudin, K. N.; Scuseria, G. E. J. Phys. Chem. B 2006, 110, 12860. (15) Baughman, R. H.; Zakhidov, A. A.; de Heer, W. A. Science 2002, 297, 787. (16) Zhang, M.; Science 2005, 309, 1215. (17) Arab, M.; Picaud, F.; Devel, M.; Ramseyer, C.; Girardet, C. Phys. ReV. B: Condens. Matter Mater. Phys. 2004, 69, 165401. (18) Langlet, R.; Arab, M.; Picaud, F.; Devel, M.; Girardet, C. J. Chem. Phys. 2004, 121, 9655. (19) Izmaylov, A. F.; Brothers, E. N.; Scuseria, G. E. J. Chem. Phys. 2006, 125, 224105. (20) See, for example, ref 8. (21) 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 E.02; Gaussian, Inc.: Wallingford, CT, 2004. (22) Kudin, K. N.; Scuseria, G. E. Phys. ReV. B: Condens. Matter Mater. Phys. 2000, 61, 16440. (23) Kudin, K. N.; Scuseria, G. E. J. Chem. Phys. 2000, 113, 7779. (24) Kudin, K. N.; Carr, R.; Resta, R. J. Chem. Phys. 2005, 122, 134907. (25) Frey, J. T.; Doren, D. J. TubeGen 3.1; University of Delaware, Newark, DE 2003. (26) Nanotubes are named on the basis of a chiral vector (n, m) which denotes which two carbons are superimposed when a sheet of graphene is rolled into a nanotube. Zigzag nanotubes are all of the form (n, 0), whereas armchair nanotubes are all of the form (n, n). Chiral nanotubes are all the nanotubes that are neither zigzag or armchair. Note that nanotubes with Mod(n - m, 3) ) 0 are metallic, and Mod(n - m, 3) ) 1 and Mod(n - m, 3) ) 2 nanotubes are semiconducting and in some instances have been considered as three separate categories of nanotubes instead of two (metallic and semiconducting). (27) The van der Waals distance is assumed to be the same between nanotubes as it is for graphene sheets in graphite, 3.35 Å. (28) Hohenberg, P.; Kohn, W. Phys. ReV. B: Condens. Matter Mater. Phys. 1964, 136, 864. Jon, W.; Sham, L. J. Phys. ReV. A: At., Mol., Opt. Phys. 1965, 140, 1133. (29) Note that this functional is a meta-GGA. Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Phys. ReV. Lett. 2003, 91, 146401. (30) Mori-Sanchez, P.; Wu, Q.; Yang, W. J. Chem. Phys. 2003, 119, 11001. (31) Roothan, C. C. J. ReV. Mod. Phys. 1951, 23, 69. (32) Strout, D. L.; Scuseria, G. E. J. Chem. Phys. 1995, 102, 8448. (33) Scuseria, G. E. J. Phys. Chem. 1999, 103, 4782. (34) Izmaylov, A. F.; Scuseria, G. E.; Frisch, M. J. J. Chem. Phys. 2006, 125, 104013. (35) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J. Chem. Phys. 1969, 51, 2657. (36) The integration grid was set to “ultrafine,” which requests a grid of 99 radial shells and 590 angular points per shell pruned for computational efficiency, and the SCF convergence criteria was set to root mean squared density change between cycles to be less than 1 × 10-7. The Gaussian default number of k points root mean squared coefficient change between cycles was used at all times, which is 79 for zigzag nanotubes. Finally, the CPSCF trial vector convergence criterion for the 2-vector was set to 1 × 10-6. (37) Tao, J.; Perdew, J. P. J. Chem. Phys. 2005, 122, 114102. (38) Kudin, K. N.; Bettinger, H. F.; Scuseria, G. E. Phys. ReV. B: Condens. Matter Mater. Phys. 2001, 63, 045413. (39) Bettinger, H. F.; Kudin, K. N.; Scuseria, G. E. J. Am. Chem. Soc. 2001, 123, 12849.
