J. Phys. Chem. 1994,98, 9773-9776
9773
Lack of Superaromaticity in Carbon Nanotubes Jun-ichi Aihara Department of Chemistry, Faculty of Science, Shizuoka University, Oya, Shizuoka 422, Japan Received: May 30, 1994; In Final Form: July 21, 1994@
The degree of aromatic stabilization caused by the cylindrical structure of a n-electron system was estimated for different infinitely long carbon nanotubes. It was found that this kind of superaromaticity is almost absent in graphitic tubules although very many conjugated circuits can be chosen along the circumference. This constitutes the main reason why all realistic graphitic tubules are as highly aromatic as graphite.
1. Introduction A graphitic tubule is formally obtainable by wrapping a single graphite sheet onto a long tubule.'-'* Carbon nanotubules are a novel allotrope of carbon, consisting of one or more concentric graphitic tubules.'-4 Like graphite, graphitic tubules can be regarded as highly condensed benzenoid systems. Many types and thicknesses are conceivable for graphitic tubule^.^-'^ We distinguish between them by using the notation (a,b), where a and b are zero or the positive integers6 Thus, tubule(a,b) is an infinitely long graphitic tubule which is constructed as shown in Figure 1. Inherent properties of a cyclic n-electron system have been referred to collectively as ar0mati~ity.l~The percentage topological resonance energy (%TRE) is useful for estimating the degree of aromatic stabili~ation.'~~'~-~* This quantity is given as 100 times the topological resonance energy (TRE),19320 divided by the total n-binding energy of the graph-theoretically defined polyene reference. However, it is impossible to obtain the exact %TRE of a very or infinitely large n-electron system. We previously proposed a method for estimating the approximate but sufficiently reliable %TREs of graphite and graphitic tubules.12 A graphite sheet proved to be highly aromatic. We confirmed that all graphitic tubules of realistic size are as highly aromatic as graphite.'* For polycyclic aromatic hydrocarbons, the conjugated circuits are the main origin of aromaticity.21122We pointed out that the macrocyclic structure of kekulene (l),formed by 12 benzene rings, scarcely contributes to the aromaticity in the entire m ~ l e c u l e . ~ ~ It - ' ~seemed that a number of macrocyclic conjugated circuits in 1 contribute little to the overall aromaticity. In this context, part of aromaticity in graphitic tubules may be attributable to the cylindrical structure of the n-electron system since numerous conjugated circuits can be chosen along the circumference. The aromaticity caused by such circumferential conjugated circuits, if any, may be called superaromaticity. Note that no such conjugated circuits can be taken from a graphite sheet. It is the purpose of this paper to acquire an insight into the degree of superaromaticity in various graphitic tubules.
Figure 1. A single graphite sheet. Tubule(a,b) is formed by rolling the graphite sheet into a cylinder with the hexagon at (a$) folding onto the hexagon at (0,O).Hexagons with bold figures correspond to metallic tubules.
1
2
2. Theory Real or realistic graphitic tubules may be called Huckel-type tubules since there are no Mobius-type or singly twisted conjugated subsystems in them. Huckel-type tubule(a,b) is metallic if a 2b = 3n, where n is the positive integer, but is more or less semiconductive ~ t h e r w i s e . ~Saito ~ ~ - et ~ ~al.~ ~
+
@
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Armchair Tubule
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-365419412098-9773$04.50/0
classified graphitic tubules into armchair, zigzag, and other ones.9 Tubule(a,O) and tubule(a,a) represent zigzag and armchair tubules, respectively. All Huckel-type armchair tubules are metallic with no band gaps. A Hiickel-type zigzag tubule is metallic if a = 3n, but is semiconductive otherwise. Most graphitic tubules are neither armchair nor zigzag ones, but are chiral in g e ~ m e t r y . ~ . ~ . ~ * ' ~ 0 1994 American Chemical Society
Aihara
9774 J. Phys. Chem., Vol. 98, No. 39, 1994
A B Figure 2. Cross sections of Hiickel- and Mobius-type graphitic tubules (A and B, respectively). Lobes denote 2pz atomic orbitals of carbon atoms located along the cross section.
We use the Huckel molecular orbital model in its simplest form. The mixing of n and a orbitals due to the curvature of the tubule can he neglected in the first approximatiot~?.~~ and thus we consider only n bands. The n-orbital energy of a single graphite sheet can be expressed as a function of a twodimensional wavenumber vector k in the f0rm:9-12.24
A
B
Figure 3. Ir-Electron systems of kekulene (A) and the hypothetical Mtibius-type isomer (B). Half the energy difference between A and B was interpreted as a superammatic stabilization energy (SSE) due to
the macrocyclic structure of the n-electron system.14 The SSE is vanishingly small for kekulene.
of graphite and Hiickel-type tubules.12 This carhon molecule resembles graphite in many respects and can be viewed as a graphitic molecule of finite size. In brief, all carhon atoms in 2 are identical, forming an edgeless hexagonal network?' The total n-binding energy per carbon atom for the polyene reference of a given carbon molecule will be called the unit reference energy for short. The unit reference energy of 2 is 1.52784181, 4 cos2g (1) )]lwhich " is very close to the corresponding value estimated by Hess and Schaad (1.5216181)?8,29 The Hess-Schaad value represents half the sum of the n-bond energies of one formal Here, 9, is the resonance integral between bonded carbon atoms, C=C and two formal C-C bonds, both observable in acyclic and c i s 1.5 times the carbon-carbon bond length. The n-orbital polyene^?^.^^ energy of Hiickel-type tubule(a,b) can be formulated by applying For the reasons mentioned above, the unit reference energy the boundary condition (ak, bkJc = 2Nn (N = 1, 2, ..., a of 2 can be used as an approximate hut common unit reference 1, a) to this relation?-1z We then obtain the formula for the energy for graphitic tubules. In principle, the polyene reference n-binding energy per carbon atom of Hiickel-type tuhule(a,b) of a given Huckel-type n-electron system is identical with that in the form:I2 of the Mobius-type isomer. Therefore, the unit reference energy of 2 applies not only to Huckel-type but also to Mobius-type I81 a tubules. For all graphitic tubules, the TRE per carbon atom E,,(a,b) = 1 4 anNZl a can be approximated as the difference between the unit n-hinding energy obtained from eq 2 or 3 and the unit reference energy of 2. The %TRE of any tubule is given as 100 times the TRE per carbon atom, divided by the same unit reference energy.12 Conversely, the TRE per carbon atom is given as The total n-binding energy per carhon atom of a graphite sheet 0.0152784181 times the %TRE. The %TRE is free from any is 1.57460181.25-26 . effect of the curvature because it is a dimensionless quantity. Next, hypothetical Mobius-type isomers of Huckel-type The method used to estimate the degree of superaromaticity graphitic tubules are defined as shown in Figure 2. Mohiusin kekulene can be applied to graphitic t u h u l e ~ . ' ~ -It' ~is for type tubules are formed by rolling a long graphilic ribbon into this purpose that we defined the hypothetical Mobius-type a tubule, in such a manner that all carbon-carbon bonds at the tubules above. On the hasis of the reasoning applied to kekulene seam have the resonance integral of -B. These n bonds are and related molecules (see Figure 3),I4-l6 half the energy supposed to be singly twisted.I4-l6 As in the case of Hiickeldifference between Hiickel- and Mobius-type tubules of the type tubules, Mobius-type ones consist of cylindrical hexagonal same size can he interpreted as a superaromatic stabilization networks. On going from a given Huckel-type tubule to the energy (SSE) per carhon atom. This quantity is attributable to Mobius-type isomer, all circuits chosen along the circumference circumferential conjugated circuits in the cylindrical n-electron of the cylinder are converted into singly twisted or Mobiussystem. The SSE per carbon atom is part of the TRE per carbon type circuits, but all other circuits remain unchanged. There atom. The pcentage supemmatic stabilization energy (%SSE) are an odd number of singly twisted n bonds along every is given by dividing 100 times the SSE per carbon atom by the circumferential circuit. In contrast, there are no or an even unit reference energy of the carbon molecule 2. number of twisted n bonds in other circuits. A formula for the unit n-binding energy of Mobius-type tuhule(a,b) can be obtained simply by applying the boundary condition (ak, bk,) 3. Results and Discussion x c = (2N 1)n to eq 1. It is expressed as The %TREs and the %SSEs estimated for Hiickel- and Mobius-type tubules are listed in Tables 1 and 2. A single 181 a 2N+1 a+2b EM(a,b) 1 4c o s ( y graphite sheet is highly aromatic with a %TRE of 3.061." For anN=l 2a reference, the %TREs of benzene, coronene, and kekulene (1) are 3.528, 2.817, and 2.340, r e s p e c t i ~ e l y . l ~It, ~is~interesting to see that all Hiickel- and Mobius-type tubules are also as aromatic as a graphite sheet. This means that every Hiickeltype tubule has essentially the same degree of aromaticity as We previously devised the polyene-like reference structure the Mobius-type isomer. As far as other conditions are the of a hypothetical 54-carhon molecule (2) to estimate the %TREs
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Superaromaticity in Carbon Nanotubes
J. Phys. Chem., Vol. 98, No. 39, 1994 9775
TABLE 1: %TREs and the %SSEs for Zigzag Tubules Mobius type
Hiickel type a
b
%TRE!
%SSE
%TRE
%SSE
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0
3.011 3.081 3.069 3.039 3.069 3.065 3.049 3.065 3.063 3.054 3.063 3.062 3.056 3.062 3.062 3.057 3.061 3.061 3.058 3.061 3.061 3.059
-0.043 0.019 0.007 -0.018 0.008 0.004 -0.010 0.004 0.002 -0.006 0.003 0.001 -0.004 0.002 0.001 -0.002 0.001 0.001 -0.002 0.001 0.001 -0.001
3.097 3.042 3.055 3.076 3.052 3.057 3.068 3.056 3.058 3.065 3.058 3.059 3.063 3.059 3.059 3.062 3.059 3.059 3.061 3.059 3.060 3.061
0.043 -0.019 -0.007 0.018 -0.008 -0.004 0.010
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-0.004 -0.002 0.006 -0.003 -0.001 0.004 -0.002 -0.001 0.002 -0.001 -0.001 0.002 -0.001 -0.001 0.001
TABLE 2: %TREs and the %SSEs for Armchair Tubules Mobius type
Hiickel type a
b
%TRE
%SSE
%TRE
%SSE
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2.999 3.025 3.038 3.046 3.050 3.053 3.055 3.056 3.057 3.057 3.058 3.058 3.059 3.059 3.059 3.059 3.059 3.059 3.059 3.060 3.060
-0.055 -0.03 1 -0.019 -0.013 -0.009 -0.006 -0.005 -0.004 -0.003 -0.002 -0.002 -0.002 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.000 -0.000
3.108 3.087 3.077 3.071 3.068 3.066 3.064 3.063 3.063 3.062 3.062 3.061 3.061 3.061 3.061 3.061 3.061 3.061 3.060 3.060 3.060
0.055 0.03 1 0.019 0.013 0.009 0.006 0.005 0.004 0.003 0.002 0.002 0.002
0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.000
same, every Huckel-type tubule must be thermodynamically as stable as a Mobius-type isomer. All metallic Huckel-type tubules (a 2b = 3n) have only slightly smaller %TREs than a graphite sheet, whereas all semiconductive Huckel-type tubules (a 2b t 3n) have slightly larger %TREs.12 Such a trend in aromaticity is reversed in Mobius-type tubules. If a given Hiickel-type tubule is metallic with no band gap, the Mobius-type isomer is slightly lower in energy than that. Conversely, if a given Huckel-type tubule is semiconductive, the Mobius-type isomer is slightly higher in energy than that. These facts really indicate that the sign of the %SSE is reversed on going from the Huckel-type to the Mobius-type isomer. Metallicity is related to the sign of superaromaticity, being responsible for the oscillation of the %SSE about zero. In Tables 1 and 2 , positive and negative %SSEs correspond to semiconductive and metallic tubules, respectively. As the energy difference between any Hiickel-type tubule and the Mobius-type isomer is very or negligibly small, all %SSEs are very small. It then follows that conjugated circuits taken
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along the circumference of the cylinder do not play an important role in stabilizing the entire n-electron system. Thus, graphitic tubules are not appreciably superaromatic at all. Negligible superaromaticity justifies the conjecture that a graphitic tubule of any thickness can be formed without loss of aromatic stabilization.12 This constitutes the reason why tubules can change their diameters so as to form a closely packed multilayered carbon nanotube.l s 2 Graphitic tubules in a multilayered nanotube are always closely Negligible superaromaticity or superantiaromaticity further decreases as the tubule becomes thicker. For example, tubule( 100,O) and tubule( 100, 100) are essentially nonsuperaromatic with the %SSEs of 0.000. Very thin graphitic tubules were observed as single-walled carbon nanotubules, whose diameters lie in the range 0.7-1.6 rm3z4 Iijima and Ichihashi speculated that such nanotubes might be the embryo for the multilayered nan~tubes.~ A typical singlewalled nanotube observed was Huckel-type tubule( 18,2) with a diameter of 1.37 nm.3 For this semiconductive tubule, the %TRE is 3.063. This tubule likewise has vanishingly small superaromatic character with the %SSE of 0.003. Thus, it has been established that not only aromaticity but also superaromaticity scarcely depends on the thickness and the chirality of the tubule.
4. Concluding Remarks In general, a cyclic n-electron system with a large HOMOLUMO energy separation is thermodynamically very stable and highly aromatic. Therefore, aromaticity in a molecule is very sensitive to the magnitude of the HOMO-LUMO gap.30 However, this is not the case with graphitic tubules. All graphitic tubules exhibit essentially the same aromatic character alothough their energy gaps vary ~ i d e l y . This ~ . ~ is ~ consistent with the lack of superaromaticity or superantiaromaticity in these carbon materials. It is in marked contrast to the fact that the cylindrical structure plays a critical role in making a tubule metallic or s e m i c o n d u ~ t i v e . ~ . ~ ~ ~ , ~ ~ Acknowledgment. This work was supported by a Grantin-Aid for Scientific Research from the Ministry of Education, Science, and Culture, Japan. Dicussion with Professor Tsuguhiro Tamaribuchi at Shizuoka University is gratefully acknowledged. References and Notes (1) Iijima, S. Nature 1991, 354, 56. (2) Ajayan, P. M.; Ichihashi, T.; Iijima, S. Chem. Phys. Lett. 1993, 202, 384. ( 3 ) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (4) Bethune, D. S.; Kiang, C. H.; de Vries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605. ( 5 ) Mintmire, J. W.; Dunlap, B. I.; White, C. T. Phys. Rev. Lett. 1992, 68, 631. (6) Hamada, N.; Sawada, S.; Oshiyama, A. Phys. Rev. Lett. 1992,68, 1579. (7) Tanaka, K.; Okahara, K.; Okada, M.; Yamabe, T. Chem. Phys. Lett. 1992, 191, 469. ( 8 ) Harigaya, K. Phys. Rev. B 1992, 45, 12071. (9) Saito, R.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Phys. Rev. B 1992, 46, 1804. (10) Klein, D. J.; Seitz, W. A,; Schmalz, T.G . J . Phys. Chem. 1993, 97, 1231. (11) Okahara, K.; Satoh, T.; Aoki, H.; Tanaka, K.; Yamabe, T. Chem. Phys. Lett. 1994, 219, 462. (12) Aihara, J.; Yamabe, T.; Hosoya, H. Synth. Met. 1994, 64, 309. (13) Aihara, J. Sci. Am. 1992, 266 (S), 62. (14) Aihara, J. J. Am. Chem. SOC.1992, 114, 865. (15) Aihara, J. Bull. Chem. SOC.Jpn. 1993, 66, 57. (16) Aihara, J. J. Chem. SOC.,Perkin Trans. 2 1994, 971. (17) Aihara, J.; Takata, S. J. Chem. Soc., Perkin Trans. 2 1994, 65. (18) Aihara, J. J . Mol. Struct. (Theochemj, in press. (19) Aihara, J. J. Am. Chem. SOC.1976, 98, 2750.
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Aihara (27) Hosoya, H.; Tsukano, Y.; Ohuchi, M.; Nakada, K. In Computer Aided Innovation of New Materials II; Doyama, M., Kihara, J., Tanaka,
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