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A theoretical study on the geometrical features of finite-length carbon nanotubes capped with fullerene hemisphere
Chemical Physics Letters 386 (2004) 38–43 www.elsevier.com/locate/cplett
A theoretical study on the geometrical features of finite-length carbon nanotubes capped with fullerene hemisphere Takashi Yumura a,*, Kaori Hirahara a, Shunji Bandow a, Kazunari Yoshizawa b, Sumio Iijima a a b
Department of Materials Science and Engineering, Meijo University, Tenpaku-ku, Nagoya 468-8502, Japan Institute for Materials Chemistry and Engineering, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan Received 2 October 2003; in final form 20 December 2003 Published online: 2 February 2004
Abstract The structures of the finite-length ð5; 5Þ and ð9; 0Þ carbon nanotubes capped with fullerene hemisphere are analyzed by quantum chemical calculations at the B3LYP DFT level of theory. DFT calculations demonstrate that the geometries of the armchair tubes depend on the number of cyclic cis-polyene chains lined up along the tube axis, whereas the zigzag tubes consist of Kekule-type networks in the cylinder, the geometries being independent of the number of component cyclic trans-polyene chains. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction A single-walled carbon nanotube (SWNT) consists of a graphene sheet wrapped into a cylinder with a nanometer-scale diameter [1,2]. The structures of SWNTs can be represented by a chiral vector C ¼ na1 þ ma2 , where a1 and a2 denote equivalent lattice vectors of the graphene sheet. Much has been studied on the electronic and structural properties of the infinite-length SWNTs [3–10]. For instance, Kertesz et al. [8,9] calculated the geometries of armchair and zigzag SWNTs using density functional theory (DFT) and predicted characteristic variations in the C–C bond lengths as a function of n–m for the (n; m) nanotube. As for finite-length nanotubes, however, information is still limited [11–14]. The present letter has been motivated by our recent observation of finite-length nanotubes capped with fullerene hemisphere, which are formed inside a SWNT by heating ÔC60 peapod,Õ in which the C60 molecules are incorporated into a SWNT [11]. An image of high-resolution transmission electron microscopy (TEM) in Fig. 1 identified nanotubes with lengths of 13.7 and *
Corresponding author. Fax: +81-52-834-4001. E-mail address: [email protected] (T. Yumura).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.12.123
which correspond to three and four C60 mole21.2 A, cules in size, respectively. To understand the mechanism of the coalescence of C60 inside a SWNT, and the structural features of finite-length nanotubes capped with fullerene hemisphere in the scope of aromaticity, we have undertaken a DFT study on the bonding structures of the finite-length nanotubes with a focus on the correlation between the C–C bond lengths and the cylinder lengths.
2. Finite-length nanotubes capped with fullerene hemisphere Two possible finite-length nanotubes with fullerene hemisphere are considered: the ð5; 5Þ armchair nanotubes with a C5 rotation axis and the ð9; 0Þ zigzag nanotubes with a C3 rotation axis. C60 can be partitioned into one cyclic cis-polyene chain with 20 carbon atoms and two corannulene-like caps along a C5 rotation axis, and it can also be viewed as one cyclic trans-polyene chain with 18 carbon atoms and two sumanene-like caps along a C3 rotation axis. When cyclic cis-polyene chains are inserted between the corannulene-like caps, the ð5; 5Þ armchair nanotubes C40 þ 20n with D5d symmetry are
T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
39
3. Results and discussion 3.1. Geometrical features of C60
Fig. 1. A TEM image of tubular structures as intermediates in the C60 coalescence inside a SWNT at 900 °C [11]. The numbers above the SWNT denote the size of the tubular structures represented as the effective numbers of C60 . The cylinder lengths are also given below the SWNT.
formed, where n is the number of the chains inserted. The ð9; 0Þ zigzag nanotubes C42 þ 18n are composed of cyclic trans-polyene chains and the sumanene-like caps. Hence, the cylinder of the armchair nanotube C40 þ 20n can be regarded as interacting cis-polyene chains, while that of the zigzag nanotube C42 þ 18n as interacting transpolyene chains. In the zigzag series, C42 þ 18n belongs to the D3h (D3d ) group, when n is even (odd). The main purpose of the present paper is to clarify the geometrical features of C40 þ 20n (n ¼ 2–10) and C42 þ 18n (n ¼ 2–11).
We first optimized the structure of C60 with Ih symmetry using the hybrid DFT method of Becke [15] and Lee, Yang, and Parr [16] (B3LYP) and the 6-31G* basis set [17] implemented in the GA U S S I A N 98 program package [18]. Fig. 2 shows the optimized Ih structure, where the 5–6 and 6–6 bonds are 1.454 and 1.395 A, respectively. These values agree well with those obtained from a neutron diffraction experiment [19]. Bond-length alternation occurs along the C5 axis within one cyclic cis-polyene chain. The C–C bonds oriented along the axis are contracted, while those around the chain are elongated, as indicated by 1 in Chart 1, where thick lines In contrast, represent C–C bonds shorter than 1.42 A. one cyclic trans-polyene chain along the C3 axis has a different bond-deformation pattern, in which short C–C appear on every two bonds, as shown bonds of 1.395 A by 2 in Chart 1.
Fig. 2. Optimized geometries of C60 (Ih ), C240 (D5d ), and C240 (D3d ) at the B3LYP DFT level of theory. Bond lengths are in A.
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T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
C60 (along C5 rotation axis)
(along C3 rotation axis)
1
2
HOMO I
HOMO II Chart 1.
These patterns of bond deformation are rationalized by the frontier orbital ideas [20,21]. Two of the fivefold degenerate highest-occupied molecular orbitals (HOMOs) in the equator of C60 are also shown in Chart 1. Along the C5 axis, HOMO I consists of in-phase and out-of-phase combinations with respect to the 6–6 and 5–6 bonds, respectively. Because the orbital pattern I interacts consistently with that of the corannulene cap, it is deemed as the essential factor determining the ge-
C80 (0.065)
ometries of the armchair nanotubes with the corannulene caps. Along the C3 rotation axis, however, HOMO II has no orbital interactions in the 5–6 bonds despite the bonding interactions in the 6–6 bonds. The geometrical features of the finite-length zigzag nanotubes also agree with the orbital pattern II due to a good interaction with the sumanene caps. 3.2. Geometrical features of the finite-length carbon nanotubes with fullerene hemisphere The structure of C240 with the armchair form, shown in Fig. 2, involves seven cyclic cis-polyene chains with pattern 1; it is also displayed in Fig. 3, where the C–C bonds are categorized according to their lengths. One at the edge, whereas C–C bond is shortened to 1.405 A the other C–C bonds are lengthened up to 1.443 and On the other hand, the chains at the center 1.455 A. and elongated have contracted C–C bonds, 1.420 A, bonds, 1.436 A. The bond-length alternation, d, at the edges is significant in comparison with those at the
C120
C100 3.636
(0.051)
(0.047) 6.117
(0.065)
(0.028)
8.572
(0.047) (0.051)
(0.050)
(0.047)
(0.049) (0.034)
C180
C160
C140
(0.024) 11.026 13.507
(0.034)
(0.017) (0.049)
15.963
(0.024) (0.047) (0.050)
C200
C240
C220
(0.046)
(0.050)
(0.048)
(0.023)
(0.030)
(0.016)
18.413 20.905
(0.016)
23.354
(0.030) (0.016) (0.046)
(0.023) (0.048) (0.050)
Fig. 3. Bond-deformation patterns in the cylindrical part of the finite-length armchair nanotubes. The thick lines indicate C–C bonds shorter than the broken lines those ranging 1.421–1.430 A, and the single lines those larger than 1.431 A. Symbol d(A) represents the bond-length 1.42 A, alternations. The cylinder lengths are also given in A.
T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
center. In contrast, the zigzag form of C240 has bonddeformation patterns with Kekule-type networks shown in Fig. 4. Framework 2 is involved in the cyclic transpolyene chains at the center of C240; the differences between the shorter and longer C–C bonds ranging 0.010–0.020 A. We see in Fig. 3 that the geometries of the armchair series strongly depend on the number of cyclic cispolyene chains. The series of C40 þ 20n can be classified into three subgroups; C20 þ 60l , C40 þ 60l , and C60 þ 60l , where l is an integer. The C20 þ 60l cylinders can be regarded as composed of cyclophenacene, in which the
C78 (0.041)
C96 2.952
(0.053)
(0.041)
5.156 (0.053)
C114
C132
(0.041)
(0.043)
(0.017) 7.297 (0.017)
(0.027)
9.421
(0.041) (0.043)
C150
C168
(0.043)
(0.043)
(0.022) (0.022)
11.560
(0.020) (0.010)
13.694
(0.020)
(0.043)
(0.043)
C186
C204
(0.043)
(0.042)
(0.020)
(0.020)
(0.012)
15.829
(0.011)
(0.012)
(0.010)
(0.020)
(0.011)
17.964
(0.020) (0.043) (0.042)
C222
C240
(0.042)
(0.042)
(0.020)
(0.020)
(0.011)
(0.011)
(0.011) (0.011)
20.101
(0.010) (0.010)
(0.011)
(0.010)
(0.020)
(0.011)
22.234
(0.020) (0.042) (0.042)
Fig. 4. Bond-deformation patterns in the cylindrical part of the finitelength zigzag nanotubes.
41
two alternating bond chains 1 are located on the edge. Bond deviations within a chain vary from 0.03 to as the bonds approach to the edge. The C60 þ 60l 0.05 A cylinders have different structural properties, in which two cyclic cis-polyene chains with pattern 1 are connected by one carbon belt lying in a plane perpendicular to the tube axis. In contrast, the C40 þ 60l cylinders have except in the edge no C–C bonds longer than 1.42 A, region, and hence one cyclic cis-polyene chain with pattern 1 is located on each edge of the cylinder. Although the C40 þ 20n cylinders are here subdivided into three categories, all armchair forms contain at least one cyclic cis-polyene chain with pattern 1; the C–C bonds and those along the tube axis are shorter than 1.42 A, around the belts are lengthened. This is a unique electronic effect appearing in a nanometer-sized system that cannot be observed in the infinite-length nanotubes [9]. In contrast to the capped armchair series, the ð5; 5Þ nanotubes terminated by H atoms have Kekule, incomplete Clar, and complete Clar networks [22]. The geometrical features in the capped nanotubes seem reasonable in view of HOMO I, but those in the nanotubes terminated by H atoms are consistent with the pattern of the highest-occupied crystal orbital (HOCO) of the carbon-ladder polymers with the phenanthreneedge structure [23,24]. In contrast to the armchair series, the bond-deformation patterns in the zigzag series do not sensitively depend on the cylinder length, as shown in Fig. 4. In the zigzag forms, cyclic trans-polyene chains with pattern 2 are contained in the cylindrical part, except for those adjacent to the edge, in which there is a nearly equidistant cyclic trans-polyene chain. The deformed chains are linked by two distinct types of C–C bonds (one being and the other longer) to generate shorter than 1.42 A Kekule-type networks in the cylinder. The magnitude of the chain deformation in the C42 þ 18n cylinder varies and converges toward that in from 0.01 to 0.02 A, the corresponding infinite-length nanotubes [9] with increasing n. However, the geometrical features of the infinite-length zigzag nanotubes, in which nearly equidistant cyclic polyene chains are involved [9], differ from those of the finite-length nanotubes. Similar equidistant cyclic polyene chains are also involved in the zigzag nanotubes terminated by H atoms, this seems reasonable from the HOCO character of the polymers with the acene-edge structure [23,24]. These differences are reflected in the cylinder lengths, particularly in those of the finite-length carbon nanotubes, as shown in Figs. 3 and 4. The cylinder lengths of C180 and C240 , calculated to be 15.963 and 23.354 A, respectively, are significantly different from those estimated from the TEM observation [11]. This discrepancy indicates that the trimer and tetramer of C60 are not generated in the initial stages of the C60 coalescence inside a SWNT. On the other hand, since the
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T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
for the tube on the left-hand experimental value (13.7 A) side in Fig. 1 is equivalent to those calculated for C160 and C168 , the tubular structure should correspond to C160 or C168 . Furthermore, the C220 and C222 cylinders have lengths close to that in the tube on the right-hand suggesting the formation of C220 side in Fig. 1 (21.2 A), or C222 . The numbers of atoms in C160 and C168 are smaller than that in the C60 trimer, and those in C220 and C222 are also smaller than in the tetramer. Accordingly, we predict that the release of carbon atoms takes place during the formation of the finite-length nanotubes capped with fullerene hemispheres. They are naturally the most stable ones in energy, but there are other possible structures because appreciable energy can be provided during the reaction. 3.3. Electronic features of the finite-length carbon nanotubes with fullerene hemisphere Finally, we discuss the gaps between the HOMO and the lowest-unoccupied molecular orbital (LUMO) in the finite-length carbon nanotubes. The HOMO–LUMO gaps for both nanotubes are shown in Fig. 5 as a function of the number of cyclic polyene chains. Although the infinite-length ð5; 5Þ and ð9; 0Þ nanotubes have a metallic character, the HOMO–LUMO gaps in the finite-length nanotubes are not zero. The gaps for the armchair series decrease with an oscillating manner having a periodicity of 3, with increasing chain width. In contrast, the gaps for the zigzag series decrease with increasing number of chains. These features are fully
consistent with those obtained at the B3LYP/6-311G* level of theory [14]. Our DFT calculations show a significant correlation between the HOMO–LUMO gaps and the bond deformation patterns in the finite-length nanotubes. In the armchair series, the C20þ60l cylinders have smaller gaps, C40 þ 60l larger gaps, and C60þ60l medium gaps. In C40 þ 60l a single cyclic cis-polyene chain with pattern 1 is located on each edge of the cylinder, whereas in C20þ60l alternating bond-length chains are distributed throughout the cylinder. Since the single alternating bond-length chain interacting with each corannulene cap is involved in C60 and C40 þ 60l , the gaps for C40 þ 60l are close to that for C60 . However, interchain interactions between the cis-polyene chains in C20þ60l should lead to smaller gaps than those for C60 . In C60þ60l , other bond-alternation patterns between C20þ60l and C40 þ 60l are contained, and accordingly the gaps have medium values. In contrast to the armchair series, the geometrical features in the zigzag series are found to be insensitive to the cylinder lengths; thus no oscillatory behavior appears in the HOMO–LUMO gaps for the zigzag series.
4. Conclusions We have investigated the finite-length ð5; 5Þ and ð9; 0Þ nanotubes by DFT calculations at the B3LYP DFT level of theory. In the finite-length armchair C40 þ 20n and zigzag C42 þ 18n nanotubes, certain numbers (n) of cyclic cis- and trans-polyene chains are lined up along the tube axis, respectively. The geometries of the armchair series depend on the number of cyclic cis-polyene chains involved, and the finite-length carbon nanotubes can be classified into three subgroups. In contrast, the zigzag nanotubes, which are insensitive to the chain width, consist of Kekule networks in the cylindrical segment. We have also analyzed the gap between the HOMO and LUMO as a function of the number of cyclic polyene chains and demonstrated a significant relationship between the HOMO–LUMO gaps and the bond deformation patterns in the finite-length nanotubes.
Acknowledgements
Fig. 5. HOMO–LUMO gaps as a function of the number of cyclic polyene chains in the finite-length nanotubes. The armchair C40 þ 20n and zigzag C42 þ 18n nanotubes are represented by square and circle dots, respectively.
This work has been supported by the 21st century COE program at Meijo University and jointly with MEXT (Ministry of Education, Culture, Sports, Science and Technology, Japan) and by the ÔNanotechnology Support ProjectÕ of MEXT. The support from the US office of Naval Research (for S.I. and K.H.), the Japan Society for the Promotion of Science (JSPS), the Murata Science Foundation, and Kyushu University P&P ÔGreen ChemistryÕ (for K.Y.), and the JSPS postdoctoral fellowship (for T.Y.) are gratefully acknowledged.
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References [1] S. Iijima, Nature 354 (1991) 56. [2] S. Iijima, T. Ichikawa, Nature 363 (1993) 603. [3] R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Appl. Phys. Lett. 60 (1992) 2204. [4] J.W. Mintmire, B.I. Dunlap, C.T White, Phys. Rev. Lett. 68 (1992) 631. [5] N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579. [6] K. Tanaka, K. Okahara, M. Okada, T. Yamabe, Chem. Phys. Lett. 191 (1992) 469. [7] K. Tanaka, H. Ago, T. Yamabe, K. Okahara, M. Okada, Int. J. Quant. Chem. 63 (1997) 637. [8] G. Sun, J. K€ urti, M. Kertesz, R.H. Baughman, J. Am. Chem. Soc. 124 (2002) 15076. [9] G. Sun, J. K€ urti, M. Kertesz, R.H. Baughman, J. Phys. Chem. B 107 (2003) 6924. [10] K. Kanamitsu, S. Saito, J. Phys. Soc. Jpn. 71 (2002) 483. [11] S. Bandow, K. Hirahara, T. Hiraoka, G. Chen, P.C. Eklund, S. Iijima, MRS bulletin, in press.
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[12] A. Rochefort, D.R. Salahub, P. Avouris, J. Phys. Chem. B 103 (1999) 641. [13] T. Sato, M. Tanaka, T. Yamabe, Synth. Met. 103 (1999) 2525. [14] J. Cioslowski, N. Rao, D. Moncrieff, J. Am. Chem. Soc. 124 (2002) 8485. [15] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [16] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [17] R. Ditchfield, W.J. Hehre, J.A. Pople, J. Chem. Phys. 54 (1971) 724. [18] M.J. Frisch et al., GA U S S I A N 98, Gaussian, Inc., Pittsburgh, PA, 1998. [19] W.I.F. David, R.M. Ibberson, I.C. Matthewman, K. Prassides, T. John, S. Dennis, J.P. Hare, H.W. Kroto, R. Taylor, D.R.M. Walton, Nature 353 (1991) 147. [20] R.B. Woodward, R. Hoffmann, The Conservation of Orbital Symmetry, Verlag Chemie, Weinheim, 1970. [21] K. Fukui, Theory of Orientation and Stereoselection, Springer, Berlin, 1975. [22] Y. Matsuo, K. Tahara, E. Nakamura, Org. Lett. 5 (2003) 3181. [23] K. Yoshizawa, K. Yahara, K. Tanaka, T. Yamabe, J. Phys. Chem. B 102 (1998) 498. [24] T. Yumura, S. Bandow, K. Yoshizawa, S. Iijima, in preparation.
A theoretical study on the geometrical features of finite-length carbon nanotubes capped with fullerene hemisphere Takashi Yumura a,*, Kaori Hirahara a, Shunji Bandow a, Kazunari Yoshizawa b, Sumio Iijima a a b
Department of Materials Science and Engineering, Meijo University, Tenpaku-ku, Nagoya 468-8502, Japan Institute for Materials Chemistry and Engineering, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan Received 2 October 2003; in final form 20 December 2003 Published online: 2 February 2004
Abstract The structures of the finite-length ð5; 5Þ and ð9; 0Þ carbon nanotubes capped with fullerene hemisphere are analyzed by quantum chemical calculations at the B3LYP DFT level of theory. DFT calculations demonstrate that the geometries of the armchair tubes depend on the number of cyclic cis-polyene chains lined up along the tube axis, whereas the zigzag tubes consist of Kekule-type networks in the cylinder, the geometries being independent of the number of component cyclic trans-polyene chains. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction A single-walled carbon nanotube (SWNT) consists of a graphene sheet wrapped into a cylinder with a nanometer-scale diameter [1,2]. The structures of SWNTs can be represented by a chiral vector C ¼ na1 þ ma2 , where a1 and a2 denote equivalent lattice vectors of the graphene sheet. Much has been studied on the electronic and structural properties of the infinite-length SWNTs [3–10]. For instance, Kertesz et al. [8,9] calculated the geometries of armchair and zigzag SWNTs using density functional theory (DFT) and predicted characteristic variations in the C–C bond lengths as a function of n–m for the (n; m) nanotube. As for finite-length nanotubes, however, information is still limited [11–14]. The present letter has been motivated by our recent observation of finite-length nanotubes capped with fullerene hemisphere, which are formed inside a SWNT by heating ÔC60 peapod,Õ in which the C60 molecules are incorporated into a SWNT [11]. An image of high-resolution transmission electron microscopy (TEM) in Fig. 1 identified nanotubes with lengths of 13.7 and *
Corresponding author. Fax: +81-52-834-4001. E-mail address: [email protected] (T. Yumura).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.12.123
which correspond to three and four C60 mole21.2 A, cules in size, respectively. To understand the mechanism of the coalescence of C60 inside a SWNT, and the structural features of finite-length nanotubes capped with fullerene hemisphere in the scope of aromaticity, we have undertaken a DFT study on the bonding structures of the finite-length nanotubes with a focus on the correlation between the C–C bond lengths and the cylinder lengths.
2. Finite-length nanotubes capped with fullerene hemisphere Two possible finite-length nanotubes with fullerene hemisphere are considered: the ð5; 5Þ armchair nanotubes with a C5 rotation axis and the ð9; 0Þ zigzag nanotubes with a C3 rotation axis. C60 can be partitioned into one cyclic cis-polyene chain with 20 carbon atoms and two corannulene-like caps along a C5 rotation axis, and it can also be viewed as one cyclic trans-polyene chain with 18 carbon atoms and two sumanene-like caps along a C3 rotation axis. When cyclic cis-polyene chains are inserted between the corannulene-like caps, the ð5; 5Þ armchair nanotubes C40 þ 20n with D5d symmetry are
T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
39
3. Results and discussion 3.1. Geometrical features of C60
Fig. 1. A TEM image of tubular structures as intermediates in the C60 coalescence inside a SWNT at 900 °C [11]. The numbers above the SWNT denote the size of the tubular structures represented as the effective numbers of C60 . The cylinder lengths are also given below the SWNT.
formed, where n is the number of the chains inserted. The ð9; 0Þ zigzag nanotubes C42 þ 18n are composed of cyclic trans-polyene chains and the sumanene-like caps. Hence, the cylinder of the armchair nanotube C40 þ 20n can be regarded as interacting cis-polyene chains, while that of the zigzag nanotube C42 þ 18n as interacting transpolyene chains. In the zigzag series, C42 þ 18n belongs to the D3h (D3d ) group, when n is even (odd). The main purpose of the present paper is to clarify the geometrical features of C40 þ 20n (n ¼ 2–10) and C42 þ 18n (n ¼ 2–11).
We first optimized the structure of C60 with Ih symmetry using the hybrid DFT method of Becke [15] and Lee, Yang, and Parr [16] (B3LYP) and the 6-31G* basis set [17] implemented in the GA U S S I A N 98 program package [18]. Fig. 2 shows the optimized Ih structure, where the 5–6 and 6–6 bonds are 1.454 and 1.395 A, respectively. These values agree well with those obtained from a neutron diffraction experiment [19]. Bond-length alternation occurs along the C5 axis within one cyclic cis-polyene chain. The C–C bonds oriented along the axis are contracted, while those around the chain are elongated, as indicated by 1 in Chart 1, where thick lines In contrast, represent C–C bonds shorter than 1.42 A. one cyclic trans-polyene chain along the C3 axis has a different bond-deformation pattern, in which short C–C appear on every two bonds, as shown bonds of 1.395 A by 2 in Chart 1.
Fig. 2. Optimized geometries of C60 (Ih ), C240 (D5d ), and C240 (D3d ) at the B3LYP DFT level of theory. Bond lengths are in A.
40
T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
C60 (along C5 rotation axis)
(along C3 rotation axis)
1
2
HOMO I
HOMO II Chart 1.
These patterns of bond deformation are rationalized by the frontier orbital ideas [20,21]. Two of the fivefold degenerate highest-occupied molecular orbitals (HOMOs) in the equator of C60 are also shown in Chart 1. Along the C5 axis, HOMO I consists of in-phase and out-of-phase combinations with respect to the 6–6 and 5–6 bonds, respectively. Because the orbital pattern I interacts consistently with that of the corannulene cap, it is deemed as the essential factor determining the ge-
C80 (0.065)
ometries of the armchair nanotubes with the corannulene caps. Along the C3 rotation axis, however, HOMO II has no orbital interactions in the 5–6 bonds despite the bonding interactions in the 6–6 bonds. The geometrical features of the finite-length zigzag nanotubes also agree with the orbital pattern II due to a good interaction with the sumanene caps. 3.2. Geometrical features of the finite-length carbon nanotubes with fullerene hemisphere The structure of C240 with the armchair form, shown in Fig. 2, involves seven cyclic cis-polyene chains with pattern 1; it is also displayed in Fig. 3, where the C–C bonds are categorized according to their lengths. One at the edge, whereas C–C bond is shortened to 1.405 A the other C–C bonds are lengthened up to 1.443 and On the other hand, the chains at the center 1.455 A. and elongated have contracted C–C bonds, 1.420 A, bonds, 1.436 A. The bond-length alternation, d, at the edges is significant in comparison with those at the
C120
C100 3.636
(0.051)
(0.047) 6.117
(0.065)
(0.028)
8.572
(0.047) (0.051)
(0.050)
(0.047)
(0.049) (0.034)
C180
C160
C140
(0.024) 11.026 13.507
(0.034)
(0.017) (0.049)
15.963
(0.024) (0.047) (0.050)
C200
C240
C220
(0.046)
(0.050)
(0.048)
(0.023)
(0.030)
(0.016)
18.413 20.905
(0.016)
23.354
(0.030) (0.016) (0.046)
(0.023) (0.048) (0.050)
Fig. 3. Bond-deformation patterns in the cylindrical part of the finite-length armchair nanotubes. The thick lines indicate C–C bonds shorter than the broken lines those ranging 1.421–1.430 A, and the single lines those larger than 1.431 A. Symbol d(A) represents the bond-length 1.42 A, alternations. The cylinder lengths are also given in A.
T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
center. In contrast, the zigzag form of C240 has bonddeformation patterns with Kekule-type networks shown in Fig. 4. Framework 2 is involved in the cyclic transpolyene chains at the center of C240; the differences between the shorter and longer C–C bonds ranging 0.010–0.020 A. We see in Fig. 3 that the geometries of the armchair series strongly depend on the number of cyclic cispolyene chains. The series of C40 þ 20n can be classified into three subgroups; C20 þ 60l , C40 þ 60l , and C60 þ 60l , where l is an integer. The C20 þ 60l cylinders can be regarded as composed of cyclophenacene, in which the
C78 (0.041)
C96 2.952
(0.053)
(0.041)
5.156 (0.053)
C114
C132
(0.041)
(0.043)
(0.017) 7.297 (0.017)
(0.027)
9.421
(0.041) (0.043)
C150
C168
(0.043)
(0.043)
(0.022) (0.022)
11.560
(0.020) (0.010)
13.694
(0.020)
(0.043)
(0.043)
C186
C204
(0.043)
(0.042)
(0.020)
(0.020)
(0.012)
15.829
(0.011)
(0.012)
(0.010)
(0.020)
(0.011)
17.964
(0.020) (0.043) (0.042)
C222
C240
(0.042)
(0.042)
(0.020)
(0.020)
(0.011)
(0.011)
(0.011) (0.011)
20.101
(0.010) (0.010)
(0.011)
(0.010)
(0.020)
(0.011)
22.234
(0.020) (0.042) (0.042)
Fig. 4. Bond-deformation patterns in the cylindrical part of the finitelength zigzag nanotubes.
41
two alternating bond chains 1 are located on the edge. Bond deviations within a chain vary from 0.03 to as the bonds approach to the edge. The C60 þ 60l 0.05 A cylinders have different structural properties, in which two cyclic cis-polyene chains with pattern 1 are connected by one carbon belt lying in a plane perpendicular to the tube axis. In contrast, the C40 þ 60l cylinders have except in the edge no C–C bonds longer than 1.42 A, region, and hence one cyclic cis-polyene chain with pattern 1 is located on each edge of the cylinder. Although the C40 þ 20n cylinders are here subdivided into three categories, all armchair forms contain at least one cyclic cis-polyene chain with pattern 1; the C–C bonds and those along the tube axis are shorter than 1.42 A, around the belts are lengthened. This is a unique electronic effect appearing in a nanometer-sized system that cannot be observed in the infinite-length nanotubes [9]. In contrast to the capped armchair series, the ð5; 5Þ nanotubes terminated by H atoms have Kekule, incomplete Clar, and complete Clar networks [22]. The geometrical features in the capped nanotubes seem reasonable in view of HOMO I, but those in the nanotubes terminated by H atoms are consistent with the pattern of the highest-occupied crystal orbital (HOCO) of the carbon-ladder polymers with the phenanthreneedge structure [23,24]. In contrast to the armchair series, the bond-deformation patterns in the zigzag series do not sensitively depend on the cylinder length, as shown in Fig. 4. In the zigzag forms, cyclic trans-polyene chains with pattern 2 are contained in the cylindrical part, except for those adjacent to the edge, in which there is a nearly equidistant cyclic trans-polyene chain. The deformed chains are linked by two distinct types of C–C bonds (one being and the other longer) to generate shorter than 1.42 A Kekule-type networks in the cylinder. The magnitude of the chain deformation in the C42 þ 18n cylinder varies and converges toward that in from 0.01 to 0.02 A, the corresponding infinite-length nanotubes [9] with increasing n. However, the geometrical features of the infinite-length zigzag nanotubes, in which nearly equidistant cyclic polyene chains are involved [9], differ from those of the finite-length nanotubes. Similar equidistant cyclic polyene chains are also involved in the zigzag nanotubes terminated by H atoms, this seems reasonable from the HOCO character of the polymers with the acene-edge structure [23,24]. These differences are reflected in the cylinder lengths, particularly in those of the finite-length carbon nanotubes, as shown in Figs. 3 and 4. The cylinder lengths of C180 and C240 , calculated to be 15.963 and 23.354 A, respectively, are significantly different from those estimated from the TEM observation [11]. This discrepancy indicates that the trimer and tetramer of C60 are not generated in the initial stages of the C60 coalescence inside a SWNT. On the other hand, since the
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T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
for the tube on the left-hand experimental value (13.7 A) side in Fig. 1 is equivalent to those calculated for C160 and C168 , the tubular structure should correspond to C160 or C168 . Furthermore, the C220 and C222 cylinders have lengths close to that in the tube on the right-hand suggesting the formation of C220 side in Fig. 1 (21.2 A), or C222 . The numbers of atoms in C160 and C168 are smaller than that in the C60 trimer, and those in C220 and C222 are also smaller than in the tetramer. Accordingly, we predict that the release of carbon atoms takes place during the formation of the finite-length nanotubes capped with fullerene hemispheres. They are naturally the most stable ones in energy, but there are other possible structures because appreciable energy can be provided during the reaction. 3.3. Electronic features of the finite-length carbon nanotubes with fullerene hemisphere Finally, we discuss the gaps between the HOMO and the lowest-unoccupied molecular orbital (LUMO) in the finite-length carbon nanotubes. The HOMO–LUMO gaps for both nanotubes are shown in Fig. 5 as a function of the number of cyclic polyene chains. Although the infinite-length ð5; 5Þ and ð9; 0Þ nanotubes have a metallic character, the HOMO–LUMO gaps in the finite-length nanotubes are not zero. The gaps for the armchair series decrease with an oscillating manner having a periodicity of 3, with increasing chain width. In contrast, the gaps for the zigzag series decrease with increasing number of chains. These features are fully
consistent with those obtained at the B3LYP/6-311G* level of theory [14]. Our DFT calculations show a significant correlation between the HOMO–LUMO gaps and the bond deformation patterns in the finite-length nanotubes. In the armchair series, the C20þ60l cylinders have smaller gaps, C40 þ 60l larger gaps, and C60þ60l medium gaps. In C40 þ 60l a single cyclic cis-polyene chain with pattern 1 is located on each edge of the cylinder, whereas in C20þ60l alternating bond-length chains are distributed throughout the cylinder. Since the single alternating bond-length chain interacting with each corannulene cap is involved in C60 and C40 þ 60l , the gaps for C40 þ 60l are close to that for C60 . However, interchain interactions between the cis-polyene chains in C20þ60l should lead to smaller gaps than those for C60 . In C60þ60l , other bond-alternation patterns between C20þ60l and C40 þ 60l are contained, and accordingly the gaps have medium values. In contrast to the armchair series, the geometrical features in the zigzag series are found to be insensitive to the cylinder lengths; thus no oscillatory behavior appears in the HOMO–LUMO gaps for the zigzag series.
4. Conclusions We have investigated the finite-length ð5; 5Þ and ð9; 0Þ nanotubes by DFT calculations at the B3LYP DFT level of theory. In the finite-length armchair C40 þ 20n and zigzag C42 þ 18n nanotubes, certain numbers (n) of cyclic cis- and trans-polyene chains are lined up along the tube axis, respectively. The geometries of the armchair series depend on the number of cyclic cis-polyene chains involved, and the finite-length carbon nanotubes can be classified into three subgroups. In contrast, the zigzag nanotubes, which are insensitive to the chain width, consist of Kekule networks in the cylindrical segment. We have also analyzed the gap between the HOMO and LUMO as a function of the number of cyclic polyene chains and demonstrated a significant relationship between the HOMO–LUMO gaps and the bond deformation patterns in the finite-length nanotubes.
Acknowledgements
Fig. 5. HOMO–LUMO gaps as a function of the number of cyclic polyene chains in the finite-length nanotubes. The armchair C40 þ 20n and zigzag C42 þ 18n nanotubes are represented by square and circle dots, respectively.
This work has been supported by the 21st century COE program at Meijo University and jointly with MEXT (Ministry of Education, Culture, Sports, Science and Technology, Japan) and by the ÔNanotechnology Support ProjectÕ of MEXT. The support from the US office of Naval Research (for S.I. and K.H.), the Japan Society for the Promotion of Science (JSPS), the Murata Science Foundation, and Kyushu University P&P ÔGreen ChemistryÕ (for K.Y.), and the JSPS postdoctoral fellowship (for T.Y.) are gratefully acknowledged.
T. Yumura et al. / Chemical Physics Letters 386 (2004) 38–43
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