## Introduction: geometric description of carbon nanotubes

A paper titled “Helical microtubules of graphitic carbon” reported the milestone discovery of carbon nanotubes (CNTs) [1]. This paper was also remarkable in that it ingeniously suggested the presence of helicity arising from the helical arrangement of sp^{2}-carbon networks, which hinted at the wealth of the structural variations of CNTs. Six months later, a proposal for the geometric description (or nomenclature) of single-wall CNTs (SWNTs) was reported by Saito, Dresselhaus, and Dresselhaus (the descriptors are referred to herein as SDD descriptors) [2], which provided an important basis for further development and discussion [3]. Among the geometric SDD descriptors, the most elemental vectors are the unit vectors *a*_{1} and *a*_{2}, which define one hexagonal unit on the carbon sheet, as shown in Fig. 1 [4]. The chiral vector *C*_{h} specifies the SWNT structures, except for their length, using a combination of unit vectors *n**a*_{1}+*m**a*_{2} where *n* and *m* are integers such that 0≤|*m*|≤*n*. The popular geometric index known as the chiral index (*n,m*) shows the allocated coordinates of the chiral vector on an oblique coordinate system of unit vectors [5]. Three SWNT classifications, i.e., armchair, helical, and zigzag, that convey the shape of the cross sections can also be identified as SWNTs with *n*=*m, n*≠*m*, and *m*=0, respectively. The recent successful isolation of optically active SWNTs has made it necessary to have an additional descriptor for handedness, and of the few proposals reported for the identification of handedness, the one proposed by Komatsu [6] conveys handedness by using the basic IUPAC terminology of helicity with (*P*)- and (*M*)-descriptors in combination with the unit vector system [7, 8].

The other important but less popular descriptor for SWNT structures is the translational vector ** T**. The translational vector is defined to describe the lattice of a carbon sheet in the axis direction. Oriented perpendicular to the chiral vector

*C*_{h}such that

**=**

*T**t*

_{1}

*a*_{1}+

*t*

_{2}

*a*_{2}, the translational vector expresses the minimum periodicity of an SWNT structure along the tubular axis. The coordinates of the translational vector, for instance, is (

*t*

_{1},

*t*

_{2})=(7,–8) for (12,8)-SWNT (Fig. 1). This translational vector for the description of periodicity is the only geometric measure along the tubular axis, and there is no geometric descriptor available for the length of a finite SWNT molecule. In this paper, we wish to propose additional geometric descriptors to describe and discuss the geometric structures of finite SWNT molecules with discrete lengths. We hope that the new geometric measures will help facilitate the investigation and discussion of finite SWNT as a molecular entity. We first consider our own examples before extending the concept to other examples.

## Background: emergence of finite carbon nanotube “molecules“

The curious lack of geometric descriptors for the finite length of SWNT shows that the CNT has not been a molecular entity defined as “any constitutionally distinct molecule identifiable as a separately distinguishable entity” [9]. Despite many studies of finite SWNT congeners based on top-down cleaving methods [10], the cleaved SWNT congeners have not been discrete and identifiable molecules but are instead classified as a chemical species consisting of “sets or ensembles of molecular entities” [11]. Therefore, the translational vector and other measures describing the length of finite SWNT have not been fully exploited.

Recently, a new class of macrocyclic compounds (i.e., molecular entities) has emerged. These compounds possess fundamental sp^{2}-carbon networks of SWNT. The macrocycles, especially those possessing persistent tubular shapes, allow the unequivocal placement of geometric vectors such as the unit vectors *a*_{1} and *a*_{2} and, consequently, the chiral and translational vectors *C*_{h} and ** T**. As a result, the geometric treatment of the macrocycles as finite models/segments/molecules of SWNT becomes possible. This, in turn, allows for their structural identification by using SDD descriptors. For instance, stereoisomers (atropisomers) of [4]cyclo-2,8-chrysenylenes ([4]CC

_{2,8}, Fig. 2) can be identified as (12,8)-, (11,9)-, and (10,10)-isomers with their chiral index identifiers [12], and the constitutional isomers, [4]cyclo-3,9-chrysenylenes ([4]CC

_{3,9}), can be identified as (16,0)-isomers [13]. The isomers with helical handedness can also be identified by Komatsu’s (

*P*)- and (

*M*)-descriptors [6]. Using the chiral index and handedness descriptors helps clarify the isomeric structural differences between tubular macrocyclic sp

^{2}-carbon networks with identical chemical compositions.

## Extension of geometric description to finite nanotube molecules

In our recent examples, we further introduced another series of tubular macrocycles, the [4]cyclo-2,8-anthanthrenylenes ([4]CA_{2,8}, Fig. 2) [14]. Although their stereoisomers can also be discriminated by chiral index and (*P*)/(*M*) handedness, a structural comparison between [4]CC_{2,8} and [4]CA_{2,8} was not possible despite the common tubular skeletons shared by these molecules. The only geometric difference between them is in their axial “length” by addition of four sp^{2}-carbon atoms at the edge of each arylene unit in [4]CA, and there is a need for appropriate descriptors. We propose descriptors called length index (*t*_{f}), bond-filling index (F_{b}), and atom-filling index (F_{a}) for the geometric description of finite SWNT molecules.

### Length index

The geometric measures for the size of SWNT, i.e., circumferential length (|*C*_{h}|=*L*), diameter (*d*_{t}), and unit length (|** T**|=

*T*), are expressed by equations based on combinations of the chiral index integers,

*n*and

*m*, and a lattice constant (|

*a*_{1}|=|

*a*_{2}|=

*a*). The lattice constant equals the length of a short diagonal line in a hexagon [3, 15, 16]. When necessary, one can convert all of the geometric measures to real chemical values by incorporating the real values of the lattice constant

*a*. Saito, Dresselhaus, and Dresselhaus used a value of 2.49 Å (=1.44×

*a*, which they determined based on the 1.44 Å C–C bond length in SWNT [3].

We propose the creation of new geometric measures for length that are also based on the SDD descriptors. As with other geometric measures, we propose expressing the finite length (*T*_{f}) using the lattice constant, *a*,

where *t*_{f} is the length index, which is an index to be used for structural comparisons (*vide infra*). In this form, the length is expressed as a multiple of the lattice constant, the unit length of a single hexagon. To obtain *t*_{f}, we allocate the edge atoms of a finite SWNT molecule to their coordinates in an oblique coordinate system defined by the unit vectors *a*_{1} and *a*_{2}. For instance, when we allocate the two edge carbon atoms to the coordinates (α_{1},α_{2}) and (β_{1},β_{2}) (Fig. 1), we can obtain the length index *t*_{f} using the chiral index integers

Using this equation and the real value of *a*, one can obtain the finite length *T*_{f} using eq. 1.

### Filling index

Considering the growing library of finite SWNT molecules, we propose two additional indexes: the bond-filling index (F_{b}) and the atom-filling index (F_{a}). These indexes will also be convenient for use in structural comparisons. The bond-filling index is defined as the percentage of connected chemical bonds within the finite length *T*_{f}, and the atom-filling index is the percentage of existing sp^{2}-carbon atoms within the same region. These measures can show the completeness of the structures of finite SWNT molecules.

### Typical examples

We demonstrate the use of the new descriptors by applying them to real finite SWNT molecules. Figure 3 shows a summary of the geometric measures for finite SWNT molecules containing chrysene and anthanthrene as the arylene base units. We first examine the [4]CC molecules with the new measures. The length indexes of the [4]CC isomers range from 1.50 to 2.08 and readily allow us to compare the finite lengths of the isomers quantitatively. For instance, the finite length of the longest isomer, (11,9)-[4]CC_{2,8} (*t*_{f}=2.08), is 1.39 times greater than that of the shortest isomer, (10,10)_{ABAB}-[4]CC_{2,8} (*t*_{f}=1.50). Filling indexes are also of considerable use for abstracting the important structural information. The values F_{b}=100 % and F_{a}=100 % for (12,8)-[4]CC_{2,8} show that the molecule possesses a complete set of chemical bonds, which are necessary for finite (12,8)-SWNT molecules of *t*_{f}=1.66, whereas the values, F_{b}=68 % and F_{a}=72 %, for (10,10)_{ABAB}-[4]CC_{2,8}, provide the quantitative measures of filling/missing bonds and atoms for finite (10,10)-SWNT molecules with *t*_{f}=2.00.

A structural comparison between [4]CC and [4]CA can easily be performed using the new measures. A few examples are described. The highest value,* t*_{f}=3.00 for (10,10)_{AABB}-[4]CA_{2,8}, corresponds to the longest finite SWNT molecule among the congeners, and the comparison with *t*_{f}=2.00 for (10,10)_{AABB}-[4]CC_{2,8} shows that, with the presence of four sp^{2}-carbon atoms added on the edge of one arylene unit, 1.5-fold lengthening has been achieved.

We apply the new geometric measures to other examples. To the best of our knowledge, the oldest applicable example is [*n*]cyclo-*ortho*-phenylene ([*n*]COP), which was synthesized long before the discovery of CNT [17, 18]. The crystallographic analysis of isomers with *n*=4 and 6 revealed the presence of alternate orientations of the phenylene units (Fig. 4), and these tubular structures can be identified with the geometric descriptors. [4]COP or [6]COP have the fundamental structures of (2,2)- or (3,3)-SWNT structures, respectively, with *t*_{f}=2.50, F_{b}=88 % and F_{a}=100 %. Note that the other conformational isomers with non-tubular structures cannot be identified with the same descriptors (*vide infra*). A more recent example of “picotube ([4]cyclo-9,10-anthracenylene)” by Herges has the fundamental structure of (4,4)-SWNT with *t*_{f}=3.00, F_{b}=90 % and F_{a}=100 % [19]. Nakamura’s cyclophenacene molecule, made by the top-down synthesis from C_{60}, has the fundamental structure of (5,5)-SWNT with *t*_{f}=1.50, F_{b}=100 % and F_{a}=100 % [20].

### Web-based applet

The new geometric measures should be useful for other researchers who synthesize their own finite SWNT molecules and, more importantly, should provide fundamental parameters for structural discussion and comparison of the many finite SWNT molecules to be supplied in the near future. To serve this demand, we developed a web-based applet that can be used by researchers in the field. The URL is http://www.orgchem2.chem.tohoku.ac.jp/finite/. The applet is easy to use, and we briefly describe the procedure. In step 1, we define the chiral vector by typing in the chiral index (*n,m*). At this stage, we roughly specify the length of CNT to be displayed in the next stage by entering the repeating numbers of the axial lattice. In step 2, we allocate the coordinates of the edge atoms. Although the coordinates can be directly entered in the input forms, the coordinates can also be allocated simply by clicking the edge atoms. In step 3, we define the structure of the finite SWNT molecule by defining the chemical bonds that are present in the finite structure. The on-screen structure can be processed by clicking the bonds to be removed or added. Through these three steps, we can easily obtain the indexes of *t*_{f}, F_{a} and F_{b}. The atoms and bonds on the left and right edges are identical, because these two vertical lines at the edges are joint lines of the carbon sheet to be rolled in the tubular structure.

### Considerations for the applications of geometric measures

For the geometric evaluation of finite SWNT molecules, we think that careful consideration is necessary before use. The most fundamental point is the persistency of the belt or tubular shapes. All of the descriptors for SWNT, including the SDD descriptors and our newly derived descriptors, start from the allocation of the unit vectors *a*_{1} and *a*_{2} [3]. Therefore, by definition, the descriptors cannot be applied to molecules with fluctuating structures or molecules without proofs of tubular structures in which the unit vectors cannot be defined unequivocally. We take our own molecule as an inappropriate example. [6]Cyclo-2,7-naphthylene ([6]CNAP) comprises 60 sp^{2}-carbon atoms within macrocyclic networks [21]. The dynamics of the macrocyclic systems have not been investigated, and one might postulate the existence of ring-flipping dynamics that could occur through an intermediate tubular structure (Fig. 5). Assuming the tubular structure, one might further assign its geometric structural parameters, such as the chiral index or *t*_{f}. In the absence of any evidences for tubular structure, however, this procedure is not at all appropriate.

An interesting special case is that of [*n*]cyclo-*para*-phenylenes ([*n*]CPP) [22–25]. The static tubular structures of [*n*]CPP, for instance, from X-ray crystallographic analysis are surely captured by the geometric descriptors of SWNT. Crystal tubular structures of [*n*]CPP (*n*=6, 8, 9, 10, and 12) can thus be identified as finite (*n,n*)-SWNT structures with *t*_{f}=1.00, F_{b}=100 % and F_{a}=100 % (Fig. 6) [22c,e, 23c,d]. However, attention should be paid to the use of descriptors for the dynamic solution-phase structures or for the structures without any steric information. Relevant studies suggested that the molecules possess fluctuating structures that involve the rapid rotation of phenylene rings [22e, 26]. Although this fluctuation does not seemingly affect the chiral index assignment, the fluctuation forces the unit vectors *a*_{1} and *a*_{2} to rotate and, therefore, inherently obscures the base carbonaceous sheet for geometric descriptors. This careful consideration is especially important for [*n*]CPP congeners with structural variations that could alter the geometric expressions among possible dynamic structures [22d,g, 23e,f, 25].

## Conclusion

In this paper, we proposed new descriptors for finite SWNT molecules. Based on the original SDD descriptors, the new descriptors provide geometric measures with the length, bond- and atom-filling indexes for finite SWNT molecules. We hope that, along with the web-based applet for this system, many researchers in this emerging field will find the descriptors useful for the development and discussion of finite SWNT molecules.

This study is supported partly by KAKENHI (24241036, 25810018). T.M. thanks JSPS for a predoctoral fellowship.

## References and notes

- [1]
S. Iijima.

*Nature***354**, 56 (1991). - [2]
R. Saito, M. Fujita, G. Dresselhaus, M. S. Dresselhaus.

*Appl. Phys. Lett.***60**, 2204 (1992). - [3]
R. Saito, G. Dresselhaus, M. S. Dresselhaus.

*Physical Properties of Carbon Nanotubes*, Imperial College Press, London (1998). - [4]
In the original proposal of Saito, Dresselhaus, and Dresselhaus, the “helical” type is referred to as the “chiral” type. However, SWNT molecules in other categories can also be “chiral”, as we demonstrated the presence of “chiral” and “zigzag” SWNT molecules in our examples (see ref. [13]). Considering these facts, we think it appropriate to describe (

*n*,*m*)-SWNT as “helical” type*(**n*≠*m**)*. - [5]
Note that the terms chiral or chirality, which are used for SWNT description, are not consistent with the definition provided in the IUPAC terminologies.

- [6]
N. Komatsu.

*Jpn. J. Appl. Phys*.**49**, 02BC01 (2010). - [7]
(a) R. S. Cahn, C. Ingold, V. Prelog.

*Angew. Chem., Int. Ed. Engl.***5**, 385 (1966); (b) G. P. Moss.*Pure Appl. Chem*.**68**, 2193 (1996). - [8]
E. L. Eliel, S. H. Wilen, L. N. Mander.

*Stereochemistry of Organic Compounds*, John Wiley, Hoboken (1994). - [9]
IUPAC.

*Compendium of Chemical Terminology*, 2nd ed. (the “Gold Book”). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://dx.doi.org/10.1351/goldbook (2006–) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. - [10]
J. Liu, A. G. Rinzler, H. J. Dai, J. H. Hafner, R. K. Bradley, P. L. Boul, A. Lu, T. Iverson, K. Shelimov, C. B. Huffman, F. Rodriguez-Macias, Y. S. Shon, T. R. Lee, D. T. Colbert, R. E. Smalley.

*Science***280**, 1253 (1998). - [11]
Only one example clarified the fundamental chemical composition (i.e., molecular weight) of SWNT agglomerates: H. Isobe, T. Tanaka, R. Maeda, E. Noiri, N. Solin, M. Yudasaka, S. Iijima, E. Nakamura.

*Angew. Chem., Int. Ed.***45**, 6676 (2006). - [12]
(a) S. Hitosugi, W. Nakanishi, T. Yamasaki, H. Isobe.

*Nat. Commun*. 2, (2011). DOI: 10.1038/ncomms1505; (b) S. Hitosugi, W. Nakanishi, H. Isobe.*Chem. Asian J.***7**, 1550 (2012). - [13]
S. Hitosugi, T. Yamasaki, H. Isobe.

*J. Am. Chem. Soc.***134**, 12442 (2012). - [14]
T. Matsuno, S. Kamata, S. Hitosugi, H. Isobe.

*Chem. Sci.***4**, 3179 (2013). - [15]
L = a n 2 + m 2 + n m ; d t = ( a n 2 + m 2 + n m ) / π . \(L\, = \,a\sqrt {{n^2}\, + \,{m^2}\, + \,nm} ;\,\,\,{d_t}\, = \,(a\sqrt {{n^2}\, + \,{m^2}\, + \,nm} )/\pi .\)

- [16]
By defining

*d*as the greatest common divisor of*n*and*m, T*can be expressed by the following equations: When*n*–*m*is not a multiple of 3*d*, T = a 3 ( n 2 + m 2 + n m ) / d ; \(T\, = \,a\sqrt {3({n^2}\, + \,{m^2}\, + \,nm)} /d;\) when*n*–*m*is a multiple of 3*d*, T = a 3 ( n 2 + m 2 + n m ) / 3 d . \(T\, = \,a\sqrt {3({n^2}\, + \,{m^2}\, + \,nm)} /3d.\) - [17]
We used a nomenclature that is consistent with recent popular examples of [

*n*]cyclo-para-phenylene. - [18]
(a) W. S. Rapson, R. G. Shuttleworth, J. N. van Niekerk.

*J. Chem. Soc.*326 (1943); (b) L. Friedman, D. F. Lindow.*J. Am. Chem. Soc.***90**, 2324 (1968); (c) H. N. C. Wong, T. C. W. Mak.*J. Chem. Soc., Chem. Commun*. 543 (1982); (d) H. N. C. Wong, T. -Y. Luh, T. C. W. Mak.*Acta Crystallogr., Sect. C***40**, 1721 (1984). - [19]
S. Kammermeier, P. G. Jones, R. Herges.

*Angew. Chem., Int. Ed.***35**, 2669 (1996). - [20]
E. Nakamura, K. Tahara, Y. Matsuo, M. Sawamura.

*J. Am. Chem. Soc.***125**, 2834 (2003). - [21]
(a) W. Nakanishi, T. Yoshioka, H. Taka, J. Y. Xue, H. Kita, H. Isobe.

*Angew. Chem., Int. Ed.***50**, 5323 (2011); (b) W. Nakanishi, J. Y. Xue, T. Yoshioka, H. Isobe.*Acta Crystallogr., Sect. E***67**, o1762 (2011); (c) J. Y. Xue, W. Nakanishi, D. Tanimoto, D. Hara, Y. Nakamura, H. Isobe.*Tetrahedron Lett.***54**, 4963 (2013). - [22]
(a) R. Jasti, J. Bhattacharjee, J. B. Neaton, C. R. Bertozzi.

*J. Am. Chem. Soc*.**130**, 17646 (2008); (b) T. J. Sisto, M. R. Golder, E. S. Hirst, R. Jasti.*J. Am. Chem. Soc.***133**, 15800 (2011); (c) J. Xia, R. Jasti.*Angew. Chem., Int. Ed.***51**, 2474 (2012); (d) T. J. Sisto, X. Tian, R. Jasti.*J. Org. Chem*.**77**, 5857 (2012); (e) J. Xia, J. W. Bacon, R. Jasti.*Chem. Sci.***3**, 3018 (2012); (f) E. R. Darzi, T. J. Sisto, R. Jasti.*J. Org. Chem*.**77**, 6624 (2012); (g) J. Xia, M. R. Golder, M. E. Foster, B. M. Wong, R. Jasti.*J. Am. Chem Soc*.**134**, 19709 (2012). - [23]
(a) H. Takaba, H. Omachi, Y. Yamamoto, J. Bouffard, K. Itami.

*Angew. Chem., Int. Ed.***48**, 6112 (2009); (b) H. Omachi, S. Matsuura, Y. Segawa, K. Itami.*Angew. Chem., Int. Ed*.**49**, 10202 (2010); (c) Y. Segawa, S. Miyamoto, H. Omachi, S. Matsuura, P. Šenel, T. Sasamori, N. Tokitoh, K. Itami.*Angew. Chem., Int. Ed*.**50**, 3244 (2011); (d) Y. Segawa, P. Šenel, S. Matsuura, H. Omachi, K. Itami.*Chem. Lett*.**40**, 423 (2011); (e) H. Omachi, Y. Segawa, K, Itami.*Org. Lett*.**13**, 2480 (2011); (f) A. Yagi, Y. Segawa, K. Itami.*J. Am. Chem. Soc*.**134**, 2962 (2012); (g) K. Matsui, Y. Segawa, K. Itami.*Org. Lett*.**14**, 1888 (2012); (h) Y. Ishii, Y. Nakanishi, H. Omachi, S. Matsuura, K. Matsui, H. Shinohara, Y. Segawa, K. Itami.*Chem. Sci*.**3**, 2340 (2012). - [24]
(a) S. Yamago, Y. Watanabe, T. Iwamoto.

*Angew. Chem., Int. Ed.***49**, 757 (2010); (b) T. Iwamoto, Y. Watanabe, Y. Sakamoto, T. Suzuki, S. Yamago.*J. Am. Chem. Soc*.**133**, 8354 (2011); (c) E. Kayahara, Y. Sakamoto, T. Suzuki, S. Yamago.*Org. Lett*.**14**, 3284 (2012); (d) E. Kayahara, T. Iwamoto, T. Suzuki, S. Yamago.*Chem. Lett*.**42**, 621 (2013). - [25]
T. Nishiuchi, X. L. Feng, V. Enkelmann, M. Wagner, K. Müllen.

*Chem.—Eur. J.***18**, 16621 (2012). - [26]
Y. Segawa, H. Omachi, K. Itami.

*Org. Lett.***12**, 2262 (2010).