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 (|a₁|=|a₂|=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×$$\sqrt{3}$$) for 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,

$$T_{\text{f}}=t_{\text{f}}\times a,\text{\hspace{1em}(1)}$$(1)

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 a₁ and a₂. For instance, when we allocate the two edge carbon atoms to the coordinates (α₁,α₂) and (β₁,β₂) (Fig. 1), we can obtain the length index t_f using the chiral index integers

$$t_{\text{f}}=\frac{\sqrt{3}\left|m({\alpha }_1-{\beta }_1)-n({\alpha }_2-{\beta }_2)\right|}{2\sqrt{n^2+nm+m^2}}.\text{\hspace{1em}(2)}$$(2)

Using this equation and the real value of a, one can obtain the finite length T_f using eq. 1.

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 sp²-carbon atoms within the same region. These measures can show the completeness of the structures of finite SWNT molecules.

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.

Fig. 3

Geometric measures for finite SWNT molecules of [4]CC and [4]CA. Except for (16,0)-[4]CA_3,9, all of the molecules have been synthesized [12–14].

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 sp²-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 C₆₀, has the fundamental structure of (5,5)-SWNT with t_f=1.50, F_b=100 % and F_a=100 % [20].

Fig. 4

Geometric measures for finite SWNT molecules of COP, picotube and cyclophenacene.

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.

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 a₁ and a₂ [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 sp²-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.

Fig. 5

Imaginary ring-flipping dynamics of [6]CNAP.

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 a₁ and a₂ 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].

Fig. 6

Geometric measures for finite SWNT molecules of [n]CPP.