Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes Chern Chuang, Yuan-Chia Fan, and Bih-Yaw Jin* Department of Chemistry and Center of Theoretical Sciences, National Taiwan University, Taipei, Taiwan, ROC Received October 26, 2008 In this study, we develop a generalized classification scheme for toroidal (TCNT) and helical carbon nanotubes (HCNT) containing both pentagons and heptagons simultaneously. We show that a particular class of TCNTs with n-fold rotational symmetry and well-defined latitude coordinates can be uniquely characterized by a set of four indices, and each of the indices can be linked to the relative arrangement of pentagons and heptagons in the corresponding torus. Chiral isomers or the corresponding helical derivatives, HCNTs, can also be readily derived either by introducing a chiral vector or dissecting a distorted TCNT through certain longitude. This generalized scheme can generate the whole family of fullerenes with genus one, ranging from giant TCNTs down to small ones containing only a few hundred atoms. To the best of our knowledge, almost all of the construction methods for TCNTs in the literature belong to special cases of our generalized classification scheme. INTRODUCTION Since the discovery of fullerene, C 60 , by Kroto et al., 1 followed by the discovery of carbon nanotube by Iijima, 2 a great quantity of studies of the structural and physical properties of this new kind of material was made. 3 In particular, sp 2 carbon structures containing negative curva- tures with nonhexagonal defects were proposed and then observed. 4-9 Among them, toroidal carbon allotropes 11 are of special physical interests due to the potential existence of peculiar persistent currents and giant paramagnetic susceptibility in these systems. 12-19 Although experimental observation of small tori (down to 100 atoms per torus) is not yet reported, theoretical studies showed that these tori may exhibit lower cohesive energy than fullerenes and are considered as thermodynamically stable compounds. 20-26 Systematic enumeration and construction of TCNTs as a function of carbon atoms are complicated graph-theoretical problems. Several groups had developed distinct strategies for the construction of TCNTs, which we only briefly summarize. The first systematic method was proposed by Dunlap, 27 in which he discovered the bending angle by joining two carbon nanotubes (CNT) of orthogonal chiral vectors is about 30 degrees from the original tube axis. Repeatedly connecting 6 pairs of zigzag and armchair CNTs can produce a D 6h torus. Itoh and Ihara’s group started from a pre-existing smaller D nh or D nd torus. 22-25 By inserting appropriate graphitic stripes into it, a series of elongated tori is obtained. Berger and Avron developed an ingenious and tricky systematic construction rule for tiling of isosceles right- angle triangles on a rectangular plane, and such tiling can be mapped to a Clifford torus in 3-D. Given a set of 4 numbers, (m, g 1 , g m , z) in their notation, one can specify a D nd torus. 20,21 Fowler et al. discovered a simple method to generate a chiral D n isomer from a D nd torus by the Stone- Wales transformation. 28,29 By choosing different positions and shapes of super cells, La ´szlo ´ et al. 37 arrived at D nd - symmetric TCNTs, and, if the edges of the super cell are chosen to be intersecting the unit cell edge vectors at suitable angles, then HCNTs can be formed. In the most up-to-date classification of polygonal TCNTs of D 6h symmetry, Tamura et al. have found that this kind of TCNTs may possess interesting magnetic properties. 12 Here, in this article, we show that the numerous TCNT construction and classification methods mentioned (with the exception of Dunlap’s), which seem to have different features or are even totally uncorrelated, can be unified in a much more generalized scheme. Starting parent structures are those TCNTs that have well-defined latitudes, i.e. they are inter- connected polyacetylene chains if viewed along certain directions with rotational symmetry. Based on these observa- tions, we have developed a general classification and systematic construction scheme for toroidal molecules that includes, to our knowledge, almost every case in the literature. Starting from the cut-and-fold idea of Tamura et al., we further define two different structure isomerization processes that finally lead to the various stable TCNTs discussed in the literature. The correspondences of our notation to previous studies are also given in this paper. Furthermore, the HCNTs as proposed by two different groups, Akagi et al. 30 and La ´szlo ´ et al., 37 are shown to be special cases of the current scheme. The helicities of these HCNTs can be derived directly from the parameters given in the isomerization procedure. BASIC CONSTRUCTION SCHEME It is well-known that inserting nonhexagons into a graphene sheet in general introduces nonzero Gaussian curvature. 9,10 Because an sp 2 carbon possesses a bond angle of approximately 120 degrees, for stability concerns, we allow only pentagons and heptagons in most parts of the current article. Polygonization of a certain surface must satisfy Euler’s formula,  ) V - E + F, where V, E, and F are the number of vertices, edges, and faces of the polygoniza- tion, respectively, and  is the Euler characteristic of the * Corresponding author e-mail: byjin@ntu.edu.tw. J. Chem. Inf. Model. 2009, 49, 361–368 361 10.1021/ci800395r CCC: $40.75  2009 American Chemical Society Published on Web 01/27/2009