Hypothetical toroidal, cylindrical, and helical analogs of C 60 Chern Chuang, Bih-Yaw Jin * Department of Chemistry, Center of Theoretical Sciences, and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC 1. Introduction In the early nineties, right after the discovery of carbon nanotubes (CNT) [1], Vanderbilt and Tersoff [2] proposed a negative-curvature analog of C 60 , which is basically a triply periodic C 168 structure composed of only hexagons and heptagons. While the pentagons in C 60 are all surrounded by hexagons and the hexagons are surrounded by alternating hexagons and pentagons, the heptagons in C 168 are surrounded by hexagons and the hexagons are surrounded by alternating hexagons and heptagons. The resulting structure for C 168 is like replacing each carbon atom in a diamond lattice with hollowed carbon nanotetrapod (superatom), and the bonding between these superatoms with a suitable nanotube (superbond). Topologically speaking, C 60 is isomorphic to a sphere and C 168 is isomorphic to the D-type triply periodic minimal surface (TPMS) [3–7]. According to the Gauss– Bonnet theorem, each unit cell in C 168 contains exactly 24 heptagons and every carbon atom in this structure belongs to a particular heptagon. This amazing analogy between the two molecular structures raises an immediate question: Is it possible to construct a graphitic structure, which may possess both positive and negative Gaussian curvatures at the same time, such that nonhexagons are completely surrounded by hexagons and hexagons are surrounded alternately by hexagons and nonhexa- gons? In the discussion made by Fowler and Pisanski [8], this is equivalent to asking for Clar type fullerenes with all their Clar rings nonhexagonal, where C 60 and C 168 are both solutions possessing positive and negative curvatures throughout, respectively. In this paper, we provide a positive answer to the question. Three different solutions are suggested where they have the same shapes and topologies of a torus, a cylinder, and a helical tube, respectively. The latter two are singly periodic infinite structures and are derived from the torus case. All three graphitic structures proposed have pentagons and heptagons in equal numbers, and each pair of pentagon and heptagon comes together with four hexagons. In other words, the ratio between the numbers of polygons in these structures are N 5 : N 6 : N 7 ¼ 1 : 4 : 1, where N i is the number of i-gons. Quantum chemical calculations showed that the heat of formation of the toroidal and cylindrical structures are comparable to that of C 60 . The structural properties and general geometric features of the three molecular structures are discussed. 2. Buckytori It is well known that any toroidal carbon nanotube (TCNT) with all faces hexagonal can be formed by bending, and connecting the ends, of a finite straight carbon nanotube. The polyhex TCNT has been of theoretical interest especially on its magnetic response because of its curious topology and geometry [9–23]. It was found that for certain magic numbers of chiral vectors and the number of unit cells the polyhex TCNT may carry extremely large para- magnetic persistent current under an external magnetic field [9]. However, for the polyhex TCNTs to be stable without seriously distorting chemical bondings therein, the number of carbon atoms Journal of Molecular Graphics and Modelling xxx (2009) xxx–xxx ARTICLE INFO Article history: Received 4 May 2009 Received in revised form 17 July 2009 Accepted 22 July 2009 Available online xxx PACS: 85.65.+h Keywords: Toroidal Helical Carbon nanotube C 60 ABSTRACT Toroidal, cylindrical, and helical analogs of C 60 buckyball are theoretically constructed and analyzed. In these structures, pentagons and heptagons are separated compactly by hexagons in analogy to pentagons in C 60 and heptagons in C 168 proposed by Vanderbilt and Tersoff (1992) [2]. Specifically, all nonhexagons therein are surrounded by hexagons and hexagons are surrounded alternatively by hexagons and nonhexagons, i.e. these structures are polyhedra of Clar type with all their Clar rings nonhexagonal. Quantum chemical calculations have been carried out which show that they possess stabilities comparative to that of C 60 . And their structural features are also investigated in detail. Buckled carbon nanotubes deriving from buckytori with periodically varying radii are suggested to be candidacies for the product of coalescing arrays of C 60 . The helicity of the buckyhelices as a function of their characterizing shifting parameters is studied. In the limit of large shifting parameter, the buckyhelices adopt an unusual geometric form that has not been reported in the literature yet. ß 2009 Elsevier Inc.. All rights reserved. * Corresponding author. E-mail address: byjin@ntu.edu.tw (B.-Y. Jin). G Model JMG-5888; No of Pages 6 Please cite this article in press as: C. Chuang, B.-Y. Jin, Hypothetical toroidal, cylindrical, and helical analogs of C 60 , J. Mol. Graph. Model. (2009), doi:10.1016/j.jmgm.2009.07.004 Contents lists available at ScienceDirect Journal of Molecular Graphics and Modelling journal homepage: www.elsevier.com/locate/JMGM 1093-3263/$ – see front matter ß 2009 Elsevier Inc.. All rights reserved. doi:10.1016/j.jmgm.2009.07.004