ARTICLES Cut-and-Unfold Approach to Fullerene Enumeration A. M. Livshits* and Yu. E. Lozovik Institute of Spectroscopy, Russian Academy of Sciences, 142190, Troitsk, Moscow Region, Russia Received November 10, 2003 A simple and effective approach for the enumeration of fullerene structures is proposed. The method combines the formalism of a fullerene graph cut-and-unfold onto a planar triangular lattice and a topological description of closed quasi-2D clusters. A tabulation of possible fullerene graphs C n is given for the number of atoms 20 e n e 150. 1. INTRODUCTION Fullerenes, nanotubes, “onions”, and other carbon nano- structures (see, e.g., refs 1,2) are currently of great interest. In particular, they are considered as a possible basis for nanoelectronic technology; 3,4 they are applied in molecular design and in the creation of new ultrastrong or supercon- ductive materials. 5 Fullerenes are demanded in medicine, 6 etc. Fullerenes C n are spherical carbon clusters with (almost) sp 2 hybridization, with σ-bonds forming only hexagonal and pentagonal rings. One can represent a fullerene by a cubic graph, or by a convex polyhedron, taking atoms as vertices, and σ-bonds as edges. Taking advantage of the Euler theorem connecting the number of faces (f), the number of vertices (V), and the number of edges (e) of a convex polyhedron one can perceive that an arbitrary fullerene always contains exactly 12 pentagonal rings (faces of the polyhedron). The number of possible fullerene structures C n quickly grows with n. While there is only one fullerene C 60 7 and only one fullerene C 70 that satisfy the isolated pentagon rule (IPR), at n ) 100 already 450 different IPR fullerenes C n can exist. 8 Hundreds of thousands of IPR fullerenes are topologically possible at n ≈ 150 (see below). Enumeration of possible fullerene structures C n is an important theoretical and applied problem. The ring-spiral algorithm, 9 top-down approach, 10 and other algorithms of numeric generation of fullerene structures 11-13 are known. Some algorithms are designed for fullerenes of high symmetry only. 14-17 Comparison of the results obtained by different approaches is mandatory, because typically algorithms cannot guarantee that all of the possible fullerene structures have been generated. 18 Methods in refs 11 and 15-17 use the idea of “cutting and unfolding” a fullerene onto a planar hexagonal or triangular lattice. In the case of a triangular lattice, the centers of pentagonal and hexagonal faces of a fullerene are projected onto the lattice points. According to the method in ref 11, an arbitrary fullerene C n can be cut and unfolded onto a planar triangular lattice. The resulting net diagram, which is a closed polygon with 22 corners, can be wholly described by 11 vectors of triangular lattice. The area of the net diagram S(n) is equal to  3/4‚n, where n is the number of atoms in the fullerene molecule. The reverse procedure allows one to construct a fullerene, starting from its net diagram. To obtain isomers of C n , one generates all of the net diagrams with the area S(n), and then, on the basis of these net diagrams, one constructs the fullerenes. The following problem arises. A fullerene can be cut and unfolded onto a planar lattice in many ways. Vice versa, a particular fullerene can be constructed from a number of different net diagrams. So, one needs a method to determine the constructed isomorphic and nonisomorphic fullerene structures. For example, one can generate approximate atomic coordinates 11 of the fullerene, corresponding to a given net diagram. Using the Hu ¨ ckel molecular orbital (HMO) theory, one can then calculate various characteristics of the structure, * Corresponding author phone: 8(095)339-4834; e-mail: livshits@ isan.troitsk.ru. f +V- e ) 2 (1) Figure 1. Net diagram of the fullerene (D 5h )C 70 . Figure 2. The structure of a cap segment. 1517 J. Chem. Inf. Comput. Sci. 2004, 44, 1517-1520 10.1021/ci0342573 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/09/2004