Topological coordinates for nanotubes Istv an L aszl  o * Department of Theoretical Physics, Institute of Physics and Center for Applied Mathematics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary Available online 3 February 2004 Abstract The topological coordinate method is a simple and effective approach for generating good initial coordinates for fullerene and nanotube carbon structures in molecular mechanics calculations. In this method some special eigenfunctions, the bi-lobal eigen- functions of the H€ uckel Hamiltonian, or the adjacency matrix are used. It is based on a special connection between the electronic and geometric structure of fullerenes and nanotubes. We have found that the most efficient nanotube initial coordinates can be obtained with the four bi-lobal eigenvector method. The three bi-lobal eigenvector method gave relative good initial coordinates only if the two ends of the tube were closed. In both cases the scaling factors based on the Schr€ odinger equation of a particle in a rectangular box gave the best result. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: A. Carbon nanotubes; A. Fullerene 1. Introduction The electronic structure of a polyhex single-walled carbon nanotube is governed by the diameter and chi- rality [1–5]. If the tube contains pentagonal and hep- tagonal faces too, its physical behavior depends on the topological properties as well [6–27], and the final geo- metrical structure can be determined only by some molecular mechanics calculation. For the relaxation process we need some good initial coordinates and various methods can be used in generating the input positions of the carbon atoms. One of them is the NiceGraph algorithm [20] and the other is the topo- logical coordinate method [21,23]. In the NiceGraph algorithm a spring-embedder routine is used for opti- mizing the 3D drawing, and the topological coordinate method is based on the bi-lobal eigenvectors of the adjacency matrix of the graph. Here we shall study the application of the topological coordinate method for nanotubes. In Ref. [25] we pre- sented a version where the nanotube coordinates are obtained from those of the corresponding torus. Why do not we use algorithm developed for the fullerenes? What are the influences of various scaling factors? These questions will be examined in this paper. 2. The topological coordinate method Let A be the adjacency matrix with elements A ij ¼ 1, if atoms i and j are adjacent and A ij ¼ 0 otherwise. From this definition follows that H ¼A, where H is the H€ uckel Hamiltonian matrix with a ¼ 0 and b ¼1, where a is the diagonal matrix element and b is the non- zero non-diagonal matrix element of H. It is assumed further that a 1 > a 2 P a 3 P  P a n ; ð1Þ if a k is the kth eigenvalue of A and c k is the corre- sponding eigenvector. First we define the bi-lobal eigenvectors [21,22]. Vectors having this bi-lobal property can be identified by the graph-disconnection test: for a candidate vector, color all vertices bearing positive coefficients black, all bearing negative coefficients white, and all bearing a zero coefficient gray; now delete all gray vertices, all edges incident on gray vertices, and all edges connecting a black to a white vertex; if the graph now consists of exactly two connected components, one of black and one of white vertices, then the eigenvector is bi-lobal type [21,22,24]. * Tel.: +36-1-463-2146; fax: +36-1-463-3567. E-mail address: laszlo@feynman.phy.bme.hu (I. L aszl o). 0008-6223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2003.12.022 Carbon 42 (2004) 983–986 www.elsevier.com/locate/carbon