The ideal nanotube structure can be obtained from a graphene sheet by rolling it up along the straight line connecting two lattice points into a seamless cylinder in such a way that the two points coincide. 11,19,20 The tube is uniquely

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Two different local-density approximations, the Gáspár-Kohn-Sham and the Perdew-Zunger approximations, of the density-functional method have been used to calculate structural and electronic properties of polyethylene systems with several different dihedral angles. For each system, the CC bond lengths and the CCC and HCH bond angles are optimized simultaneously. All the parameters appear to be strongly coupled with torsional freedom and vary with the change in dihedral angle in a pattern similar to that of the total energy. The total energy has an absolute minimum for the planar zigzag conformation but a distinct local minimum for the quasistable helical conformation. Another minimum occurs in the energy curve close to this gauche minimum. The calculated valence and conduction bands are discussed and compared with other theoretical calculations and experiment. ͓S0163-1829͑96͒02239-4͔