Size-dependent HOMO-LUMO gap oscillation of carbon nanotube with a finite length

The electronic structure of carbon nanotube(5, 5) with a finite length, is studied on the basis of Hückel molecular orbital method. HOMO-LUMO gap oscillates and decreases as the length l of tube increases, where l is the number of cis-polyene rings of the circumference of carbon nanotube. For tube(5,5, l) without a cap the gap vanishes at l = 3n + 1, where n is a positive integer. On the other hand, the minimum appears at l = 3n for capped tube(5,5, l). Therefore the gap oscillates with the periodicity of 3 for either capped one or one without a cap. Such behaviors are similar to the one-dimensional graphite systems starting from polyphenanthrene.