  We have shown how the interplay between the broken symmetry associated with
the finite single-wall nanotubes (SWNT) and the boundary conditions affects the
electronic properties of SWNTs in a profound way. For finite SWNTs (p,q)
characterized by p=k+l,q=k-l,p-q=2l,l=0,1...,k, and k=1,2..., we found that the
band gaps of finite SWNTs belonging to a certain k exhibit similar well-defined
oscillating patterns but with diminishing amplitudes from the armchair (l=0) to
the zigzag (l=k) SWNTs. These profound changes hold intiguing implications in
the potential utilization of these finite NTs as the basic component of
molecular scale devices.
