  The electronic structure of finite-length armchair carbon nanotubes has been
studied using several ab-initio and semi-empirical quantum computational
techniques. The additional confinement of the electrons along the tube axis
leads to the opening of a band-gap in short armchair tubes. The value of the
band-gap decreases with increasing tube length, however, the decrease is not
monotonic but shows a well defined oscillation in short tubes. This oscillation
can be explained in terms of periodic changes in the bonding characteristics of
the HOMO and LUMO orbitals of the tubes. Finite size graphene sheets are also
found to have a finite band-gap, but no clear oscillation is observed. As the
length of the tube increases the density of states (DOS) spectrum evolves from
that characteristic of a zero-dimensional (0-D) system to that characteristic
of a delocalized one-dimensional (1-D) system. This transformation appears to
be complete already for tubes 5-10 nm long. The chemical stability of the
nanotubes, expressed by the binding energy of a carbon atom, increases in a
similar manner.
