  Using density-functional theory calculations, we investigate the addition
energy (AE) of quantum dots formed of fullerenes or closed single-wall carbon
nanotubes. We focus on the connection between symmetry and oscillations in the
AE spectrum. In the highly symmetric fullerenes the oscillation period is large
because of the large level degeneracy and Hund's rule. For long nanotubes, the
AE oscillation is fourfold. Adding defects destroys the spatial symmetry of the
tubes, leaving only spin degeneracy; correspondingly, the fourfold behavior is
destroyed, leaving an even/odd behavior which is quite robust. We use our
symmetry results to explain recent experiments. [Phys. Rev. Lett. 91, 116803
(2003).]
