  The $C_{60}$ molecule and its fragments are studied using Configuration
Interaction (CI) and Variational Monte Carlo (VMC) techniques, within the
Hubbard model. Using benzene as a test case, we compare the results of the
approximate calculations with exact calculations. The fragments of $C_{60}$
studied are pyracylene, fluoranthene and corannulene. The energies, bond
orders, spin-spin and charge-correlation functions of these systems are
obtained for various values of the Hubbard parameter, $U$. The analysis of bond
orders and correlation functions of these individual molecules allow us to
visualise pyracylene as a naphthalene unit with two ethylenic moieties and
fluoranthene as weakly bridged benzene and naphthalene units. Corannulene is
the largest fragment of $C_{60}$ that we have studied. The hexagon-hexagon(h-h)
bond orders are slightly larger than those of the hexagon-pentagon bonds(h-p),
a feature also found in other fragments. We also find bonds between two
co-ordinated carbon sites to be stronger than bonds involving three coordinated
carbon sites. In $C_{60}$, the h-h bonds are stronger than in corannulene and
the h-p bonds weaker than in corannulene for all correlation strengths.
Introducing bond alternation in the buckyball enhances this difference.
