  We develop a new $\it {symmetrized}$ version of the Projector Quantum Monte
Carlo (SPQMC) method which preserves the symmetries of the system by
simultaneously sampling all symmetry related Ising configurations at each MC
step and use it to study the effect of electron correlations on a single
$C_{60}$ molecule and its structural motifs, within the Hubbard model. The
SPQMC method allows more accurate estimates of correlation functions as seen
from calculations on small systems. We study the ground state properties of a
single fullerene molecule, with and without bond alternation. We construct
projections of the bond orders and spin-spin correlation functions on the space
of irreducible representations of the icosahedral point group. These
projections, analogous to the structure factors of translationally invariant
systems, give the amplitudes for distortions which transform as the irreducible
representations of the point group. The amplitude for the $H_{g}$ distortion is
the largest while the spin structure has large weights in the $T_{2g}$ and
$G_{u}$ representations.

Condensed Matter