  One of the long standing problems in quantum chemistry had been the inability
to exploit full spatial and spin symmetry of an electronic Hamiltonian
belonging to a non-Abelian point group. Here we present a general technique
which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to
obtain its full eigenvalue spectrum. This is a hybrid method based on Valence
Bond basis and the basis of constant z-component of the total spin. This
technique is applicable to systems with any point group symmetry and is easy to
implement on a computer. We illustrate the power of the method by applying it
to a model icosahedral half-filled electronic system. This model spans a huge
Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group.
The $C_{60}$ molecule has this symmetry and hence our calculation throw light
on the higher energy excited states of the bucky ball. This method can also be
utilized to study finite temperature properties of strongly correlated systems
within an exact diagonalization approach.
