  We study the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on a series of
finite-size clusters with features inspired by the fullerenes. Frustration due
to the presence of pentagonal rings makes such structures challenging in the
context of quantum Monte-Carlo methods. We use an exact diagonalization
approach combined with a truncation method in which only the most important
basis states of the Hilbert space are retained. We describe an efficient
variational method for finding an optimal truncation of a given size which
minimizes the error in the ground state energy. Ground state energies and
spin-spin correlations are obtained for clusters with up to thirty-two sites
without the need to restrict the symmetry of the structures. The results are
compared to full-space calculations and to unfrustrated structures based on the
honeycomb lattice.
