  We report on an effective Hubbard Hamiltonian approach for the study of
electronic correlations in C$_{20}$ isomers, cage, bowl and ring, with quantum
Monte Carlo and exact diagonalization methods. The tight-binding hopping
parameter, $t$, in the effective Hamiltonian is determined by a fit to density
functional theory calculations, and the on-site Coulomb interaction, $U/t$, is
determined by calculating the isomers' affinity energies, which are compared to
experimental values. For the C$_{20}$ fullerene cage we estimate $t_{\rm
cage}\simeq 0.68-1.36$ eV and $(U/t)_{\rm cage} \simeq 7.1-12.2$. The resulting
effective Hamiltonian is then used to study the shift of spectral peaks in the
density of states of neutral and one-electron-doped C$_{20}$ isomers. Energy
gaps are also extracted for possible future comparison with experiments.
