  We investigate the accuracy of first-principles many-body theories at the
nanoscale by comparing the low energy excitations of the carbon fullerenes
C_20, C_24, C_50, C_60, C_70, and C_80 with experiment. Properties are
calculated via the GW-Bethe-Salpeter Equation (GW-BSE) and diffusion Quantum
Monte Carlo (QMC) methods. We critically compare these theories and assess
their accuracy against available photoabsorption and photoelectron spectroscopy
data. The first ionization potentials are consistently well reproduced and are
similar for all the fullerenes and methods studied. The electron affinities and
first triplet excitation energies show substantial method and geometry
dependence. These results establish the validity of many-body theories as
viable alternative to density-functional theory in describing electronic
properties of confined carbon nanostructures. We find a correlation between
energy gap and stability of fullerenes. We also find that the electron affinity
of fullerenes is very high and size-independent, which explains their tendency
to form compounds with electron-donor cations.
