  The binding energies of n < 100 carbon clusters are calculated using the
ab-initio density functional theory code Quantum Espresso. Carbon cluster
geometries are determined using several levels of classical techniques and
further refined using density functional theory. The resulting energies are
used to compute the work of cluster formation and the nucleation rate in a
saturated, hydrogen-poor carbon gas. Compared to classical calculations that
adopt the capillary approximation, we find that nucleation of carbon clusters
is enhanced at low temperatures and depressed at high temperatures. This
difference is ascribed to the different behavior of the critical cluster size.
We find that the critical cluster size is at n = 27 or n = 8 for a broad range
of temperatures and saturations, instead of being a smooth function of such
parameters. The results of our calculations can be used to follow carbonaceous
cluster/grain formation, stability, and growth in hydrogen poor environments,
such as the inner layers of core-collapse supernova and supernova remnants.
