Why is the bond multiplicity in C2 so elusive?

We reexamine the full GVB (or spin-coupled) description of the bonding in the ground state of C2 at its equilibrium geometry, prompted by recent controversy as to whether or not this system exhibits four bonds. We show that two apparently different interpretations (namely a single sigma bond plus antiferromagnetic coupling of high spin units on the two atoms as opposed to a conventional set of covalent sigma and pi bonds) do in fact have a high overlap. Neither description is adequate. Further insights into the nature of the bonding in this system emerge from an analysis of spin correlation matrix elements. We also analyze domain-averaged Fermi holes and values of the QTAIM-generalized Wiberg-Mayer index. It proves important to assess the significance of the various numerical results by means of direct comparison with analogous calculations for HCCH. We suggest that an enhanced weight for triplet spin-coupling modes, relative to HCCH, increases the difficulty of describing the electronic structure of C2 in terms of conventional bonding models that feature a whole number of two-center two-electron bonds.