Inspired by a blog post of Henry Rzepa (see here) Shaik and co-workers examined the C2 species with an eye towards the nature of the bond between the two carbon atoms.1 Using both a valence bond approach and a full CI approach, they end up at the same place: there is a quadruple bond here!
The argument rests largely on a definition of of an in situ bond energy. For the VB approach, this requires choosing as a reference a non-bonding interaction between the atoms with regards to a pair of electrons. For the CI approach, the bond energy is half the energy of the singlet-triplet gap. So, for C2, the VB/6-31G* estimate of the bond energy of the putative fourth bond is 14.3 kcal mol-1. For the full CI/6-31G* computations of the singlet-triplet gap, the bond energy estimate is 14.8 kcal mol-1, and using the experimental value of the gap, the estimate is 13.2 kcal mol-1. Not a strong bond, but certainly meaningful!
In the VB approach, the fourth bond is a weighted sum of the antibonding 2σu and bonding 3σg orbitals – a combination that gives rise to small constructive overlap between the two C atoms. In the CI model, the wavefunction is dominated by the first two configurations; the first configuration, with a coefficient of C0=0.828 has 2σu doubly occupied and the second coefficient, with CD=0.324, has the 3σg orbital doubly occupied. Considering that 3σg is a bonding orbital, the significant contribution of this configuration gives rise to the fourth bond.
(1) Shaik, S.; Danovich, D.; Wu, W.; Su, P.; Rzepa, H. S.; Hiberty, P. C., "Quadruple bonding in C2 and analogous eight-valence electron species," Nat. Chem., 2012, 4, 195-200, DOI: 10.1038/nchem.1263.