  Bonding in the C$_2$ molecule is investigated with CAS(8,8) wave functions
using canonical MOs. In a subsequent step, orthogonal atomic orbitals are
constructed by localizing the CASSCF MOs on the two carbon atoms with an
orthogonal transformation. This orbital transformation causes an orthogonal
transformation of the configuration state functions (CSF) spanning the function
space of the singlet ground state of C$_2$. Instead of CSFs built from
canonical MOs one gets CSFs of orthogonal deformed atomic orbitals (AO). This
approach resembles the orthogonal valence bond methods (OVB) CSFs which are
very different from conventional VB, based on non-orthogonal AOs. To get used
to the different argumentation, the bonding situation in ethane (single bond),
ethene (double bond), and the nitrogen molecule (triple bond) are also studied.
The complex bonding situation in C$_2$ is caused by the possibility to excite
an electron with spin flip from the doubly occupied 2s AO into the 2p subshell,
the resulting high-spin $^5S_u$ state of the carbon atom allows for a better
reduction of the Pauli repulsion. But the electron structure around the
equilibrium distance does not allow to say that C$_2$ in its ground state has a
double, or triple, or even a quadruple bond.
