A quqdruple bond is a type of chemical bond between two atoms involving 8 electrons. This bond is an existension of the more familiar types double bond and triple bond. Stable quatruple bonds are most common among the transition metal , in the middle of the d block such as tungsten and chromium. Typically the ligands that support quadruple bonds are pi donors . not in pi acceptors.
Valence bond theory predicts a quatruple bond as the only way to satisfy the octet rule for carbon in dicarbon. However MOT shows that three are two sets of paired electrons in the sigma system. (one bonding and one anti bonding) and two sets of paired electrons in a degenerate pi bonding set of orbitals. This adds up to give a bond order of 2 , meaning that three exist a double bond between the 2 carbon in a C2 molecule. This is surprising because the MO diagram of diatomic carbon would show that three are 2 pi bonds and no sigma bonds.