 Solution

The ions are acetylide ${\mathrm{C}}_2^{2-}$ and linear tricarbide ${\mathrm{C}}_3^{4-}$. Charge balance gives $\mathrm{M}{\mathrm{g}}^{2+}[{\mathrm{C}}_2{]}^{2-}$ and $(\mathrm{M}{\mathrm{g}}^{2+}{)}_2[{\mathrm{C}}_3{]}^{4-}$.

The acetylide has ten valence electrons, two more than neutral ${\mathrm{C}}_2$. They fill the p-derived sigma bonding orbital, giving the formal configuration ${\pi }_u^4{\sigma }_g(2p{)}^2$ above the cancelling $2s$ pair. Its bond order is three, with the Lewis structure ${}^{-}\mathrm{C}\equiv {\mathrm{C}}^{-}$ and a lone pair at each end; each carbon is approximately $sp$ hybridized.

The tricarbide has $3(4)+4=16$ valence electrons and is isoelectronic with carbon dioxide. The cumulene contributor ${}^{2-}\mathrm{C}=\mathrm{C}={\mathrm{C}}^{2-}$ has a central $sp$ carbon, two sigma bonds and two perpendicular pi bonds. The end carbons each have two lone pairs and can be described with an $s{p}^2$ localized basis. Its two C–C bonds have formal order two in this model; the central C–C–C angle is $180^{\circ }$. Other resonance contributors help describe its protonation chemistry.