  Evolutionary structure searches are used to predict stable phases with unique
stoichiometries in the hydrogen-rich region of the magnesium/hydrogen phase
diagram under pressure. MgH$_4$, MgH$_{12}$ and MgH$_{16}$ are found to be
thermodynamically stable with respect to decomposition into MgH$_2$ and H$_2$
near 100 GPa, and all lie on the convex hull by 200 GPa. MgH$_4$ contains two
H$^-$ anions and one H$_2$ molecule per Mg$^{2+}$ cation, whereas the
hydrogenic sublattices of MgH$_{12}$ and MgH$_{16}$ are composed solely of
H$_2^{\delta -}$ molecules. The high-hydrogen content stoichiometries have a
large density of states at the Fermi level, and the $T_c$ of MgH$_{12}$ at 140
GPa is calculated to be nearly three times greater than that of the classic
hydride, MgH$_2$, at 180 GPa.
