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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. 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