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Ionic compounds are formed when an atom $\ce{A}$ donates one of it's electron to another atom $\ce{B}$ to form an ionic compound $\ce{A+B-}$. This is significantly more likely to happen as the electronegativity difference of $\ce{A}$ and $\ce{B}$ increases.

Hydrogen has a Pauling electronegativity of $2.20$. So, it appears to me that $\ce{H_2}$ should be able to form ionic bonds both as $$\ce{H+ + A- -> H+A-}$$ $$\ce{B+ + H- -> B+H-}$$

What are compounds that contain $\ce{H-}$ called and what are some examples?

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    $\begingroup$ H+ can't form a true ionic bond. H- can and does. $\endgroup$
    – Ivan Neretin
    Jun 11, 2021 at 15:35

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Hydrogen does form ionic bonds. Although not as bare $\ce{H+}$, it can form ionic bonds in the form of $\ce{H-}$ (hydride anion). $\ce{H-}$ forms ionic bonds with alkali metal ions and alkaline earth metal ions (except $\ce{Be^{2+}}$). These type of ionic compounds formed by hydride ion are called saline hydrides, as they are salt type. These hydrides like $\ce{CaH2}$ (also known as hydrolith) can also be used for production of dihydrogen, as reducing agents, desiccant.

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    $\begingroup$ The lattice structures of lithium and magnesium hydrides feature six-coordination of the metal which has only four valence orbitals, and they readily hydrolyze to give the hydroxide plus hydrogen. On this basis I would call lithium and magnesium hydrides predominantly ionic. $\endgroup$
    – Oscar Lanzi
    Jun 11, 2021 at 16:30
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    $\begingroup$ @OscarLanzi, corrected. $\endgroup$
    – Adithya
    Jun 11, 2021 at 16:35

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