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**Hint :**The composition of $ H_{2}^{+} $ and $ H_{2}^{-} $ defines the stability order. $ H_{2}^{+} $ does not include any electron in the antibonding molecular orbital. $ H_{2}^{-} $ includes electrons in antibonding molecular orbital.The stability can be easily identified from the electron presence in bonding and antibonding molecular orbital.

**Complete Step By Step Answer:**

The electron presence in anti-bonding orbital result in repulsion and decrease of stability

The stability order follows the molecular orbital theory

The formula for bonding order can be regarded as follows

Bonding order $ =\dfrac{1}{2} $ (Number of bonding electrons -Number of antibonding electrons)

The number of antibonding electrons in $ H_{2}^{+} $ is 0

The number of antibonding electrons in $ H_{2}^{-} $ is 1

The bonding order of $ H_{2}^{+} $ is $ ~ $ $ =\dfrac{1}{2}(1-0)=0.5 $

The bonding order of $ H_{2}^{-} $ is $ =\dfrac{1}{2}(2-1)=0.5 $

$ H_{2}^{+} $ and $ H_{2}^{-} $ molecule have the same bond order

Presence of anti-bonding electron decreases the stability of the molecule

Therefore, $ H_{2}^{+} $ is more stable than $ H_{2}^{-} $

Both $ H_{2}^{+} $ and $ H_{2}^{-} $ are equally stable is incorrect statement

Both $ H_{2}^{+} $ and $ H_{2}^{-} $ do not exist is incorrect statement

$ H_{2}^{-} $ is more stable than $ H_{2}^{+} $ is incorrect statement

The correct statement is $ H_{2}^{+} $ is more stable than $ H_{2}^{-} $

Due to the electron-electron repulsion in $ H_{2}^{-} $ , the stability of $ H_{2}^{-} $ is low

The configuration of $ H_{2}^{+} $ is $ 1 $ electron in $ 1 $ s bonding orbital

The configuration of $ H_{2}^{-} $ is $ 1 $ electron in $ 1 $ s bonding orbital and $ 1 $ electron in $ 1 $ s antibonding orbital

Even though the bond order of both $ H_{2}^{+} $ and $ H_{2}^{-} $ are same, the stability is different

Therefore, $ H_{2}^{+}>H_{2}^{-} $ .

**Note :**

The stability order follows the molecular orbital theory. The electron presence in anti-bonding orbital results in repulsion and decrease of stability. Electron-electron repulsion in $ H_{2}^{-} $ , the stability of $ H_{2}^{-} $ is low.

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