Which of the following are predicted by the molecular orbital model to be stable diatomic species? a. $\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}$ b. \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\)

Count the total number of valence electrons in the given species. Remember that the charge of the species affects the number of electrons: a. \(\mathrm{H_{2}^{+}}\) has \(1 + 1 - 1 = 1\) electron \(\mathrm{H_{2}}\) has \(1 + 1 = 2\) electrons \(\mathrm{H_{2}^{-}}\) has \(1 + 1 + 1 = 3\) electrons \(\mathrm{H_{2}^{2-}}\) has \(1 + 1 + 2 = 4\) electrons b. \(\mathrm{He_{2}^{2+}}\) has \(2 + 2 - 2 = 2\) electrons \(\mathrm{He_{2}^{+}}\) has \(2 + 2 - 1 = 3\) electrons \(\mathrm{He_{2}}\) has \(2+2 = 4\) electrons

Apply the bond order formula: \(Bond\: order = \frac{(Number\:of\: electrons\: in\: bonding\: orbitals) - (Number\: of\: electrons\: in\: antibonding\:orbitals)}{2}\). We consider only sigma orbitals for the given species: For H, \(\sigma_{1s}\) is a bonding orbital, and \(\sigma_{1s}^{*}\) is an antibonding orbital. For He, \(\sigma_{1s}\), \(\sigma_{1s}^{*}\), \(\sigma_{2s}\), and \(\sigma_{2s}^{*}\) are the orbitals, where asterisks indicate antibonding orbitals. a. \(\mathrm{H_{2}^{+}}\) has 1 electron in the bonding orbital (\(\sigma_{1s}\)): Bond order \(= \frac{1 - 0}{2} = 0.5\) \(\mathrm{H_{2}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)): Bond order \(= \frac{2 - 0}{2} = 1\) \(\mathrm{H_{2}^{-}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 1 electron in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 1}{2} = 0.5\) \(\mathrm{H_{2}^{2-}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 2 electrons in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 2}{2} = 0\) b. \(\mathrm{He_{2}^{2+}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)): Bond order \(= \frac{2 - 0}{2} = 1\) \(\mathrm{He_{2}^{+}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 1 electron in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 1}{2} = 0.5\) \(\mathrm{He_{2}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 2 electrons in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 2}{2} = 0\)

If the bond order is greater than 0, the molecule is considered to be stable. a. Stable diatomic species among \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) are: \(\mathrm{H}_{2}^{+}\) (bond order \(=0.5\)) \(\mathrm{H}_{2}\) (bond order \(=1\)) \(\mathrm{H}_{2}^{-}\) (bond order \(=0.5\)) b. Stable diatomic species among \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\) are: \(\mathrm{He}_{2}^{2+}\) (bond order \(=1\)) \(\mathrm{He}_{2}^{+}\) (bond order \(=0.5\)) Hence, the stable diatomic species are: \(\mathrm{H}_{2}^{+}\), \(\mathrm{H}_{2}\), \(\mathrm{H}_{2}^{-}\), \(\mathrm{He}_{2}^{2+}\), and \(\mathrm{He}_{2}^{+}\).