The following is part of a molecular orbital energy-level diagram for MOs constructed from 1 s atomic orbitals. (a) What labels do we use for the two MOs shown? (b) For which of the following molecules or ions could this be the energy-level diagram: $$ \mathrm{H}_{2} \mathrm{He}_{2}, \mathrm{H}_{2}^{+}, \mathrm{He}_{2}^{+}, \mathrm{or} \mathrm{H}_{2}^{-} ? $$ (c) What is the bond order of the molecule or ion? (d) If an electron is added to the system, into which of the MOs will it be added? [Section 9.7\(]\)

We need to identify the two MOs shown in the given energy-level diagram. In this case, the MOs are formed from 1s atomic orbitals. These two MOs are σ (sigma) and σ* (sigma-star) molecular orbitals. The σ orbital is the bonding orbital that is lower in energy and is formed due to the constructive combination of atomic orbitals, while the σ* orbital is the anti-bonding orbital that is higher in energy and is formed due to the destructive combination of atomic orbitals.

We should choose one of the molecules/ions from the list, which is: \(\mathrm{H}_{2}, \mathrm{He}_{2}, \mathrm{H}_{2}^{+}, \mathrm{He}_{2}^{+}, \) or \(\mathrm{H}_{2}^{-}\). For this step, it is essential to remember the electron configurations and occupancy of molecular orbitals. For instance: 1. \(\mathrm{H}_{2}\): Both hydrogen atoms have 1 electron in their 1s orbitals, and the molecule has a total of 2 electrons, filling the σ bonding orbital but not the σ* anti-bonding orbital. 2. \(\mathrm{He}_{2}\): Each helium atom has 2 electrons in its 1s orbital, and the molecule has a total of 4 electrons. This means that both the σ and σ* orbitals will be filled with 2 electrons each, canceling each other in terms of bond formation. 3. \(\mathrm{H}_{2}^{+}\): The molecule has a total of 1 electron which will fill the σ orbital (with the other σ* orbital remaining empty), resulting in bond formation. 4. \(\mathrm{He}_{2}^{+}\): The ion has a total of 3 electrons, resulting in 2 electrons occupying the σ orbital and 1 electron in the σ* anti-bonding orbital. 5. \(\mathrm{H}_{2}^{-}\): The ion has a total of 3 electrons, filling the σ orbital with 2 electrons, while the other electron goes into the σ* anti-bonding orbital. The energy-level diagram depicts a molecule/ion with both the σ and σ* orbitals occupied by an equal number of electrons, which results in no bond formation. Hence, the correct molecule/ion is \(\mathrm{He}_{2}\).

To calculate the bond order, use the formula: Bond order = \(0.5 \times (\textit{number of electrons in bonding orbitals} - \textit{number of electrons in anti-bonding orbitals})\) For \(\mathrm{He}_{2}\), both the σ and σ* orbitals contain 2 electrons each: Bond order = \(0.5 \times (2 - 2) = 0\)

If an electron were to be added to the system, it would enter the molecular orbital with the lowest available energy level. Since both the σ and σ* orbitals are already filled, adding an electron to the \(\mathrm{He}_{2}\) system would result in an increased energy level. More specifically, it will be added to the next available orbital, which in this case, is a higher-lying σ* anti-bonding orbital.