Question

Consider the \(\mathrm{H}_{2}^{+}\) ion. (a) Sketch the molecular orbitals of the ion and draw its energy-level diagram. (b) How many electrons are there in the \(\mathrm{H}_{2}^{+}\) ion? (c) Draw the electron configuration of the ion in terms of its MOs. (d) What is the bond order in \(\mathrm{H}_{2}{ }^{+}\) ? (e) Suppose that the ion is excited by light so that an electron moves from a lower-energy to a higherenergy MO. Would you expect the excited-state \(\mathrm{H}_{2}^{+}\) ion to be stable or to fall apart? Explain.

Short answer

The H2+ ion has one electron, and its electron configuration is σ1s^1. The bond order is 1/2, indicating a weak bond. When excited by light, the electron moves to the antibonding molecular orbital (σ1s*), resulting in a bond order of -1/2. This suggests that the excited-state H2+ ion will be unstable and likely fall apart.

Step-by-step solution

(a) Sketching molecular orbitals and energy-level diagram for H2+ ion

To sketch the molecular orbitals and energy-level diagram of the H2+ ion, we start with the atomic orbitals of the individual H atoms. Each H atom has one electron in the 1s orbital. When two H atoms come together to form the H2+ ion, the 1s orbitals of the two atoms interact with each other and form two molecular orbitals: one bonding (σ1s) and one antibonding (σ1s*). The bonding molecular orbital (σ1s) is lower in energy and is formed due to constructive interference of the atomic orbitals, whereas the antibonding molecular orbital (σ1s*) is higher in energy and is formed due to destructive interference of the atomic orbitals. In the energy-level diagram, the y-axis represents the energy, and the molecular orbitals are represented as horizontal lines, with the bonding orbital below the antibonding orbital. The atomic orbitals of the individual H atoms are represented at the same energy level as the bonding orbital.

(b) Number of electrons in H2+ ion

The H2+ ion is formed by the combination of two hydrogen atoms, with one electron being removed from the system. As each hydrogen atom has one electron, the total number of electrons in the H2+ ion is 2 - 1 = 1 electron.

(c) Electron configuration of H2+ ion

The electron configuration of the H2+ ion is determined by placing the electrons in the available molecular orbitals according to the Aufbau principle (filling orbitals with lower energy first). Since there is only one electron in the H2+ ion, it will occupy the lower energy bonding molecular orbital (σ1s). Therefore, the electron configuration of the H2+ ion is σ1s^1.

(d) Bond order in H2+ ion

Bond order is calculated as follows: Bond order = (Number of electrons in bonding orbitals - Number of electrons in antibonding orbitals) / 2 In the case of the H2+ ion, there is one electron in the bonding molecular orbital (σ1s) and no electrons in the antibonding molecular orbital (σ1s*). Therefore, the bond order of the H2+ ion is (1 - 0) / 2 = 1/2.

(e) Stability of excited-state H2+ ion

When the H2+ ion is excited by light, its single electron can move from the bonding molecular orbital (σ1s) to the antibonding molecular orbital (σ1s*). In this excited state, there will be no electrons in the bonding molecular orbital and one electron in the antibonding molecular orbital. The bond order in this excited state can be calculated as: Bond order = (Number of electrons in bonding orbitals - Number of electrons in antibonding orbitals) / 2 Bond order = (0 - 1) / 2 = -1/2 The negative bond order indicates that the excited-state H2+ ion will not have a stable bond and is likely to fall apart.

Key concepts

Energy-Level Diagram

An energy-level diagram is a visual representation that helps understand the energies of atomic or molecular orbitals and how electrons populate these states. For the hydrogen molecular ion (\textsc{H}\(_2^+\)), starting with the atomic orbitals of hydrogen is essential. Each hydrogen atom contributes a 1s orbital. When these atoms bond, their orbitals combine to form molecular orbitals: a lower-energy bonding orbital (\textsc{σ}1s) and a higher-energy antibonding orbital (\textsc{σ}1s*).

The bonding orbital results from constructive interference between the atomic orbitals, which lowers the energy, while the antibonding orbital is a product of destructive interference, raising the energy.

In the diagram, these orbitals are depicted as horizontal lines; the \textsc{σ}1s line is placed lower than the \textsc{σ}1s* to reflect their energy difference. The electron in the \textsc{H}\(_2^+\) ion occupies the \textsc{σ}1s bonding orbital, which is indicated by an upward arrow on the lower line of the energy-level diagram.

H2+ Ion Electron Configuration

Understanding the electron configuration of the \textsc{H}\(_2^+\) ion provides insight into its chemical behavior. There is just a single electron in this ion, as it is formed by two hydrogen atoms sharing their electrons and one electron is lost during ion formation. Following the Aufbau principle, we fill the orbital with the lowest energy first.

The \textsc{H}\(_2^+\) electron configuration is thus \textsc{σ}1s\(^1\). This signifies that the lone electron resides in the lower-energy bonding molecular orbital (\textsc{σ}1s), depicted by a single arrow (or a single electron dot) in the energy-level diagram for \textsc{H}\(_2^+\). The absence of any electrons in the antibonding orbital (\textsc{σ}1s*) is key to understanding the ion's stability and chemical propensity.

Bond Order Calculation

Calculating the bond order gives insight into the strength and stability of a chemical bond in a molecule. The formula for bond order is: \textsc{Bond Order} = (Number of electrons in bonding orbitals - Number of electrons in antibonding orbitals) / 2.

For the \textsc{H}\(_2^+\) ion, there is one electron in the bonding molecular orbital and no electrons in the antibonding orbital. This makes the bond order (1 - 0) / 2 = 0.5. A positive bond order implies a stable bonding interaction, though in the case of \textsc{H}\(_2^+\), the bond order is low at 0.5, indicating a bond weaker than a typical single bond.

Conversely, when the ion is in an excited state with its electron in the antibonding orbital, the bond order becomes negative ((0 - 1) / 2 = -0.5), suggesting an unstable bond that is likely to dissociate. This explains why the excited state of \textsc{H}\(_2^+\) is predicted to be short-lived.