Question

(a) Sketch the molecular orbitals of the \(\mathrm{H}_{2}^{-}\) ion and draw its energy-level diagram. (b) Write the electron configuration of the ion in terms of its MOs. (c) Calculate the bond order in \(\mathrm{H}_{2}^{-}\). (d) Suppose that the ion is excited by light, so that an electron moves from a lower-energy to a higher- energy molecular orbital. Would you expect the excited-state \(\mathrm{H}_{2}^{-}\) ion to be stable? Explain.

Short answer

The molecular orbitals of the 𝐻2− ion consist of a lower-energy bonding σ1s orbital and a higher-energy antibonding σ1s* orbital. The energy-level diagram displays these orbitals on the y-axis. The electron configuration for this ion is σ1s^2 σ1s*^1, and the bond order is 1/2. In the excited state, the electron configuration is σ1s^1 σ1s*^2, and the bond order is -1/2, meaning that the excited 𝐻2− ion is unstable and will dissociate into individual hydrogen atoms and an electron.

Step-by-step solution

Sketch Molecular Orbitals of the 𝐻2− ion

We have to construct the molecular orbitals of the 𝐻2− ion using atomic orbitals of hydrogen atoms. For each hydrogen atom, there is one 1s orbital. When the two 1s orbitals of the two hydrogen atoms combine, they form two new molecular orbitals, one constructive (lower-energy bonding σ1s orbital) and one destructive (higher-energy antibonding σ1s* orbital).

Draw the Energy-Level Diagram

Next, draw the energy-level diagram for the 𝐻2− ion. On the y-axis, you will have energy increasing upwards. Add the σ1s orbital at a lower energy level and the σ1s* orbital at a higher energy level. Label these orbitals and the individual atomic orbitals.

Write the Electron Configuration of the 𝐻2− ion

For a 𝐻2− ion, there are 3 electrons present. To write the electron configuration in terms of the MOs, we follow the Aufbau principle and fill the orbitals with the lowest energy first. The electron configuration will be as follows: σ1s^2 σ1s*^1

Calculate the Bond Order in 𝐻2− ion

The bond order can be calculated using the formula: Bond Order = (Bonding electrons - Antibonding electrons) / 2 For the 𝐻2− ion, there are 2 bonding electrons in σ1s and 1 antibonding electron in σ1s*. Bond Order = (2 - 1) / 2 = 1/2

Determine the Stability of the Excited-State 𝐻2− ion

In the excited state, an electron is promoted from a lower-energy molecular orbital to a higher-energy molecular orbital (from σ1s to σ1s* in this ion). The electron configuration of the excited state would be: σ1s^1 σ1s*^2 Now, we need to calculate the bond order for the excited state: Bond Order (excited) = (1 - 2) / 2 = -1/2 Since the bond order is negative for the excited state, this indicates that the excited 𝐻2− ion would be unstable. The repulsion between the nuclei will be greater than the bonding force resulting from the electrons, so the ion would dissociate into individual hydrogen atoms and an electron.

Key concepts

H2- ion

The \(\mathrm{H}_{2}^{-}\) ion, simply put, is a negatively charged molecule formed by two hydrogen atoms and an extra electron. In atomic terms, each hydrogen atom has a 1s orbital. When these two atoms bond together, they combine their 1s orbitals to form molecular orbitals.
These newly formed molecular orbitals are central to understanding the ion's structure and properties.

• The bonding molecular orbital is called \( \sigma 1s \) and it has lower energy, making it more stable.
• The antibonding molecular orbital is \( \sigma 1s^* \) which carries higher energy, and thus, is less stable.

Energy-level diagram

An energy-level diagram is an essential tool for visualizing the molecular orbitals of the \(\mathrm{H}_{2}^{-}\) ion.
Picture it as a ladder. Lower rungs represent lower energy, while higher rungs represent higher energy.

• The lowest rung is the \(\sigma 1s \) orbital. Electrons here are at a lower energy state, contributing to stronger, more stable bonds.
• Above this is the \(\sigma 1s^* \) orbital, where energy is higher, causing weaker bonds due to less stability.

Label both the bonding and antibonding orbitals clearly to understand where electrons are likely to be found. Remember that in the \(\mathrm{H}_{2}^{-}\) ion, one electron is in this higher energy orbital.

Bond Order

The bond order is a numerical indication of bond strength and stability in the \(\mathrm{H}_{2}^{-}\) ion.
A higher bond order typically means a stronger and more stable bond.
Use the simple formula: \[ \text{Bond Order} = \frac{\text{Number of bonding electrons} - \text{Number of antibonding electrons}}{2} \] In the \(\mathrm{H}_{2}^{-}\) ion, there are two bonding electrons in the \(\sigma 1s \) orbital, and one antibonding electron in the \(\sigma 1s^* \) orbital.
So, for \(\mathrm{H}_{2}^{-}\), the formula calculates as: \[ \text{Bond Order} = \frac{2 - 1}{2} = \frac{1}{2} \] This result points to a relatively weak bond present in the ion.

Electron Configuration

The electron configuration describes how electrons are distributed in the molecular orbitals of an ion.
Applying the Aufbau principle, which states that electrons populate the lowest available energy levels first, gives us this configuration for the \(\mathrm{H}_{2}^{-}\) ion:

• First, fill the lower energy bonding orbital \(\sigma 1s\): \(\sigma 1s^2\)
• Next, place the remaining electron in the higher energy antibonding orbital \(\sigma 1s^*\): \(\sigma 1s^*^1\)

Thus, the electron configuration for the \(\mathrm{H}_{2}^{-}\) ion is \(\sigma 1s^2 \sigma 1s^*^1\), accurately reflecting how the ion's electrons are organized.