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 H₂⁻ ion has an electron configuration of σ₁s²σ₁s*¹ with a bond order of 0.5, indicating weak bonding. Upon excitation with light, the electron configuration becomes σ₁s¹σ₁s*², yielding a bond order of -0.5. This negative bond order means the excited-state H₂⁻ ion is unstable due to a greater number of electrons in antibonding MOs, resulting in repulsive forces between the hydrogen atoms.

Step-by-step solution

Sketch the molecular orbitals and energy-level diagram of H₂⁻ ion

To sketch the molecular orbitals and energy-level diagram of the H₂⁻ ion, we need to consider the constructive and destructive overlap of two hydrogen atom orbitals (1s orbitals). 1. When two 1s orbitals overlap constructively, they form a bonding molecular orbital called the σ₁s orbital. 2. When two 1s orbitals overlap destructively, they form an antibonding molecular orbital called the σ₁s* orbital. The energy-level diagram will have the σ₁s orbital at a lower energy level than the σ₁s* orbital.

Write the electron configuration of H₂⁻ ion in terms of its MOs

A neutral H₂ molecule has two electrons, and since the H₂⁻ ion has a -1 charge, it has three electrons in total. The electron configuration follows the aufbau principle, meaning that the lower energy orbitals are filled before the higher energy orbitals. So, the electron configuration of H₂⁻ is: σ₁s²σ₁s*¹

Calculate the bond order in H₂⁻

Bond order is defined as `(number of electrons in bonding MOs - number of electrons in antibonding MOs)/2`. Using the electron configuration derived in step 2, we can calculate the bond order: Bond order = (2 - 1)/2 = 0.5

Discuss the stability of the excited-state H₂⁻ ion

Suppose that the ion is excited by light, and an electron moves from a lower-energy molecular orbital (σ₁s) to a higher-energy molecular orbital (σ₁s*). In this case, the electron configuration would become: σ₁s¹σ₁s*² Calculating the bond order for this excited-state configuration: Bond order = (1 - 2)/2 = -0.5 Since the bond order is negative in the excited state, the excited-state H₂⁻ ion would not be stable. This is because a negative bond order indicates that there are more electrons in antibonding MOs than bonding MOs, which leads to a repulsive force between the two hydrogen atoms, destabilizing the molecule.

Key concepts

Understanding the H2- Ion

The \(\mathrm{H}_{2}^{-}\) ion is a molecular species formed by the combination of two hydrogen atoms along with an extra electron. This extra electron gives the ion a negative charge, differentiating it from the neutral \(\mathrm{H}_{2}\) molecule. \The primary requirement to form an \(\mathrm{H}_{2}^{-}\) ion is the mutual interaction and overlap of the individual atomic orbitals of each hydrogen atom. \Each hydrogen atom contributes one 1s orbital, which can overlap to form molecular orbitals. \Therefore, the resultant molecular orbitals include a bonding orbital and an antibonding orbital, which are the core focus when examining this ion. By understanding the behavior and arrangement of this extra electron, you can determine the chemical bonding, stability, and other properties of \(\mathrm{H}_{2}^{-}\) ion. \This ion is an exciting example illustrating electron behavior in molecular orbital theory.

Calculating Bond Order

The concept of bond order is fundamental in molecular chemistry, providing insights into the strength and stability of a bond. It relates directly to the number of electrons shared between atoms. \When referring to \(\mathrm{H}_{2}^{-}\), the bond order can be calculated using the formula: \[\text{Bond Order} = \frac{\text{(Number of electrons in bonding MOs) - (Number of electrons in antibonding MOs)}}{2}\] Substituting in the number of electrons, we have 2 electrons in the bonding orbital and 1 in the antibonding orbital, \which results in a bond order of 0.5: \[\frac{(2 - 1)}{2} = 0.5\]

• A bond order of 0.5 indicates a fairly weak bond, suggesting limited stability under normal conditions.
• This fractional bond order implies that while there is a bond between the two hydrogen atoms, it is not as strong as in a diatomic hydrogen molecule (\(\mathrm{H}_{2}\)).

Understanding Electron Configuration

Electron configuration describes the arrangement of electrons in a molecular entity, which determines both its properties and chemical behavior. \In the case of the \(\mathrm{H}_{2}^{-}\) ion, the presence of three electrons (due to the extra electron beyond the neutral \(\mathrm{H}_{2}\)) demands meticulous electron arrangement. \The electron configuration follows the Aufbau principle where lower energy orbitals are filled before moving on to higher energy levels. \For \(\mathrm{H}_{2}^{-}\), this configuration is noted as \(\sigma_{1s}^{2}\sigma_{1s}^{*1}\).

• The \(\sigma_{1s}\) orbital hosts 2 electrons, maximizing its lower energy capacity.
• The \(\sigma_{1s}^{*}\) orbital, being higher in energy, contains the extra single electron.

This configuration plays a crucial role in dictating the molecular properties and influences the stability and bond characteristics of the ion.

Delving into Molecular Orbital Theory

Molecular Orbital Theory (MOT) is an essential concept that helps in understanding the bonding and properties of molecules like \(\mathrm{H}_{2}^{-}\). \It offers a more comprehensive description than simple valence bond theory, and explains how atomic orbitals combine to form new orbitals - molecular orbitals. \In the case of \(\mathrm{H}_{2}^{-}\), two types of molecular orbitals are formed from the overlap of the 1s orbitals of each hydrogen:

• Bonding Molecular Orbital (\(\sigma_{1s}\)): - This orbital is formed by constructive interference, leading to increased electron density between the hydrogen nuclei.
• Antibonding Molecular Orbital (\(\sigma_{1s}^{*}\)): - Formed by destructive interference, associated with a node between nuclei where electron density is minimal.

The framework of MOT allows chemists to rationalize why certain molecules hold together and the energies involved, thus addressing questions of stability and reactivity in molecular systems.