Question

Show the molecular orbital structure of \(\mathrm{H}_{2}\). Explain the molecular orbital structure both in terms of electron configuration and LCAO. Show the bonding orbital and antibonding orbital, and calculate the bond order.

Short answer

The molecular orbital structure of H₂ consists of one bonding orbital (ψ_bond) formed by the constructive addition of the 1s orbitals from both hydrogen atoms and one antibonding orbital (ψ_antibond) formed by the destructive interference of these orbitals. The electron configuration of H₂ is (ψ_bond)². The calculated bond order of H₂ is 1, indicating that it has a single bond between the two hydrogen atoms.

Step-by-step solution

Identify atomic orbitals involved

Hydrogen atoms have only one electron in their 1s orbital. Since we need to explain the molecular orbital structure of H₂ molecule, we must consider two 1s atomic orbitals (orbitals from both the hydrogen atoms).

Combine atomic orbitals using LCAO

To form molecular orbitals, we will use Linear Combination of Atomic Orbitals (LCAO) to combine 1s orbitals of both hydrogen atoms. LCAO involves adding or subtracting the wave functions of the original atomic orbitals. When we add the wave functions, we get a bonding molecular orbital (ψ_bond). When we subtract the wave functions, we get an antibonding molecular orbital (ψ_antibond). ψ_bond = ψ(1s)_A + ψ(1s)_B ψ_antibond = ψ(1s)_A - ψ(1s)_B

Electron Configuration

As each hydrogen atom has only one electron, the H₂ molecule will have a total of two electrons. These electrons will occupy the lowest energy molecular orbitals (Aufbau principle). In the case of H₂, the bonding molecular orbital (ψ_bond) will have lower energy than the antibonding molecular orbital (ψ_antibond). Thus, both electrons will occupy the bonding orbital, or the molecular orbital formed from the constructive addition of both atomic orbitals. Electron configuration: (ψ_bond)²

Bonding and Antibonding Orbitals

The bonding molecular orbital (ψ_bond) is the result of the constructive interference (addition) of the 1s orbitals from both hydrogen atoms, and this leads to increased electron density between the two nuclei, stabilizing the molecule. The antibonding molecular orbital (ψ_antibond) is the result of the destructive interference (subtraction) of the 1s orbitals, leading to decreased electron density between the two nuclei and an unstable region due to the presence of a nodal plane.

Calculate Bond Order

Bond order is a measure of the bond strength and stability of a molecule. It can be determined using the following formula: Bond Order = (Number of electrons in bonding orbitals - Number of electrons in antibonding orbitals) / 2 For H₂: Bond Order = (2 - 0) / 2 = 1 In conclusion, the molecular orbital structure of H₂ consists of one bonding orbital (ψ_bond) formed by the constructive addition of the 1s orbitals and one antibonding orbital (ψ_antibond) formed by the destructive interference of these orbitals. The electron configuration of H₂ is (ψ_bond)². The calculated bond order of H₂ is 1, indicating that it has a single bond between the two hydrogen atoms.

Key concepts

LCAO (Linear Combination of Atomic Orbitals)

The Linear Combination of Atomic Orbitals (LCAO) is a key concept in understanding how molecules form from individual atoms. It involves the combination of atomic orbitals from adjacent atoms to create new molecular orbitals. These molecular orbitals can house electrons in a molecule.

• LCAO basically combines wave functions of atomic orbitals from two or more atoms.
• The combination can be constructive or destructive.

In the case of two hydrogen atoms forming a hydrogen molecule (H₂), we start with the 1s orbitals from each hydrogen atom. Using LCAO, these 1s orbitals merge to provide two possible outcomes. The constructive combination leads to a bonding molecular orbital, while the destructive combination gives rise to an antibonding molecular orbital. This approach is vital for comprehending how chemical bonds form at a quantum mechanical level.

Bonding and Antibonding Orbitals

When atomic orbitals combine through LCAO, they form two types of molecular orbitals: bonding and antibonding.

• Bonding Orbitals (ψ_bond): These orbitals result from the constructive interference of the atomic orbitals. In the case of H₂, the addition of wave functions of the 1s orbitals results in a molecular orbital with increased electron density between the nuclei. This increased density acts like a glue, holding the two nuclei together, therefore stabilizing the molecule.
• Antibonding Orbitals (ψ_antibond): These are formed through destructive interference, represented by the subtraction of wave functions. For H₂, this causes a nodal plane where there is decreased electron density between the nuclei, rendering this orbital destabilizing and of higher energy compared to the bonding orbital.

Bonding orbitals are always lower in energy and thus filled first, contributing to the overall stability of the molecule.

Bond Order

Bond order is a numerical value that provides insight into the bond strength and stability of a molecule. It can be calculated as:\[\text{Bond Order} = \frac{\text{Number of electrons in bonding orbitals} - \text{Number of electrons in antibonding orbitals}}{2}\]For the hydrogen molecule (H₂):

• There are two electrons in the bonding orbital.
• Zero electrons in the antibonding orbital.
• Thus, the bond order is calculated as \(\frac{2 - 0}{2} = 1\).

A bond order of 1 indicates a single bond between the two hydrogen atoms. Higher bond orders typically suggest stronger and more stable bonds, while a bond order of 0 would imply that no stable bond formation exists. Understanding bond order is essential for predicting how different molecules will behave chemically.