Question

Consider the following molecular orbitals formed from the combination of two hydrogen 1s orbitals: a. Which is the bonding molecular orbital and which is the antibonding molecular orbital? Explain how you can tell by looking at their shapes. b. Which of the two molecular orbitals is lower in energy? Why is this true?

Short answer

a. The sigma (σ) molecular orbital, which has an elongated cloud of electrons connecting the two hydrogen nuclei, is the bonding orbital. The sigma star (σ*) molecular orbital, with two separate clouds of electrons and a node between the two hydrogen nuclei, is the antibonding orbital. b. The bonding molecular orbital (σ) is lower in energy due to its stabilizing effect resulting from increased electron density between the hydrogen nuclei, whereas the antibonding molecular orbital (σ*) is higher in energy because of its destabilizing effect caused by decreased electron density.

Step-by-step solution

Identify the bonding and antibonding molecular orbitals

When two atomic orbitals combine, they form molecular orbitals. The combination of two hydrogen 1s orbitals produces two molecular orbitals: a sigma (σ) bonding molecular orbital and a sigma (σ*) antibonding molecular orbital. The bonding molecular orbital (σ) is formed when the electron wave functions of the two 1s orbitals have the same phase, which means they constructively interfere with each other, leading to increased electron density between the two nuclei. The increased electron density creates a stabilizing effect on the molecule. The shape of this orbital looks like a single elongated cloud of electrons connecting the two hydrogen nuclei. The antibonding molecular orbital (σ*) is formed when the electron wave functions of the two 1s orbitals have opposite phases, which means they destructively interfere with each other, leading to decreased electron density between the two nuclei. The decreased electron density creates a destabilizing effect on the molecule. The shape of this orbital looks like two separate clouds of electrons, one around each hydrogen nucleus and a node (a region with zero electron density) between them.

Determine the lower-energy molecular orbital

The bonding molecular orbital (σ) has lower energy than the antibonding molecular orbital (σ*). This is because the bonding molecular orbital has increased electron density between the two nuclei, which leads to a stabilizing effect on the molecule. This stabilizing effect results in the electrons in the bonding molecular orbital being attracted to both nuclei, thus lowering the energy. In contrast, the antibonding molecular orbital has decreased electron density between the two nuclei, creating a destabilizing effect on the molecule. This destabilizing effect means that the electrons in the antibonding molecular orbital are less strongly attracted to both nuclei and, as a result, have a higher energy. In summary: a. The bonding molecular orbital (σ) is the one with increased electron density between the two hydrogen nuclei, and the antibonding molecular orbital (σ*) is the one with decreased electron density and a node between the nuclei. b. The bonding molecular orbital (σ) is lower in energy due to its stabilizing effect on the molecule, while the antibonding molecular orbital (σ*) is higher in energy because of its destabilizing effect.

Key concepts

Bonding Molecular Orbital

When two hydrogen atoms come together, their 1s orbitals can join to form molecular orbitals. One of these is known as the bonding molecular orbital, indicated as \(\sigma\). This bonding molecular orbital results from the in-phase overlap of the electron wave functions from the hydrogen atoms' 1s orbitals.

• This in-phase overlap means the electrons are in sync, constructively building onto each other, which increases the electron density between the two hydrogen nuclei.
• This increased electron density means the electrons are more likely to be found between the nuclei, effectively pulling them closer together.
• This pull creates a stabilizing effect on the system, which is why the bonding molecular orbital is considered a source of stability in molecules.

As a result of these characteristics, the shape of the bonding molecular orbital resembles a single, elongated electron cloud stretching from one hydrogen nucleus to the other, visually connecting them.

Antibonding Molecular Orbital

In contrast with the bonding molecular orbital, there is also the antibonding molecular orbital, represented as \(\sigma^*\). This molecular orbital forms when the electron wave functions from the hydrogen 1s orbitals overlap out of phase.

• An out-of-phase overlap means the electron densities cancel to some extent, leading to a reduced electron density between the hydrogen nuclei.
• Such conditions result in a node—a region where the probability of finding an electron is zero—right between the nuclei.
• This decreased electron presence leads to a destabilizing effect, as there isn't enough pull to keep the nuclei close to each other.

Thus, the shape of the antibonding molecular orbital appears as two separate electron clouds, each localized around a hydrogen atom, with a clear division or node in the center.

Energy Levels

Understanding energy levels is crucial when analyzing molecular orbitals. In general, for hydrogen atoms, the bonding molecular orbital (\(\sigma\)) is lower in energy than the antibonding molecular orbital (\(\sigma^*\)).

• Lower energy correlates with a more stable configuration because the electron density is concentrated between the nuclei.
• This concentration facilitates a stronger attraction between the electrons and both atomic nuclei, stabilizing the molecule.

On the flip side, the antibonding molecular orbital, having less electron density between the nuclei and a destabilizing node, is higher in energy. The electrons in an antibonding molecular orbital don't contribute to the stability of the system, making such configurations energetically expensive for the molecule.

Hydrogen 1s Orbitals

The hydrogen 1s orbital is the fundamental energy level where you find the single electron of a hydrogen atom when it is not excited. In forming molecular orbitals, these 1s atomic orbitals from two hydrogen atoms combine to create the molecular orbitals of a hydrogen molecule.

• Fundamentally, the 1s orbital is spherical, with uniform electron density around the nucleus.
• When these orbitals overlap, the interaction can lead to the formation of one bonding and one antibonding molecular orbital.

This orbital interaction is fundamental in the simplest molecule formation, like \( H_2 \), providing insights into more complex bonding scenarios in chemistry. Through this combination and subsequent electron distribution, one understands how atoms bond and how the stability of molecules arises from these interactions.