Molecular Orbitals

About

Molecular orbitals (MOs) are mathematical functions that describe the behavior of electrons in a molecule. They are formed by the combination of atomic orbitals (AOs) from the individual atoms in a molecule when they bond. Molecular orbitals provide a way to understand how electrons are distributed in a molecule and how they contribute to bonding, anti-bonding, and non-bonding interactions.

1. Formation of Molecular Orbitals

When atoms come together to form a molecule, their atomic orbitals (like (1s), (2p), etc.) overlap to form molecular orbitals. These molecular orbitals extend over the entire molecule and describe the regions where electrons are most likely to be found.

The combination of two atomic orbitals can result in two types of molecular orbitals:

• Bonding molecular orbitals: Lower in energy, these are formed by constructive interference (in-phase overlap) of atomic orbitals, leading to electron density concentrated between the nuclei of the atoms. Electrons in bonding orbitals help hold the atoms together.

• Antibonding molecular orbitals: Higher in energy, these result from destructive interference (out-of-phase overlap) of atomic orbitals, leading to electron density located away from the bond. Electrons in antibonding orbitals tend to destabilize the bond.

• Non-bonding molecular orbitals: These orbitals do not contribute to bonding or antibonding interactions and are typically localized on atoms that don’t directly participate in bonding (such as lone pairs).

2. Types of Molecular Orbitals

There are different types of molecular orbitals depending on the symmetry of the atomic orbitals and the bonding type:

• Sigma (σ) orbitals: Formed by the head-on overlap of atomic orbitals along the internuclear axis. These orbitals are symmetric about the bond axis. The bonding sigma orbital is typically the strongest bond in a molecule.

  • σ-bonding MO: Formed from constructive overlap.

  • σ-antibonding MO*: Formed from destructive overlap.

• Pi (π) orbitals: Formed by the sideways overlap of p orbitals. The electron density in these orbitals lies above and below the internuclear axis. Pi bonds are usually weaker than sigma bonds.

  • π-bonding MO: Formed by the constructive overlap of two p orbitals.

  • π-antibonding MO*: Formed by the destructive overlap of two p orbitals.

3. Molecular Orbital Energy Diagram

The relative energies of the molecular orbitals formed from atomic orbitals can be represented in a molecular orbital energy diagram. The diagram shows:

• Bonding orbitals lower in energy than the original atomic orbitals.

• Antibonding orbitals higher in energy than the atomic orbitals.

• Non-bonding orbitals (if any) at the same energy level as the atomic orbitals they come from.

For example, in a homonuclear diatomic molecule (like H₂ or O₂):

• The two (1s) orbitals combine to form one σ-bonding orbital and one σ*-antibonding orbital.

• The (2p) orbitals can combine to form both σ- and π-type bonding and antibonding orbitals.

Example: Molecular Orbitals of H₂

In the hydrogen molecule (H₂), the two (1s) atomic orbitals combine to form:

• A σ (bonding) orbital: Lower in energy and containing the two bonding electrons, leading to a stable bond between the hydrogen atoms.

• A σ (antibonding) orbital*: Higher in energy, which remains unoccupied in the ground state.

Thus, the electron configuration for H₂ is:

$\sigma(1s)^2$

This describes two electrons occupying the σ-bonding orbital, resulting in a stable bond.

4. Molecular Orbital Theory and Bond Order

The bond order of a molecule can be calculated from the molecular orbital theory by considering the number of electrons in bonding and antibonding orbitals:

$\text{Bond order} = \frac{(\text{number of electrons in bonding orbitals}) - (\text{number of electrons in antibonding orbitals})}{2}$

• A bond order of 1 indicates a single bond.

• A bond order of 2 indicates a double bond.

• A bond order of 0 means no bond (the molecule is unstable).

For example, in the oxygen molecule (O₂):

• The electronic configuration in terms of molecular orbitals is $\sigma(2s)^2 \sigma^{*}(2s)^2 \sigma(2p_z)^2 \pi(2p_x)^2 \pi(2p_y)^2 \pi^*(2p_x)^1 \pi^*(2p_y)^1$

• The bond order is:

• $\text{Bond order} = \frac{(8 \text{ bonding electrons}) - (4 \text{ antibonding electrons})}{2} = 2$

• This indicates a double bond in O₂.

5. Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO)

• HOMO: The highest occupied molecular orbital is the molecular orbital that contains the highest-energy electrons in the molecule. It represents the outermost filled energy level, and these electrons are often involved in chemical reactions.

• LUMO: The lowest unoccupied molecular orbital is the molecular orbital that is closest in energy to the HOMO but is empty. The LUMO is where an electron would go if the molecule were excited (e.g., during absorption of light or electron transfer).

• The energy difference between the HOMO and LUMO is known as the HOMO-LUMO gap, which is important for understanding the reactivity and properties of the molecule, such as its color and electrical conductivity.

6. Visualization of Molecular Orbitals

Molecular orbitals can be visualized as regions in space where electrons are likely to be found. These regions are typically depicted as shapes or clouds around the nuclei, indicating the distribution of electron density:

• Bonding orbitals tend to have electron density concentrated between the nuclei.

• Antibonding orbitals have nodes (regions of zero electron density) between the nuclei, making the bond less stable.

7. Role of Molecular Orbitals in Chemical Bonding

• Bonding: Electrons in bonding MOs stabilize the molecule and lead to the formation of chemical bonds.

• Antibonding: Electrons in antibonding MOs destabilize the molecule and weaken chemical bonds.

• Non-bonding: Electrons in non-bonding MOs do not contribute to bonding or antibonding but can still influence the reactivity of the molecule (e.g., lone pairs).

Summary:

• Molecular orbitals describe how electrons are distributed in a molecule.

• They result from the combination of atomic orbitals, forming bonding, antibonding, and non-bonding orbitals.

• The energy levels and filling of these orbitals determine the molecule's bond order, stability, and reactivity.

• HOMO and LUMO are key orbitals in determining how a molecule interacts with light and other chemicals.

Method

The molecular orbitals are calculated in xTB 6.6.0 with the --molden argument.

Find

The Molecular Orbital can be found under the Scalar Field category in the property tree. Once selected, the orbital and isovalue can be set.