Question

Show how a \(d_{x z}\) atomic orbital and a \(p_{z}\) atomic orbital combine to form a bonding molecular orbital. Assume the \(x\) -axis is the internuclear axis. Is a \(\sigma\) or a \(\pi\) molecular orbital formed? Explain.

Short answer

A \(d_{xz}\) atomic orbital and a \(p_z\) atomic orbital combine to form a bonding molecular orbital through the overlap of their lobes with the same sign, which occurs in the same plane. There is no direct interaction along the internuclear axis (x-axis). Therefore, a π (pi) molecular orbital is formed in this process.

Step-by-step solution

Visualize the orbitals involved

Firstly, let's visualize the \(d_{xz}\) and \(p_z\) atomic orbitals. The \(d_{xz}\) orbital lies in the xz-plane and has four lobes. In contrast, the \(p_z\) orbital has two lobes along the z-axis. Keep in mind that the internuclear axis is the x-axis.

Determine the possible overlap

To form a molecular orbital, orbitals must overlap in space. In this case, since the \(d_{xz}\) orbital lies in the xz-plane and the \(p_z\) orbital is aligned along the z-axis, there is no direct overlap along the internuclear (x) axis. The overlap occurs between \(d_{xz}\) and \(p_z\) lobes lying in the same plane.

Analyze the orbital overlap

The bonding molecular orbital formed results from the constructive interference of wave functions. This constructive interference happens when the wave functions of the orbitals have the same sign (either positive or negative). In this case, an overlap between one lobe of the \(d_{xz}\) and one lobe of the \(p_z\) orbital with the same sign forms the bonding molecular orbital.

Determine the type of molecular orbital formed

Now we need to decide if the molecular orbital formed is a σ (sigma) or a π (pi) orbital. A σ molecular orbital occurs when there is symmetry around the internuclear axis, while a π molecular orbital occurs when there is no symmetry around the internuclear axis. Since the overlap occurs only between lobes lying in the same plane and there is no direct interaction along the internuclear axis, the molecular orbital formed is a π (pi) molecular orbital.

Conclusion

In summary, a \(d_{xz}\) atomic orbital and a \(p_z\) atomic orbital can combine to form a bonding molecular orbital through the overlap of their lobes with the same sign. Since there is no direct interaction along the internuclear axis (x-axis), a π (pi) molecular orbital is formed in this process.