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

The \(d_{xz}\) and \(p_{z}\) atomic orbitals overlap along the z-axis, resulting in a bonding molecular orbital. The combination of these orbitals has cylindrical symmetry around the internuclear x-axis, leading to the formation of a \(\sigma\) molecular orbital.

Step-by-step solution

Understanding the shapes and orientations of atomic orbitals

In order to understand how the two atomic orbitals combine, we need to review the shapes and orientations of each atomic orbital. The \(d_{xz}\) atomic orbital is one of the five d orbitals, with its lobes lying along the x and z axes, bisecting the xz plane. Its maximum electron density is found along these axes, meaning that it is aligned in a way that can overlap with another orbital along those axes. The \(p_{z}\) atomic orbital is one of the three p orbitals and has a dumbbell shape, with its electron density concentrated along the z-axis. It has two lobes, one above and one below the x-axis.

Overlap of atomic orbitals

Now that we have an understanding of the individual orbitals, we can analyze the overlap between the \(d_{xz}\) and \(p_{z}\) atomic orbitals. The \(d_{xz}\) atomic orbital has electron density along both the x and z axes, while the \(p_{z}\) atomic orbital has electron density along the z-axis. Thus, the overlap between the orbitals occurs along the z-axis.

Formation of molecular orbital

When atomic orbitals combine, they can form bonding or anti-bonding orbitals depending on the alignment of their wave functions. In this case, a bonding molecular orbital will result, as the electron density from the \(d_{xz}\) and \(p_{z}\) orbitals is aligned along the z-axis, allowing for constructive interference and reinforcing the electron density in that region. As such, a new molecular orbital is formed.

Determining the type of molecular orbital

To determine if the molecular orbital is a \(\sigma\) or a \(\pi\) orbital, we need to consider the symmetry of the bond with respect to the internuclear axis. As previously mentioned, the internuclear axis is the x-axis in this case. A \(\sigma\) bond is cylindrically symmetric around the internuclear axis, while a \(\pi\) bond has cylindrical symmetry on planes parallel to the internuclear axis but not around it. Given that the combination between the \(d_{xz}\) and \(p_{z}\) atomic orbitals occurs along the z-axis and has cylindrical symmetry around the internuclear x-axis, the resulting molecular orbital is a \(\sigma\) molecular orbital. In conclusion, the overlap between a \(d_{xz}\) atomic orbital and a \(p_{z}\) atomic orbital along the z-axis forms a bonding \(\sigma\) molecular orbital.