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.

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.

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.

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.

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.