Trigonal Bipyramidal Geometry and d Orbitals

In coordination chemistry, the interaction between the central metal atom's d orbitals and the ligand orbitals is crucial for understanding bonding and electronic structure. For a metal complex with trigonal bipyramidal (TBP) geometry, the central metal is surrounded by five ligands. Three ligands are in the equatorial plane (forming a triangle), and two ligands are in the axial positions (along the z-axis, perpendicular to the equatorial plane).

Sigma Bonding in TBP Complexes

When considering only sigma (σ) bonding, the ligand orbitals point directly towards the central metal atom. The metal d orbitals must have appropriate symmetry to overlap with these sigma ligand orbitals to form sigma bonding and antibonding molecular orbitals. The five d orbitals are:

• \$d_{z^2}\$
• \$d_{x^2-y^2}\$
• \$d_{xy}\$
• \$d_{xz}\$
• \$d_{yz}\$

Interaction of d Orbitals with Sigma Ligands

Let's analyze how each d orbital interacts with the sigma ligands in TBP geometry:

• \$d_{z^2}\$: This orbital has electron density along the z-axis and a torus in the xy plane. It has significant overlap with the two axial ligands (along the z-axis) and also interacts with the equatorial ligands due to the torus shape. This orbital participates in sigma bonding.
• \$d_{x^2-y^2}\$ and \$d_{xy}\$: These orbitals lie in the xy plane, oriented along the axes (\$d_{x^2-y^2}\$) or between the axes (\$d_{xy}\$). They have significant overlap with the three equatorial ligands. These orbitals participate in sigma bonding.
• \$d_{xz}\$ and \$d_{yz}\$: These orbitals lie in the xz and yz planes, respectively. In the TBP geometry, the axial ligands are along the z-axis and the equatorial ligands are in the xy plane. The lobes of the \$d_{xz}\$ and \$d_{yz}\$ orbitals point between the ligand positions. Therefore, they have very poor or no effective sigma overlap with the sigma ligand orbitals in this geometry.

Non-bonding d Orbitals

Based on the symmetry analysis of sigma interactions, the d orbitals that do not have effective overlap with the sigma ligand orbitals will remain non-bonding orbitals when only sigma bonding is considered. These are the orbitals whose symmetry does not match that of the sigma ligand combination orbitals. In the TBP geometry, considering only sigma interactions, the \$d_{xz}\$ and \$d_{yz}\$ orbitals do not participate in sigma bonding.

Thus, the d orbitals that remain non-bonding in the MO diagram of a metal complex with trigonal bipyramidal (TBP) geometry, considering only σ - bonding, are \$d_{xz}\$ and \$d_{yz}\$.