Question

Predict the crystal field energy-level diagram for a square pyramidal \(\mathrm{ML}_{s}\) complex that has two ligands along the \(\pm x\) and \(\pm y\) axes but only one ligand along the \(z\) axis. Your diagram should be intermediate between those for an octahedral \(\mathrm{ML}_{6}\) complex and a square planar \(\mathrm{ML}_{4}\) complex.

Short answer

In square pyramidal \(\text{ML}_5\), \(d_{z^2}\) is intermediate, \(d_{x^2-y^2}\) is highest, other \(d\) orbitals are slighly higher than in octahedral.

Step-by-step solution

Understand the Geometrical Arrangement

A square pyramidal \( \text{ML}_5 \) complex has a geometry where four ligands form a square plane around the metal ion on the \( xy \) plane, and one ligand is on the \( z \)-axis coordinating to the metal atom. This is intermediate between an octahedral \( \text{ML}_6 \) complex and a square planar \( \text{ML}_4 \) complex.

Determine the Electron Distribution for Octahedral Geometry

In an octahedral \(\text{ML}_6\) complex, the \(d\) orbitals split into two sets, \(t_{2g}\) (\(d_{xy}, d_{xz}, d_{yz}\)) and \(e_g\) (\(d_{z^2}, d_{x^2-y^2}\)), with the \(e_g\) set higher in energy.

Consider the Influence of the Square Planar Component

In a square planar \( \text{ML}_4 \) complex, the \(d_{z^2}\) orbital becomes non-bonding, experiencing less repulsion due to the absence of axial ligands, leaving orbitals \(d_{xy}\) and \(d_{x^2-y^2}\) in the \(xy\) plane as higher energy due to strong ligand interaction.

Predict the Crystal Field Splitting for Square Pyramidal

For the \( \text{ML}_5 \) complex, the \(d_{z^2}\) orbital will be intermediate in energy as one axial ligand is present. The \(d_{x^2-y^2}\) orbital should remain at a higher energy from plane repulsion. The \(d_{xy}, d_{xz}, \text{ and } d_{yz}\) will likely shift slightly higher than in octahedral arrangement but not as high as square planar full rearrangement.

Sketch the Energy-Level Diagram

Arrange the \(d_{xy}, d_{xz},\) and \(d_{yz}\) orbitals slightly higher than found in octahedral, but lower than square planar. The \(d_{z^2}\) falls between the energy levels of the fully octahedral and square planar settings, and \(d_{x^2-y^2}\) should be the highest energy orbital because of repulsion with in-plane ligands.

Key concepts

Square Pyramidal Complexes

In a square pyramidal complex, imagine placing four ligands at the corners of a square on the \(xy\) plane, while the fifth ligand sits above the square, along the \(z\) axis. This creates a distinct pyramid shape. The spatial arrangement affects the energy of the \(d\) orbitals.

The \(d_{z^2}\) orbital experiences less repulsion compared to an octahedral complex because there is only one axial ligand. This places it at an intermediate energy level. The \(d_{x^2-y^2}\) orbital faces more repulsion due to direct interaction with ligands in the square plane. It generally becomes the highest in energy among the \(d\) orbitals.

On the other hand, the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals are slightly higher in energy than in a typical octahedral complex but not to the extent seen in a square planar arrangement. This happens because they also have to interact with the ligands along the square plane, albeit to a lesser degree than \(d_{x^2-y^2}\).

• The energy levels of these orbitals can predict the electronic distribution.
• Understanding this arrangement is crucial for interpreting spectroscopic properties and magnetic behavior.

Octahedral Complexes

Octahedral complexes are common in coordination chemistry. Here, six ligands surround a central metal ion in a symmetrical fashion, positioned at the corners of an octahedron.

In this geometry, the \(d\) orbitals split into two distinct energy sets due to the ligands' electric field:

• \(t_{2g}\): \(d_{xy}\), \(d_{xz}\), \(d_{yz}\) - these are lower in energy because they lie between the axes where ligands are positioned.
• \(e_g\): \(d_{z^2}\), \(d_{x^2-y^2}\) - these are higher in energy since they directly interact with the ligands along the axes.

The difference in energy between these sets is known as the crystal field splitting energy. The splitting influences the color, magnetism, and reactivity of the complex.

In this arrangement, each ligand’s repulsion causes the \(e_g\) orbitals to face higher energy levels. It's the foundation for understanding more complex geometries and their electronic properties.

Square Planar Complexes

Square planar complexes are a type of four-coordinate complex where ligands occupy the corners of a square. This geometry significantly affects the distribution and energy of the \(d\) orbitals.

The absence of axial ligands leads to the \(d_{z^2}\) orbital being non-bonding, sitting at a lower energy level because it faces minimal ligand interaction. Meanwhile, the \(d_{x^2-y^2}\) orbital experiences maximum repulsion from the in-plane ligands, making it the highest in energy.

The \(d_{xy}\) orbital is slightly higher in energy compared to \(d_{z^2}\) but lower than \(d_{x^2-y^2}\), due to its orientation in the plane. The \(d_{xz}\) and \(d_{yz}\) orbitals are generally degenerate, facing less interaction and sitting at intermediate energy levels.

• Square planar complexes are often low-spin and involve metals that promote \(d^8\) configurations.
• This arrangement strongly influences the chemical and physical properties, affecting reaction mechanisms and stability.