Question

Which of the following d-orbitals experience more repulsion in the crystal field splitting of octahedral complex ?

• A. \( d_{xy}, d_{yz}, d_{xz} \)
• B. \( d_{x^2-y^2}, d_{z^2} \)
• C. \( d_{xy}, d_{x^2-y^2} \)
• D. \( d_{xz}, d_{z^2} \)

Hint:

In Octahedral splitting: \( e_{g} \) (axial) is high energy, \( t_{2g} \) (non-axial) is low energy.
In Tetrahedral splitting, the order is reversed because ligands approach between the axes.

Answer 1

The Correct Option is B

Concept:

• Crystal Field Theory (CFT) describes the splitting of degenerate d-orbitals of a metal ion in the presence of a ligand field.

• In an octahedral complex, the central metal ion is surrounded by six ligands located at the corners of a regular octahedron.

• These ligands approach the metal ion along the Cartesian axes (\( x, y, \) and \( z \) axes).

• Electrostatic repulsion occurs between the lone pairs of the ligands and the electrons in the metal's d-orbitals.
Step 1: Classify the geometry of d-orbitals
The five d-orbitals are oriented differently in 3D space:

1. \( d_{xy}, d_{yz}, d_{xz} \): These are called non-axial orbitals because their lobes lie between the axes at \( 45^\circ \) angles.
2. \( d_{x^2-y^2}, d_{z^2} \): These are called axial orbitals because their lobes point directly along the \( x, y, \) and \( z \) axes.

Step 2: Analyze the direction of ligand approach
In an octahedral geometry, the six ligands approach specifically along the \( \pm x, \pm y, \) and \( \pm z \) directions.
This means the ligands head directly toward any orbital that is oriented along these axes.

Step 3: Evaluate the magnitude of repulsion
Orbitals pointing directly at the ligands (\( d_{x^2-y^2} \) and \( d_{z^2} \)) experience strong electrostatic repulsion.
Orbitals pointing between the ligands (\( d_{xy}, d_{yz}, d_{xz} \)) experience relatively less repulsion.

Step 4: Conclusion on splitting
Due to higher repulsion, the energy of the axial orbitals (\( e_{g} \) set) increases more than the energy of the non-axial orbitals (\( t_{2g} \) set).
Therefore, \( d_{x^2-y^2} \) and \( d_{z^2} \) experience more repulsion.