Question

On the basis of crystal field theory write the electronic configuration for \( d^4 \) ion if \( \Delta_0 \lt P \).

Hint:

If \( \Delta_0 \lt P \), it forms a High Spin complex (\( t_{2g}^3 \, e_g^1 \)). If \( \Delta_0 \gt P \), it forms a Low Spin complex, where electrons pair up instead of jumping (\( t_{2g}^4 \, e_g^0 \)).

Answer 1

Concept: Crystal Field Theory (CFT) explains the splitting of degenerate d-orbitals into two sets of different energies when surrounded by ligands in an octahedral field:

• \( t_{2g} \): The lower energy set consisting of 3 orbitals (\(d_{xy}, d_{yz}, d_{zx}\)).

• \( e_g \): The higher energy set consisting of 2 orbitals (\(d_{x^2-y^2}, d_{z^2}\)).
The energy gap between them is the crystal field splitting energy (\( \Delta_0 \)). The term \( P \) stands for pairing energy (the energy required to force two electrons into the same orbital). Step 1: Evaluating the given condition (\( \Delta_0 \lt P \)).
The condition \( \Delta_0 \lt P \) indicates that the crystal field splitting energy is less than the energy required to pair up electrons.

• This situation arises in the presence of a weak field ligand.

• Because the energy gap (\( \Delta_0 \)) is small, it requires less energy for an electron to jump to the higher \( e_g \) orbitals than to pair up with another electron in the lower \( t_{2g} \) orbitals.

Step 2: Writing the electronic configuration for \( d^4 \).
According to Hund's rule and the Aufbau principle under these specific conditions:

• The first three electrons will singly occupy the three lower energy \( t_{2g} \) orbitals (Configuration: \( t_{2g}^3 \)).

• For the fourth electron, since \( \Delta_0 \lt P \), it is energetically more favorable to jump to the higher \( e_g \) orbital rather than pairing up in the \( t_{2g} \) level.

• Therefore, the fourth electron occupies an \( e_g \) orbital.
Final Answer: The electronic configuration for a \( d^4 \) ion under the condition \( \Delta_0 \lt P \) is \( t_{2g}^3 \, e_g^1 \).