Match Tetrahedral Metal Ion Configurations with Crystal Field Stabilization Energy

To determine the Crystal Field Stabilization Energy (CFSE) for tetrahedral metal ion configurations, we use Crystal Field Theory (CFT) for tetrahedral geometry.

In a tetrahedral crystal field:

• The $d$-orbitals split into two sets: lower energy $e$ orbitals (comprising $d_{x^2-y^2}$ and $d_{z^2}$) and higher energy $t_2$ orbitals (comprising $d_{xy}, d_{yz}, d_{xz}$).
• The energy of each electron in an $e$ orbital is lowered by $-0.6 \Delta_t$.
• The energy of each electron in a $t_2$ orbital is raised by $+0.4 \Delta_t$.

Tetrahedral complexes are predominantly high-spin due to the relatively small splitting energy ($\Delta_t$).

The general formula for CFSE in units of $\Delta_t$ is: $\text{CFSE} = \left[ n_e (-0.6) + n_{t_2} (+0.4) \right] \Delta_t$ where $n_e$ is the number of electrons in $e$ orbitals and $n_{t_2}$ is the number of electrons in $t_2$ orbitals.

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Step-by-Step Calculation for List-I Configurations:

1. A. $d^2$ configuration:

   • Electronic configuration: $e^2 \, t_2^0$
   • $\text{CFSE} = [2 \times (-0.6) + 0 \times (+0.4)] \Delta_t = -1.2 \Delta_t$
   • Matches with III.

2. B. $d^4$ configuration:

   • Electronic configuration (high-spin): $e^2 \, t_2^2$
   • $\text{CFSE} = [2 \times (-0.6) + 2 \times (+0.4)] \Delta_t = (-1.2 + 0.8) \Delta_t = -0.4 \Delta_t$
   • Matches with IV.

3. C. $d^6$ configuration:

   • Electronic configuration (high-spin): $e^3 \, t_2^3$
   • $\text{CFSE} = [3 \times (-0.6) + 3 \times (+0.4)] \Delta_t = (-1.8 + 1.2) \Delta_t = -0.6 \Delta_t$
   • Matches with I.

4. D. $d^8$ configuration:

   • Electronic configuration: $e^4 \, t_2^4$
   • $\text{CFSE} = [4 \times (-0.6) + 4 \times (+0.4)] \Delta_t = (-2.4 + 1.6) \Delta_t = -0.8 \Delta_t$
   • Matches with II.

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Conclusion:

The correct matching is: $\text{A - III, B - IV, C - I, D - II}$

This corresponds to Option C.