For an octahedral complex, which of the following d electron configuration will give maximum crystal-field stabilisation energy?

Options

A. High spin with `d^6` configuration
B. Low spin with `d^4` configuration
C. Low spin with `d^5` configuration
D. High spin with `d^7` configuration

Answer

To determine which d-electron configuration in an octahedral complex provides the maximum crystal-field stabilization energy (CFSE), we need to analyze the configurations based on how electrons fill the t2g and eg orbitals.

### Step-by-Step Solution:

1. **Understanding Octahedral Splitting**:
- In octahedral complexes, the d-orbitals split into two sets: the lower energy t2g (dxy, dyz, dzx) and the higher energy eg (dx2-y2, dz2) orbitals. The energy difference between these two sets is denoted as Δo.

2. **High-spin vs Low-spin Configurations**:
- High-spin configurations fill the orbitals singly before pairing occurs, while low-spin configurations pair electrons in the lower energy t2g orbitals before occupying the eg orbitals.

3. **Crystal Field Stabilization Energy (CFSE) Formula**:
- The CFSE can be calculated using the formula:
\[ \text{CFSE} = (n_{t2g} \times -\frac{2}{5} \Delta_o) + (n_{eg} \times \frac{3}{5} \Delta_o) \]
- Where \( n_{t2g} \) is the number of electrons in the t2g orbitals and \( n_{eg} \) is the number of electrons in the eg orbitals.

4. **Calculating CFSE for Different Configurations**:
- Let's evaluate the CFSE for the given configurations:

**Configuration 1: d6 (t2g4 eg2)**
- CFSE = \( (4 \times -\frac{2}{5} \Delta_o) + (2 \times \frac{3}{5} \Delta_o) \)
- CFSE = \( -\frac{8}{5} \Delta_o + \frac{6}{5} \Delta_o = -\frac{2}{5} \Delta_o \)

**Configuration 2: d4 (t2g4 eg0)**
- CFSE = \( (4 \times -\frac{2}{5} \Delta_o) + (0 \times \frac{3}{5} \Delta_o) \)
- CFSE = \( -\frac{8}{5} \Delta_o \)

**Configuration 3: d5 (t2g5 eg0)**
- CFSE = \( (5 \times -\frac{2}{5} \Delta_o) + (0 \times \frac{3}{5} \Delta_o) \)
- CFSE = \( -2 \Delta_o \)

**Configuration 4: d7 (t2g5 eg2)**
- CFSE = \( (5 \times -\frac{2}{5} \Delta_o) + (2 \times \frac{3}{5} \Delta_o) \)
- CFSE = \( -2 \Delta_o + \frac{6}{5} \Delta_o = -\frac{4}{5} \Delta_o \)

5. **Comparing CFSE Values**:
- From the calculations:
- Configuration 1 (d6): CFSE = \(-\frac{2}{5} \Delta_o\)
- Configuration 2 (d4): CFSE = \(-\frac{8}{5} \Delta_o\)
- Configuration 3 (d5): CFSE = \(-2 \Delta_o\)
- Configuration 4 (d7): CFSE = \(-\frac{4}{5} \Delta_o\)

- The configuration that provides the maximum CFSE (least negative value) is **d6 (t2g4 eg2)** with a CFSE of \(-\frac{2}{5} \Delta_o\).

### Conclusion:
The d-electron configuration that gives maximum crystal-field stabilization energy in an octahedral complex is **d6 (t2g4 eg2)**.