Low spin complex of `d^6`-cation in an octahedral field will have the following energy:

Options

A. (a) `(-12)/(5)Delta_0+P`
B. (b) `(-12)/(5)Delta_0+3P`
C. (c) `(-2)/(5)Delta_0+2P`
D. (d) `(-2)/(5)Delta_0+P`

Explanation

To solve the question regarding the energy of a low-spin complex of a \(d^6\) cation in an octahedral field, we can follow these steps:

### Step 1: Understand the Electron Configuration
In an octahedral field, the \(d\) orbitals split into two sets: the lower energy \(t_{2g}\) (which can hold up to 6 electrons) and the higher energy \(e_g\) (which can hold up to 4 electrons). For a \(d^6\) configuration, we need to determine how the electrons will be arranged.

**Hint:** Remember that in a low-spin complex, strong field ligands cause pairing of electrons in the lower energy orbitals before filling the higher energy orbitals.

### Step 2: Filling the Orbitals
For a low-spin \(d^6\) complex, the electrons will first fill the \(t_{2g}\) orbitals and pair up before any electrons occupy the \(e_g\) orbitals. The arrangement will be:
- \(t_{2g}\): 6 electrons (3 pairs)
- \(e_g\): 0 electrons

**Hint:** Visualize the orbital filling and pairing process. The \(t_{2g}\) orbitals will be completely filled with paired electrons.

### Step 3: Calculate the Crystal Field Stabilization Energy (CFSE)
The CFSE can be calculated using the formula:
\[ \text{CFSE} = (n_{t_{2g}} \times -\frac{2}{5} \Delta_o) + (n_{e_g} \times \frac{3}{5} \Delta_o) \]
Where:
- \(n_{t_{2g}} = 6\) (number of electrons in \(t_{2g}\))
- \(n_{e_g} = 0\) (number of electrons in \(e_g\))
- \(\Delta_o\) is the crystal field splitting energy.

Substituting the values:
\[ \text{CFSE} = (6 \times -\frac{2}{5} \Delta_o) + (0 \times \frac{3}{5} \Delta_o) = -\frac{12}{5} \Delta_o \]

**Hint:** Pay attention to the signs in the CFSE calculation. The \(t_{2g}\) orbitals provide stabilization, while the \(e_g\) orbitals do not contribute since they are unoccupied.

### Step 4: Consider the Pairing Energy
In addition to the CFSE, we need to consider the pairing energy. For each pair of electrons in the \(t_{2g}\) orbitals, there is a pairing energy cost. Since we have 3 pairs, the total pairing energy will be \(3P\), where \(P\) is the pairing energy.

**Hint:** Remember that pairing energy is an additional energy cost that must be accounted for when calculating the total energy of the complex.

### Step 5: Total Energy Calculation
The total energy \(E\) of the low-spin complex can be expressed as:
\[ E = \text{CFSE} + \text{Pairing Energy} \]
Substituting the values:
\[ E = -\frac{12}{5} \Delta_o + 3P \]

**Hint:** This total energy reflects both the stabilization from the CFSE and the destabilization from the pairing energy.

### Conclusion
The energy of the low-spin \(d^6\) cation in an octahedral field is given by:
\[ E = -\frac{12}{5} \Delta_o + 3P \]

This expression shows how the strong field ligands lead to a significant stabilization effect through CFSE, while also considering the energy cost of pairing.