Step 1: Understanding the Question:
We need to determine the magnetic behavior and d-d transition characteristics of an octahedral complex with a \(t_{2g}^4 e_g^0\) electronic configuration.
This configuration represents a low-spin \(d^4\) system in an octahedral crystal field.
Step 2: Key Formula or Approach:
1. For magnetic properties: Distribute the 4 electrons in the split d-orbitals (\(t_{2g}\) and \(e_g\)) using Hund's rule of maximum multiplicity to count the number of unpaired electrons.
2. For d-d transitions: Use the spin selection rule, which states that electronic transitions are spin-allowed if there is no change in the net spin multiplicity (\(\Delta S = 0\)).
Step 3: Detailed Explanation:
1. In an octahedral field, the five d-orbitals split into the lower-energy \(t_{2g}\) triplet and the higher-energy \(e_g\) doublet.
2. The given configuration is \(t_{2g}^4 e_g^0\).
3. Distributing the 4 electrons in the three \(t_{2g}\) orbitals:
- Place one electron in each of the three orbitals: \(\uparrow, \uparrow, \uparrow\)
- The fourth electron is forced to pair up in the first available orbital: \(\uparrow\downarrow, \uparrow, \uparrow\)
4. Counting the unpaired electrons:
- Number of unpaired electrons (\(n\)) = 2.
- Since \(n = 2\), the complex is paramagnetic.
5. Now analyze the selection rule for d-d transitions:
- A d-d transition involves promoting an electron from a \(t_{2g}\) orbital to an \(e_g\) orbital.
- Since there are paired electrons in the ground state, an electron can undergo excitation to the empty \(e_g\) level without changing its spin direction.
- This allows the excited state to have the same number of unpaired electrons (and thus the same total spin) as the ground state.
- Since a transition can occur with \(\Delta S = 0\), the transition is spin-allowed.
6. This matches the properties described in Option (B).
Step 4: Final Answer:
The complex is paramagnetic with 2 unpaired electrons and exhibits spin-allowed d-d transitions, which corresponds to Option (B).