$K_4[Fe(CN)_6]$ Complex and $Fe^{2+}$ Ion Analysis

The complex under consideration is potassium hexacyanoferrate(II), with the formula $K_4[Fe(CN)_6]$. This compound contains the iron ion ($Fe$) in the +2 oxidation state.

• Oxidation state of Fe: $+2$
• Electronic configuration of $Fe^{2+}$: $d^6$

Ligand Field Strength and Spin State

The cyanide ligand ($CN^-$) is a strong field ligand. In an octahedral coordination environment, as in the $[Fe(CN)_6]^{4-}$ complex ion, a $d^6$ metal ion with strong field ligands adopts a low-spin configuration.

Ground State Term Symbol $^1A_{1g}$

For a $d^6$ ion in a strong octahedral field, the electron configuration is $t_{2g}^6 e_g^0$. This configuration results in a single, doubly degenerate ground state.

• Electron configuration: $t_{2g}^6 e_g^0$
• Ground state term symbol: $^1A_{1g}$

Allowed Transitions ($^1A_{1g}$ to Excited States)

Allowed transitions in coordination complexes are typically spin-allowed ($d-d$ transitions). For the $^1A_{1g}$ ground state, spin-allowed transitions must lead to excited states with the same spin multiplicity (singlet, S=0).

Promotion of an electron from the $t_{2g}$ orbital to the $e_g$ orbital ($t_{2g} \rightarrow e_g$) leads to a $t_{2g}^5 e_g^1$ configuration. The term symbols derived from this configuration include singlet states ($^1T_{1g}$ and $^1T_{2g}$) and triplet states ($^3T_{1g}$, $^3T_{2g}$, $^3E_g$, $^3A_{2g}$).

The spin-allowed transitions from the ground state $^1A_{1g}$ are to the excited singlet states:

• $^1A_{1g} \rightarrow ^1T_{1g}$
• $^1A_{1g} \rightarrow ^1T_{2g}$

These transitions correspond to the correct options B and C.