Step 1: Understanding the Concept:
Outer orbital complexes (high spin) use outer d-orbitals (\(nd\)) for hybridization, while inner orbital complexes (low spin) use inner (\(n-1\))d orbitals. This depends on the ligand field strength.
Step 2: Detailed Explanation:}}
\begin{itemize}
\item (A) \([Fe(CN)_6]^{4-}\): Fe\(^{2+}\) (d\(^6\)), CN\(^-\) is strong field ligand, pairing occurs, uses inner d orbitals (d\(^2sp^3\)), inner orbital complex.
\item (B) \([Mn(CN)_6]^{4-}\): Mn\(^{2+}\) (d\(^5\)), CN\(^-\) strong field, pairing occurs, uses inner d orbitals, inner orbital complex.
\item (C) \([Co(NH_3)_6]^{3+}\): Co\(^{3+}\) (d\(^6\)), NH\(_3\) is strong field for Co\(^{3+}\), pairing occurs, uses inner d orbitals (d\(^2sp^3\)), inner orbital complex.
\item (D) \([Ni(NH_3)_6]^{2+}\): Ni\(^{2+}\) (d\(^8\)), NH\(_3\) is moderate field but for Ni\(^{2+}\), it forms outer orbital complex using sp\(^3\)d\(^2\) hybridization (outer d orbitals).
\end{itemize}
Step 3: Final Answer:
\([Ni(NH_3)_6]^{2+}\) is an outer orbital complex, option (D).