The number of spin-allowed d-d transitions in a complex ion depends on its electronic configuration, geometry, and the ligand field splitting. For a complex ion to exhibit three spin-allowed d-d transitions, there must be at least three distinct energy levels accessible from the ground state via transitions between split d-orbitals, without changing the spin multiplicity.

Analyzing Complex Ions for Three Spin-Allowed d-d Transitions

We analyze each complex ion presented in the options:

1. $[V(H₂O)₆]³⁺$

• Metal ion: V³⁺
• Electronic configuration: $d²$
• Geometry: Octahedral (O_h)
• Splitting: $t_2g$ and $e_g$ levels
• Ground state term symbol (O_h): $³T_1g(F)$
• Transitions: Three spin-allowed transitions occur: $³T_1g(F)$ $\rightarrow$ $³T_2g$, $³T_1g(F)$ $\rightarrow$ $³T_1g(P)$, and $³T_1g(F)$ $\rightarrow$ $³A_2g$.
• Verdict: Exhibits three spin-allowed d-d transitions.

2. $[Ni(H₂O)₆]²⁺$

• Metal ion: Ni²⁺
• Electronic configuration: $d⁸$
• Geometry: Octahedral (O_h)
• Splitting: $t_2g$ and $e_g$ levels
• Ground state term symbol (O_h): $³A_2g$
• Transitions: Three spin-allowed transitions occur: $³A_2g$ $\rightarrow$ $³T_2g$, $³A_2g$ $\rightarrow$ $³T_1g(F)$, and $³A_2g$ $\rightarrow$ $³T_1g(P)$.
• Verdict: Exhibits three spin-allowed d-d transitions.

3. $[Cr(H₂O)₆]³⁺$

• Metal ion: Cr³⁺
• Electronic configuration: $d³$
• Geometry: Octahedral (O_h)
• Splitting: $t_2g$ and $e_g$ levels
• Ground state term symbol (O_h): $⁴A_2g$
• Transitions: Three spin-allowed transitions occur: $⁴A_2g$ $\rightarrow$ $⁴T_2g$, $⁴A_2g$ $\rightarrow$ $⁴T_1g(F)$, and $⁴A_2g$ $\rightarrow$ $⁴T_1g(P)$.
• Verdict: Exhibits three spin-allowed d-d transitions.

4. $[NiCl₄]^2-$

• Metal ion: Ni²⁺
• Electronic configuration: $d⁸$
• Geometry: Tetrahedral (T_d)
• Splitting: $A₂$, $T₁$, $T₂$, $E$ levels (different order than O_h)
• Ground state term symbol (T_d): $³A₂$
• Transitions: Three spin-allowed transitions occur: $³A₂$ $\rightarrow$ $³T₂$, $³A₂$ $\rightarrow$ $³T₁(F)$, and $³A₂$ $\rightarrow$ $³T₁(P)$.
• Verdict: Exhibits three spin-allowed d-d transitions.

5. $[Ti(H₂O)₆]³⁺$

• Metal ion: Ti³⁺
• Electronic configuration: $d¹$
• Geometry: Octahedral (O_h)
• Transitions: Only one spin-allowed transition possible: $t_2g$ $\rightarrow$ $e_g$ ($²T_2g$ $\rightarrow$ $²E_g$).
• Verdict: Exhibits only one spin-allowed d-d transition.

6. $[Mn(H₂O)₆]²⁺$

• Metal ion: Mn²⁺
• Electronic configuration: $d⁵$ (high spin)
• Geometry: Octahedral (O_h)
• Ground state term symbol (O_h): $⁶A_1g$
• Transitions: Only one spin-allowed transition: $⁶A_1g$ $\rightarrow$ $⁶T_1g$. Other transitions are spin-forbidden.
• Verdict: Exhibits only one spin-allowed d-d transition.

7. $[FeF₆]^3-$

• Metal ion: Fe³⁺
• Electronic configuration: $d⁵$ (high spin)
• Geometry: Octahedral (O_h)
• Ground state term symbol (O_h): $⁶A_1g$
• Transitions: Only one spin-allowed transition: $⁶A_1g$ $\rightarrow$ $⁶T_1g$. Other transitions are spin-forbidden.
• Verdict: Exhibits only one spin-allowed d-d transition.

8. $[FeCl₄]^2-$

• Metal ion: Fe²⁺
• Electronic configuration: $d⁶$
• Geometry: Tetrahedral (T_d)
• Ground state term symbol (T_d): $⁵E$
• Transitions: Two spin-allowed transitions: $⁵E$ $\rightarrow$ $⁵T₂$ and $⁵E$ $\rightarrow$ $⁵T₁$.
• Verdict: Exhibits two spin-allowed d-d transitions.

Conclusion

Based on the analysis:

• Option A: $[V(H₂O)₆]³⁺$ (3 transitions) and $[Ni(H₂O)₆]²⁺$ (3 transitions) are both correct.
• Option B: $[Cr(H₂O)₆]³⁺$ (3 transitions) and $[NiCl₄]^2-$ (3 transitions) are both correct.
• Option C: $[Ti(H₂O)₆]³⁺$ (1 transition) and $[Mn(H₂O)₆]²⁺$ (1 transition) are incorrect.
• Option D: $[FeF₆]^3-$ (1 transition) and $[FeCl₄]^2-$ (2 transitions) are incorrect.

Therefore, the correct pairs are those in options A and B.