The complex with highest magnitude of crystal field splitting energy $$(\Delta_0)$$ is

The magnitude of Crystal Field Splitting Energy in octahedral complexes ($$Δ_o$$) depends on several key parameters:

• Oxidation State of the Metal Ion: Higher positive charges increase the electrostatic attraction for ligands, bringing them closer and raising $$Δ_o$$. Here, all options contain a $$+3$$ central metal ion ($$Ti^{3+},Cr^{3+},Mn^{3+},Fe^{3+}$$), so oxidation state cannot differentiate them.
  
• Nature of the Ligand: All options contain the identical weak-field aqua ligand ($$H_{2}O$$).
  
• Transition Series (3d vs 4d vs 5d): All ions belong to the same 3d transition series.
  
• Number of d-electrons ($$d^n$$ configuration): For identical oxidation states and ligands in the 3d series, $$Δ_o$$generally varies based on the size of the metal radius and the stabilization profile of the electronic configuration.

Electronic Configuration Profiles

Let's inspect the 3d configurations for each +3 central metal ion:

Why Chromium ($$Cr^{3+}$$) possesses the Maximum $$Δ_o$$

As we move from left to right along the 3d period ($$Ti^{3+} → V^{3+} → Cr^{3+}$$), the effective nuclear charge (Zeff) increases steadily while the ionic radius decreases. This contraction pulls the $$H_2O$$ ligands closer to the metal nucleus, resulting in progressively stronger repulsions and a larger splitting value ($$Δ_o$$).

However, past $$Cr^{3+}$$ (d3), electrons begin entering the high-energy eg orbitals (as seen in $$Mn^{3+}$$ (d4) and $$Fe^{3+}$$ (d5)). Electrons in eg orbitals point directly along the axes toward the incoming ligands, providing shielding that increases the ionic radius and dampens ligand-metal proximity. Thus, $$Δ_o$$drops sharply after the d3 point.

Therefore, the d3 configuration of Cr3+ marks the absolute peak value for $$Δ_o$$among hexaaqua trivalent ions of the first transition series.