For a given metal ion and set of ligands, is the crystal-field splitting energy larger for a tetrahedral or an octahedral geometry?

Crystal field theory is a model used to explain the electronic structure and properties of transition metal complexes. In this theory, the interaction between a metal ion and the ligands is considered to be purely electrostatic. When ligands approach a metal ion, they interact with the metal's d-orbitals and create an uneven pattern of repulsion, leading to a splitting of the d-orbitals into higher and lower energy levels.

Tetrahedral geometry has four ligands surrounding the metal ion in a tetrahedral arrangement. In this geometry, the d-orbitals split into two sets: the lower energy d_xy, d_yz, and d_xz, and the higher energy d_x^2-y^2 and d_z^2 orbitals. Octahedral geometry has six ligands surrounding the metal ion in an octahedral arrangement. In this geometry, the d-orbitals also split into two sets: the lower energy d_x^2-y^2 and d_z^2 orbitals, and the higher energy d_xy, d_yz, and d_xz orbitals.

The crystal-field splitting energy is the energy difference between the lower and higher energy d-orbitals. The splitting energy for tetrahedral and octahedral geometries depends on the position and electrostatic effect of the ligands surrounding the metal ion. For tetrahedral geometry, the splitting is less intense because the ligands are farther from the metal ion and interact less with the d-orbitals compared to an octahedral geometry. The magnitude of the crystal-field splitting energy for a tetrahedral geometry is typically about 4/9 times that of an octahedral geometry for the same set of ligands and metal ion. Therefore, the crystal-field splitting energy is larger for an octahedral geometry than for a tetrahedral geometry for a given metal ion and set of ligands.