(a) What is the crystal field splitting energy \((\Delta) ?\) (b) How does it arise for an octahedral field of ligands? (c) How is it different for a tetrahedral field of ligands?

In an octahedral field, six ligands surround the metal ion at the vertices of an octahedron. This configuration affects the energy levels of the metal’s d orbitals. In the presence of these ligands, the initially degenerate d orbitals split into two distinct sets:

• The lower-energy set, known as the t_2g set includes the d_xy, d_xz, and d_yz orbitals
• The higher-energy set, known as the e_g set includes the d_z² and d_x²-y² orbitals

The energy difference between these two sets is the crystal field splitting energy, denoted as \(\text{Δ}_0\). The splitting occurs due to the electrostatic interactions between the ligands' electrons and the electrons in the metal ion's d orbitals. Because octahedral fields have more ligand interactions, the splitting is typically larger.

In a tetrahedral field, four ligands surround the metal ion at the vertices of a tetrahedron. This arrangement causes the d orbitals to split differently compared to an octahedral field. Here, the splitting results in:

• The higher-energy t₂ set includes the d_xy, d_xz, and d_yz orbitals
• The lower-energy e set includes the d_z² and d_x²-y² orbitals

The energy difference in a tetrahedral field is denoted as \(\text{Δ}_t\). This difference is smaller than \(\text{Δ}_0\) because the ligands in a tetrahedral arrangement interact less directly with the d orbitals, leading to a weaker splitting effect.

Ligands are ions or molecules that surround a central metal ion in a complex. Their interactions with the metal ion's d orbitals are the foundation of Crystal Field Theory. Strong-field ligands, such as CN⁻ or CO, cause a larger splitting of the d orbitals. Weak-field ligands, like I⁻ or Br⁻, result in smaller splitting.
The arrangement of these ligands around the metal ion determines how the d orbitals split and significantly influences the chemical and physical properties of the complex.

The d orbitals (d_xy, d_xz, d_yz, d_z², d_x²-y²) are five degenerate (having the same energy) orbitals in a free ion. However, when a metal ion is in a complex, the spatial arrangement and electrostatic interactions of surrounding ligands cause these orbitals to split into sets of different energies.
In an octahedral field, they split into a lower-energy t_2g set (d_xy, d_xz, d_yz) and a higher-energy e_g set (d_z², d_x²-y²). Conversely, in a tetrahedral field, the higher-energy set includes the t₂ orbitals (d_xy, d_xz, d_yz) and the lower-energy set includes the e orbitals (d_z², d_x²-y²).

Transition metal complexes consist of a central transition metal ion surrounded by ligands. These complexes exhibit a wide range of colors and magnetic properties due to the d orbitals' splitting. The crystal field splitting energy (\text{Δ}) varies depending on the geometry (octahedral or tetrahedral) and the nature of the ligands.
The value of \(\text{Δ}\) affects various properties such as the complex’s stability, color, and magnetic behavior. For instance, large \(\text{Δ}_0\) values in octahedral complexes often result in low-spin configurations, while smaller \(\text{Δ}_t\) values in tetrahedral complexes typically lead to high-spin configurations. These properties make the study of such complexes critically important in fields like catalysis, materials science, and bioinorganic chemistry.