For the same type of ligands, explain why the crystal field splitting for an octahedral complex is always greater than that for a tetrahedral complex.

In crystal field theory, an octahedral complex represents a molecular structure where a central metal ion is surrounded by six ligands positioned at the corners of an octahedron. This geometry creates a highly symmetrical arrangement that significantly influences the behavior of the metal's d-orbitals. Due to the octahedral arrangement, the ligands induce an electrostatic field that affects the energies of these d-orbitals, causing them to split into different energy levels.
An easy way to visualize this is by imagining a sphere with six points touching its surface from all directions along the coordinate axes. These points represent the positions of the ligands. The metal ion at the center experiences repulsions from these ligands, which leads to a unique distribution in the energy levels of its d-orbitals, specifically into a lower-energy set (t2g) and a higher-energy set (eg). The crystal field splitting energy (Δoct) in an octahedral complex is a crucial factor in determining its color, magnetic properties, and stability.

In contrast to an octahedral complex, a tetrahedral complex features a central metal ion surrounded by four ligands at the corners of a tetrahedron. This arrangement generates less symmetry compared to the octahedral geometry.
Imagine a pyramid with the metal ion at the center; the ligands are positioned at each of the four corners. When compared to an octahedral complex, the ligands in a tetrahedral complex do not approach the metal ion directly along the coordinate axes, resulting in less severe interactions with the d-orbitals. This translates to a different kind of splitting pattern, where three d-orbitals rise in energy to form the set labeled as t2, and the remaining two drop in energy to form the e set. The splitting energy for a tetrahedral complex (Δtet) is always smaller than that of an octahedral complex. Notably, the difference in the crystal field splitting energy is an essential factor in interpreting the physical and chemical properties of these complexes.

D-orbitals are a set of five orbitals found in the d-subshell of an atom's electron configuration, known conventionally as dxy, dxz, dyz, dx2-y2, and dz2. These orbitals play a significant role in transition metal chemistry due to their involvement in bond formation with ligands.
Under the influence of ligand fields, these five degenerate (meaning they have the same energy) d-orbitals can undergo splitting into groups of differing energies. The exact pattern of splitting and the number of orbitals in each energy level depend on the geometry of the metal complex. The characteristic shapes of d-orbitals dictate how they will interact with incoming ligands. The d-orbitals also determine the electronic configuration of the metal ion, which in turn affects the magnetic properties and possible electronic transitions that can occur, influencing the color of the complex.

Ligand field theory is an advanced conceptual model that helps explain the splitting of d-orbitals in transition metal complexes. It can be considered an extension of crystal field theory, taking into account not just the electrostatic interactions but also the covalent aspects between the ligands and the metal ion.
Ligand field theory is incredibly useful in understanding the electronic structure of complexes, predicting their magnetic properties, and elucidating the mechanisms of their chemical reactions. By describing how ligands affect the energy levels of d-orbitals, the theory aids in comprehending the absorption of light and the resulting color changes observed in various metal complexes. In essence, the theory helps bridge the gap between the physical observations and the intricate electronic interactions within coordination complexes.