Why are there both high-spin and low-spin octahedral complexes but only high- spin tetrahedral complexes?

Octahedral complexes are one of the most common types found in coordination chemistry. In these complexes, six ligands surround a central metal ion, producing an octahedral shape. This geometry causes the d-orbitals of the metal ion to split into two sets of energy levels due to the electrostatic field interactions between the ligands and the metal ion's d-orbitals. The resulting split places two orbitals, known as the \( e_g \) set (dx2-y2 and dz2), at a higher energy level, and the remaining three orbitals, known as the \( t_{2g} \) set (dxy, dxz, dyz), at a lower energy level. This energy difference is termed crystal field splitting energy (\( \Delta_{oct} \)). Whether a complex is high-spin or low-spin depends on the relative magnitudes of \( \Delta_{oct} \) and the electron pairing energy.

Tetrahedral complexes have four ligands surrounding the central metal ion, forming a tetrahedral shape. In this geometry, the d-orbitals split into different sets than in octahedral complexes. The splitting results in three orbitals (dxy, dxz, dyz) being at a higher energy level (known as the \( t_{2g} \) set), while two orbitals (dx2-y2 and dz2) are at a lower energy level (known as the \( e_g \) set). However, the crystal field splitting energy in tetrahedral complexes (\( \Delta_{tet} \)) is significantly smaller, roughly 4/9 of \( \Delta_{oct} \). Due to the smaller energy difference, the pairing energy required to force electrons into the lower energy orbitals is usually much higher than the crystal field splitting energy, leading to tetrahedral complexes generally adopting high-spin configurations.

The terms high-spin and low-spin refer to the number of unpaired electrons in a coordination complex. High-spin complexes have the maximum number of unpaired electrons because the crystal field splitting energy is not sufficient to overcome the pairing energy. This situation typically arises with weak field ligands. These weak field ligands produce a small \( \Delta_{oct} \) or \( \Delta_{tet} \), causing electrons to fill higher energy levels rather than pair.

Low-spin complexes, on the other hand, have fewer unpaired electrons. This happens when strong field ligands generate enough crystal field splitting energy to overcome the pairing energy, causing electrons to pair up in the lower energy orbitals. Only octahedral complexes can exhibit both high-spin and low-spin configurations because the \( \Delta_{oct} \) is more comparable to the pairing energy. Tetrahedral complexes, with their much smaller \( \Delta_{tet} \), do not have a sufficiently high field to promote low-spin configurations, thus they are always high-spin.