Sketch the \(d\) -orbital energy level diagram for a typical octahedral complex.

Crystal field splitting occurs when metal ions form complexes with ligands, molecules or ions that can donate a pair of electrons. In an octahedral complex, the d-orbitals split into two energy levels due to the electrostatic interactions with ligands positioned along the axes.

This splitting results in three d-orbitals (the t2g set: d_xy, d_xz, and d_yz) having lower energy and two (the e_g set: d_z², d_x²-y²) with higher energy. The energy difference between these sets is denoted as Δ_o (octahedral splitting energy).

This concept is key to understanding the behavior of transition metal complexes, including their magnetic properties and the color they exhibit in solutions. It's caused by the specific geometry of the octahedral arrangement, where ligands are symmetrically arranged around the central metal ion.

An octahedral complex forms when six ligands symmetrically surround a central metal ion at equal distances. The shape resembles an octahedron, a polyhedron with eight faces.

The understanding of an octahedral complex is pivotal as it frequently occurs in coordination chemistry. It influences the crystal field splitting and dictates the arrangement of an electron in d-orbitals. Due to the octahedral ligand field, electron repulsion is minimized when lower energy t2g orbitals are filled before higher energy e_g orbitals.

Electron Distribution in Orbtials

Hund's rule addresses the distribution of electrons across orbitals of the same energy. It states that electrons will fill degenerate orbitals singly as far as possible before pairing up.

Within an octahedral complex, once the t2g orbitals are singly filled, additional electrons then pair up in these orbitals to minimize electron-electron repulsions. Following Hund's rule results in the most stable arrangement with the highest possible spin states for these electrons.

The Pauli exclusion principle is a fundamental concept in quantum mechanics that restricts the placement of electrons in orbitals.

It states that no two electrons can have the same set of quantum numbers within an atom, meaning each orbital can hold a maximum of two electrons with opposite spins. When filling d-orbitals in a crystal field such as an octahedral complex, this principle ensures that electrons are distributed properly across the orbitals, maintaining the uniqueness of their quantum states.