# Ligand K-edge

Ligand K-edge spectroscopy is a spectroscopic technique used to study the electronic structures of metal-ligand complexes.[1] This method measures X-ray absorption caused by the excitation of ligand 1s electrons to unfilled p orbitals (principal quantum number n <= 4) and continuum states, which creates a characteristic absorption feature called the K-edge.

## Pre-edges

Transitions at energies lower than the edge can occur, provided they lead to orbitals with some ligand p character; these features are called pre-edges. Pre-edge intensities (D0) are related to the amount of ligand (L) character in the unfilled orbital:

${\displaystyle D_{0}(L\ 1s\rightarrow \psi ^{*})=const\ \vert \langle L\ 1s\vert \mathbf {r} \vert \psi ^{*}\rangle \vert ^{2}=\alpha ^{2}\ const\ \vert \langle L\ 1s\vert \mathbf {r} \vert L\ np\rangle \vert ^{2}}$

where ${\displaystyle \psi ^{*}}$ is the wavefunction of the unfilled orbital, r is the transition dipole operator, and ${\displaystyle \alpha ^{2}}$ is the "covalency" or ligand character in the orbital. Since ${\displaystyle \psi ^{*}={\sqrt {1-\alpha ^{2}}}\vert M_{d}\rangle -\alpha \vert L_{np}\rangle }$, the above expression relating intensity and quantum transition operators can be simplified to use experimental values:

${\displaystyle D_{0}={\frac {\alpha ^{2}h}{3n}}I_{s}}$

where n is the number of absorbing ligand atoms, h is the number of holes, and Is is the transition dipole integral which can be determined experimentally. Therefore, by measuring the intensity of pre-edges, it is possible to experimentally determine the amount of ligand character in a molecular orbital.