Maximum coverage problem

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Ligand K-edge XAS is a spectroscopic technique that allows the direct study of metal-ligand bonding. In a XAS experiment, electrons in ligand 1s orbitals are excited to unfilled p (principal quantum number n <= 4) and continuum states. This results in a large spectral feature known as a rising edge. Transitions at energies lower than the edge can occur, provided they are to orbitals with some ligand p character, and are called pre-edges. Such transitions are often observed in the data of complexes with metal-ligand bonding.

Pre-edge intensities (D₀) are related to the amount of ligand (L) character in the unfilled orbital:

$D_0(L1s\to {\psi }^{*})=const|⟨L1s|\mathbf{r}|{\psi }^{*}⟩{|}^2={\alpha }^{2}const|⟨L1s\left|\mathbf{r}\right|Lnp⟩{|}^2$

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

$D_0=\frac{{\alpha }^{2}h}{3n}{I}_s$

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