Quadrupole Splitting
The quadrupole splitting ΔE Q occurs when magnetic interactions between nuclear quadrupole moment and electric field gradient at the nuclear position are present:
\(\Delta E_{Q}=\frac{1}{2}\text{eQ}V_{3}\left[1+\frac{\left(V_{1}-V_{2}\right)^{2}}{3V_{3}^{2}}\ \right]^{1/2}\)(3)
Here, Q is the nuclear electric quadrupole moment for the nuclearI =3/2 state and Vi are the eigenvalues of the tensor representing the environmental electric field gradient,20 which arises when the field at the nuclear position is inhomogeneous due to deviations of the valence electron distribution from cubic symmetry.72
The sign of ΔE Q depends on the relative energy of the magnetically split substates of the nuclear excited state; in the case of Fe, these are states with I z = ±1/2 andI z = ±3/2. Lippard and coworkers have noted that due to the convention that V 3 be the largest eigenvalue, the sign of the quadrupole splitting can be predicted incorrectly in cases where V 1 is small andV 2 and V 3 are very close in magnitude.20 On the other hand, Pápai and Vankó later showed in their extensive correlation study that the sign is predicted correctly in all instances where it is known experimentally (19 of 66 complexes).21 In an MO picture, a perfectly symmetrical t2g3 configuration will produce no quadrupole splitting, while any asymmetry in the ligand sphere and, in the case of ionic species, counter ions will lead to an increase in quadrupole splitting assigned as lattice contributions. The nature of the ligand influences the relative magnitude and orientation of V 1–3, and thereby the sign of ΔE Q.
The variations in the valence electronic structure are quite subtle, and although evidently they can be measured experimentally, DFT calculations often appear not quite sensitive enough to represent the finer nuances of the asymmetry in electron density. The prediction of quadrupole splittings are therefore associated with larger errors than obtained for the isomer shift: correlation lines with R2-values of ca. 0.95 and mean absolute errors of 0.22 mm s−1 have been obtained with DFT.21 Furthermore, the quadrupole splitting value can be much more sensitive to changes in temperature, rendering low-temperature measurements important for an adequate comparison between experiment and theory.