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.