A simple proof that third-order quadrupole perturbations of the NMR central transition of half-integral spin nuclei are zero

It has been known for a long time that the third-order quadrupole corrections to transitions from mz=-n/2 to mz=+n/2 are zero in the NMR of half-integer nuclei. However, the derivation has relied on deriving the corrections to the energy levels through somewhat laborious calculations. Only when the transitions between the levels were calculated was it revealed that the corrections to the transition frequency were zero. In this paper, we use Liouville-space methods to work with the transitions directly. Application of a recently published [A.D. Bain, Exact calculation, using angular momentum, of combined Zeeman and quadrupolar interactions in NMR, Mol. Phys. 101 (2003) 3163-3175] selection rule for the quadrupole coupling leads to a very simple proof that third-order corrections to the central and other symmetrical transitions are zero. The simplicity of the proof suggests there is a fundamental symmetry involved.