Theoretical Description of the Proton Magnetic Resonance Line Shapes of Ammonium Ions under the Influence of Tunneling

A theoretical treatment for the NMR absorption line shape of a tetrahedral group of protons under the influence of tunneling has been developed for the particular case of N${\mathrm{H}}_4^{+}$ ions. For the applied field parallel to a twofold axis of the ion the derived line shapes depend on a single splitting parameter $J$. This is a measure of the tunneling splitting of the torsional ground state in comparison to the dipolar energy of the ion. As $J$ varies from 0 to $\infty$ the calculated line shapes are found to vary between the limiting cases of the distinguishable proton (four spin ½) and indistinguishable proton (spin isomeric) situations, respectively. These theoretical line shapes are compared with the observed "rigid-lattice" line shapes reported for the halides N${\mathrm{H}}_4$Cl, N${\mathrm{H}}_4$Br, and N${\mathrm{H}}_4$I. N${\mathrm{H}}_4$Cl and N${\mathrm{H}}_4$Br are found to be consistent with $J=0\mathrm{}$ although the line shape of the latter exhibits an unexplained departure near the center of the resonance. N${\mathrm{H}}_4$I is found to exhibit observable splitting effects ($J=3.3\mathrm{}$) in which tunneling has displaced one absorption component sufficiently into the wings to be resolved.