Figure presents the constant-pressure heat capacity (panel (a)) and the temperature dependent phonon dispersion of α-Li 3 N (panel (b)) calculated within the QHA from 0 K to 690 K, and the resulting properties are shown in Figure . The constant pressure heat capacity, C P , shown in Figure (a) is extrapolated from the free energies, while the thermal dependence of the phonon dispersion, shown in Figure (b) is calculated using the mode Grüneisen parameters defined in Equation . C P is linear around T s ≈ 678 K. There is no peak or discontinuity in the C P close to T s , which reflects the inability of the QHA to capture the α-Li 3 N superionic transition. The calculated values of the weighted average of the mode Grüneisen parameters range from -1.6 to 0.9, which results in mild softening of the phonon dispersion, as shown in Figure (b), but, again, gives no distinct mode softening at 678 K that might correspond to the superionic transition. In contrast, a QHA treatment of cubic CeO 2 does predict the emergence of imaginary modes at high temperature. For the case of Li 3 N, thermal expansion effects alone cannot be the physical mechanism driving the superionic transition observed at 678 K. This conclusion stresses that Grüneisen parameters, a common measure of anharmonicity, are not a general descriptor for superionic transitions. and power spectra, G qλ , using Equation . In contrast to the harmonic and quasi-anharmonic models discussed above, where we found no signature of the superionic transition, here a phonon breakdown occurs at 678 K (Figure ). Above 600 K there is a drastic increase in the average phonon linewidth (inversely proportional to the mode lifetime), which is associated with the large MSD of Lu ions in the supercell.