Beyond free energies, phonon calculations come with an additional indicator of crystal stability. Displacements that result in a lower energy structure manifest as imaginary phonon frequencies. Thus, a material with no imaginary frequencies is locally stable. If the instability involves a phonon mode away from the centre of the Brillouin zone, this structural transition will require an expansion of the crystallographic unit cell along a particular direction. For example, most cubic perovskites feature instabilities associated with octahedral tilting that require expansions of the unit cell to describe. In practice, many structures, including a wide range of cubic perovskites, predicted to be dynamically unstable from harmonic phonon theory are in fact stable at finite temperatures. This results from thermal renormalisation of the phonon frequencies, which will be considered in detail below. Furthermore, the vibrational free energy in Eq. 12 becomes ill-defined for imaginary modes. Thus, harmonic phonon theory will fundamentally fail to predict the thermodynamic stability of these structures at elevated temperatures-also when the HA is only used to compare free energies. To model such cases, anharmonic contributions are essential.