Ab initio molecular dynamics (AIMD) simulations of cubic zirconia were performed with VASP. In all cases a time step of 2 ps and a Noose thermostat was used in an NVT emsemble. For standard AIMD simulations, a plane wave energy cuto↵ of 700 eV, a 4⇥4⇥4 primitive supercell and -point sampling was used. For the con-strained MD simulations, oxygen atoms were displaced along the X-mode and one coordinate for each oxygen was kept fixed. These simulations were performed with a plane wave energy cuto↵ of 400 eV, a 2⇥2⇥2 conventional supercell and 2⇥2⇥2 k-point sampling. The free energy gradient was extracted using the Blue Moon ensemble, and the free energy profile was determined through numerical integration .