Free-energy calculations for the cubic ZrO2 crystal as an example of a system with a soft mode.

We calculate the free energy for a crystalline ZrO(2) with a soft mode by the first-principles method, using the double-well energy-displacement relation. The soft-mode branch is considered as an ensemble of independent anharmonic oscillators of the parabola-plus-Gaussian or of the 2-4 polynomial forms. The anharmonic contributions are included to reproduce the cubic-to-tetragonal phase transition, however, it appears that the cubic phase does not become the most stable within the framework of the independent oscillators approach.