Fast free-energy calculations for unstable high-temperature phases

We present a fast and accurate method to calculate vibrational properties for mechanically unstable high-temperature phases that suffer from imaginary frequencies at zero temperature. The method is based on standard finite-difference calculations with optimized large displacements and is significantly more efficient than other methods. We demonstrate its application for calculation of phonon dispersion relations, free energies, phase-transition temperatures, and vacancy-formation energies for body-centered-cubic high-temperature phases of Ti, Zr, and Hf.