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.