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.

Figure 1

(a) The energy vs phonon amplitude curve and (b) its second derivative for an $N$-point frozen phonon cell[10] displacement of the bcc cell of Hf (solid) and Mo (dashed).

Figure 2

Large-displacement calculated phonon dispersions for (a) Ti, (b) Zr, and (c) Hf vs experimental phonon energies from [1, 2, 3]. Solid lines are calculated with $4\times 4\times 4$ phonon supercells, and the dashed line for Hf is calculated with an $8\times 8\times 8$ supercell.

Figure 3

Calculated temperature dependence of bulk modulus for (a) Ti, (b) Zr, and (c) Hf, normalized to the transition temperature of each metal, with corresponding experimental values from Refs. 2, 3, and 19.