We can further use the phonon calculations to compute the vibrational contributions to the temperature-dependent constant-volume (Helmholtz) free energy 𝐴: 𝐴(𝑇) = 𝑈 latt + 𝐴 vib (𝑇) = 𝑈 latt + 𝑈 vib (𝑇) -𝑇𝑆 vib (𝑇) (Eq. 2) where 𝐴 vib = 𝑈 vib (𝑇) -𝑇𝑆 vib (𝑇) is the contribution to the free energy from the phonon modes and 𝑈 vib and 𝑆 vib are the phonon internal energy and entropy respectively. When two phases are close in energy, differences in phonon spectra and 𝐴 vib can lead to qualitative changes in stability at 0 K and/or to temperature-induced phase transitions. 67 Bi2S3 and the R3 ̅ m phase for Bi2Se3 and Bi2Te3 (c.f. Table ).