where α, β, and γ are the Cartesian indices, and Φ 0 , Φ α , Φ αβ , and Φ αβγ are zeroth, first, second, and third-order force constants, respectively. In this study, the secondorder, or "harmonic", force constants were calculated using the finite-difference method implemented in the postprocessing software PHONOPY . Within the harmonic approximation, the Helmholtz free energy of a crystal at constant volume can be expressed as