Thermal Expansion: Thus far, for renormalization, volume expansion is not taken into account. If renormalization predicts completely real phonons at some temperature (T r ), the Gruneisen parameters and the coefficients of thermal expansion (CTE) are calculated straight from third-order IFCs. Once CTE values are accrued at multiple temperatures where phonon is real, the linear thermal expansion fraction at a given T * can be computed by numerically integrating