The third set of improvements will focus on incorporating contributions from other high-symmetry points when determining the quantum covariance matrix (QCV). This would help us to capture a more comprehensive view of phase transition by extending our analysis to include phase transitions that occur at high-symmetry points beyond the Γ point, such as the X-W-K and P-Γ-N paths in cubic ZrO 2 and BCC Zr respectively. This enhancement will involve robust thermal displacement sampling, utilizing QCV generated at these relevant high-symmetry points. By accounting for thermal displacements in these critical regions, we expect to improve our understanding of phase transitions and further refine our renormalization scheme.