The second set of improvements could come by adopting more reliable physical methods to implement anharmonic free energy correction. At the level of perturbation theory, explicit forms of free energy correction due to 3rd-order anharmonicity have been implemented and used in the literature . This would serve as a higher-order free energy correction to what is currently implemented in this workflow. At a different level, a method implementing thermodynamic integration (TI) using only IFCs with no Figure | Vibrational free energy differentials (∆F) determine phase transition temperatures for Zr, ZrO2 and GeTe (a) HCP-to-BCC phase transition in Zr, (b) tetragonal-to-cubic phase transition in ZrO2, (c) rhombohedral-to-cubic phase transition in GeTe. The blue lines represent free energies calculated using the quasi-harmonic approximation, denoted as ∆F(QHA), whereas the green lines correspond to free energies that include an additional anharmonic correction, indicated as ∆F(QHA+anh). The experimental phase transition temperatures (Expt. Tc) are marked in magenta squares. We separate all temperatures into three regions depending on whether the phonon modes are imaginary or not according to Figure . Temperatures are categorized into three distinct zones based on the phonon modes, as illustrated in Figure . The "imaginary zone" encompasses the temperature range where phonon modes are imaginary. Conversely, the "real zone" refers to where phonon modes are entirely real. The region between the highest imaginary-mode temperature and the lowest real-mode temperature is defined as the "transition zone" here. In this zone, the imaginary modes gradually vanish and become real. The width of this "transition zone" is determined by the interval of renormalization temperatures.