There have been some theoretical studies of high pressure phase transition behavior of BaF ₂, while in most cases the attention is paid mainly to the optical and electrical properties of BaF ₂under increasing pressure. To date, there has been still a lack of theoretical explanation for the hysteresis phenomenon of high-pressure phase of BaF ₂when the pressure is released. In addition, the pressure-dependent behavior of the BaF ₂band gap is still under controversy, and there are few studies of its high-pressure Raman spectra. Therefore, first principle is used to make a supplementary calculation of the high pressure behavior of BaF ₂. For a given pressure Pand temperature T, the thermodynamic stable phase has the lowest Gibbs free energy. The calculations are performed at zero temperature and hence, the Gibbs free energy becomes equal to the enthalpy. Thus, the variation of enthalpy is calculated as a function of pressure to study the high-pressure phase stability of BaF ₂based on density functional theory as implemented in the Vienna ab initio simulation package (VASP). The results show that the BaF ₂undergoes two structural phase transitions from Fm3 m(cubic) to Pnma(orthorhombic) and then to P6 ₃/ mmc(hexagonal) with increasing pressure, and their corresponding transition pressures are 3.5 and 18.3 GPa, respectively. By calculating the evolution of lattice constant with pressure, it is found that at about 15 GPa (near the second phase transition pressure), the lattice constants of the Pnmastructure show abnormal behavior (a slight increase in b _oand a slight decrease in a _o). We suggest that this behavior leads the band gap to decrease, indicated by analyzing the calculated results of Pnmastructure of other materials. The Pnmastructure completely transforms into P6 ₃/ mmcstructure at about 20 GPa. By analyzing the phonon dispersion curves of BaF ₂as a function of pressure, the structural stability information of the material can also be obtained. Then the density functional perturbation theory (DFPT) is used to calculate the phonon dispersion curves of BaF ₂by VASP code and Phonopy code. The hysteresis phenomenon of the P6 ₃/ mmcstructure, when the pressure is released, is explained by the kinetic stability. The results predict that the P6 ₃/ mmcstructure can be stabilized at least to 80 GPa.