  We report the first principles investigations on the phonons of three
polytypes of BaTiO3 (BTO): paraelectric (PE) cubic Pm-3m and two ferroelectric
(FE) phases, tetragonal P4mm and rhombohedral R3m. The phonon frequencies
calculated using various exchange-correlation functionals, including density
functional theory, Hartree-Fock approximation, and their hybrids were reviewed.
The pressure-induced interplays between the modes form individual phases were
explored by calculating the phonon modes as a function of pressure, P from -15
to 230 GPa. The pressure-sensitive modes of the FE phases showed softening and
converged to the modes of the PE phase at pressures below ~ 10 GPa. These
results on the FE phases can be interpreted as phonon-precursors for a change
in symmetry from low- to high-symmetry and partly as a theoretical explanation
for the pressure-induced mode-coupling behaviors reported by Sood et al. [Phys.
Rev. B 51, 8892 (1995)]. As pressure is applied further beyond ~ 50 GPa to the
cubic PE phase, the lowest F1u mode softens again and diverges into two
separate modes of tetragonal FE P4mm at above ~ 150 GPa. These phonon-branching
behaviors at high pressures provide a clear re-confirmation of the re-entrant
ferroelectricity predicted in [Phys. Rev. Lett. 95, 196804 (2005); Phys. Rev. B
74, 180101 (2006); ibid. 85, 054108 (2012)]. The high-pressure-re-entrant FE
polarization was not found in the rhombohedral structure. Instead, the
centosymmetric R-3m phase was favored at above ~ 30 GPa. The phonon modes
calculated for the phonon-propagation vectors in the high-symmetry directions
show that the Pm-3m phase exhibits polar instability at the \Gamma point and
non-polar instability at the X, M, and R points under high pressure.
