Lattice dynamics and origin of ferroelectricity in ${\mathrm{BaTiO}}_3$: Linearized-augmented-plane-wave total-energy calculations

Phonon frequencies and eigenvectors have been computed from first principles for the three optic ${\mathit{F}}_{1\mathit{u}}$ modes in ${\mathrm{BaTiO}}_3$ using the full-potential linearized-augmented-plane-wave method. We find that the ferroelectric instability in ${\mathrm{BaTiO}}_3$ can be understood from calculations for a perfect crystal with periodic boundary conditions. The energy wells for the soft-mode distortion are deeper for rhombohedral [111] displacements relative to tetragonal [001] displacements, but they are relatively shallow and comparable to the transition temperature. The nonrigid part of the charge-density distortion is centered around the Ti ion rather than the O, and the Ti charge is closer to 2.9+ than 4+. There is significant hybridization between the Ti and O, but the Ba is quite ionic and is well described as a ${\mathrm{Ba}}^{2+}$ ion. The Ti-O hybridization is essential to the ferroelectric instability.