Density Functional Study of Cubic to Rhombohedral Transition in $\alpha$-AlF$_3$

Under heating, $\alpha$-AlF$_3$ undergoes a structural phase transition from rhombohedral to cubic at temperature $T$ around 730 K. The density functional method is used to examine the $T$=0 energy surface in the structural parameter space, and finds the minimum in good agreement with the observed rhombohedral structure. The energy surface and electronic wave-functions at the minimum are then used to calculate properties including density of states, $\Gamma$-point phonon modes, and the dielectric function. The dipole formed at each fluorine ion in the low temperature phase is also calculated, and is used in a classical electrostatic picture to examine possible antiferroelectric aspects of this phase transition.