  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.
