Simulations of the superionic conductor CaF2 have been undertaken, with a model for the interionic interactions derived directly from electronic structure calculations. The model includes such many-body effects as polarization and the adaptation of the size of the anion to its instantaneous environment ("compression"). Physical properties of CaF2 accurately reproduced include the phonon frequencies, the fluorite-->alpha-PbCl2-structure phase transition and the solid-state diffusion. Analysis shows that the compression effect is much less important for fluorides than oxides but, nevertheless, significantly affects the phase transition pressure. Polarization effects, particularly of the cations, play a crucial role in allowing the anion diffusion, and the importance of a short-range contribution to the cation polarization is highlighted. These findings are contrasted with previous work using empirically parametrized shell and pair potentials, which suggest that other interactions play the dominant role.