Dynamical Jahn--Teller effect and antiferromagnetism in Cs$_3$C$_{60}$

The dynamical Jahn--Teller effect on fullerene sites in insulating Cs$_3$C$_{60}$ is investigated fully ab initio. The vibronic excitations of rotational type are at $\ge$65 cm$^{-1}$ while the net kinetic contribution to the Jahn--Teller stabilization energy constitutes ca 90 meV. This means that no localization of distortions by intermolecular interactions is possible in these fullerides, therefore, free rotations of deformations take place independently on each C$_{60}$. The latter destroy the orbital ordering and establish a conventional exchange interaction between $S=1/2$ on fullerene sites. The corresponding exchange model is derived and predicts the N\'{e}el temperature for A15 Cs$_3$C$_{60}$ close to experiment.