  The dynamical Jahn-Teller effect of C$_{60}^{n-}$ anions ($n = $ 1-5) is
studied using the numerical diagonalization of the linear $p^n \otimes 8d$
Jahn-Teller Hamiltonian with the currently established coupling parameters. It
is found that in all anions the Jahn-Teller effect stabilizes the low-spin
states, resulting in the violation of Hund's rule. The energy gain due to the
Jahn-Teller dynamics is found to be comparable to the static Jahn-Teller
stabilization. The Jahn-Teller dynamics influences the thermodynamic properties
via strong variation of the density of vibronic states with energy. Thus, the
large vibronic entropy in the low-spin states enhances the effective spin gap
of C$_{60}^{3-}$ quenching the spin crossover. From the calculations of the
effective spin gap in function of the Hund's rule coupling, we found that the
latter should amount 40 $\pm$ 5 meV in order to cope with the violation of
Hund's rule and to reproduce the large spin gap. With the obtained numerical
solutions the matrix elements of electronic operators for the low-lying
vibronic levels and the vibronic reduction factors are calculated for all
anions.
