  We find the exact semiclassical (strong coupling) zero-point energy shifts
applicable to the $e\otimes (n E)$ and $t\otimes (n H)$ dynamic Jahn-Teller
problems, for an arbitrary number $n$ of discrete vibrational modes
simultaneously coupled to one single electronic level. We also obtain an
analytical formula for the frequency of the resulting normal modes, which has
an attractive and apparently general Slater-Koster form. The limits of validity
of this approach are assessed by comparison with O'Brien's previous
effective-mode approach, and with accurate numerical diagonalizations.
Numerical values obtained for $t\otimes (n H)$ with $n =8$ and coupling
constants appropriate to C$_{60}^-$ are used for this purpose, and are
discussed in the context of fullerene.
