  We study the survival probability associated with a semi-classical matrix
Shr\"odinger operator that models the predissociation of a general molecule in
the Born-Oppenheimer approximation. We show that it is given by its usual
time-dependent exponential contribution, up to a reminder term that is
exponentially small (in the semiclassical parameter) with arbitrarily large
rate of decay. The result applies in any dimension, and in presence of a number
of resonances that may tend to infinity as the semiclassical parameter tends to
0.
