We present the construction of an exponentially accurate time-dependent
Born-Oppenheimer approximation for molecular quantum mechanics. We study
molecular systems whose electron masses are held fixed and whose nuclear masses
are proportional to $\epsilon^{-4}$, where $\epsilon$ is a small expansion
parameter. By optimal truncation of an asymptotic expansion, we construct
approximate solutions to the time-dependent Schr\"odinger equation that agree
with exact normalized solutions up to errors whose norms are bounded by $\ds C
\exp(-\gamma/\epsilon^2)$, for some C and $\gamma>0$.