  We study the dynamics of a molecule's nuclear wave-function near an avoided
crossing of two electronic energy levels, for one nuclear degree of freedom. We
derive the general form of the Schroedinger equation in the n-th superadiabatic
representation for all n, and give some partial results about the asymptotics
for large n. Using these results, we obtain closed formulas for the time
development of the component of the wave function in an initially unoccupied
energy subspace, when a wave packet crosses the transition region. In the
optimal superadiabatic representation, which we define, this component builds
up monontonically. Finally, we give an explicit formula for the transition wave
function away from the crossing, which is in excellent agreement with high
precision numerical calculations.
