  We explain the concept of superadiabatic approximations and show how in the
context of the Born- Oppenheimer approximation they lead to an explicit formula
that can be used to predict transitions at avoided crossings. Based on this
formula, we present a simple method for computing wave packet dynamics across
avoided crossings. Only knowledge of the adiabatic electronic energy levels
near the avoided crossing is required for the computation. In particular, this
means that no diabatization procedure is necessary, the adiabatic energy levels
can be computed on the fly, and they only need to be computed to higher
accuracy when an avoided crossing is detected. We test the quality of our
method on the paradigmatic example of photo-dissociation of NaI, finding very
good agreement with results of exact wave packet calculations.
