  We investigate two well-known approaches for extending the fewest switches
surface hopping (FSSH) algorithm to periodic time-dependent couplings. The
first formalism acts as if the instantaneous adiabatic electronic states were
standard adiabatic states, which just happen to evolve in time. The second
formalism replaces the role of the usual adiabatic states by the
time-independent adiabatic Floquet states. For a set of modified Tully model
problems, the Floquet FSSH (F-FSSH) formalism gives a better estimate for both
transmission and reflection probabilities than the instantaneous adiabatic FSSH
(IA-FSSH) formalism. More importantly, only F-FSSH predicts the correct final
scattering momentum. Finally, in order to use Floquet theory accurately, we
find that it is crucial to account for the interference between Floquet states.
Our results should be of interest to all those interested in laser induced
molecular dynamics.
