  We explore an asymmetric two-fermion Hubbard dimer to test the accuracy of
the adiabatic approximation of time-dependent density functional theory in
modelling time-resolved charge transfer. We show that the model shares
essential features of a ground state long-range molecule in real-space, and by
applying a resonant field we show that the model also reproduces essential
traits of the CT dynamics. The simplicity of the model allows us to propagate
with an "adiabatically-exact" approximation, i.e. one that uses the exact
ground-state exchange-correlation functional, and compare with the exact
propagation. This allows us to study the impact of the time-dependent
charge-transfer step feature in the exact correlation potential of real
molecules on the resulting dynamics. Tuning the parameters of the dimer allows
a study both of charge-transfer between open-shell fragments and between
closed-shell fragments. We find that the adiabatically-exact functional is
unable to properly transfer charge, even in situations where the
adiabatically-exact resonance frequency is remarkably close to the exact
resonance, and we analyze why.
