  We investigate the origin of a disagreement between the two-mode model and
the exact Gross-Pitaevskii dynamics applied to double-well systems. In general
this model, even in its improved version, predicts a faster dynamics and
underestimates the critical population imbalance separating Josephson and
self-trapping regimes. We show that the source of this mismatch in the dynamics
lies in the value of the on-site interaction energy parameter. Using simplified
Thomas-Fermi densities, we find that the on-site energy parameter exhibits a
linear dependence on the population imbalance, which is also confirmed by
Gross-Pitaevskii simulations. When introducing this dependence in the two-mode
equations of motion, we obtain a reduced interaction energy parameter which
depends on the dimensionality of the system. The use of this new parameter
significantly heals the disagreement in the dynamics and also produces better
estimates of the critical imbalance.
