  When describing the low-energy physics of bosons in a double-well potential
with a high barrier between the wells and sufficiently weak atom-atom
interactions, one can to a good approximation ignore the high energy states and
thereby obtain an effective two-mode model. Here, we show that the regime in
which the two-mode model is valid can be extended by adding an on-site
three-body interaction term and a three-body interaction-induced tunneling term
to the two-mode Hamiltonian. These terms effectively account for virtual
transitions to the higher energy states. We determine appropriate strengths of
the three-body terms by an optimization of the minimal value of the wave
function overlap within a certain time window. Considering different initial
states with three or four atoms, we find that the resulting model accurately
captures the dynamics of the system for parameters where the two-mode model
without the three-body terms is poor. We also investigate the dependence of the
strengths of the three-body terms on the barrier height and the atom-atom
interaction strength. The optimal three-body interaction strengths depend on
the initial state of the system.
