  We study the semiclassical distribution of resonances of a $2 \times 2$
matrix Schr\"odinger operator, obtained as a reduction of an Hamiltonian when
studying molecular predissociation models under the Born-Oppenheimer
approximation. The energy considered is above the energy-level crossing of the
two associated classical trajectories, and is respectively trapping and
non-trapping for those trajectories. Under a condition between the contact
order $m$ of the crossings and the vanishing order $k$ of the interaction term
at the crossing points, we show that, asymptotically in the semiclassical limit
$h \to 0^+$, the imaginary part of the resonances is of size
$h^{1+2(k+1)/(m+1)}$ in the general case and shrinks to $h^{1+2(k+2)/(m+1)}$
when both $k$ and $m$ are odd. We also compute the first term of the associated
asymptotic expansions.
