  Using the diabatic formalism, which generalizes the adiabatic approximation
in the Born-Oppenheimer formalism, we apply well-known Hamiltonian methods to
calculate the effect of open di-meson thresholds that lie well below the mass
of elementary $c\bar c q\bar q^\prime$, $c\bar c s\bar s$, and $c \bar c q \bar
s$ tetraquark bound states. We compute the resulting mass shifts for these
states, as well as their decay widths to the corresponding meson pairs. Each
mass eigenstate, originally produced using a bound-state approximation under
the diabatic formalism, consists of an admixture of a compact
diquark-antidiquark configuration (an eigenstate of the original dynamical
diquark model) with an extended di-meson configuration induced by the nearest
threshold. We compare our results with those from our recent work that employs
a scattering formalism, and find a great deal of agreement, but also comment
upon interesting discrepancies between the two approaches.
