  For a gas of N bosons interacting through a two-body Morse potential a
variational bound of the free energy of a confined system is obtained. The
calculation method is based on the Feynman-Kac functional projected on the
symmetric representation. Within the harmonic approximation a variational
estimate of the effect of the interaction range on the existence of
many-particle bound states, and on the N-T phase diagram is obtained.
