  Recent experimental and theoretical results have shown the existence of a
liquid-liquid phase transition in isotropic systems, such as biological
solutions and colloids, whose interaction can be represented via an effective
potential with a repulsive soft-core and an attractive part. We investigate how
the phase diagram of a schematic general isotropic system, interacting via a
soft-core squared attractive potential, changes by varying the parameters of
the potential. It has been shown that this potential has a phase diagram with a
liquid-liquid phase transition in addition to the standard gas-liquid phase
transition and that, for a short-range soft-core, the phase diagram resulting
from molecular dynamics simulations can be interpreted through a modified van
der Waals equation. Here we consider the case of soft-core ranges comparable
with or larger than the hard-core diameter. Because an analysis using molecular
dynamics simulations of such systems or potentials is too time-demanding, we
adopt an integral equation approach in the hypernetted-chain approximation.
Thus we can estimate how the temperature and density of both critical points
depend on the potential's parameters for large soft-core ranges. The present
results confirm and extend our previous analysis, showing that this potential
has two fluid-fluid critical points that are well separated in temperature and
in density only if there is a balance between the attractive and repulsive part
of the potential. We find that for large soft-core ranges our results satisfy a
simple relation between the potential's parameters.
