  We use molecular dynamics simulations to test integral equation theory
predictions for the structure of fluids of spherical particles with eight
different piecewise-constant pair interaction forms comprising a hard core and
a combination of two shoulders and/or wells. Since model pair potentials like
these are of interest for discretized or coarse-grained representations of
effective interactions in complex fluids (e.g., for computationally intensive
inverse optimization problems), we focus here on assessing how accurately their
properties can be predicted by analytical or simple numerical closures
including Percus-Yevick, hypernetted chain, reference hypernetted chain,
first-order mean spherical approximation, and a modified first-order mean
spherical approximation. To make quantitative comparisons between the predicted
and simulated radial distribution functions, we introduce a cumulative
structural error metric. For equilibrium fluid state points of these models, we
find that the reference hypernetted chain closure is the most accurate of the
tested approximations as characterized by this metric or related thermodynamic
quantities.
