  We study the equilibrium properties of a single flexible homopolymers where
consecutive monomers are represented by impenetrable hard spherical beads
tangential to each other, and non-consecutive monomers interact via a
square-well potential. To this aim, we use both replica exchange canonical
simulations and micro-canonical Wang-Landau techniques. We perform a close
comparative analysis of the corresponding results, and find perfect agreement
between the two methods as well as the past results. The model is then refined
in two different directions. By allowing partial overlapping between
consecutive beads, we break the spherical symmetry and thus provide a severe
constraint on the possible conformations of the chain. This leads to a single
helix or a double helix ground state, depending on the range of the
interactions. Alternatively, we introduce additional spherical beads at
specific positions to represent the steric hindrance of the side chains in real
proteins. The ground state in this case is found to be again a globule, thus
showing that contrary to some recent suggestions, side chains alone are not
sufficient to insure a protein-like structure. Finally, we show that a
combination of the above two effects increases the stability of the obtaining
secondary structures, in agreement with previous results. The fundamental role
played by the range of the square-well attraction is highlighted, and it is
shown that they play a role similar to that found in simple liquids and
polymers. Perspectives in terms of protein folding are finally discussed.
