$q_{ij}(r)$ corresponding to closures belonging to the class given by Eq. (\[eq6\]) have the same functional form as the PY solution – Eq. (\[eq2\]) – , each closure is characterized by its own approximation to $y_{ij}(\sigma _{ij})$, which is involved in the expressions of the parameters $a_{i}$, $b_{i}$, $\Lambda _{ij}$. For instance, the $C_{0}$ and $C_{1}$ approximations correspond to

$$\begin{array}{ccc}
\begin{array}{c}
y_{ij}(\sigma _{ij})=1 \\ 
\end{array}
&  & 
\begin{array}{c}
\text{(}C_{0}\text{ closure)} \\ 
\end{array}
\\ 
y_{ij}(\sigma _{ij})=1+y_{ij}^{(1)}(\sigma _{ij})\ \eta  &  & \text{(}C_{1}%
\text{ closure),}%
\end{array}%$$

which are, respectively, the zeroth– and first–order truncations of the density expansion for the exact cavity function at contact (see Ref. [Gazzillo04]{} for details).

While a brute-force truncation of the above-mentioned density expansions leads to analytical expressions simple enough to be applied to the multi-component (polydisperse) case, one should reasonably expect less accuracy, expecially in the high-density regime. In the one-component case, we can carefully check this point.

In Figure \[fig1\] coexistence curves obtained from the $C_0$ and $C_{1}$ approximations are compared with the PY ones (using both compressibility and energy routes), and with Monte Carlo simulations from Ref. [@Miller03]. It is apparent how the PY energy route (PYE) yields a rather precise representation of the MC 