giving rise to a Dirac delta function in the Boltzmann factor [@Baxter68], i. e.$$e^{-\beta \phi _{ij}^{\mathrm{SHS}}\left( r\right) }=\theta \left( r-\sigma
_{ij}\right) +t_{ij}\ \sigma _{ij}\delta \left( r-\sigma _{ij}\right) 
\label{eq1}$$where $\beta =\left( k_{B}T\right) ^{-1}$, while $\theta $ and $\delta $ are the Heaviside step function and the Dirac delta function, respectively.

The advantage of the sticky limit is that one effectively deals with a single parameter $\tau _{ij}$ for each pair, rather than a combination of energy and length scales (as occurs in the square-well model, for which no analytical solution is known). On the one hand, this particular limit has the disadvantage of introducing some pathologies in the model, notably in the one-component case [@Stell91]. On the other hand, Baxter’s model represents the simplest paradigmatic example of a combination of steep repulsion and short-range attraction which entails a complete analytical solution in the one-component case, within a robust approximation such as the PY closure.

In the multicomponent case, the PY solution of the OZ equation in terms of Baxter’s factor correlation function reads [@Perram75; @Barboy79]$$q_{ij}(r)=\left\{ 
\begin{array}{l}
\frac{1}{2}a_{i}(r-\sigma _{ij})^{2}+(b_{i}+a_{i}\ \sigma _{ij})(r-\sigma
_{ij})+\Lambda _{ij},\qquad (\sigma _{i}-\sigma _{j})/2\leq r\leq \sigma
_{ij} \\ 
\qquad 0,\qquad \qquad \qquad \qquad \text{\ \ \ \ \ \ \ 