Exact solution of the Percus Yevick integral equation for hard spheres

The exact solution for the Percus-Yevick integral equation for hard spheres was derived by M. S. Wertheim in 1963 \cite{PRL_1963_10_000321} (See also \cite{JMP_1964_05_00643}) (and for mixtures by in Lebowitz 1964 \cite{PR_1964_133_00A895}). The direct correlation function is given by (\cite{PRL_1963_10_000321} Eq. 6) \begin{equation} C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4} \end{equation} where \begin{equation} \eta = \frac{1}{6} \pi R^3 \rho \end{equation} and $R$ is the hard sphere diameter.\\ The equation of state is (\cite{PRL_1963_10_000321} Eq. 7) \begin{equation} \beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \end{equation} Everett Thiele (1963 \cite{JCP_1963_39_00474}) also studied this system, resulting in (Eq. 23) \begin{equation} h_0(r) = ar+ br^2 + cr^4 \end{equation} where (Eq. 24) \begin{equation} a = \frac{(2x+1)^2}{(x-1)^4} \end{equation} and \begin{equation} b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4} \end{equation} and \begin{equation} c= \frac{x(2x+1)^2}{2(x-1)^4} \end{equation} and where $x=\rho/4$.\\ The pressure via the pressure route (Eq.s 32 and 33) is \begin{equation} P=nkT\frac{(1+2x+3x^2)}{(1-x)^2} \end{equation} and the compressibility route is \begin{equation} P=nkT\frac{(1+x+x^2)}{(1-x)^3} \end{equation}