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)

$C(r/R)=-{\frac {(1+2\eta )^{2}-6\eta (1+{\frac {1}{2}}\eta )^{2}(r/R)+\eta (1+2\eta )^{2}{\frac {(r/R)^{3}}{2}}}{(1-\eta )^{4}}}$

where

$\eta ={\frac {1}{6}}\pi R^{3}\rho$

and $R$ is the hard sphere diameter.\\ The equation of state is (\cite{PRL_1963_10_000321} Eq. 7)

$\beta P\rho ={\frac {(1+\eta +\eta ^{2})}{(1-\eta )^{3}}}$

Everett Thiele (1963 \cite{JCP_1963_39_00474}) also studied this system, resulting in (Eq. 23)

$h_{0}(r)=ar+br^{2}+cr^{4}$

where (Eq. 24)

$a={\frac {(2x+1)^{2}}{(x-1)^{4}}}$

and

$b=-{\frac {12x+12x^{2}+3x^{3}}{2(x-1)^{4}}}$

and

$c={\frac {x(2x+1)^{2}}{2(x-1)^{4}}}$

and where $x=\rho/4$.\\ The pressure via the pressure route (Eq.s 32 and 33) is

$P=nkT{\frac {(1+2x+3x^{2})}{(1-x)^{2}}}$

and the compressibility route is

$P=nkT{\frac {(1+x+x^{2})}{(1-x)^{3}}}$