Mean spherical approximation

The Lebowitz and Percus mean spherical approximation (MSA) (1966) \cite{PR_1966_144_000251} closure is given by

$c(r)=-\beta \omega (r),~~~~r>\sigma .$

The {\bf Blum and H$\o$ye} mean spherical approximation (MSA) (1978-1980) \cite{JSP_1978_19_0317_nolotengoSpringer,JSP_1980_22_0661_nolotengoSpringer} closure is given by

${\rm {g}}_{ij}(r)\equiv h_{ij}(r)+1=0~~~~~~~~r<\sigma _{ij}=(\sigma _{i}+\sigma _{j})/2$

and

$c_{ij}(r)=\sum _{n=1}{\frac {K_{ij}^{(n)}}{r}}e^{-z_{n}r}~~~~~~\sigma _{ij}<r$

where $h_{ij}(r)$ and $c_{ij}(r)$ are the total and the direct correlation functions for two spherical molecules of $i$ and $j$ species, $\sigma_i$ is the diameter of $i$ species of molecule.\\ Duh and Haymet (Eq. 9 \cite{JCP_1995_103_02625}) write the MSA approximation as

$g(r)={\frac {c(r)+\beta \Phi _{2}(r)}{1-e^{\beta \Phi _{1}(r)}}}$

where $\Phi_1$ and $\Phi_2$ comes from the WCA division of the LJ potential.\\ By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625})

$s(r)=h(r)-c(r)-\beta \Phi _{2}(r)$

one can arrive at (Eq. 11 \cite{JCP_1995_103_02625})

$B(r)\approx B^{\rm {MSA}}(s)=\ln(1+s)-s$

The Percus Yevick approximation may be recovered from the above equation by setting $\Phi _{2}=0$.