Verlet modified

The Verlet modified closure relation for hard sphere fluids, in terms of the cavity correlation function, is (Eq. 3)

$Y(r)=\gamma (r)-[\frac{A{\gamma }^2(r)/2}{1+B\gamma (r)/2}]$

where the radial distribution function is expressed as (Eq. 1)

$\mathrm{g}(r)=e^{-\beta \mathrm{\Phi }(r)}+Y(r)$

and where several sets of values are tried for A and B (Note, when A=0 the hyper-netted chain is recovered).

Later Verlet used a Padé (2/1) approximant ( Eq. 6) fitted to obtain the best hard sphere results by minimising the difference between the pressures obtained via the virial and compressibility routes:

$Y(r)=\gamma (r)-\frac{A}{2}{\gamma }^2(r)[\frac{1+\lambda \gamma (r)}{1+\mu \gamma (r)}]$

with $A=0.80$, $\lambda =0.03496$ and $\mu =0.6586$ where the radial distribution function for hard spheres is written as (Eq. 1)

$\mathrm{g}(r)=\exp [Y(r)]\mathrm{f}\mathrm{o}\mathrm{r}r\ge d$

where $d$ is the hard sphere diameter.