Verlet modified

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

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

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

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

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:

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

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

${\mathrm {g} }(r)=\exp[Y(r)]~~~~\mathrm {for} ~~~~r\geq d$

where $d$ is the hard sphere diameter.