Verlet modified

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The Verlet modified [1] 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 ([2] 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 \ge d

where d is the hard sphere diameter.