Verlet modified

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The Verlet modified (1980) (Ref. 1) closure relation for hard sphere fluids, in terms of the cavity correlation function, is (Eq. 3)

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

where several sets of values are tried for A and B (Note, when A=0 the hyper-netted chain is recovered). Later (Ref. 2) 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:

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