Verlet modified

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:

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

with $A=0.80$, $\lambda =0.03496$ and $\mu =0.6586$.

References

1. Loup Verlet "Integral equations for classical fluids I. The hard sphere case", Molecular Physics 41 pp. 183-190 (1980)
2. Loup Verlet "Integral equations for classical fluids II. Hard spheres again", Molecular Physics 42 pp. 1291-1302 (1981)