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The '''Verlet modified''' <ref>[http://dx.doi.org/10.1080/00268978000102671 Loup Verlet "Integral equations for classical fluids I. The hard sphere case", Molecular Physics '''41''' pp. 183-190 (1980)]</ref> [[Closure relations | closure relation]] for [[hard sphere model | hard sphere]] fluids,
The '''Verlet modified''' (1980) (Ref. 1) [[Closure relations | closure relation]] for [[hard sphere model | hard sphere]] fluids,
in terms of the [[cavity correlation function]], is (Eq. 3)
in terms of the [[cavity correlation function]], is (Eq. 3)


:<math> Y(r) = \gamma (r) -   \left[ \frac{A \gamma^2(r)/2}{1+ B \gamma(r) /2} \right]</math>
:<math>\ln y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[ \frac{1}{1+ B \gamma(r) /2} \right]</math>


where the [[radial distribution function]] is expressed as (Eq. 1)
where several sets of values are tried for ''A'' and ''B''  (Note, when ''A=0'' the [[HNC| hyper-netted chain]] is recovered).
 
Later (Ref. 2)  Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[hard sphere model | hard sphere]] results
:<math>{\mathrm g}(r)  = e^{-\beta \Phi(r)} + Y(r)</math>
 
and where several sets of values are tried for ''A'' and ''B''  (Note, when ''A=0'' the [[HNC| hyper-netted chain]] is recovered).
 
Later   Verlet used a Padé (2/1) approximant (<ref>[http://dx.doi.org/10.1080/00268978100100971 Loup Verlet "Integral equations for classical fluids II. Hard spheres again", Molecular Physics '''42''' pp. 1291-1302 (1981)]</ref> Eq. 6) fitted to obtain the best [[hard sphere model | hard sphere]] results
by minimising the difference between the pressures obtained via the [[Pressure equation | virial]] and [[Compressibility equation | compressibility]] routes:
by minimising the difference between the pressures obtained via the [[Pressure equation | virial]] and [[Compressibility equation | compressibility]] routes:


:<math> Y(r) = \gamma (r) - \frac{A}{2} \gamma^2(r) \left[  \frac{1+ \lambda \gamma(r)}{1+ \mu \gamma(r)} \right]</math>
:<math>\ln y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[  \frac{1+ \lambda \gamma(r)}{1+ \mu \gamma(r)} \right]</math>
 
with <math>A= 0.80</math>, <math>\lambda=  0.03496</math> and <math>\mu = 0.6586</math>
where the radial distribution function for hard spheres is written as (Eq. 1)


:<math>{\mathrm g}(r)  =  \exp[Y(r)] ~~~~ \mathrm{for} ~~~~ r \ge d</math>
with <math>A= 0.80</math>, <math>\lambda0.03496</math> and <math>\mu = 0.6586</math>.


where <math>d</math> is the hard sphere diameter.
==References==
==References==
<references/>
#[http://dx.doi.org/10.1080/00268978000102671 Loup Verlet "Integral equations for classical fluids I. The hard sphere case", Molecular Physics '''41''' pp. 183-190 (1980)]
 
#[http://dx.doi.org/10.1080/00268978100100971 Loup Verlet "Integral equations for classical fluids II. Hard spheres again", Molecular Physics '''42''' pp. 1291-1302 (1981)]
 
[[Category: Integral equations]]
[[Category: Integral equations]]
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