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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(r) = \gamma (r) - \frac{A}{2} \gamma^2(r) \left[ \frac{1+ \lambda \gamma(r)}{1+ \mu \gamma(r)} \right]}

with $A=0.80$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda= 0.03496} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = 0.6586} where the radial distribution function for hard spheres is written as (Eq. 1)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathrm g}(r) = \exp[Y(r)] ~~~~ \mathrm{for} ~~~~ r \ge d}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is the hard sphere diameter.