Relationship between the Hard-Sphere Fluid and Fluids with Realistic Repulsive Forces

We consider the equilibrium statistical mechanics of classical fluids in which the potential energy is decomposable into repulsive pair interactions. A generalized cluster expansion is derived relating the thermodynamic and structural properties of such systems to those of the hard-sphere fluid. The expansion is ordered by a softness parameter $\xi$ which is essentially the range of intermolecular distances in which the difference between the Mayer $f$ functions for the repulsive potential and an appropriate reference hard-sphere potential is nonzero. The first (lowest-order) approximation generated by the expansion equates the free energy and $y\left(r\right)\mathrm{}$ for the fluid to the respective functions appropriate to a system of hard spheres with diameter $d$. Here $y\left(r\right)=g\left(r\right)\mathrm{} e^{+\beta u\left(r\right)\mathrm{}}$, where $g\left(r\right)\mathrm{}$ and $u\left(r\right)\mathrm{}$ denote the radial distribution function and repulsive pair potential, respectively. A prescription is given for choosing a temperature- and density-dependent diameter $d$ in the reference hard-sphere fluid so that the first approximation for the free energy contains errors of order ${\xi }^4$ only, and the corrections to the first approximation for $g\left(r\right)\mathrm{}$ are of order ${\xi }^2$. The method is used to calculate the properties of a fluid whose intermolecular potential varies as $r^{-12\mathrm{}}$. The repulsive potential that produces the repulsive forces in the Lennard-Jones potential is also studied. Since the properties of the hard-sphere fluid are known from the results of computer calculations and conveniently summarized by analytic equations, the application of the first approximation is numerically very simple. With this approximation, the results obtained for both model systems agree closely with those obtained by Monte Carlo calculations.