Perturbation Theory for Fluids

Recently, Barker and Henderson have introduced a semimacroscopic approximation to the second order in the expansion of the configuration integral by considering the pair potential as the sum of a strong (repulsive) part and a weaker (long-range) part. We analyze this approximation and show that the essential part of it is to reduce the higher-order distribution functions to a second-order nonuniform distribution function, the nonuniformity coming from fixing a particle at the origin. The approximation can be done to all orders, and the series can be summed to give an expression for the free energy and pressure. The expression involves the grand partition function for the nonuniform system taken at the chemical potential for the reference system minus the perturbing potential. The functions of the nonuniform system are approximated by taking the functional form they have at uniformity and by using the product of the density and the radial distribution function of the reference system for the nonuniform density. The summed series can then be computed. For the three-dimensional square-well fluid it is confirmed that the convergence is very rapid down to reduced temperatures about 0.5. Comparison with the one-dimensional square-well fluid shows that below this temperature the convergence is slow and it is necessary to use the whole series.