Determining liquid structure from the tail of the direct correlation function

TL;DR

This paper extends Stell's method to determine liquid structure from the tail of the direct correlation function for soft-core potentials, introducing a new function T(r) and a variational solution approach.

Contribution

It generalizes the tail-based determination of pair correlation functions to soft-core potentials and proposes a new variational method for solving the Ornstein-Zernike equation.

Findings

The method accurately predicts pair correlations for Lennard-Jones and Yukawa fluids.

A new variational approach improves efficiency in solving the Ornstein-Zernike equation.

Reinterpretation of integral equation closures based on the tail of c(r).

Abstract

In important early work, Stell showed that one can determine the pair correlation function h(r) of the hard sphere fluid for all distances r by specifying only the "tail" of the direct correlation function c(r) at separations greater than the hard core diameter. We extend this idea in a very natural way to potentials with a soft repulsive core of finite extent and a weaker and longer ranged tail. We introduce a new continuous function T(r) which reduces exactly to the tail of c(r) outside the (soft) core region and show that both h(r) and c(r) depend only on the "out projection" of T(r): i.e., the product of the Boltzmann factor of the repulsive core potential times T(r). Standard integral equation closures can thus be reinterpreted and assessed in terms of their predictions for the tail of c(r) and simple approximations for its form suggest new closures. A new and very efficient variational method is proposed for solving the Ornstein-Zernike equation given an approximation for the tail of c. Initial applications of these ideas to the Lennard-Jones and the hard core Yukawa fluid are discussed.