Representation of the direct correlation function of the hard-sphere fluid

An accurate representation of the structural pair correlation functions of the hard sphere (HS) fluid up to the freezing density is proposed which combines the pole expression for the total correlation function h(r), the Ornstein-Zernike equation, and molecular dynamics (MD) computer simulation data. In the scheme, h(r) is expressed in terms of a set of pole parameters, which reveals how the tail of the Fourier transform of h(r) contains information on the discontinuities in the derivatives of the direct correlation function (DCF). This formulation leads to a DCF expressed as the sum of a numerically obtainable part and an analytic part which consists of elementary integral terms, some of which are found to give rise to the discontinuities. An exact formula for the magnitude of these discontinuities is derived, which indicates that there is a particular density (ρ≅0.133) below which the magnitude of the discontinuities decreases with increasing order of the derivative. With the accurate MD simulation data the set of parameters that specifies h(r) was determined. These can be used to obtain the different structural functions of the HS fluid, and following the calculation stages of the pole structure scheme the DCF is obtained. From this route to the DCF the second pole of the HS fluid can be determined and the non-negligible role of the "out of core" part of the DCF at high densities is revealed. The density-dependent separation range where the two pole approximation represents well the h(r) has been determined.