Numerical solution of the HNC equation for fluids of non-spherical particles. An efficient method with application to dipolar hard spheres

Abstract

An efficient algorithm is developed for solving the HNC and related equations for fluids characterized by non-spherical, angle-dependent pair interactions by choosing the dipolar hard spheres as an example system. The algorithm is a hybrid of the Picard type and Newton-Raphson (NR) methods. We note that the convergence properties are governed mainly by a few leading projections in the rotational-invariant expansion of η(12) (=h(12) - c(12); h and c are the total and direct correlation functions respectively) for the domain of r < d (d is the hard-sphere diameter) where the analytical expansion of the HNC closure reduces to a much simpler form, and apply the NR technique only to these variables. It is shown for the dipolar hard spheres that the Jacobian matrix can be split into two smaller independent matrices resulting in a considerable saving of computing time.

A slightly modified version of the MSA procedure is proposed for setting the initial values of the iteration variables. It is demonstrated that our version prevents severe instability often caused by the MSA. By using the modified version, our hybrid algorithm is compared with the Picard type method and is shown to be more robust and over an order of magnitude more efficient. The basic strategies for incorporating the NR technique in the iteration cycle and for setting the initial values of the iteration variables are expected also to be applicable to fluids of non-spherical particles other than the dipolar hard spheres.