Numerical solution of the hypernetted chain equation for a solute of arbitrary geometry in three dimensions

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Abstract
The average solvent distribution near complex solid substrates of arbitrary geometry is calculated by solving the hypernetted chain (HNC) integral equation on a three-dimensional discrete cubic grid. A numerical fast Fourier transform in three dimensions is used to calculate the spatial convolutions appearing in the HNC equation. The approach is illustrated by calculating the average solvent density in the neighborhood of small clusters of Lennard-Jones particles and inside a periodic array of cavities representing a simplified model of a porous material such as a zeolite. Molecular dynamics simulations are performed to test the results obtained from the integral equation. It is generally observed that the average solvent density is described accurately by the integral equation. The results are compared with those obtained from a superposition approximation in terms of radial pair correlation functions, and the reference interaction site model (RISM) integral equations. The superposition approximation significantly overestimates the amplitude of the density peaks in particular cases. Nevertheless, the number of the nearest neighbors around the clusters is well reproduced by all approaches. The present calculations demonstrate the feasibility of a numerical solution of HNC-type integral equations for arbitrarily complex geometries using a three-dimensional discrete grid.