Guide to efficient solution of PC-SAFT classical Density Functional Theory in various Coordinate Systems using fast Fourier and similar Transforms

Abstract

Classical density functional theory (DFT) is a powerful tool for studying solvation or problems where resolution of interfacial domains or interfacial properties among phases (or thin films) is required. Many interesting problems necessitate multi-dimensional modeling, which calls for robust and efficient algorithmic implementations of the Helmholtz energy functionals. A possible approach for achieving efficient numerical solutions is using the convolution theorem of the Fourier transform. This study is meant to facilitate research and application of DFT methods, by providing a detailed guide on solving DFT problems in multi-dimensional domains. Methods for efficiently solving the convolution integrals in Fourier space are presented for Cartesian, cylindrical, and spherical coordinates. For cylindrical and spherical coordinate systems, rotational and spherical symmetry is exploited, respectively. To enable easy implementation, our approach is based on fast Fourier, fast Hankel, fast sine and cosine transforms on equidistant grids, all of which can be applied using off-the-shelf algorithms. Subtle details for implementing algorithms in cylindrical and spherical coordinate systems are emphasized. The work covers functionals based on weighted densities exemplarily. Functionals according to fundamental measure theory (FMT) as well as a Helmholtz energy functional based on the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state are worked out in detail (and given as Supporting Information).