An efficient algorithm for classical density functional theory in three dimensions: Ionic solutions

Classical density functional theory (DFT) of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses O(N log N) operations and O(N) memory, where N is the number of grid points. This system-size scaling is significant because of the very large N required for three-dimensional systems. The algorithm uses fast Fourier transforms (FFTs) to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard (iterative substitution) iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT, we use fundamental measure theory. For the electrostatic DFT, we present two algorithms. One is for the "bulk-fluid" functional of Rosenfeld Y. Rosenfeld, J. Chem. Phys. 98, 8126 (1993) that uses O(N log N) operations. The other is for the "reference fluid density" (RFD) functional D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 (2002). This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires O(N(2)) operations.