Explicitly accounting for background charges in a fast multipole method to simulate periodically replicated non neutral microscopic systems

We present two schemes coupling a Fast Multipole Method (FMM) (devoted to standard and polarizable force fields) and an approach to explicitly account for uniformly distributed background charges to simulate periodically replicated molecular systems with a net charge. These schemes rely on recent analytical relations allowing one to compute the electrostatic potential generated by a uniformly charged cube at any point in space. Whereas the first scheme considers the exact relations, the second one is based on grid interpolation of precomputed data for equal precision. Contrary to available approaches, our coupled schemes prevent the use of Ewald summation techniques as usually proposed in periodic FMM approaches. For a polarizable force field (based on induced dipole moments), we show the ability of our schemes to model molecular neutral and charged systems at the same level of accuracy as the Smooth Particle Mesh Ewald (SPME) approach, to predict usual properties. Moreover, our most efficient scheme, based on interpolating precomputed grid data, is already more efficient than SPME to simulate 3000 atom size periodic systems and one order of magnitude more efficient to compute electrostatic terms of periodic systems at the 100k atom size and above.