Optimizing the Accuracy and Efficiency of Fast Hierarchical Multipole Expansions for MD Simulations

Based on <i>p</i>’th order Cartesian Taylor expansions of Coulomb interactions and on hierarchical decompositions of macromolecular simulation systems into hierarchies of nested, structure-adapted, and adaptively formed clusters of increasing size, fast multipole methods are constructed for rapid and accurate calculations of electrostatic interactions. These so-called SAMM_<i>p</i> algorithms are formulated through totally symmetric and traceless tensors describing the multipole moments and the coefficients of local Taylor expansions. Simple recursions for the efficient evaluation and shifting of multipole moments are given. The required tensors are explicitly given up to order <i>p</i> = 4. The SAMM_<i>p</i> algorithms are shown to guarantee the reaction principle. For systems with periodic boundaries, a reaction field (RF) correction is applied, which introduces at distances beyond the “minimum image convention” boundary a dielectric continuum surrounding each cluster at the top level of coarse graining. The correctness of the present SAMM_<i>p</i> implementation is demonstrated by analyzing the scaling of the residuals and by checking the numerical accuracy of the reaction principle for a pair of distant molecular ions in vacuum. Molecular dynamics simulations of pure water and aqueous solutions containing artificial ions, which are enclosed by periodic boundaries, demonstrate the stability and low-noise behavior of SAMM_<i>p</i>/RF