Implementing the cell multipole method for dipolar and charged dipolar systems

Abstract

Recently, substantial progress has been made in reducing the cost of computing Coulomb sums in fields such as astrophysics and molecular dynamics (MD). This has culminated in the fast multipole method (FMM), and its Cartesian counterpart, the cell multipole method (CMM). In MD simulations, these methods provide efficient and accurate algorithms for computing the energy and forces of charge-charge interactions. However, dipolar interactions are as important and as challenging to compute. In this paper we implement the cell multipole method (CMM) for dipolar domains. This CMM implementation is an accurate and efficient approach for computing interactions in million-particle dipolar systems. In addition, we use CMM to treat systems of permanent dipoles and induced dipoles, permanent charges and permanent dipoles, and permanent charges and induced dipoles. We derive and discuss expressions for accurate and efficient computation of all interactions in these systems. The approach is also applicable to polarizable charged dipolar systems. Finally, we apply one-level CMM and hierarchical CMM to arbitrary dipolar systems and discuss accuracy and speed. The hierarchical implementation of CMM is shown to scale linearly with the number of dipoles.