ABSTRACT
Simulation of condensed-phase phenomena requires the use of periodic boundary conditions. This chapter presents a periodic Fast Multipole Method based on a renormalization approach that uses a cubic central unit cell geometry. Construction of the hierarchy tree involves assembling progressively larger central super-cells then replicating them to the near- and far-fields; this keeps the number of multipole translation operations at each hierarchy level constant across the entire tree. Once multipole moments of the central unit cell are known, a recurrence relation, which uses the regular-to-regular and regular-to-irregular multipole translations, sums up the far-field contribution from the hierarchy levels resulting in the lattice sum moments quickly converging to double-precision accuracy. The chapter provides derivations for the lattice sum for the energy, stress tensor, and force acting on particles in the central unit cell, and explains the method to achieve the proper scaling by the cell index of the image cell contribution to the stress tensor.
