Extension of the fast multipole method for the rectangular cells with an anisotropic partition tree structure

The fast multipole method (FMM) is an order N method for the numerically rigorous calculation of the electrostatic interactions among point charges in a system of interest. The FMM is utilized for massively parallelized software for molecular dynamics (MD) calculations. However, an inconvenient limitation is imposed on the implementation of the FMM: In three-dimensional case, a cubic MD unit cell is hierarchically divided by the octree partitioning under isotropic periodic boundary conditions along three axes. Here, we extended the FMM algorithm adaptive to a rectangular MD unit cell with different periodicity along the axes by applying an anisotropic hierarchical partitioning. The algorithm was implemented into the parallelized general-purpose MD calculation software designed for a system with uniform distribution of point charges in the unit cell. The partition tree can be a mixture of binary and ternary branches, the branches being chosen arbitrarily with respect to the coordinate axes at any levels. Errors in the calculated electrostatic interactions are discussed in detail for a selected partition tree structure. The extension enables us to execute MD calculations under more general conditions for the shape of the unit cell, partition tree, and boundary conditions, keeping the accuracy of the calculated electrostatic interactions as high as that with the conventional FMM. An extension of the present FMM algorithm to other prime number branches, such as 5 and 7, is straightforward.