Cell multipole method for molecular simulations in bulk and confined systems

Abstract

One of the bottlenecks in molecular simulations is to treat large systems involving electrostatic interactions. Computational time in conventional molecular simulation methods scales with O(N 2 ), where N is the number of atoms. With the emergence of new simulations methodologies, such as the cell multipole method ͑CMM͒, and massively parallel supercomputers, simulations of 10-million atoms or more have been performed. In this work, the optimal hierarchical cell level and the algorithm for Taylor expansion were recommended for fast and efficient molecular dynamics simulations of three-dimensional ͑3D͒ systems. CMM was then extended to treat quasi-two-dimensional ͑2D͒ systems, which is very important for condensed matter physics problems. In addition, CMM was applied to grand canonical ensemble Monte Carlo simulations for both 3D and 2D systems. Under the optimal conditions, our results show that computational time is approximately linear with N for large systems, average error in total potential energy is about 0.05% for 3D and 0.32% for 2D systems, and the RMS force error is 0.27% for 3D and 0.43% for 2D systems when compared with the Ewald summation.

Long ranged interactions in computer simulations and for quasi-2D systems

“…The multipole expansion in terms of Cartesian tensors is obtained by expanding 1/|r −r i | in a Taylor series in r i and the spherical tensors representation is obtained as projections on the spherical harmonics. Both representations can be used to implement multipole methods: in the Fast Multipole Method (FMM) the representation with spherical tensor is used [355,356,198,366] and in the Cell Multipole Method (CMM) the Taylor series is represented in Cartesian coordinates [367][368][369]. It is also worthwhile to outline that the FMM aims to compute the electrostatic energy or forces due to the interaction of charges in the box L 0 , the contributions due to the periodic images (if any) are computed by using Schmidt and Lee [356], Nijboer-de Wette [156,189], Challacombe et al [198] and Smith [199] or renormalization group methods [370,371].…”

Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE)

“…While this technique was introduced for kernels of the form R m , the proposed technique has been extended to frequency domain sub-wavelength kernels [9], Yukawa (or shielded Coulomb) potentials and Gauss transforms. While similar methodologies have been introduced earlier [18,19], they are either not generalizable or offer only some of the advantages of this scheme. In what follows, a brief overview of ACE algorithm and the relevant definitions and theorems are presented.…”

“…From Eqs. (18) and (16) it can be inferred that the number of upward tree traversals (multipole-to-multipole and multipole-to-local translations) equals N max K, where N max cD t is the diameter of the sphere encompassing the entire low-frequency region X. These constraints mandate a new definition be used when developing interaction lists in the oct-tree as follows:…”

Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials

“…Further, ACE exploits the fact that these tensors are totally symmetric, and derives an exact algorithm for upward and downward tree traversals. It must be noted that, similar methodologies have been introduced earlier [25,26] but they are either not generalizable or offer only some of the advantages of this scheme. ACE has been successfully used to accelerate a number of different kernels, including London [24], low frequency Helmholtz (extensible to Yukawa) [27,28], and retarded wave [29].…”