A particle–particle particle-multigrid method for long-range interactions in molecular simulations

Abstract

A fast method of order $\mathcal{O}(N)$ is proposed to calculate interaction energies and forces in molecular systems with open boundaries, exerted by long range Coulomb interactions. The method consists of a fast multigrid Poisson solver for the far field smooth part of the potential and a particle–particle based method for the near field contribution. Boundary conditions are calculated with a multipole expansion method. Test cases are performed for the performance of the method.

Introduction

Coulomb interactions often play a crucial role for static and dynamical properties in a variety of complex systems, characterized by polar or charged system components, e.g., polar liquids, proteins, DNA, membranes, polyelectrolytes or plasmas. Due to the long range nature of these interactions their determination is computationally very demanding. Since pair interactions of all particles in the system have to be taken into account, the problem has complexity $\mathcal{O}(N^2)$. Therefore the size of the systems or length of the system trajectory is mainly limited by the computational overhead induced by electrostatic interactions. Due to the great interest in systems, dominated by Coulomb interactions, there was great effort spent in developing faster and more efficient methods with a lower complexity. For systems with periodic boundary conditions the most widely used method is the Ewald summation technique which is formally an exact analogue of an infinite lattice sum. Practically a small, controllable truncation error is accepted and it could be shown that for a given error, the method scale like $\mathcal{O}(N^{3/2})$ [1]. A faster variant of this method was later on developed, using fast Fourier transform techniques to reduce the complexity to $\mathcal{O}(N\log (N))$ [2]. A modification of the Ewald summation consists in the splitting of long range and short range contributions to the electrostatic energy. The so-called particle–particle particle-mesh method (P³M) solves the field equation with a fast Fourier method, using a modified Green's function, which adjusts the solution closely to the continuum solution. The method splits the field into near and far field contributions, where the short range part is calculated by an explicit particle–particle summation. Due to the fast Fourier transform the complexity is again reduced to $\mathcal{O}(N\log (N))$ [3].

For open systems, mainly two fast methods are in use, which both profit from a multipole expansion of far field contributions of charges to the local field. The Barnes–Hut tree algorithm splits the contributions hierarchically to end up in an $\mathcal{O}(N\log (N))$ complexity [4], while the Fast Multipole Method (FMM) reduces this complexity to $\mathcal{O}(N)$ by taking into account interactions of multipoles [5].

In the present article a method for open systems is proposed which goes in line with the idea of the P³M method, i.e. the near- and far-field contributions are treated in separate ways. The idea here is to apply the multigrid (MG) method to the solution of Poisson's equation to calculate a global potential energy surface. This problem would consist in an $\mathcal{O}(N^2)$ complexity. However, multigrid methods reduce this complexity to $\mathcal{O}(N)$, making them very attractive to many body problems. Having the global solution it is required to correct for the self-energy of the particles and the contributions from the near field part. This is performed via subtracting the grid based Green's function. In a last step the near field part is calculated as a pair-sum over all neighbored particles. In the following the mathematical basis is shown and performance measurements are compared with explicit pair-wise calculations.