Multilevel summation method for electrostatic force evaluation

The multilevel summation method (MSM) offers an efficient algorithm utilizing convolution for evaluating long-range forces arising in molecular dynamics simulations. Shifting the balance of computation and communication, MSM provides key advantages over the ubiquitous particle–mesh Ewald (PME) method, offering better scaling on parallel computers and permitting more modeling flexibility, with support for periodic systems as does PME but also for semiperiodic and nonperiodic systems. The version of MSM available in the simulation program NAMD is described, and its performance and accuracy are compared with the PME method. The accuracy feasible for MSM in practical applications reproduces PME results for water property calculations of density, diffusion constant, dielectric constant, surface tension, radial distribution function, and distance-dependent Kirkwood factor, even though the numerical accuracy of PME is higher than that of MSM. Excellent agreement between MSM and PME is found also for interface potentials of air–water and membrane–water interfaces, where long-range Coulombic interactions are crucial. Applications demonstrate also the suitability of MSM for systems with semiperiodic and nonperiodic boundaries. For this purpose, simulations have been performed with periodic boundaries along directions parallel to a membrane surface but not along the surface normal, yielding membrane pore formation induced by an imbalance of charge across the membrane. Using a similar semiperiodic boundary condition, ion conduction through a graphene nanopore driven by an ion gradient has been simulated. Furthermore, proteins have been simulated inside a single spherical water droplet. Finally, parallel scalability results show the ability of MSM to outperform PME when scaling a system of modest size (less than 100 K atoms) to over a thousand processors, demonstrating the suitability of MSM for large-scale parallel simulation.