Fast summation methods

There is a growing number of approaches to treat the essentially infinite reach of charge-charge interactions. To mention just a few of the more traditional numerical ones which are well adapted to the requirements of MD, we have charge group cut-off [63], Ewald [72] summation, smooth particle Ewald [66] summation and particle-particle-particle-mesh (P M) [73]. There are also several variations of hierarchical methods [74] a few examples are the method of Bames and Hut (BH) [75], the fast multipole method (EMM), with [76] and without [77] multipoles, and the cell multipole method [78]. 

A. Arnold and C. Holm (2002) MMM2D A fast and accurate summation method for electrostatic interactions in 2D slab geometries. Comp. Phys. Comm. 148(3), pp. 327-348... 

A variety of methods are available for computing electrostatic energies (and forces), including the Fast Multipole Method [38], the Particle-Particle-Particle-Mesh Method [179], and methods based on the technique of Ewald Summation [125, 367] we only discuss a particular variant of the latter approach here and not... 

In this section we give an introduction to the Ewald summation, collecting the important equations for energy and forces. We also discuss briefly the fast multipole method and MMM. In the next section we present some recent results of our research on how to deal with partially periodic boundary conditions. Finally, we briefly discuss a new lattice method due to Maggs [21]. The material has been mainly collected from the sources [10,15,22-27). As good textbooks for background material we recommend the second edition of Frenkel and Smit [28] and the book by AUen and Tildesley [29]. 

Three efficient approaches for electrostatic modeling of inhomogeneous systems are the fast multipole method the Ewald summation... 

Part V, by Andrey Dobrynin, focuses on simulations of charged polymer systems (polyelectrolytes, polyampholytes). Chains at infinite dilution are examined first, and how electrostatic interactions at various salt concentrations affect conformation is discussed, according to scaling theory and to simulations. Simulation methods for solutions of charged polymers at finite concentration, including explicitly represented ions, are then presented. Summation methods for electrostatic interactions (Ewald, particle-particle particle mesh, fast multipole method) are derived and discussed in detail. Applications of simulations in understanding Manning ion condensation and bundle formation in polyelectrolyte solutions are presented. This chapter puts the recent simulations results, and methods used to obtain them, in the context of the state of the art of the polyelectrolyte theory. 

The second DFT LCAO linear-scaling method by Scuseria and Kudin (SK method) [379] uses Gaussian atomic orbitals and a fast multipole method, which achieves not only linear-scaling with system size, but also very high accuracy in aU infinite summations [397]. This approach allows both all-electron and pseudopotential calculations and can be applied also with hybrid HF-DFT exchange-correlation functionals. 

Molecular dynamics was performed at constant temperature with AMBER 4.1 all-atom force field [121] and Particle Mesh Ewald method (PME) was used for the calculation of electrostatic interactions [122]. This is a fast implementation of the Ewald summation method for calculating the full electrostatic energy of a unit cell in a macroscopic lattice of repeating images. The PME grid spacing was 1.0A. It was interpolated on a cubic B-spline, with the direct set tolerance set to 0.000001. Periodic boundary conditions were imposed in all directions. All solute-solute non-bonded interactions were calculated without jmy cut-off distance, while a non-bonded residue based cutoff distance of 9A was used for the solvent-solvent and for the solute-solvent interactions. The non-bonded pair list was updated every 20 steps and the... 

Gelle A, Lepetit MB. Fast calculation of the electrostatic potential in ionic crystals by direct summation method. J Chem Phys. 2008 128 244716. 

The past 20 years have seen a renewal of interest in lattice summation methods, catalyzed by the advances in high-performance computing and the ability thereby provided to approach molecular dynamics and condensed-phase structural problems that had previously seemed inaccessible. In this respect, an important development was the so-called fast multipole method (FMM) [5]. With its help, the electrostatic energies of arrays of charged particles can be evaluated in computing times that are nearly linear in the number of particles. One of the strengths of FMM is that the charge distribution need not be periodic, and methods of Ewald character can be combined with FMM concepts for studies of periodic systems [6]. 

---