Electrostatic energies and forces computed without explicit interparticle interactions: A linear time complexity formulation

Abstract

A rapid method for the calculation of the electrostatic energy of a system without a cutoff is described in which the computational time grows linearly with the number of particles or charges. The inverse of the distance is approximated as a polynomial, which is then transformed into a function whose terms involve individual particles, instead of particle pairs, by a partitioning of the double sum. In this way, the electrostatic energy that is determined by the interparticle interactions is obtained without explicit calculation of these interactions. For systems of positive charges positioned on a face-centered cubic lattice, the calculation of the energy by the new method is shown to be faster than the calculation of the exact energy, in many cases by an order of magnitude, and to be accurate to within 1-2%. The application of this method to increase the accuracy of conventional truncation-based calculations in condensed-phase systems is also demonstrated by combining the approximated long-range electrostatic interactions with the exact short-range interactions in a "hybrid" calculation. For a 20-A sphere of water molecules, the forces are shown to be six times as accurate using this hybrid method as those calculated with conventional truncation of the electrostatic energy function at 12 A. This is accomplished with a slight increase in speed, and with a sevenfold increase in speed relative to the exact all-pair calculation. Structures minimized with the hybrid function are shown to be closer to structures minimized with an exact all-pair electrostatic energy function than are those minimized with a conventional 13-A cutoff-based electrostatic energy function. Comparison of the energies and forces calculated with the exact method illustrate that the absolute errors obtained with standard truncation can be very large. The extension of the current method to other pairwise functions as well as to multibody functions, is described.

The zero-multipole summation method for estimating electrostatic interactions in molecular dynamics: Analysis of the accuracy and application to liquid systems

“…It has recently been used for a new generation of coarsegrained potentials that account for a simplified electrostatic description of soluble proteins. 55 Other important non-Ewald electrostatics methods have been developed, including the single sum technique, 56 local molecular field theory, 57 the fast multipole method, 58,59 a fast multipole method combined with a reaction field, 60 the lattice-sum-emulated reaction-field method, 61 an image-charge reaction field method, 62,63 and a model of electrostatic and liquid-structure forces. 64 On the basis of the development of the ZC principle, 42,[65][66][67][68][69][70][71] the zero-dipole (ZD) summation method 72 provides the energy derived by counting the interactions for a neutralized subset regarding the dipoles as well as the charges.…”

Molecular Modeling of Nucleic Acid Structure: Electrostatics and Solvation

“…Wu and Brooks (2005) introduced an isotropic periodic sum (IPS) where periodic images are used to define the long-range interactions. Petrella and Karplus (2005) used Euler sums for calculating electrostatic interactions between particles without a cutoff. For larger systems (over ß50K atoms), periodic fast multipole methods can be applied (Schmidt and Lee, 1997), although these are slightly more complicated than standard fast multipole methods and do not appear to be supported in the standard and widely used community MD codes.…”

Algorithm improvements for molecular dynamics simulations

“…This is comparable with PME in terms of speed and accuracy, but in contrast, it can also be used for nonperiodic systems,31 which is a potentially powerful advantage. Multigrid methods can also be combined with multiple time steps, further enhancing the usefulness of the algorithms30 Petrella and Karplus32 describe a method for the calculation of electrostatic interactions between particles that replaces the double sum over particles with the product of two simpler sums.…”