Electrostatic energies and forces computed without explicit interparticle interactions: a linear time complexity...

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.