An N ·log(N) method for evaluating electrostatic energies and forces of large periodic systems is presented. The method is based on interpolation of the reciprocal space Ewald sums and evaluation of the resulting convolutions using fast Fourier transforms. Timings and accuracies are presented for three large crystalline ionic systems.

The previously developed particle mesh Ewald method is reformulated in terms of efficient B-spline interpolation of the structure factors. This reformulation allows a natural extension of the method to potentials of the form 1/r p with pу1. Furthermore, efficient calculation of the virial tensor follows. Use of B-splines in place of Lagrange interpolation leads to analytic gradients as well as a significant improvement in the accuracy. We demonstrate that arbitrary accuracy can be achieved, independent of system size N, at a cost that scales as N log(N). For biomolecular systems with many thousands of atoms this method permits the use of Ewald summation at a computational cost comparable to that of a simple truncation method of 10 Å or less.

Simulations of the HIV-l protease unit cell using a 9 A cutoff, 9/18 A "twin-range" cutoff, and full Ewald sums have been carried out to 300 ps. The results indicate that long-range electrostatic interactions are essential for proper representation of the HIV-1 protease crystal structure. The 9 A simulation did not converge in 300 ps. Inclusion of a 9/18 A "twin-range" cutoff showed significant improvement. Simulation using the Ewald summation convention gave the best overall agreement with x-ray crystallographic data, and showed the least internal differences in the time average structures of the asymmetric units. The Ewald simulation represents an efficient implementation of the Particle Mesh Ewald method [Darden et aI., J. Chern. Phys. 98, 10 089 (1993)], and illustrates the importance of including long-range electrostatic forces in large macromolecular systems.

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