Ewald Summation - Particle Mesh Ewald (PME) Method

Particle Mesh Ewald (PME) Method

Ewald summation was developed as a method of theoretical physics, long before the advent of computers. However, the Ewald method has enjoyed widespread use since the 1970s in computer simulations of particle systems, especially those interacting via an inverse square force law such as gravity or electrostatics. Applications include simulations of plasmas, galaxies and molecules.

As in normal Ewald summation, a generic interaction potential is separated into two terms - a short-ranged part that sums quickly in real space and a long-ranged part that sums quickly in Fourier space. The basic idea of particle mesh Ewald summation is to replace the direct summation of interaction energies between point particles

with two summations, a direct sum of the short-ranged potential in real space

(that is the particle part of particle mesh Ewald) and a summation in Fourier space of the long-ranged part

where and represent the Fourier transforms of the potential and the charge density (that's the Ewald part). Since both summations converge quickly in their respective spaces (real and Fourier), they may be truncated with little loss of accuracy and great improvement in required computational time. To evaluate the Fourier transform of the charge density field efficiently, one uses the Fast Fourier transform, which requires that the density field be evaluated on a discrete lattice in space (that's the mesh part).

Due to the periodicity assumption implicit in Ewald summation, applications of the PME method to physical systems require the imposition of periodic symmetry. Thus, the method is best suited to systems that can be simulated as infinite in spatial extent. In molecular dynamics simulations this is normally accomplished by deliberately constructing a charge-neutral unit cell that can be infinitely "tiled" to form images; however, to properly account for the effects of this approximation, these images are reincorporated back into the original simulation cell. The overall effect is called a periodic boundary condition. To visualize this most clearly, think of a unit cube; the upper face is effectively in contact with the lower face, the right with the left face, and the front with the back face. As a result the unit cell size must be carefully chosen to be large enough to avoid improper motion correlations between two faces "in contact", but still small enough to be computationally feasible. The definition of the cutoff between short- and long-range interactions can also introduce artifacts.

The restriction of the density field to a mesh makes the PME method more efficient for systems with "smooth" variations in density, or continuous potential functions. Localized systems or those with large fluctuations in density may be treated more efficiently with the fast multipole method of Greengard and Rokhlin.