One of the challenges facing the molecular dynamics simulation (MD) community is the study of
biologically important molecules especially in the presence of a solvent (typically water).
Biological systems of interest (e.g. enzymes, proteins, DNA strands, membranes)
range in size from a few tens to millions of atoms. When solvent molecules are added, system
sizes of interest to MD range from about a thousand atoms on up, with a few tens of thousands
of atoms being the largest sizes routinely studied today due to the computational requirements
of the simulations.
Periodic Boundary Conditions (PBC) have long been employed to minimize surface effects in a variety
of calculations [11]. In infinite PBC, the
simulation box is infinitely replicated in all directions to form a lattice.
In practice, most MD simulations evaluate potentials using some cutoff scheme
for computational efficiency. In these cutoff schemes, each particle interacts with
the nearest images of the other N-1 particles (minimum-image convention), or only
with those minimum images contained in a sphere of radius 
centered at the particle.
The use of cutoff methods, however, has been shown to introduce significant errors
and artificial behavior in a simulation [5,45,54].
To meet the objective of improving the quality and efficiency of MD simulations, it is important
to develop algorithms that compute the N-body problem for systems with PBC at a cost that is
not much greater than that of cutoff schemes but with a better accuracy.
The total Coulomb energy of a system of N particles in a cubic box of size L and their infinite replicas in PBC is given by:
where 
is the charge of particle i. The cell-coordinate vector is
, where x, y, z
are the cartesian coordinate unit vectors.
The origin cell is located at 
with image cells located at
intervals in all three dimensions as n goes to infinity, see Fig.(1).
Figure 1: In a 2-D system (a) The unit cell coordinates; (b) A 3x3 periodic lattice built from unit cells.
The first sum is primed to indicate that terms with i=j are omitted when 
.
The distance between a particle in the origin cell and another at an image cell
is 
. The above sum is conditionally convergent, which
means that the result depends on the order of summation [3].
Although other shapes are possible, the infinitely periodic system, by convention, is conceptually
built in roughly spherical layers for proper convergence.
In most MD simulations, the long-range interactions (Coulomb interactions) are
the most time consuming. This paper provides an account of the various methods used
to reduce the overhead involved in computing the Coulomb interactions with PBC.
