In a charge-neutral system, 
, the Ewald sum method transforms the potential energy of Eq.(1),
into a sum of two rapidly converging series in real and reciprocal space of this form:
The rapid convergence of the two series stems from the fact that
as 
, the function 
decays rapidly and hence the first series
(real-space) converges rapidly. In addition, 
in the second series (reciprocal-space) is a smooth function and hence its Fourier transform
decays rapidly.
A physical interpretation of this decomposition of the lattice sum follows. Each point charge in the system is viewed as being surrounded by a Gaussian charge distribution of equal magnitude and opposite sign, Fig.(2), with charge density [3] :
where 
is a positive parameter that determines the width of the distribution, and
is the position relative to the center of distribution.
This introduced charge distribution screens the interaction between neighboring point-charges,
effectively limiting them to a short range.
Consequently, the sum over all charges and their images in real space converges rapidly.
To counteract this induced Gaussian distribution, a second Gaussian charge distribution of the same sign and
magnitude as the original distribution is added for each point charge. This time the sum is performed in
the reciprocal space using Fourier transforms to solve the resulting Poisson's equation.
It is worth pointing out that the choice of a charge distribution, conveniently taken to be Gaussian, is arbitrary.
The Ewald sum has been cast with non-Gaussian charge distributions, see [30].
Figure 2: The Ewald sum components of one-dimensional point-charge system.
The vertical lines are 
unit charges, and the Gaussians are also normalized to unity.