Ewald summation was introduced in 1921 [23] as a technique to sum the long-range interactions between particles and all their infinite periodic images efficiently. For brevity, Ewald summation and Ewald sum will be used interchangeably. Ewald recast the potential energy of Eq.(1), a single slowly and conditionally convergent series, into the sum of two rapidly converging series plus a constant term.
The Ewald sum is therefore written as the sum of these three parts, namely, the real (direct) space sum
(
),
the reciprocal (imaginary, or Fourier) sum (
), and the constant term (
), known as the
self-term.
V is the volume of the simulation box, 
is a reciprocal-space vector, and 
was defined earlier.
The self-term 
is a correction term that cancels out the interaction of each of the
introduced artificial counter-charges with itself as will be explained in section(2.2).
The complimentary error function decreases monotonically as x increases and is defined by:
The theory of Ewald summation is described in more detail by Kittel [33] and
Tosi [51].
De Leeuw et al. in [17] and Deem et al. in [16], pointed out that a dipole term that is a function of the medium
surrounding the system needs to be added to the above sums.
Systems that are surrounded by vacuum experience a dipolar layer on their surface which does not exist
in systems surrounded by a conductor [3,16,17].
The dipole term includes the effects of the total dipole moment of the unit cell, the shape of the
macroscopic lattice, and the dielectric constant of the surrounding medium.
