Three parameters control the convergence of the sums in Eq.(2):
, an integer which defines the range of the
real-space sum and controls its maximum number of vectors (i.e. image cells),
similarly 
, an integer defining the summation range
in the reciprocal-space and its number of vectors, and 
, the Ewald convergence parameter,
which determines the relative rate of convergence between the real and reciprocal
sums. Note that in Eq.(3), a large value of 
, i.e. a narrow Gaussian
distribution, makes the real-space sum converge
faster, that is as 
, the 
.
This means that a small number of n-vectors (i.e. 
small) is sufficient
for a rapid convergence. On the other hand, a small 
in Eq.(4)
causes the reciprocal-space sum
to converge faster since as 
, the 
, i.e.
a small 
will suffice.
Traditionally for small systems, 
is chosen large enough so that the real-space sum
extends no further than the nearest neighbors of the original cell, while a choice of 
,
may be sufficient for reasonable convergence of the reciprocal sum.
This choice of parameters leads to a reciprocal sum
of the order N and a real sum of the order 
which is formidable for large systems.
Nijbeor [38] showed that for 
, the direct and reciprocal-space
terms converge at the same rate. However, this is of limited use as emerging methods [14,53], are capable of performing the reciprocal sum more efficiently
than the real sum.
For large systems, 
, even the minimum-image convention becomes costly (in CPU cycles)
and a fixed cutoff radius, 
, is generally used with a large 
.
Typical values of 
are between 8-12Å, e.g. AMBER [12] uses 9Å.
The choice of Ewald parameters should be based on several considerations:
and/or 
to limit the number of pair-wise interactions such that the real-space sum converges faster;
, or 
will yield
more accurate results, however it may be inefficient;
means less work done in the real sum, which is
traditionally the time consuming part; and
, the larger 
needs to be for the real
space sum to converge rapidly with a reasonable number of n-vectors.
is generally chosen to minimize the real sum and thus dictates the value of 
.
The choice of the Ewald sum parameters is system dependent and is subject to trade-offs
between accuracy and speed which in turn is influenced by the algorithmic implementation.
Rycerz and Jacobs [42] suggested choosing an 
given by this equation,
for systems with 
:
where 
is a constant that depends on the system, e.g. 
for Sodium Chloride.
Smith [50] suggested using 
while Perram
[39] proposed choosing 
to ensure
that the maximum term neglected is 
.
The above estimates for 
have been obtained by experimenting with the parameters such that
the error is minimized.
This approach may be feasible for small systems, but for large systems this method is very
inefficient.
Realizing this optimization problem, Kolafa and Perram [34] introduced simple formulas
to estimate 
, given a desired error in force (or acceleration) and other Ewald parameters.
Other Ewald methods, e.g. Fourier-based methods examined in section(4),
rely on choosing a relatively large 
.
Since Fourier-based Ewald methods utilize fast Fourier transforms to evaluate
the reciprocal sum, it is more efficient to ``bias'' the Ewald summation towards the reciprocal sum
and limit the real-space sum within a small cutoff radius [14].
It should be noted that the potential energy is invariant to the choice of 
.