Computational Physics

8.7 Long-range interactions 243

We have not yet made any progress as we have only replaced the infinite sum overRby
another infinite sum overK. It might seem that this sum diverges for smallK, but this is
not the case for charge-neutral systems: this neutrality is responsible for the exclusion
of theK= 00 0 term, and it ensures convergence of the small-Kterms. Surprisingly,
the divergences in the original real-space sum(8.131)were associated with the long
range character of the force whereas the divergence in(8.134)is due to the short range
(largeK) part. In reality, the ions have a finite size, which means that they will repel
each other at short distances. This implies that the Coulomb interaction has to be taken
into account for ranges beyond some small cut-offrcoreonly, and we can neglect the
K-values forK> 2 π/rcore. Of course, this does not yield an exactly spherical cut-off
as the reciprocal lattice is cubic, but if the cut-off radius is sufficiently small this will
cause no significant errors. Moreover, the core radius can be chosen much smaller
than the range of repulsive interaction (which is always present in realistic models)
so that this error can be reduced arbitrarily.
In case one insists on having delta-function distributions, or if the cut-off radius
is so small that calculating the Fourier sum is still inconveniently demanding, it is
possible first to replace the delta-charges by artificial, extended charge distributions
with some finite radius and then correcting for this replacement. This is done in
the so-called Ewald summation technique. We shall not give a full derivation of
the Ewald summation method since it is quite lengthy – it is described elsewhere
[55, 56] – but sketch briefly the idea behind this technique. In the Ewald method,
the extended charge distribution is taken to be a Gaussian:

ρi(r)=qi

(α

π

) 3 / 2

exp(−α|r−ri|^2 ) (8.135)

where the normalisation factor is for the three-dimensional case. This charge distri-
bution results in a convergingK-sum, and this extension is corrected for by adding
the potential resulting from the difference between the point-charge and Gaus-
sian distribution. Since this difference is neutral, it generates a rapidly decaying
potential, which can then be treated by the minimum image convention. The total
interaction potential for chargesqilocated atriis then given as

UPBC=

2 π

V

∑

K
= 000

∣∣

∣∣

∣

∑

i

qieiK·ri

∣∣

∣∣

∣

2

e−K

(^2) /( 4 α)
K^2

+

∑

i<j

qiqj

erfc(

√

αrij)

rij

−

(α

π

) 1 / 2 ∑N

i= 1

q^2 i

(8.136)
where the function erfc is the complementary error function defined in(4.116):
erfc= 1 −erf. The first term of the Ewald sum converges rapidly due to the
exp[−K^2 /( 4 α)]term resulting from the Gaussian charge distribution. The second
term in the sum is short ranged, so it can be treated in a minimum image convention.
The forces can be found by differentiation. The Ewald sum can also be generalised
for dipolar interactions (Ewald–Kornfeld method).