c CCLRC
Section 2.4
which arises from a gaussian acting on its own site, and is constant. Ewald's method, therefore,
replaces a potentially infinite sum in real space by two finite sums: one in real space and one in
reciprocal space; and the self energy correction.
For molecular systems, as opposed to systems comprised simply of point ions, additional modifica-
tions ewald excl forces are necessary to correct for the excluded (intra-molecular) coulombic
interactions. In the real space sum these are simply omitted. In reciprocal space however, the
effects of individual gaussian charges cannot easily be extracted, and the correction is made in real
space. It amounts to removing terms corresponding to the potential energy of an ion
due to the
gaussian charge on a neighbouring charge m (or vice versa). This correction appears as the final
term in the full Ewald formula below. The distinction between the error function erf and the more
usual complementary error function erf c found in the real space sum, should be noted.
The same considerations and modifications ewald frozen forces are taken into account for
frozen atoms, which mutual coulombic interaction must be excluded.
The total electrostatic energy is given by the following formula.
U
c
=
1
2V
o 0
k
= vek0
exp(-k
2
/4
2
)
k
2
|
N
j
q
j
exp(-ik � r
j
)|
2
+
1
4
0
N
n<j
q
j
q
n
r
nj
erf c(r
nj
) -
1
4
0 molecules
M
m
q q
m
m
+
erf (r
m
)
r
1-
m
m
-
(2.146)
1
4
0
F
m
q q
m
m
+
erf (r
m
)
r
1-
m
m
-
1
4
0
1
V
o
2
N
j
q
j
2
,
where N is the number of ions in the system and N
the same number discounting any excluded
(intramolecular and frozen) interactions. M
represents the number of excluded atoms in a given
molecule. F
represents the number of frozen atoms in the MD cell. V
o
is the simulation cell
volume and k is a reciprocal lattice vector defined by
k = u + mv + nw ,
(2.147)
where , m, n are integers and u, v, w are the reciprocal space basis vectors. Both V
o
and u, v, w are
derived from the vectors (a, b, c) defining the simulation cell. Thus
V
o
= |a � b � c|
(2.148)
and
u = 2
b � c
a � b � c
v = 2
c � a
a � b � c
(2.149)
w = 2
a � b
a � b � c
.
With these definitions, the Ewald formula above is applicable to general periodic systems. The last
term in the Ewald formula above is the Fuchs correction [
] for electrically non-neutral MD cells
which prevents the build-up of a charged background and the introduction of extra pressure due
to it.
In practice the convergence of the Ewald sum is controlled by three variables: the real space cutoff
r
cut
; the convergence parameter and the largest reciprocal space vector k
max
used in the reciprocal
37