c CCLRC
Section 2.4
where the second term on the right is the reaction field correction to the explicit sum, with R
c
the
radius of the cavity. The constant B
0
is defined as
B
0
=
2(
1
- 1)
(2
1
+ 1)
,
(2.140)
with
1
the dielectric constant outside the cavity. The effective pair potential is therefore
U (r
ij
) =
1
4
0
q
i
q
j
1
r
ij
+
B
0
r
2
ij
2R
3
c
.
(2.141)
This expression unfortunately leads to large fluctuations in the system coulombic energy, due to the
large `step' in the function at the cavity boundary. In DL POLY 3 this is countered by subtracting
the value of the potential at the cavity boundary from each pair contribution. The term subtracted
is
1
4
0
q
i
q
j
R
c
1 +
B
0
2
.
(2.142)
The effective pair force on an atom j arising from another atom n within the cavity is given by
f
j
=
q
i
q
j
4
0
1
r
3
ij
-
B
0
R
3
c
r
ij
.
(2.143)
The contribution of each effective pair interaction to the atomic virial is
W = -r
ij
· f
j
(2.144)
and the contribution to the atomic stress tensor is
= r
ij
f
j
,
(2.145)
where , are x, y, z components. The atomic stress tensor is symmetric.
In DL POLY 3 the reaction field is handled by the subroutine coul rfp forces.
2.4.5
Smoothed Particle Mesh Ewald
] is the best technique for calculating electrostatic interactions in a periodic (or
pseudo-periodic) system.
The basic model for a neutral periodic system is a system of charged point ions mutually interacting
via the Coulomb potential. The Ewald method makes two amendments to this simple model. Firstly
each ion is effectively neutralised (at long range) by the superposition of a spherical gaussian cloud
of opposite charge centred on the ion. The combined assembly of point ions and gaussian charges
becomes the Real Space part of the Ewald sum, which is now short ranged and treatable by the
methods described above (Section
. The second modification is to superimpose a second set of
gaussian charges, this time with the same charges as the original point ions and again centred on
the point ions (so nullifying the effect of the first set of gaussians). The potential due to these
gaussians is obtained from Poisson's equation and is solved as a Fourier series in Reciprocal Space.
The complete Ewald sum requires an additional correction, known as the self energy correction,
3
Strictly speaking, the real space sum ranges over all periodic images of the simulation cell, but in the DL POLY 3
implementation, the parameters are chosen to restrict the sum to the simulation cell and its nearest neighbours i.e.
the minimum images of the cell contents.
36