c CCLRC
Section 2.4
space sum. These are discussed more fully in Section
. DL POLY 3 can provide estimates if
requested (see CONTROL file description
As its name implies the Smoothed Particle Mesh Ewald (SPME) method is a modification of the
standard Ewald method. DL POLY 3 implements the SPME method of Essmann et al. [
Formally, this method is capable of treating van der Waals forces also, but in DL POLY 3 it
is confined to electrostatic forces only. The main difference from the standard Ewald method
is in its treatment of the the reciprocal space terms. By means of an interpolation procedure
involving (complex) B-splines, the sum in reciprocal space is represented on a three dimensional
rectangular grid. In this form the Fast Fourier Transform (FFT) may be used to perform the
primary mathematical operation, which is a 3D convolution. The efficiency of these procedures
greatly reduces the cost of the reciprocal space sum when the range of k vectors is large. The
method (briefly) is as follows (for full details see [
1. Interpolation of the exp(-i k · r
j
) terms (given here for one dimension):
exp(2i u
j
k/L) b(k)
=-
M
n
(u
j
- ) exp(2i k /K) ,
(2.150)
in which k is the integer index of the k vector in a principal direction, K is the total number
of grid points in the same direction and u
j
is the fractional coordinate of ion j scaled by a
factor K (i.e. u
j
= Ks
x
j
) . Note that the definition of the B-splines implies a dependence
on the integer K, which limits the formally infinite sum over . The coefficients M
n
(u) are
B-splines of order n and the factor b(k) is a constant computable from the formula:
b(k) = exp(2i (n - 1)k/K)
n-2
=0
M
n
( + 1) exp(2i k /K)
-1
.
(2.151)
2. Approximation of the structure factor S(k):
S(k) b
1
(k
1
) b
2
(k
2
) b
3
(k
3
) Q
(k
1
, k
2
, k
3
) ,
(2.152)
where Q
(k
1
, k
2
, k
3
) is the discrete Fourier transform of the charge array Q(
1
,
2
,
3
) defined
as
Q(
1
,
2
,
3
) =
N
j=1
q
j
n
1
,n
2
,n
3
M
n
(u
1j
-
1
- n
1
L
1
) × M
n
(u
2j
-
2
- n
2
L
2
) ×
M
n
(u
3j
-
3
- n
3
L
3
) ,
(2.153)
in which the sums over n
1,2,3
etc are required to capture contributions from all relevant
periodic cell images (which in practice means the nearest images).
3. Approximating the reciprocal space energy U
recip
:
U
recip
=
1
2V
o 0 k
1
,k
2
,k
3
G
(k
1
, k
2
, k
3
) Q(k
1
, k
2
, k
3
) ,
(2.154)
where G
is the discrete Fourier transform of the function
G(k
1
, k
2
, k
3
) =
exp(-k
2
/4
2
)
k
2
B(k
1
, k
2
, k
3
) (Q
(k
1
, k
2
, k
3
))
,
(2.155)
38