Chemistry Reference
In-Depth Information
6.3.5 T
ransForms
Nichols, Moore, and Wheeler (2009) developed a method using finite Fourier-series
expansions of molecular concentration fluctuations in order to reduce systematic
errors from the simulation boundary conditions. The procedure was validated and
compared to a truncation method for a nonideal binary liquid mixture of LJ particles
tuned to imitate the system CF
4
and CH
4
. A fluctuation expression is applied to a
portion of the total volume within a closed simulation such as
NVT
. Rather than the
sampling volume being spherical and centered on a single moving molecule, the
sampling volume is a region with one or more rectangular slabs that is stationary
with respect to the simulation cell. This leads to two alternative expressions,
∞
sin(
qr
qr
)
∫
2
dr
Sq
()
=+
x
δ
x x
ρ
gr
()
−
14
π
r
(6.25)
ij
i j
i
j
ij
0
and
1
S
()
q
=
δ
Nt
(,)
q
⋅
δ
Nt
(
−=
q
,)
N
ψψ
t
)
(,)
q
t
⋅
(
−
q
,
(6.26)
ij
i
i
i
j
N
where ψ
i
(
q
,
t
) is the Fourier mass coefficient at the time
t
of the component
i
, and the
wave vector
q
has components that are integer multiples of 2π/
L
box
. The TCFIs are
found from the structure factors via ρ
G
ij
=
S
ij
(0) − 1.
For an infinite system, the definitions of
S
ij
in Equations 6.25 and 6.26 are equiva-
lent. However, for finite systems, truncation of the integrals can lead to errors in the
integral of Equation 6.25. The advantage of Equation 6.26 relative to Equation 6.25
is that the structure factor can be based on a discrete Cartesian-based Fourier
transform, rather than a continuous, spherically symmetric Fourier transform;
Equation 6.26 is evaluated at various values of
q
, which are most accurate at larger
values. The
q
- dependent
S
ij
are then extrapolated to
q
= 0 by fitting them to polyno-
mials, the range of
q
and the polynomial order being selected empirically. Thus, it is
not
g
ij
(
r
) that is corrected; it is done via the structure factors related to the RDFs using
the radial Fourier transform. The sampling volumes of the method do not truncate
intermolecular correlations at a particular radial distance and no assumption is made
that
g
ij
→ 1 at large intermolecular separations. In effect, this approach is consistent
with periodic boundary conditions, but immune to long-range truncation effects. The
inaccuracies in simulation RDFs at large
r
yield unreliable
S
ij
(0) because of system
size, but extrapolation to
q
= 0 provides the correction. Good results are obtained for
LJ mixtures, but the method has apparently not yet been tested for molecular fluids.
6.3.6 F
iniTe
-s
ize
s
caling
The approach of Schnell et al. (2011) is to sample small nonperiodic systems in a
(periodic) simulation box (reservoir) and then scale the results. The simulation sys-
tem has sides of
L
t
in each dimension. Small systems are randomly selected subvol-
umes, denoted by
L
n
-1
,
L
n
, and
L
n
+1
, and can exchange energy and particles with the
