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interactions, such as the mean-spherical approximation and the generalized mean
field theory (Gray and Gubbins 1984). Second, Wang et al. (1973) showed from MC
simulations of Lennard-Jones (LJ) particles with significant dipole and quadrupole
moments that anisotropic forces have limited effects on
h
ij
(
r
). Also, Gubbins and
O'Connell (1974) showed that, for dense fluids, compressibility data for water and
argon could be scaled with only two parameters, meaning that the anisotropic effects
were not apparent in the water data. In addition, several studies show successful
corresponding-states scaling for the direct correlation function integrals (DCFIs)
(Brelvi and O'Connell 1972, 1975a, 1975b; Campanella, Mathias, and O'Connell
1987; Huang and O'Connell 1987; Abildskov, Ellegaard, and O'Connell 2009, 2010a,
2010b), as described in detail by O'Connell (1994). Finally, the approximation of
Equation 6.8 is the first term of the spherical harmonic expansions of the molecu-
lar correlation functions. Equation 6.8 can be systematically improved by consid-
ering the spherical harmonic expansions of the orientation-dependent TCFs and
DCFs (Gray and Gubbins 1984). The significance of this improvement is currently
unknown, however.
6.2.1 e
quivalence
oF
e
nsemBles
FST is based on the μ
VT
ensemble, so the Kirkwood-Buff integrals (KBIs) are inte-
grals over RDFs for an open system. However, simulations are most conveniently
performed in the
NpT
,
NVT
, or
NVE
ensembles. MD simulations in the μ
VT
ensemble
are possible (Çagin and Pettitt 1991) but nontrivial due to the problems associated
with inserting new particles (Beutler et al. 1994). For this and other reasons, simu-
lations are normally done on closed systems, though rigorously, the corresponding
KBIs are equal to 0 for unlike pairs and -1 for like pairs. The RDFs for μ
VT
and
NpT
simulations differ by a term of the order of 1/
N
, and the principle that
g
ij
(
r
→ ∞) = 1
is violated for closed systems (Ben-Naim 1990a). Fortunately, as illustrated in previ-
ous computational studies, RDFs in open and closed systems are extremely similar
(Weerasinghe and Pettitt 1994). This means that while the original Kirkwood-Buff
(KB) theory cannot be rigorously applied to a closed system, calculations converge to
correct results with increasing
N
. Thus, it has become standard to use the
equivalence
of ensembles
and to determine TCFIs using the MD simulations in the (
NpT
/
NVT
)
ensembles rather than the μ
VT
ensemble.
6.2.2 i
inTegraTion
The usual approach to determine the RDF
g
ab
(
r
) between the centers of mass of
particle 1 of species
a
and particle 2 of species
b
separated by the distance
r
, is the
accumulation of the number of particles
b
lying in the interval [
r
,
r
+
dr
] from a given
particle
a
, and for all available values of
r
within the central box. Numerical integra-
tion of molecular simulation RDFs is less straightforward. Theoretically,
h
ij
(
r
) goes
to zero when
r
goes to infinity. However, because the integral is evaluated numeri-
cally, convergence requires that
h
ij
(
r
) goes faster to zero than
r
2
goes to infinity. For
practicality, since the upper limit of the integral is infinite, for a convergent integral,
