Chemistry Reference
In-Depth Information
∫
(
)
(
)
=
aurhr
(),()
f r
( )
1
+
ρ
f
rr fr dr
−
′
(
′
)
′
(6.13)
Note that these theories for
c
are all consistent with the asymptotic result
c
(
r
) →
u
(
r
)/
kT
, when
r
→ ∞ (Lebowitz and Percus 1963). As shown in Wedberg et al. (2010)
,
Equation 6.11 through Equation 6.13 usually yield similar results and there is no rig-
orous basis for selecting one over another. Calculations of isothermal compressibili-
ties by integration of the extended pair correlation functions were reported only for
three state conditions by Verlet (1968), and the statistical uncertainties were large. It
is not clear whether these uncertainties were due to the quality of the simulations and
their analysis, or to the assumptions made by the extension method. Furthermore,
while the compressibilities were fairly reasonable for noble gases, agreement with
experiment was not reported in quantitative terms, due to the focus of the paper
being on other properties of the correlation functions.
For some time, the numerical solution of the liquid structure integral equations
remained a challenge. Progress along these lines was made by the numerical imple-
mentation of Verlet's method, based on the Newton-Raphson method, known as the
Gillan scheme
for solving the OZ equation (Gillan 1979; Abernethy and Gillan 1980;
Enciso 1985). Also, a factorization approach to the OZ equation was demonstrated
by Jolly, Freasier, and Bearman (1976). Before describing applications of our variant
of the Verlet method to systems resembling real molecular mixtures in Section 6.4,
we review other approaches designed to improve convergence.
6.3.2 T
TruncaTion
One of the simplest strategies is called
truncation
(Weerasinghe and Smith 2003b).
With this method, the RDF from μ
VT
simulations is approximated by the RDF from
NpT
simulations truncated at
R
lim
. This distance is chosen to be, “the range over
which the intermolecular forces dominate the distribution of particles.” Ideally,
the truncated
NpT
RDF captures the major features of the μ
VT
RDF, and its inte-
gral provides a good approximation to the desired KBIs. Truncated RDFs obtained
from
NpT
simulations have been obtained for several different mixtures, and used
to obtain properties, such as partial molar volumes (Lin and Wood 1996), and to
express the KB equation in terms of local compositions (Mansoori and Ely 1985).
As described in Chapter 5, Smith and coworkers have employed the truncation
method in order to develop accurate force fields for solutions, especially those with
biochemicals and proteins. Over the past decade, a series of force-ield development
and validation studies have been published using KBIs (Chitra and Smith 2001a;
Weerasinghe and Smith 2003d, 2004; Gee et al. 2011). The truncation approach can
be successful if the TCFIs converge within the range of distance sampled by simula-
tion. If not, the results will depend sensitively on the choice of
R
lim
. It is therefore
common to average
H
(
R
lim
) with
R
lim
varying over a selected interval. There seem
to be no general rules for selecting the interval, other than suggesting that it should
cover one oscillation of the TCF.
