Chemistry Reference
In-Depth Information
a consequence of Equation 6.31, Equation 6.39, and Equation 6.41. The partial deriv-
atives are obtained from three linear systems derived from Equation 6.31 by differ-
entiation with respect to
ĥ
11,β
,
ĥ
12,β
, and
ĥ
22,β
. The results are
∂
∂
∂
∂
∂
∂
∂
c
h
c
h
c
h
∂
c
h
c
h
c
h
c
h
c
h
c
h
11
,
β
11
,
β
11
,
β
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
11
,
β
12
,
β
22
,
β
1
1
−
xc
ρ
−
−
−
xc
xc
xc
ρ
ρ
ρ
0
0
11
,
β
2 2
,
β
[
]
−
1
12
,
β
12
,
β
12
,
β
2
=+
I
ρβ
(
⋅
∆
k
)
−
xc
ρ
1
(6.45)
1 2
,
β
2 2
,
β
11
,
β
12
,
β
22
,
β
0
1
−
xc
ρ
11
,
β
2
22
,
β
22
,
β
22
,
β
22
,
β
11
,
β
12
,
β
22
,
β
At each iteration step, these systems are solved for the partial derivatives, which then
are used to evaluate the Jacobians in Equation 6.44.
The short-range parts of the calculated DCFs are not used within the iteration
scheme, though the short-range part of the DCF obtained from the final iteration is
considered in selecting the parameters
R
ij
. Initially, the discretized TCFs are set to
h
(0)
ij
,α
=
h
MD,
ij
(
r
α
) for all
r
α
within the sampling range for
h
MD
,ij
(
r
), and
h
(0)
ij
,α
= 0 for
larger α. The iteration is carried out until
N
∑
∑
2
∆
c
ij
t
()
r
2
<
η
(6.46)
α
α
,
ij
,
α
=+
n
1
ij
with η = 10
−4
or less. Typically, this is achieved after 5 to 15 iterations. For some
systems, in particular those at high density where the functions
h
ij
(
r
) have significant
structure beyond the sampling range, the tail model by Christensen et al. (2007a,
2007b, 2007c) was used to estimate the long-range behavior for the initial guess
h
(0)
ij,
α
. Using this approach, the Newton iterations have converged for all systems
we have studied to date. Issues of how to select the matching distance and the angle
averaging of potentials are discussed in detail by Wedberg (2011).
6.4.2 r
esulTs
We now discuss results from MD simulations that test the capabilities of the method.
The KBIs are primarily verified by comparing the derivative properties obtained
from the integration procedure with the same properties obtained from alternative
analyses, or from simulation results in the literature. For the simulations of water/
organic solvent mixtures, the derivative properties obtained by integration are also
compared against values derived from correlations of experimental data. In this last
case, consistency depends not only on the accuracy of the integration procedure, but
also on the accuracy of the force field.
