Chemistry Reference
In-Depth Information
ˆ
ˆ
ˆ
h
()
k
x
h
()
k
x
h
()
k
0
ˆ
c
c
c
()
k
k
k
11
1
11
2
12
11
ˆ
ˆ
ˆ
ˆ
h
()
k
=
h
()
k
,
c
(
k
)
=
()
,
H
( )
k
=
0
xk x
h
()
h
()
k
12
12
1
11
2
12
ˆ
ˆ
()
ˆ
ˆ
h
()
k
0
x
h
2
()
k
x
h
()
k
22
22
1
1
222
(6.32)
where
I
denotes the identity matrix. Equations 6.32 assume that the fluid mixture
has at most two components. Equation 6.29 through Equation 6.31 provides a route
for computing
c
ij
(
r
) given
h
ij
(
r
). The function
h
ij
(
r
) is Hankel-transformed to yield
ĥ
ij
(
k
). The linear system in Equation 6.31 is then solved for
ĉ
ij
(
k
) at each
k
, followed
by applying the inverse Hankel transform to obtain
c
ij
(
r
). Solution of the problem of
Equation 6.10 requires that the long-range part of
h
ij
(
r
) be adjusted until the long-
range part of
c
ij
(
r
) matches a trial function
t
ij
(
r
). This is accomplished by a Newton
iteration scheme, for which grids in
r
and
k
space are introduced,
r
≡⋅
α
∆
r
,
α
=…
0
,
,
N
α
(6.33)
k
≡⋅
β
∆
k
,
β
=…
0
,
,
N
β
The upper cutoffs are
R
c
=
N
· Δ
r
and
K
c
=
N
· Δ
k
for the integrals in Equations 6.29
and 6.30, respectively. Note that
R
c
is not the sampling limit set by the simulation
box dimensions, but is typically much larger. The TCFs, DCFs, and their Hankel
transforms at an iteration step
t
are represented by discrete vectors,
hhr
()
t
≡
()
t
( ,
c
()
t
≡
cr
()
t
(
),
α
= 1,
…
,
N
α
α
ij
,
α
ij
ij
,
α
ij
(6.34)
()
t
()
t
()
t
(
t
)
( ,
hhk
≡
( ,
c
≡
c
k
β=
1
,
…
,
N
ij
β
ij
β
ij
,
β
ij
,
β
Equation 6.29 for the TCF is approximated by truncating the integral at
R
c
and
using the trapezoidal rule,
hTh
ij
()
t
()
t
=⋅
(6.35)
ij
Elements of the matrix
T
are
sin
kr
kr
δδ
+
2
βα
βα
0
α
N
α
T
=⋅
4
π
∆
r r
⋅
⋅
1
-
(6.36)
βα
α
2
with α = 1, …,
N
, β = 1, …,
N
, with sin(
kr
)/(
kr
) being unity if either
k
or
r
is zero.
Equation 6.30 for the DCF is approximated in a similar way by truncating the inte-
gral at
K
c
,
c Uc
ij
()
t
()
t
=⋅
(6.37)
ij
