Chemistry Reference
In-Depth Information
In this new molecular new reference frame, the MOZ equation becomes a com-
pact matrix equation (Kusalik and Patey 1988),
−
1
ˆ
ˆ
ˆ
−
()
χ
t
HCI
=
1
C
(7.42)
χ
χ
c
where each matrix is a
n
c
×
n
c
block matrix where
n
c
is the number of species in the
mixture, and the new matrix
C
t
is related to the initial matrix
C
χ
by the following
relation between their matrix elements,
ˆ
() ()
ˆ
tmn
;
χ
µ
mn
χ
Ck
=−1
Ck
()
(7.43)
ij
;
µν
ij
;
µν
For a two-component system, the typical 2×2 block matrix reads
ˆ
ˆ
HH
11
12
ˆ
H
χ
=
(7.4 4)
ˆ
ˆ
HH
12
22
where each submatrix
H
ij
is defined by elements
ˆ
(
;
. Equation 7.42 can
be used to derive the small-
k
form of the MOZ equation. The relation equivalent to
Equation 7.6 holds in the molecular case,
mn
hk
χ
ρρ
ij ij
µν
c
(, )
12
−(
→
β
u
12
, )
(7.45)
ij
r
→∞
which implies that the partial DCFs are Taylor expandable around
k
= 0, since the
pair interactions are integrable at large separations. In order to manage the expan-
sion, we need to take a look at the Fourier-Hankel transforms of the projections
(Blum and Torruella 1972; Fries and Patey 1985). These are defined as
+∞
∫
ˆ
mnl
l
2
mnl
ck
()
=
4
π
i
rr jkrc
()
(
r
)
(7.4 6)
ij
;
µν
l
ij
;
µν
0
Since the spherical Bessel functions have the expansion (Abramowitz and
Stegun 1970)
x
ln
l
+
2
n
()
!
−
1
n
∑
jx
()
=
(7.47)
l
2
n
(
2
++
1
)!!
2
n
n
=
0
We get the following small-
k
expansion for the DCFs,
∑
ˆ
mnl
l
+
2
t
mnlt
()
ck
()
=
kc
(7.48)
ij
;
µν
ij
;
µν
t
=
0
