Chemistry Reference
In-Depth Information
where
I
is the identity matrix,
S
is the matrix of the structure factors with elements
S
ˆ
ij
(
k
) = δ
ij
+
H
ij
(
k
), and the matrices
H
and
C
are generically defined through
ˆ
ij
(
k
) =
ˆ
()
. Equation 7.13 looks just like Equation 7.4 and this can be used to get to
the conclusions similar to those of the one component simple liquid in the previ-
ous paragraph.
First of all, Equation 7.6 holds for any interaction pairs. Therefore, one can expand
the DCFs similarly to Equation 7.7,
ρ
ij
ak
()
()
ˆ
ck=c
2
4
6
+k c+kc +k
o
(7.14)
ij
0;
ij
2;
ij
4;
ij
which can be written in matrix form as
o
()
C=C+kC +k C+ k
0
2
4
6
(7.15)
2
4
with the moment matrix defined through elements
()
(
n
/
2
1
∫
(
r
)
n
C=
ρρ
c=
ρρ
dc
rr
(7.16)
n;ij
ijn;ij
ij
)
ij
n+
1
!
which is analogous to Equation 7.8. When applied to a two-component system, one
gets from Equation 7.13 the same form as Equation 7.8 for the small-
k
behavior of
the structure factors,
A
k+
ˆ
()
lim
Sk=
(7.17)
2
−
2
ξ
k
→
0
where
ˆ
0
δ
−
−
c
ij
;i j
A=
(7.18)
ij
γ
with the notation
i
for swapping species (i.e.,
1
2
=
and
2
1
=
),
−
I
-
γ
ξ
=
(7.19)
0
and
(
)
(
)
211
γ
=CC+ CC+C
1
−
1
−
C
(7.2 0)
011222
;
;
022
;
;
0 12
;
212
;
