Chemistry Reference
In-Depth Information
5 cos
2
5 cos
2
2 sin
2
Y
j
cos
2
f
QQ
ðY
i
; Y
j
; f
i
; f
j
Þ¼
1
Y
i
Y
j
þ
ðf
i
f
j
Þ
(16)
16
sin
Y
i
cos
Y
i
sin
Y
j
cos
Y
j
cos
ðf
i
f
j
Þ :
However, since for many cases of practical interest the absolute strength of the
multipolar interactions at typical nearest and next-nearest neighbor distances in
the fluid is much weaker than the LJ interactions, one can follow the idea of
M
¨
ller and Gelb [
208
] to treat the multipolar interaction only in spherically
averaged approximation:
U
eff
ð
r
Þ¼ð
k
B
T
Þ
ln exp
h
½
U
ð
r
; fY
i
; f
i
gÞ=
k
B
T
i
fY
i
;f
i
g
:
(17)
The isotropically averaged dipolar interaction can then be cast into the form:
12
6
s
ss
r
s
ss
r
U
eff
D
¼
4
e
ss
ð
1
þ l
Þ
;
(18)
with:
4
1
12
m
l ¼
ss
k
B
T
l
c
T
c
=
T
;
(19)
e
ss
s
where
m
is the dipole moment (cf. (
6
)). Similarly, the isotropically averaged
quadrupolar interaction becomes:
12
6
10
s
ss
r
s
ss
r
20
q
s
ss
7
U
eff
Q
¼
4
e
ss
;
(20)
r
where:
¼
Q
4
10
ss
k
B
T
q
¼
= e
ss
s
q
c
ð
T
c
=
T
Þ :
(21)
Note that (
18
) (
21
) can also be justified in terms of a perturbation expansion of
the dipole dipole or quadrupole quadrupole part of the interaction in second order
in inverse temperature.
Obviously (
18
) can be interpreted as a LJ potential with renormalized
parameters:
h
i
12
6
U
ef
D
¼
4
e ðs=
r
Þ
ðs=
r
Þ
;
(22)
2
,
6
ss
with
e ¼ e
ss
ð
1
þ lÞ
s
¼ s
=ð
1
lÞ
. Notice
that
l
is proportional
to
inverse temperature and hence
s
are temperature-dependent. Using the
knowledge of critical properties for the standard LJ model in three dimensions,
e
and
