Chemistry Reference
In-Depth Information
.
B,
µ
B
θ
B
R
x
ϕ
.
.
.
θ
A
θ
B
θ
A
R
ω
z
B
A,
µ
A
A
ϕ
y
Figure 11.10 Different coordinate systems for two interacting dipoles
where T is the absolute temperature and k the Boltzmann constant. This
can be explained as follows.
With reference to Figure 11.10, let us first give some alternative
expressions for the interaction between dipoles (Coulson, 1958):
V
¼
m
A
m
B
R
3
3
ð
m
A
R
Þð
m
B
R
Þ
R
5
¼
m
A
m
B
R
3
v
3
ðm
A
Rcos
u
Þðm
B
R cos
u
Þ
A
B
cos
R
5
ð
11
:
63
Þ
¼
m
A
m
B
R
3
ð
cos
v
3cos
u
A
cos
u
Þ
B
¼
m
A
m
B
R
3
ð
sin
u
A
sin
u
B
cos
w
2 cos
u
A
cos
u
Þ
B
since, by the addition theorem (MacRobert, 1947):
cos
v ¼
cos
u
A
cos
u
B
þ
sin
u
A
sin
u
B
cos
w
ð
11
:
64
Þ
w ¼ w
A
w
B
is the dihedral angle between the planes specified by
where
m
A
,
m
B
and R. The last expression in (11.63) is the most convenient for us,
giving the dipole interaction in terms of the spherical coordinates R,
u
A
,
u
B
and
.
It is convenient to put
w
W ¼ u
A
; u
B
; w
F
ðWÞ¼
sin
ð
11
:
65
Þ
u
A
sin
u
B
cos
w
2 cos
u
A
cos
u
B
