Chemistry Reference
In-Depth Information
Therefore, the determination of the elementary dispersion constants
(the quantum mechanical relevant part of the calculation) allows for a
detailed analysis of the angle-dependent dispersion coefficients between
molecules.
An equivalent, yet explicit, expression of the C
6
angle-dependent
dispersion coefficient for the homodimer of two linear molecules as a
function of the three independent dispersion constants was derived by
Briggs et al. (1971) in their attempt to determine the dispersion coeffi-
cients of two H
2
molecules in terms of nonlinear
1
u
and
1
S
P
u
pseu-
dostates:
; wÞ¼ð
2B
þ
4D
Þþ
3
ð
B
D
Þð
cos
2
þ
cos
2
C
6
ðu
; u
u
u
Þ
A
B
A
B
2
þð
A
2B
þ
D
Þð
sin
u
A
sin
u
B
co
w
2cos
u
A
cos
u
B
Þ
ð
11
:
47
Þ
Since (cos
u ¼
x)
ð
ð
1
2
p
dxx
2
cos
2
d
w
w
1
3
;
2
3
;
1
2
1
0
h
sin
2
h
cos
2
h
cos
2
ui¼
¼
ui¼
wi¼
¼
ð
ð
1
2
p
dx
d
w
1
0
ð
11
:
48
Þ
averaging (11.47) over angles and noting that only squared terms con-
tribute to the average, we obtain the following for the isotropic C
6
dispersion coefficient:
!
þð
A
2B
þ
D
Þ
!
1
3
þ
1
3
2
3
2
3
1
2
þ
4
1
3
1
3
h
C
6
i¼ð
2B
þ
4D
Þþ
3
ð
B
D
Þ
2
3
ð
A
þ
4B
þ
4D
Þ¼
C
6
¼
ð
11
:
49
Þ
in accord with the result of the first row of Table 11.2. Magnasco et al.
(1990b) gave an alternative interesting expression for C
6
ðu
;u
;wÞ
in terms
A
B
of frequency-dependent isotropic polarizabilities
a
(iu)andpolarizability
