Chemistry Reference
In-Depth Information
where
4
X
i
X
1
j
a
i
a
j
«
i
«
j
C
11
¼
ð
11
:
30
Þ
«
i
þ«
j
is the dipole dispersion constant, the typical quantum mechanical part of
the calculation of the dispersion coefficient, while 6 is a geometrical
factor
2
. Therefore:
C
6
¼
6C
11
ð
11
:
31
Þ
is the C
6
London dispersion coefficient for the long-range interaction
between two ground-state H atoms.
In this way, the previously calculated dipole pseudospectra {
a
i
,
«
i
},
i
¼
1
; ...;
N, for each H atom can be used to obtain better and better
values for the C
6
London dispersion coefficient for the H-H interaction: a
molecular (two-centre) quantity, C
6
, can be evaluated in terms of atomic
(one-centre),
;
2
alone is useless). The
coupling between the different components of the polarizabilities occurs
through the denominator in the London formula (11.30), so that we
cannot sumover i or j to get the full, observable,
3
nonobservable, quantities,
a
i
(
a
B
. An alternative,
yet equivalent, formula for the dispersion constant is due to Casimir and
Polder (1948) in terms of the frequency-dependent polarizabilities at
imaginary frequencies of A and B:
A
or
a
a
<
:
¥
1
2
A
B
C
11
¼
du
a
ð
iu
Þ a
ð
iu
Þ
p
0
ð
iu
Þ¼
X
k
mð
0
kÞmðk
0
Þ
2
A
A
A
A
a
«
k
; a
ð
static
Þ¼a
ð
0
Þ¼
lim
u
!
0
a
ð
iu
Þ
«
2
k
þ
u
2
ð
11
:
32
Þ
where u is a real quantity.
2
Depending on the spherical symmetry of the ground-state H atoms.
3
That is, measurable.
