Chemistry Reference
In-Depth Information
where C
6
is the London dispersion coefficient, and:
4
X
i
X
«
i
«
j
«
i
þ«
j
1
a
i
a
j
C
11
¼
ð
4
:
19
Þ
j
is the dipole-dipole dispersion constant in London form. It involves
knowledge of the pseudostate components
7
of the static dipole polariz-
abilities of A and B:
X
a
A
a
i
A
¼
ð
4
:
20
Þ
i
X
a
B
a
j
B
¼
ð
4
:
21
Þ
j
where:
2
m
i
«
i
a
i
¼
ð
4
:
22
Þ
is the ith pseudostate contribution to the static dipole polarizability of
atom A, and:
2
m
j
«
j
a
j
¼
ð
4
:
23
Þ
is the jth pseudostate contribution to the static dipole polarizability of
atomB.
m
i
and
m
j
are the transition dipolemoments onA and B,
«
i
>
0 and
«
j
0 are the excitation energies from the ground states to the excited
pseudostates i and j. We notice that the pseudostate components of
the polarizabilities are not observable quantities, so that they cannot
be measured.
An alternative, yet equivalent, expression for the dipole dispersion
constant is the Casimir-Polder formula (Casimir and Polder, 1948):
>
¥
1
2
du
a
A
Þ a
B
C
11
¼
ð
iu
ð
iu
Þ
ð
4
:
24
Þ
p
0
which involves integration over the frequency u of the frequency-
dependent polarizabilities (FDPs) at imaginary frequencies of the two
atoms
8
.
7
See Section 1.3 of Chapter 1
8
u is the real frequency and i the imaginary unit
ð
i
2
¼
1
Þ
.
